![\"velocity](\"https://raw.githubusercontent.com/UCLAMAEThreads/MAE103/main/notebook/Couette.svg\")
\n\nBecause of the **no-slip condition**, the fluid next to each wall moves with it. The\nfluid next to the upper wall moves at velocity $U$, the fluid next to the lower wall\nis at rest.\n\nThe velocity profile depicts $u(y)$, the horizontal component of velocity as a\nfunction of vertical position, $y$. This increases linearly from the lower to the\nupper wall. In fact, the function is just\n\n$$u(y) = Uy/H$$\n\nThe arrows indicate the direction that the fluid is moving, and the lengths of\nthe arrows indicate the relative speed at that $y$ position.\n=#\n#-\n#=\n### Plotting velocity profiles\nThe arrows are helpful, but you can also plot a velocity profile without them.\nFor example, consider the following velocity:\n\n$$u(y) = \\frac{4U_c}{H^2} y (H - y)$$\n\nThe coefficient $U_c$ is a speed, and $H$ is the gap height.\n=#\n#=\nLet's define a function that evaluates this velocity. Here, `y`, `Uc`, and `H`\nare to be given as arguments to the function.\n=#\nu(y,Uc,H) = 4*Uc/H^2*y*(H-y)\n\n#=\nSuppose the gap height $H$ is 1 cm and the speed $U_c$ is 1 m/s. We will evaluate\nthis velocity at a range of locations between 0 and $H$:\n=#\nH = 1u\"cm\" # 1 cm = 0.01 m\nUc = 1u\"m/s\"\ny = range(0u\"cm\",H,length=101) # 101 points to evaluate at, just to make it look smooth.\n\n#=\nNow we evaluate the velocity function at the range of $y$ locations. (Remember that\nthe `.` vectorizes the evaluation of a function.)\n=#\nv = Velocity.(u.(y,Uc,H))\n\n#=\nNotice that $u$ is 0 at the beginning and end of the range. Let's plot it. But\nlet's plot it as a velocity profile, which means we make $u$ the 'x' axis and $y$\nthe 'y' axis.\n=#\n\nplot(v,y,xlim=(0u\"m/s\",2Uc),ylim=(0u\"cm\",H),\n legend=false,xlabel=\"u(y)\",ylabel=\"y\")\n#=\nThe top and bottom of this plot suggest that these are stationary walls where the\nflow is at rest. In fact, this is the velocity profile associated with pressure-driven\nflow through the gap.\n=#\n", "meta": {"hexsha": "7813e3c1e2ce0882a5b8c1b8b07746f760330e71", "size": 2597, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test/literate/1-VelocityProfiles.jl", "max_stars_repo_name": "UCLAMAEThreads/ThermofluidQuantities", "max_stars_repo_head_hexsha": "8d2c2ed6e692ef18e899c96a477b87a6cbd24ec3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-03-31T23:54:37.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-29T21:08:40.000Z", "max_issues_repo_path": "test/literate/1-VelocityProfiles.jl", "max_issues_repo_name": "UCLAMAEThreads/ThermofluidQuantities", "max_issues_repo_head_hexsha": "8d2c2ed6e692ef18e899c96a477b87a6cbd24ec3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-04-01T21:01:56.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-12T01:23:08.000Z", "max_forks_repo_path": "test/literate/1-VelocityProfiles.jl", "max_forks_repo_name": "UCLAMAEThreads/ThermofluidQuantities", "max_forks_repo_head_hexsha": "8d2c2ed6e692ef18e899c96a477b87a6cbd24ec3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.7272727273, "max_line_length": 144, "alphanum_fraction": 0.7181363111, "num_tokens": 739, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632976542184, "lm_q2_score": 0.9136765287024915, "lm_q1q2_score": 0.8473100786468016}}
{"text": "using Turing, MCMCChains, Distributions, StatsPlots, CSV\n\n@model die_model(observations) = begin\n # Set prior distribution for p (probability of rolling 1)\n p ~ Uniform(0,1)#Beta(5,1)\n\n # Probability of rolling one of 2 - 6\n q = (1-p)/5\n\n # Define how samples are obtained\n for i=1:length(observations)\n observations[i] ~ Categorical([p, q, q, q, q, q])\n end\n\n return p\nend\n\n# Define the true probabilities\ntrue_p = 0.25\ntrue_q = (1-true_p)/5\n\n## Simulate observed data\nusing DataFrames\nusing CSV\ndata = rand(Categorical([true_p, true_q, true_q, true_q, true_q, true_q]), 100)\ndf = DataFrame(X = data)\nCSV.write(\"pp_ex_1_data.csv\", df)\n\n# Load observed data\ndata = CSV.read(\"pp_ex_1_data.csv\")[:,1]\n\n# Use HMC sampler to obtain posterior samples\nnum_posterior_samples = 1000\nϵ = 0.05 # Leapfrog step size\nτ = 10 # Number of leapfrog iterations\n\n# Obtain posterior samples\nchain = sample(die_model(data), HMC(ϵ, τ, :p), num_posterior_samples, progress=false)\n\n# Extract summary of p parameter and plot histogram\np_summary = chain[:p]\nplot(p_summary, seriestype = :histogram, bg = RGB(247/255, 236/255, 226/255))\n\n\nsavefig(\"../images/pp_ex_1_hist.svg\")\n\n# Extract 95% interval\nprintln(quantile(Array(p_summary), [0.025, 0.975]))\n", "meta": {"hexsha": "3561208cd9f1df746514db5c3a0bb31b4919df73", "size": 1261, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "content/bayesian-inference-and-graphical-models/code/pp_ex_1.jl", "max_stars_repo_name": "seanrattana/courses", "max_stars_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2019-08-21T07:33:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-24T15:46:58.000Z", "max_issues_repo_path": "content/bayesian-inference-and-graphical-models/code/pp_ex_1.jl", "max_issues_repo_name": "seanrattana/courses", "max_issues_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-23T06:04:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-26T12:47:12.000Z", "max_forks_repo_path": "content/bayesian-inference-and-graphical-models/code/pp_ex_1.jl", "max_forks_repo_name": "seanrattana/courses", "max_forks_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-08-18T21:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T19:14:33.000Z", "avg_line_length": 25.7346938776, "max_line_length": 85, "alphanum_fraction": 0.7042030135, "num_tokens": 389, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9637799482964023, "lm_q2_score": 0.8791467564270271, "lm_q1q2_score": 0.84730401545419}}
{"text": "#Tutorial 2\n#=\nDaniel\n=#\n# ## is a break on the code\n\n# control+c to stop the process\nmy_integer=5\nmy_float=5.\nmy_range=1:7\nmy_vector=[1, 3, 5]\nmy_string=\"hello\"\nmy_character='o'\nmy_boolean=true\n\ntypeof(my_integer)\ntypeof(my_float)\n\ntypeof(\n my_character\n)\n\n# use \\ to write simbols like alpha\nα=5;\n#show the command\n@show α + my_integer\nprint(\"α = $α \\n\", \"My integer is: $(my_integer)\")\n\n#println automatically saparated linea\n\n#column vector\nA = [1, 3, 5]\n#row vector\nA = [1 2 3]\n\nones(3)\nzeros(5)\nC=[1 3; 5 \"test\"]\nC=[1 3; 5 9]\n\nA = 1:9\n# number with a distance o 0.5\nB= 1:0.5:9\n# secon element of break\nB[3]\n\ntypeof(C)\neltype(C)\nsize(C)\nlength(C)\nndims(C)\n\n#indexing\nC[end]\nC[1,2]\n@show C\n@show C[2,:] #show select row*\n\nusing LinearAlgebra\n\nC*I\nC^2\nC.^2 #by element operation\n\nsin.([1 2 3])\n\ninv(C)\ninv(C)*C # too costly\nC\\C # this notation is better\n\n# to installa a package type ], add pkg name\n\nusing Random\nrand(1)\nrand(3)\nrandn(3)\n\nRandom.seed!(1234) #name of the seed\nrand(1)\nrand(1)\n\nRandom.seed!(1234)\nrand(1)\n\n#if/else statement\na=3\nif a==3\n print(\"a is 3\")\nend\n\na=true\nif a\n print(\"a is true\")\nend\n\na ? print(\"a is true\") : nothing\n\nif a \n print(\"a is true\")\nelse \n print(\"a is no true\")\nend\n\nb=true\n\nif a | b \n print(\"a or b is true (or both)\")\nend\n\nif a & b # double & makes faster the process, just evaluate the firs statement. \n print(\"a and b are true\")\nend\n\n#for loops\nvals = [\"a\" \"b\" \"c\"]\n\nfor val in vals\n println(val)\nend\n\n#global\na = 3\nfor val in vals\n a = a + 1\n return a \nend\n\n\nfor i in eachindex(vals)\n println(vals[i])\nend\n\n# index and value \nfor (index, value) in enumerate(vals)\n println(\"Element $index of vals is $value\")\nend\n\n# Functions\nfunction cobbdouglas(l, k, α, β = 1 - α; A=1)\n y=A*l^α*k^β\n return y\nend\n\ncobbdouglas(1, 3, 0.5)\n\n@show output = cobbdouglas(1, 3, 0.5)\n\ncobbdouglas.([1 2], 3, 0.5)\n\ncobbdouglas(l, k, α) = A*l^α*k^(1-α)\ncobbdouglas(1, 3, 0.5)\n\n", "meta": {"hexsha": "1c36c59f55b21bd85cf2692e8d52b004320b43be", "size": 1943, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "demo.jl", "max_stars_repo_name": "DanielGuzmanGiron/Julia_macro", "max_stars_repo_head_hexsha": "ff013189629b515c672b03bb2f0a5cca7403a177", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "demo.jl", "max_issues_repo_name": "DanielGuzmanGiron/Julia_macro", "max_issues_repo_head_hexsha": "ff013189629b515c672b03bb2f0a5cca7403a177", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "demo.jl", "max_forks_repo_name": "DanielGuzmanGiron/Julia_macro", "max_forks_repo_head_hexsha": "ff013189629b515c672b03bb2f0a5cca7403a177", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 12.6993464052, "max_line_length": 81, "alphanum_fraction": 0.6356150283, "num_tokens": 732, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178919837706, "lm_q2_score": 0.9324533130837862, "lm_q1q2_score": 0.8472437637074727}}
{"text": "\"Square the sum of the first `n` positive integers\"\nfunction square_of_sum(n)\n sum(1:n)^2\nend\n\n\"Sum the squares of the first `n` positive integers\"\nfunction sum_of_squares(n)\n sum((1:n).^2)\nend\n\n\"Subtract the sum of squares from square of the sum of the first `n` positive ints\"\nfunction difference(n)\n square_of_sum(n) - sum_of_squares(n)\nend\n", "meta": {"hexsha": "406d554fe88af2c332e1372b35df378cdd66ce8e", "size": 353, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "difference-of-squares/difference-of-squares.jl", "max_stars_repo_name": "koladilip/julia-exercism", "max_stars_repo_head_hexsha": "1e67c319ae866a26365da8380659724c5cd472fc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "difference-of-squares/difference-of-squares.jl", "max_issues_repo_name": "koladilip/julia-exercism", "max_issues_repo_head_hexsha": "1e67c319ae866a26365da8380659724c5cd472fc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "difference-of-squares/difference-of-squares.jl", "max_forks_repo_name": "koladilip/julia-exercism", "max_forks_repo_head_hexsha": "1e67c319ae866a26365da8380659724c5cd472fc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.5333333333, "max_line_length": 83, "alphanum_fraction": 0.7280453258, "num_tokens": 99, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9566342037088041, "lm_q2_score": 0.8856314662716159, "lm_q1q2_score": 0.8472253525162079}}
{"text": "\"\"\"\n LinearKernel(; c = 0.0)\n\nThe linear kernel is a Mercer kernel given by\n```\n κ(x,y) = xᵀy + c\n```\nWhere `c` is a real number\n\"\"\"\nstruct LinearKernel{Tc<:Real} <: SimpleKernel\n c::Vector{Tc}\n function LinearKernel(;c::T=0.0) where {T}\n new{T}([c])\n end\nend\n\nkappa(κ::LinearKernel, xᵀy::Real) = xᵀy + first(κ.c)\n\nmetric(::LinearKernel) = DotProduct()\n\nBase.show(io::IO, κ::LinearKernel) = print(io, \"Linear Kernel (c = \", first(κ.c), \")\")\n\n\"\"\"\n PolynomialKernel(; d = 2.0, c = 0.0)\n\nThe polynomial kernel is a Mercer kernel given by\n```\n κ(x,y) = (xᵀy + c)^d\n```\nWhere `c` is a real number, and `d` is a shape parameter bigger than 1. For `d = 1` see [`LinearKernel`](@ref)\n\"\"\"\nstruct PolynomialKernel{Td<:Real, Tc<:Real} <: SimpleKernel\n d::Vector{Td}\n c::Vector{Tc}\n function PolynomialKernel(; d::Td=2.0, c::Tc=0.0) where {Td<:Real, Tc<:Real}\n @check_args(PolynomialKernel, d, d >= one(Td), \"d >= 1\")\n return new{Td, Tc}([d], [c])\n end\nend\n\nkappa(κ::PolynomialKernel, xᵀy::Real) = (xᵀy + first(κ.c))^(first(κ.d))\n\nmetric(::PolynomialKernel) = DotProduct()\n\nBase.show(io::IO, κ::PolynomialKernel) = print(io, \"Polynomial Kernel (c = \", first(κ.c), \", d = \", first(κ.d), \")\")\n", "meta": {"hexsha": "09e02ba31dfbaede9ebb3165afd57d2baa12a35b", "size": 1233, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/basekernels/polynomial.jl", "max_stars_repo_name": "tirthasheshpatel/KernelFunctions.jl", "max_stars_repo_head_hexsha": "86d430c8e275a8b41b0a4a68aeb31e488a075b83", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/basekernels/polynomial.jl", "max_issues_repo_name": "tirthasheshpatel/KernelFunctions.jl", "max_issues_repo_head_hexsha": "86d430c8e275a8b41b0a4a68aeb31e488a075b83", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/basekernels/polynomial.jl", "max_forks_repo_name": "tirthasheshpatel/KernelFunctions.jl", "max_forks_repo_head_hexsha": "86d430c8e275a8b41b0a4a68aeb31e488a075b83", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.8043478261, "max_line_length": 116, "alphanum_fraction": 0.600973236, "num_tokens": 449, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9615338079816756, "lm_q2_score": 0.8807970685907242, "lm_q1q2_score": 0.8469161594211362}}
{"text": "\"\"\"\r\n romberg(f, a, b; maxi = 10, delta = 10^-9)\r\nberisi program untuk mencari integral numerik dari fungsi `f(x)` pada\r\ninterval `[a,b]`. Program ini secara default berisi 3 masukan, yaitu fungsi `f(x)`, dan titik\r\nujung interval `[a,b]`. Secara default memiliki nilai maksimum iterasi `maxi=10` dan\r\ntoleransi error sebesar `delta = 1e-9`\r\n# Examples\r\n```jl\r\njulia> f(x) = (x^2+x+1)*cos(x)\r\nf (generic function with 1 method)\r\n\r\njulia> sol, flag, err, R = romberg(f, 0, pi/2);\r\n\r\njulia> sol\r\n2.0381974270672245\r\n\r\njulia> flag\r\n0\r\n\r\njulia> err\r\n1.213065203842234e-10\r\n\r\njulia> R\r\n6×6 Array{Float64,2}:\r\n 0.785398 0.0 0.0 0.0 0.0 0.0\r\n 1.72681 2.04062 0.0 0.0 0.0 0.0\r\n 1.96053 2.03844 2.0383 0.0 0.0 0.0\r\n 2.01879 2.03821 2.0382 2.0382 0.0 0.0\r\n 2.03335 2.0382 2.0382 2.0382 2.0382 0.0\r\n 2.03698 2.0382 2.0382 2.0382 2.0382 2.0382\r\n\r\n julia> sol, flag, err, R = romberg(f, 0, pi/2, maxi=20, delta=1e-14);\r\n\r\n julia> sol\r\n 2.038197427067236\r\n\r\n julia> flag\r\n 0\r\n\r\n julia> err\r\n 0.0\r\n\r\n julia> R\r\n 8×8 Array{Float64,2}:\r\n 0.785398 0.0 0.0 0.0 0.0 0.0 0.0 0.0\r\n 1.72681 2.04062 0.0 0.0 0.0 0.0 0.0 0.0\r\n 1.96053 2.03844 2.0383 0.0 0.0 0.0 0.0 0.0\r\n 2.01879 2.03821 2.0382 2.0382 0.0 0.0 0.0 0.0\r\n 2.03335 2.0382 2.0382 2.0382 2.0382 0.0 0.0 0.0\r\n 2.03698 2.0382 2.0382 2.0382 2.0382 2.0382 0.0 0.0\r\n 2.03789 2.0382 2.0382 2.0382 2.0382 2.0382 2.0382 0.0\r\n 2.03812 2.0382 2.0382 2.0382 2.0382 2.0382 2.0382 2.0382\r\n```\r\nreturn nilai solusi integral numerik `sol`, status solusi\r\n`flag`, hampiran galat `err`, dan matriks `R` yaitu matriks iterasi Romberg.\r\n\"\"\"\r\nfunction romberg(f, a, b; maxi = 10, delta = 10^-9)\r\n flag = 1;\r\n M = 1;\r\n h = b-a;\r\n err = 1;\r\n R = h/2*(f(a)+f(b));\r\n for J = 1:maxi\r\n # Rekursif Trapesium\r\n h = h/2;\r\n M = 2*M;\r\n s = 0;\r\n for p = 1:M/2\r\n x = a+h*(2*p-1);\r\n s = s+f(x);\r\n end\r\n R = [R zeros(size(R,1),1) ; R[J,1]/2 + h*s zeros(1,size(R,1))]\r\n # Aturan Romberg\r\n for K=1:J\r\n R[J+1,K+1]=R[J+1,K]+(R[J+1,K]-R[J,K])/(4^K-1);\r\n end\r\n err = abs.(R[J,J]-R[J+1,J+1]);\r\n if err\n#' B) Between w = -1:0
\n#' C) Between w = 0:.6\n#'\n#' *Hint*: It may be easier to see what's going on if you only plot,\n#' for example, every 15th step.\n\n# Initial guess = 0.7 so the plot range needs to be smaller to see\n(w, fw) = gradient_descent(L1, -2:0.1:1.5; tol=0.000001)\npl = plot(L1, 0.692, 0.70, xlabel=\"w\", ylabel=\"L1(w)\", legend=false)\nws = reverse(w[end:-50:1]) # Make sure end point is included\nfws = reverse(fw[end:-50:1]) \nscatter!(pl, ws,fws)\n\n# Initial guess = -2.0\n(w, fw) = gradient_descent(L1, -2:0.1:1.5; tol=0.000001, w_init=-2)\npl = plot(L1, -2, 1.5, xlabel=\"w\", ylabel=\"L1(w)\", legend=false)\nws = reverse(w[end:-50:1]) # Make sure end point is included\nfws = reverse(fw[end:-50:1]) \nscatter!(pl, ws,fws)\n\n#' ## Functions of 2 variables and their derivatives\n#' So far, we have been looking at the minimizing a loss function\n#' $L_1(w)$ that depends on a single parameter, $w$. Now let's turn to\n#' the cost function $L_2(w, b)$ from a previous notebook, that is a\n#' function of *two* parameters, $w$ and $b$, and try to minimize it.\n#'\n#' As we've seen, we get a **surface**, instead of a curve,\n#' when we graph $L_2$ as a function of both of its parameters.\n\n#' **Exercise 10**\n#'\n#' Draw a surface plot of $L_2$, given by\n#'\n#' $$L_{2}(w, b) = \\sum_i(y_i - g(x_i, w, b))^2$$\n#'\n#' using the `surface` function from `Plots.jl`. For this plot, use the\n#' values of `xs` and `ys` from notebook 5:\n#'\n#' We can get a nice interactive 3D plot by using the Plotly backend\n#' of `Plots.jl` by executing the command\n#'\n#' plotly()\n# if !(@isdefined WEAVE_ARGS)\n# plotly()\n# end\n\nxs = [2, -3, -1, 1]\nys = [0.8, 0.3, 0.4, 0.4]\n\nsigma(x) = 1/(1 + exp(-x))\ng(x,w,b) = sigma(w*x) + b\n\nL2(w,b) = sum(abs2, ys .- g.(xs,w,b))\n\npl = plot()\nsurface!(pl, -2:0.02:2, -2:0.02:2, L2, alpha=0.6, xlabel=\"w\",\n ylabel=\"b\", zlabel=\"L2(w,b)\",\n size=(600,600))\n#'\n\n#+ echo=false\nif !@isdefined WEAVE_ARGS # exe when not from Weave\n gui(pl)\nend\n\n#' ### Finding the minimum\n#' We can just about see, by rotating the graph, that $L_2$ has a single\n#' minimum. We want to find the values of $w$ and $b$ where this\n#' minimum value is reached.\n\n#' Following what we did for the function $L_1$ above, we expect that\n#' we will need to calculate derivatives of $L_2$. Since the function is\n#' more complicated, though, the derivatives are too!\n#'\n#' It turns out that the right concept is that of the\n#' [**gradient**](https://en.wikipedia.org/wiki/Gradient) of\n#' $L_2$, denoted $\\nabla L_2(w, b)$. This is a **vector** consisting\n#' of $2$ numbers if there are $2$ parameters \n#'[or, in general, $n$ numbers if there are $n$ parameters].\n#'\n#' The numbers that form the gradient $\\nabla L_2(w, b)$ are called the\n#' **partial derivatives** of $L_2$ with respect to $w$ and $b$, written as\n#'\n#' $$\\frac{\\partial L_2}{\\partial w} \\quad \\text{and} \\quad \\frac{\\partial L_2}{\\partial b}.$$\n#'\n#' Although this notation might look complicated, all it means is that we\n#' calculate derivatives just like we did before, except that we fix the value\n#' of the other variable.\n#' For example, to calculate $\\frac{\\partial L_2}{\\partial w}$,\n#' the partial derivative of $L_2$ with respect to $w$, we fix the value of\n#' $b$ and think of the resulting function as a function of the single\n#' variable $w$; then we use the formula for derivatives of functions of a\n#' single variable.\n#'\n#' [Note that $\\frac{\\partial L_2}{\\partial w}$ is itself a function of $w$ and $b$; we could write $\\frac{\\partial L_2}{\\partial w}(w, b)$.]\n\n#' #### Exercise 11\n#'\n#' Write functions that will allow you to calculate the partial derivatives of\n#' a function $f$ of two variables.\n#'\n#' In particular, declare functions called `partial_w` and `partial_b`.\n#' Each should take four inputs - a function $f$ of two variables, the first\n#' input argument to $f$, the second input argument to $f$, and a step size\n#' `h` with default value `0.001`. `partial_w` should return the partial\n#' derivative of $f$ with respect to its first input argument and `partial_b`\n#' should return the partial derivative of $f$ with respect to its second\n#' input argument.\n\n#' #### Solution\n#'\n#' The partial derivatives are\npartial_w(f, w, b, h=0.001) = (f(w+h,b) - f(w,b))/h\npartial_b(f, w, b, h=0.001) = (f(w,b+h) - f(w,b))/h\n\n#' **Test**\npartial_x = partial_w\npartial_y = partial_b\nfn(x,y) = 3*x*y\n#' Test default step\n@assert isapprox(partial_x(fn, 0.3, 0.3), 0.9; atol=1.e-6)\n@assert isapprox(partial_y(fn, 0.3, 0.3), 0.9; atol=1.e-6)\n#' Test custom step\nfn(x,y) = 3 * x^2 * y^3\n@assert isapprox(partial_x(fn, 2.3, 10.2, 1), 17828.2944; atol=1.e-2)\n@assert isapprox(partial_y(fn, 2.3, 10.2, 1), 5454.84; atol=1.e-2)\n\n\n#' #### Exercise 12\n#'\n#' Use `partial_b` from the last exercise to find the partial derivative of\n#' $L_2$ with respect to $w$ at b = 0.3,\n#' $\\frac{\\partial L_2}{\\partial w}|_{b = 0.3}$ for `w = -2:0.01:1`\n\nwrange = -2:0.01:1\npartialw = partial_w.(L2, wrange, 0.3)\n\n#' #### Exercise 13\n#'\n#' Plot the cross section of the surface of $L_2(w, b)$ at $b = 0.3$.\n#' Make this plot interactive to show that the function `partial_w` gives\n#' the slope of the tangent to this cross section for any point `w` in\n#' the range `-2:0.01:1`.\n#'\n#' For what value of $w$ in this range is the slope of the cross section\n#' closest to -1?\n\nL2_b(w) = L2(w, 0.3) # b = 0.3\n\nw = 0.0 # Change me\npl = plot(L2_b, wrange) #plot(wrange, L2_b)\nscatter!([w], [L2_b(w)])\n\nslope = partial_w(L2, w, 0.3)\nwrange_value = slope .* wrange .- slope*w .+ L2.(w,0.3)\nplot!(pl, wrange, wrange_value)\nylims!(0.3, 2.0)\ntitle!(\"Slope = $slope @ $(w)\")\n#'\n#' ## **Optional**: Functions with $n$ inputs\n#' If a function $f$ takes $n$ input arguments, we can write them as\n#' $p_1, \\ldots, p_n$, where $p_i$ means the \"$i$th parameter\".\n#' In Julia, we can wrap them up into a single **vector**. Now we can\n#' calculate the partial derivative $\\frac{\\partial L_2}{\\partial p_i}$ with\n#' respect to the $i$th variable.\n\n#' #### Exercise 14\n#'\n#' For the next exercise, you will need to use the splat command, `...`. You can use this command to \"open up\" a collection and pass all the elements of that collection as inputs to a function.\n#'\n#' For example, say you have an array, `numbers`,\n#'\n#' ```julia\n#' numbers = [4, 3, 2]\n#' ```\n#'\n#' and you want to use `numbers` to create a $4\\times3\\times3$ randomly populated array via `rand`. `rand(numbers)` will not do what you want. You could index into `numbers` to grab the values you want and pass them to `rand`, as in\n#'\n#' ```julia\n#' rand(numbers[1], numbers[2], numbers[3])\n#' ```\n#'\n#' or you could use a splat:\n#'\n#' ```julia\n#' rand(numbers...)\n#' ```\n#'\n#' Use `...` to pass the contents of `inputs`\n#'\n#' ```julia\n#' inputs = [30, 12, \"cats\"]\n#' ```\n#'\n#' to the function `dreams`\n#'\n#' ```julia\n#' dreams(i, j, perfect_mammal) = \"I wish I had $(i + j) $perfect_mammal.\"\n#' ```\n\ninputs = [30, 12, \"cats\"]\ndreams(i, j, perfect_mammal) = \"I wish I had $(i + j) $perfect_mammal.\"\ndreams(inputs...)\n\n# **Tests**\n\n@assert dreams(inputs...) == \"I wish I had 42 cats.\"\n\n#' #### Exercise 15:\n#'\n#' Write a function, `partial`, to calculate the $i$th partial derivative of a\n#' function. This function should have four inputs\n#'\n#' * a function, $f$, for which you want to compute the partial derivative\n#' * an array, *p*, specifying the values of all input arguments to $f$ at\n#' the point where you want $\\frac{\\partial f}{\\partial p_i}$ computed\n#' * the index, $i$, of the variable with respect to which you want to calculate the partial derivative of $f$\n#' * a step size with default value 0.001\n#'\n#' Hint: you will need to `copy` and modify `p` within `partial`.\n\nfunction partial(fn, p, i, h=0.001)\n p = copy(p)\n a = fn(p...)\n p[i] += 1*h\n b = fn(p...)\n Ans = (b - a)/(1*h)\n\n return Ans\nend\n\nfn1(x,y,z) = x^2 * y * z\nfn1(xyz) = fn1(xyz...)\n#' $\\frac{\\partial fn_1}{\\partial x}$\nfn1px(x,y,z) = 2*x*y*z\nfn1py(x,y,z) = x^2*1*z\nfn1pz(x,y,z) = x^2*y*1\n\n\n\nx = 1.0\ny = 2.0\nz = 1.0\nxyz = [x, y, z]\n\nactual = partial(fn1, xyz, 1, 1.e-6)\nexpected = fn1px(xyz...)\n@assert isapprox(actual, expected; atol=1.e-4)\n\n\nusing FiniteDifferences\n# what the? Why do I have to use 2nd order central difference for\n# first derivate?\nactual = central_fdm(2,1)(x->fn1(x,y,z), 1.0) \n@assert isapprox(actual, expected; atol=1.e-6)\nactual = forward_fdm(2,1)(x->fn1(x,y,z), 1.0)\n@assert isapprox(actual, expected; atol=1.e-6)\n\nusing FiniteDiff\n# This is right\nactual = FiniteDiff.finite_difference_derivative(x->fn1(x,y,z), 1.0)\n@assert isapprox(actual, expected; atol=1.e-6)\n\n# GradientFiniteDiff.finite_difference_gradient(fn1, xyz)\n(gx,gy,gz) = FiniteDiff.finite_difference_gradient(fn1, xyz)\n\nw = 1.1\nx = 2.2\ny = 3.3\nz = 4.4\nfn2(w,x,y,z) = w * x^2 * y^3 * z^4\nfn2pw(w,x,y,z) = x^2 * y^3 * z^4\nfn2px = w * 2 * x * y^3 * z^4\nfn2py = w * x^2 * 3 * y^2 * z^4\nfn2pz = w * x^2 * y^3 * 4 * z^3\n\n# Test partial()\nexpected = fn2pw(w,x,y,z)\n\nactual = partial(fn2, [w,x,y,z], 1, 1.e-6)\n@assert isapprox(actual, expected; rtol=1.e-6)\n\nactual = FiniteDiff.finite_difference_derivative(\n w->fn2(w,x,y,z), w)\n@assert isapprox(actual, expected; rtol=1.e-9)\n\n#' ## Gradient descent in 2 dimensions\n#' It turns out that the gradient vector of the function $L_2(w, b)$ gives the\n#' direction in the plane $(w, b)$ in which the function $L_2$\n#' **increases fastest**.\n#'\n#' In other words, if you start at the position $(w_0, b_0)$ and take a small\n#' step of length $\\eta$ to new values $(w', b')$, the value of the function\n#' will change to a new value $L_2(w', b')$. How do we get to the minimum of\n#' $L_2(w, b)$ as fast as possible? We want to step in the direction where\n#' $L_2$ decreases fastest!\n#'\n#' In multivariable calculus courses, it is shown that $L_2$ will *increase*\n#' fastest if you take a step **in the direction of the gradient\n#' $\\nabla L_2(w, b)$**! To decrease $L_2$ the fastest, we should take a step\n#' in the *opposite* direction, $-\\nabla L_2(w, b)$.\n#'\n#' Let's now generalize the gradient descent algorithm that we wrote previously\n#' to work our 2-dimensional function.\n\n#' #### Exercise 16\n#'\n#' Extend your 1D implementation of `gradient_descent` so that it will minimize the function $L_2$.\n#'\n#' Requirements:\n#'\n#' * Your new method for `gradient_descent` will take four input arguments: the function $f$ for which you seek the minimum, the range of values for $f$'s first input argument that you will consider, the range of values for $f$'s second input argument that you will consider, and a tolerance that will specify the maximum allowable step size, $\\sum_i \\eta \\frac{\\partial f}{\\partial p_i}$\n#'\n#' * Use $\\eta = .01$. For example, for a function $f(w, b)$, update $w$ such that $w_{n+1} = w_n - 0.01 * \\frac{\\partial f}{\\partial w_n}$\n#'\n#' * Seed `gradient_descent` with the starting coordinates [-2.0, -2.0], i.e. $w_0 = -2.0$ and $b_0 = -2.0$.\n#'\n#' * Return all steps (their coordinates) taken during gradient descent and the values of the loss function at these coordinates.\n#'\n#' Once you have done this, execute\n#'\n#' ```julia\n#' gradient_descent(L2, -2:0.02:2, -2:0.02:2, .001)\n#' ```\n#'\n#' How many steps were taken by gradient descent?\n#'\n#' Hint: Do not count your starting coordinates `[-2.0, -2.0]` as a step.\n\nfunction gradient_descent(fn, wrange, brange; tol=0.001, eta=0.01)\n w = wrange[1]\n b = brange[1]\n\n w_steps = []\n b_steps = []\n f_steps = []\n\n while true\n (gfw,gfb) = FiniteDiff.finite_difference_gradient(fn, [w,b])\n w = w - eta * gfw\n b = b - eta * gfb\n f = fn(w,b)\n\n push!(w_steps, w)\n push!(b_steps, b)\n push!(f_steps, f)\n\n if abs(eta*gfw + eta*gfb) < tol\n return (w_steps,b_steps,f_steps)\n end\n \n end\n\n #return (w_steps, b_steps, f_steps)\nend\n\nxs = [2, -3, -1, 1]\nys = [0.8, 0.3, 0.4, 0.4]\n\nsigma(x) = 1/(1 + exp(-x))\ng(x,w,b) = sigma(w*x) + b\n\nL2(w,b) = sum(abs2, ys .- g.(xs,w,b))\nL2(xb) = L2(xb...)\nwrange = -2.0:0.02:2.0\nbrange = -2.0:0.02:2.0\n(w,b,fv) = gradient_descent(L2, wrange, brange)\n\n#' #### Exercise 17\n#'\n#' Use the `surface` and `scatter!` commands to illustrate the path taken by\n#' `gradient_descent` from [-2.0, -2.0] to wherever the algorithm terminates.\n#'\n#' Where do the scattered points representing the steps of gradient descent\n#' appear the most dense?\n#'\n#' A) Near the starting point at [-2.0, -2.0]
\n#' B) Near the point [-1.8, 0] where $C_2$ appears nearly flat
\n#' C) Near the minimum of $C_2$\n\n#' #### Solution\n#'\n#' B) The following code will plot the function `C2` with an overlay of the\n#' steps taken in gradient descent\n#'\n#' You can change the frequency with which steps are plotted by changing the\n#' iterator in the `for` loop from `1:length(values)` to\n#' `1:n:length(values)` for varying values of `n`. This will allow you to\n#' see more clearly where steps are densely or sparsely packed.\n\nfig = surface(-2:0.02:2, -2:0.02:2, L2, alpha=0.6, xlabel = \"w\",\n ylabel = \"b\", zlabel = \"C2(w, b)\", size=(800,800))\nfor i in 1:10:length(fv)\n scatter!(fig, [w[i]], [b[i]], [fv[i]], markersize=1)\nend \nfig", "meta": {"hexsha": "f3f4911112028faac4a666b09d4f41f3afd28a85", "size": 23807, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "JuliaAcademy/FundMachLearn/ch0900quiz.jl", "max_stars_repo_name": "ykyang/org.allnix.julia", "max_stars_repo_head_hexsha": "58933a5848dec81c53d591b4163e9a70df62ddd8", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "JuliaAcademy/FundMachLearn/ch0900quiz.jl", "max_issues_repo_name": "ykyang/org.allnix.julia", "max_issues_repo_head_hexsha": "58933a5848dec81c53d591b4163e9a70df62ddd8", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "JuliaAcademy/FundMachLearn/ch0900quiz.jl", "max_forks_repo_name": "ykyang/org.allnix.julia", "max_forks_repo_head_hexsha": "58933a5848dec81c53d591b4163e9a70df62ddd8", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.4346049046, "max_line_length": 388, "alphanum_fraction": 0.6441802831, "num_tokens": 7700, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9362850004144266, "lm_q2_score": 0.9032942047513692, "lm_q1q2_score": 0.8457408148699849}} {"text": "### A Pluto.jl notebook ###\n# v0.15.1\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 4cb24885-2f27-400e-a20d-b8152f4c59d3\nusing LinearAlgebra\n\n# ╔═╡ 5e4375cd-a2a4-4027-88a0-a7ae49fbde1c\n# we write a function that returns a root of func in the\n# interval [low, high] with relative error tol, assuming such a root exists\nfunction binarySearch(func, low, high, tol)\n\n\t# we repeat until the tolorance condition is met\n\twhile abs(high - low)/2 > tol * abs(high + low)/2\n\n\t\t# we define mid as the midpoint of the interval [low, high]\n\t\tmid = (low + high)/2\n\n\t\t# we check if the value of func at high and mid have the same sign\n\t\tif func(high) * func(mid) > 0\n\n\t\t\t# and reassign [low, high] accordingly\n\t\t high = mid\n\t\telse\n\t\t low = mid\n\t\tend\n\tend\n\n\t# we return the avarage of low and high as the approximate root\n\treturn (low + high)/2;\nend\n\n# ╔═╡ 4533a6ef-85b5-44e2-a381-f60182886903\n# we write an improved version of binarySearch that evaluates func fewer times\n# and checks if there is definitely a root in the initial interval\nfunction binarySearchV2(func, low, high, tol)\n\n # we now also store the values of func at low and high\n f_low = func(low)\n f_high = func(high)\n\n # we check if the initial interval definitely contains a root\n if f_low * f_high > 0\n error(\"func(low) and func(high) must have different sign\")\n end\n \n # the while loop is similar to before\n while abs(high - low)/2 > tol * abs(low + high)/2\n mid = (low + high)/2\n\n # we only have one evaluation of func per loop\n f_mid = func(mid)\n \n if f_high * f_mid > 0\n high = mid\n f_high = f_mid\n # in this case f_low stays the same\n else\n low = mid\n f_low = f_mid\n # in this case f_high stays the same\n end\n end\n \n return (low + high)/2\nend\n\n# ╔═╡ 977bcb22-8089-49f3-931f-be39cbcd950c\nf(x) = exp(x)-4x;\n\n# ╔═╡ 25a35c4d-76aa-406b-b6ba-6cbd960c08b9\nzeta = 1e-7;\n\n# ╔═╡ d6406dcf-fda4-4fc3-ba27-8f33ac760c80\nmd\"Both implementations find the same approxiamte root in the interval $(0,1)$.\"\n\n# ╔═╡ 067c0a97-da55-46b9-9476-396b7d60968f\nbinarySearch(f, 0, 1, zeta)\n\n# ╔═╡ 8a15ea59-e2fc-4534-84dd-52b3c11abeac\nbinarySearchV2(f, 0, 1, zeta)\n\n# ╔═╡ 031bddca-90a9-43cd-b5d4-e6ccb41f501e\nmd\"Let's test the case where there is no root in the initial interval. Note that binarySearch does not test whether there is a root it gives a bad answer: it is an **unreliable** function.\"\n\n# ╔═╡ e53a4200-0d48-4d0a-9b60-ac447eff62d4\nbinarySearch(f, 1, 2, zeta)\n\n# ╔═╡ 57435047-508b-419a-a1b2-8e5cfa4cbfaf\nmd\"BinarySearchV2 on the other hand includes the initial check.\"\n\n# ╔═╡ a9fd5fbd-8c6e-45b5-9821-ba64175ec3e0\nbinarySearchV2(f, 1, 2, zeta)\n\n# ╔═╡ aaf5575d-fb60-43b7-858c-46babc9401b1\nmd\"Let's try it for a more complicated function that involves large matrices. We construct a $n\\times n$ matrix with random numbers.\"\n\n# ╔═╡ ebe48c7f-bae4-4262-b792-0581a0757e0d\nn = 4000;\n\n# ╔═╡ 5827d97a-5cf3-4578-90ba-11d07afb84d6\nA = rand(n,n);\n\n# ╔═╡ ba659a53-9947-4e2d-b615-c3dcd71b3c18\nmd\"Make a symmetric matrix (for real matrices, M' is the transpose of M).\"\n\n# ╔═╡ f997e7f0-26d2-4f90-939b-4776a29d5f66\nB = A + A';\n\n# ╔═╡ 25142ed0-b0d2-440a-874b-e49fb7fd198c\ng(x) = eigmax(exp.(x*B))-2n;\n\n# ╔═╡ 8dfb5f2b-95ef-4701-b90e-2727aa2b2a32\nmd\"\"\"\nThis function computes a matrix with elements\n$A_{ij} = exp(x B_{ij} )$, computes the largest eigenvalue $\\lambda$ of $A$ and returns $\\lambda - 2n$.\n\"\"\"\n\n# ╔═╡ f0fdc7d6-9aa3-4a6c-b2e0-f45706d4ae30\ng(0), g(1)\n\n# ╔═╡ fe08f690-93a7-46bb-a110-2cb917d12991\nmd\"\"\"\nSince $g$ is continuous and $sign(g(0)) \\neq sign(g(1))$, it must have a root in the interval $(0,1)$. \n\"\"\"\n\n# ╔═╡ 0ff24d6c-9032-4743-9400-f90a70b0bc11\nbinarySearchV2(g, 0, 1, zeta)\n\n# ╔═╡ 00000000-0000-0000-0000-000000000001\nPLUTO_PROJECT_TOML_CONTENTS = \"\"\"\n[deps]\nLinearAlgebra = \"37e2e46d-f89d-539d-b4ee-838fcccc9c8e\"\n\"\"\"\n\n# ╔═╡ 00000000-0000-0000-0000-000000000002\nPLUTO_MANIFEST_TOML_CONTENTS = \"\"\"\n# This file is machine-generated - editing it directly is not advised\n\n[[Libdl]]\nuuid = \"8f399da3-3557-5675-b5ff-fb832c97cbdb\"\n\n[[LinearAlgebra]]\ndeps = [\"Libdl\"]\nuuid = \"37e2e46d-f89d-539d-b4ee-838fcccc9c8e\"\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╠═5e4375cd-a2a4-4027-88a0-a7ae49fbde1c\n# ╠═4533a6ef-85b5-44e2-a381-f60182886903\n# ╠═977bcb22-8089-49f3-931f-be39cbcd950c\n# ╠═25a35c4d-76aa-406b-b6ba-6cbd960c08b9\n# ╟─d6406dcf-fda4-4fc3-ba27-8f33ac760c80\n# ╠═067c0a97-da55-46b9-9476-396b7d60968f\n# ╠═8a15ea59-e2fc-4534-84dd-52b3c11abeac\n# ╟─031bddca-90a9-43cd-b5d4-e6ccb41f501e\n# ╠═e53a4200-0d48-4d0a-9b60-ac447eff62d4\n# ╟─57435047-508b-419a-a1b2-8e5cfa4cbfaf\n# ╠═a9fd5fbd-8c6e-45b5-9821-ba64175ec3e0\n# ╟─aaf5575d-fb60-43b7-858c-46babc9401b1\n# ╠═ebe48c7f-bae4-4262-b792-0581a0757e0d\n# ╠═5827d97a-5cf3-4578-90ba-11d07afb84d6\n# ╟─ba659a53-9947-4e2d-b615-c3dcd71b3c18\n# ╠═f997e7f0-26d2-4f90-939b-4776a29d5f66\n# ╠═4cb24885-2f27-400e-a20d-b8152f4c59d3\n# ╠═25142ed0-b0d2-440a-874b-e49fb7fd198c\n# ╟─8dfb5f2b-95ef-4701-b90e-2727aa2b2a32\n# ╠═f0fdc7d6-9aa3-4a6c-b2e0-f45706d4ae30\n# ╟─fe08f690-93a7-46bb-a110-2cb917d12991\n# ╠═0ff24d6c-9032-4743-9400-f90a70b0bc11\n# ╟─00000000-0000-0000-0000-000000000001\n# ╟─00000000-0000-0000-0000-000000000002\n", "meta": {"hexsha": "13f5833c72c3abf841bbdd38fbede6be8d8021f2", "size": 5133, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "1a/Root Finding/binaryTest.jl", "max_stars_repo_name": "sje30/catam-julia", "max_stars_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2021-07-13T12:55:37.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T14:40:23.000Z", "max_issues_repo_path": "1a/Root Finding/binaryTest.jl", "max_issues_repo_name": "sje30/catam-julia", "max_issues_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2021-07-11T21:35:47.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-25T12:10:58.000Z", "max_forks_repo_path": "1a/Root Finding/binaryTest.jl", "max_forks_repo_name": "sje30/catam-julia", "max_forks_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-10-13T21:00:47.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-13T21:00:47.000Z", "avg_line_length": 29.3314285714, "max_line_length": 189, "alphanum_fraction": 0.7108903176, "num_tokens": 2335, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070133672955, "lm_q2_score": 0.9294404008810105, "lm_q1q2_score": 0.8457043392685422}} {"text": "# Examples presented in class - Lecture 1\n# Example 4 - Classification\n\nusing JuMP, Ipopt\n\n# Generating random multivariate data.\ndata11 = rand(Normal(8, 4), 100)\ndata12 = rand(Normal(12, 2), 100)\ndata1 = [data11 data12]\n\ndata01 = rand(Normal(2, 6), 100)\ndata02 = rand(Normal(5, 2), 100)\ndata0 = [data01 data02]\n\nscatter(data1[:,1], data1[:,2], label=\"Pos. obs.\")\nscatter!(data0[:,1], data0[:,2], label=\"Neg. obs.\")\n\nfunction classify(data1, data0, δ)\n m = Model(IpoptOptimizer())\n\n\n @variable(m, u[1:100] >= 0)\n @variable(m, v[1:100] >= 0)\n @variable(m, a[1:2])\n @variable(m, b)\n\n @constraint(m, error1[i=1:100], sum(a[j]*data1[i,j] for j = 1:2) - b - u[i] <= -1)\n @constraint(m, error0[i=1:100], sum(a[j]*data0[i,j] for j = 1:2) - b + v[i] >= 1)\n\n# This parameter trades off accuracy and robustness for the classifier\n @objective(m, Min, sum(u[i] for i=1:100) + sum(v[i] for i=1:100)\n + δ*(sum(a[j]^2 for j=1:2)))\n\n solve(m);\n\n return getvalue(a), getvalue(b), getvalue(u), getvalue(v)\nend\n\na, b, u1, v1 = classify(data1, data0, 0.01);\n\n# Calculating the points on plane defined by a and b\nx1 = linspace(-10,20,100);\nx2 = (b - a[1]*x1)/ a[2];\n# boundaries of the slab\nx2u = (b + 1 - a[1]*x1)/ a[2];\nx2d = (b - 1 - a[1]*x1)/ a[2];\nplot!(x1, [x2u x2 x2d],\n xlabel = \"x_1\",\n ylabel = \"x_2\",\n label= [ \"\", \"Classifier\", \"\"] ,\n color=[:green :green :green],\n linestyle=[ :dot :solid :dot],\n )\n\n# another classifier, with δ=50\na, b, u2, v2 = classify(data1, data0, 100);\nx2 = (b - a[1]*x1)/ a[2];\nx2u = (b + 1 - a[1]*x1)/ a[2];\nx2d = (b - 1 - a[1]*x1)/ a[2];\nplot!(x1, [x2u x2 x2d],\n xlabel = \"x_1\",\n ylabel = \"x_2\",\n legend = false,\n color=[:purple :purple :purple],\n linestyle=[:dot :solid :dot],\n )\n", "meta": {"hexsha": "5a49622ffb163acd8d2bc86cb32c050568168670", "size": 1771, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/lectures/Lecture_1/Other/L1_Ex4-Classification.jl", "max_stars_repo_name": "gamma-opt/CourseParser.jl", "max_stars_repo_head_hexsha": "be59cf09c2c8b34373a6cd0f972f46528c4233dc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/lectures/Lecture_1/Other/L1_Ex4-Classification.jl", "max_issues_repo_name": "gamma-opt/CourseParser.jl", "max_issues_repo_head_hexsha": "be59cf09c2c8b34373a6cd0f972f46528c4233dc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/lectures/Lecture_1/Other/L1_Ex4-Classification.jl", "max_forks_repo_name": "gamma-opt/CourseParser.jl", "max_forks_repo_head_hexsha": "be59cf09c2c8b34373a6cd0f972f46528c4233dc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.4328358209, "max_line_length": 86, "alphanum_fraction": 0.5748164879, "num_tokens": 695, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.956634196290671, "lm_q2_score": 0.8840392848011834, "lm_q1q2_score": 0.8457022107051597}} {"text": "\"\"\"\n LogUniform(a,b)\n\nA positive random variable `X` is log-uniformly with parameters `a` and `b` if the logarithm of `X` is `Uniform(log(a), log(b))`.\nThe *log uniform* distribution is also known as *reciprocal distribution*.\n```julia\nLogUniform(1,10)\n```\nExternal links\n\n* [Log uniform distribution on Wikipedia](https://en.wikipedia.org/wiki/Reciprocal_distribution)\n\"\"\"\nstruct LogUniform{T<:Real} <: ContinuousUnivariateDistribution\n a::T\n b::T\n LogUniform{T}(a::T, b::T) where {T <: Real} = new{T}(a, b)\nend\n\nfunction LogUniform(a::T, b::T; check_args::Bool=true) where {T <: Real}\n check_args && @check_args(LogUniform, 0 < a < b)\n LogUniform{T}(a, b)\nend\n\nLogUniform(a::Real, b::Real; check_args::Bool=true) = LogUniform(promote(a, b)...; check_args=check_args)\n\nconvert(::Type{LogUniform{T}}, d::LogUniform) where {T<:Real} = LogUniform(T(d.a), T(d.b); check_args=false)\nBase.minimum(d::LogUniform) = d.a\nBase.maximum(d::LogUniform) = d.b\n\n#### Parameters\nparams(d::LogUniform) = (d.a, d.b)\npartype(::LogUniform{T}) where {T<:Real} = T\n\n#### Statistics\n\nfunction mean(d::LogUniform)\n a, b = params(d)\n (b - a) / log(b/a)\nend\nfunction var(d::LogUniform)\n a, b = params(d)\n log_ba = log(b/a)\n (b^2 - a^2) / (2*log_ba) - ((b-a)/ log_ba)^2\nend\nmode(d::LogUniform) = d.a\nmodes(d::LogUniform) = partype(d)[]\n\nfunction entropy(d::LogUniform)\n a,b = params(d)\n log(a * b) / 2 + log(log(b / a))\nend\n#### Evaluation\nfunction pdf(d::LogUniform, x::Real)\n x1, a, b = promote(x, params(d)...) # ensure e.g. pdf(LogUniform(1,2), 1f0)::Float32\n res = inv(x1 * log(b / a))\n return insupport(d, x1) ? res : zero(res)\nend\nfunction cdf(d::LogUniform, x::Real)\n x1, a, b = promote(x, params(d)...) # ensure e.g. cdf(LogUniform(1,2), 1f0)::Float32\n x1 = clamp(x1, a, b)\n return log(x1 / a) / log(b / a)\nend\nlogpdf(d::LogUniform, x::Real) = log(pdf(d,x))\n\nfunction quantile(d::LogUniform, p::Real)\n p1,a,b = promote(p, params(d)...) # ensure e.g. quantile(LogUniform(1,2), 1f0)::Float32\n exp(p1 * log(b/a)) * a\nend\n\nfunction kldivergence(p::LogUniform, q::LogUniform)\n ap, bp, aq, bq = promote(params(p)..., params(q)...)\n finite = aq <= ap < bp <= bq\n res = log(log(bq / aq) / log(bp / ap))\n return finite ? res : oftype(res, Inf)\nend\n", "meta": {"hexsha": "da4ef3209fd66949f32f6deee120a536f47a80e8", "size": 2300, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/univariate/continuous/loguniform.jl", "max_stars_repo_name": "pdeffebach/Distributions.jl", "max_stars_repo_head_hexsha": "8aea3cc82ee2f8ffe1e8cd754e7fcd99369c7a1c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/univariate/continuous/loguniform.jl", "max_issues_repo_name": "pdeffebach/Distributions.jl", "max_issues_repo_head_hexsha": "8aea3cc82ee2f8ffe1e8cd754e7fcd99369c7a1c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/univariate/continuous/loguniform.jl", "max_forks_repo_name": "pdeffebach/Distributions.jl", "max_forks_repo_head_hexsha": "8aea3cc82ee2f8ffe1e8cd754e7fcd99369c7a1c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.2631578947, "max_line_length": 129, "alphanum_fraction": 0.6273913043, "num_tokens": 780, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.933430805473952, "lm_q2_score": 0.9059898114992677, "lm_q1q2_score": 0.8456787994989554}} {"text": "using Plots\nusing Random\nusing Pkg; Pkg.add(\"LaTeXStrings\")\nusing LaTeXStrings\n\n# Set seed\nRandom.seed!(123)\n\n# Generate MC points\nn = 2000\npoints_x = rand(n)\npoints_y = rand(n)\n\n# Label points in circle\nlabels = [x^2 + y^2 <= 1 for (x,y) in zip(points_x, points_y)]\n\n# Estaimte of pi\nπ_estimate = 4*sum(labels)/n\n\n# Create plot of MC points\nplot(points_x, points_y, seriestype = :scatter,\n group = .!labels,\n title = L\"\\textrm{Monte Carlo } \\pi \\textrm{ estimation}\",\n legend = false,\n bg = RGB(247/255, 236/255, 226/255))\n\n# Add semi-circle quadrant plot\ncircle_x = 0:0.01:1\ncircle_y = sqrt.(1 .- circle_x.^2)\nplot!(circle_x, circle_y, seriestype= :line, legend = false, lc = :yellow)\n\n# Generate plots comparing π estimate across multiple n\nn_values = [5000, 10000, 100000, 1e6, 2e6]\nπ_estimates_error = zeros(5)\n\nfor i in 1:5\n n_value = Int64(n_values[i])\n points_x = rand(n_value)\n points_y = rand(n_value)\n estimate = 4*sum([x^2 + y^2 <= 1 for (x,y) in zip(points_x, points_y)])/\n n_value\n π_estimates_error[i] = 100*abs(π - estimate)/π\nend\n\nprintln(π_estimates_error)\n\n# savefig(\"../images/pi-monte-carlo.svg\")\n\n# Plot area under circle curve\nplot(circle_x, circle_y, legend = false,\n fillrange = [zeros(length(circle_x)), circle_y],\n alpha = .5,\n lc = :yellow)\n", "meta": {"hexsha": "98e474f6ec8aa60fbfc56719354b09a047a6d07c", "size": 1316, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "content/stochastic-approximations/code/pi_mc_plot.jl", "max_stars_repo_name": "seanrattana/courses", "max_stars_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2019-08-21T07:33:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-24T15:46:58.000Z", "max_issues_repo_path": "content/stochastic-approximations/code/pi_mc_plot.jl", "max_issues_repo_name": "seanrattana/courses", "max_issues_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-23T06:04:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-26T12:47:12.000Z", "max_forks_repo_path": "content/stochastic-approximations/code/pi_mc_plot.jl", "max_forks_repo_name": "seanrattana/courses", "max_forks_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-08-18T21:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T19:14:33.000Z", "avg_line_length": 24.3703703704, "max_line_length": 76, "alphanum_fraction": 0.6717325228, "num_tokens": 432, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9465966732132748, "lm_q2_score": 0.8933094152856196, "lm_q1q2_score": 0.8456037206594632}} {"text": "using NumericalMethodsforEngineers\nusing Test\n\n@sym begin\n ClearAll(xi, yi, Y, Ydot, Ydotdot, ytilde1, ytilde2, R)\n xi = [0, 1//3, 2//3, 1]\n yi = [0, a, b, 1]\n Y(x_) := LagrangePolynomial(xi, yi)\n #\n # Can be re-formulated as Y(x) = F(x) + C1(a) * Ψ1(x) + C2(a) * Ψ2(x)\n #\n # F(x_) := (1/2)*(9x^3 - 9x^2 + 2x)\n # C1(a_) := (27/2)*(a-b)\n # Ψ1(x_) := x^3 - x^2\n # C2(a_) := -(9/2)*(2a-b)\n # Ψ2(x_) := x^2 - x\n #\n # Y(x_) := F(x) + C1(a)*Ψ1(x) + C2(a)*Ψ2(x)\n #\n R(x_) = Simplify(D(Y(x), x, 2) - 3*x - 4*Y(x))\n sol = Solve([R(xi[2]), R(xi[3])], [a, b])\n ytilde(x_) = Y(x) ./ Flatten(sol)\n SetJ(r, ToString(Simplify(R(x))))\n SetJ(s, \"[$(sol[1][1][2]), $(sol[1][2][2])]\")\n SetJ(t, ToString(Simplify(ytilde(x))))\nend\n\nprintln(\"\\n\\nExample 7.15b: y'' = 3x + 4y, y(0)=0, y(1)=1\")\nprintln(\"by 2-point collocation Weighted Residual Method\")\n@sym Println(\"\\nY(x): \", Y(x), \"\\n\")\n@sym Println(\"R(x) = \", R(x), \"\\n\")\nprintln(\"(a, b) = $(s)\\n\")\n@sym Println(\"ytilde_2pt_collocation(x) = \", Simplify(ytilde(x)), \"\\n\")\nprintln()\n\n@eval rf_2pt_collocation_2(x, a, b) = $(Meta.parse(r))\n@eval ytilde_2pt_collocation(x) = $(Meta.parse(t))\n@eval (a, b) = $(Meta.parse(s))\n\n@test t == \"(1/806)*x*(239 - 252x + 819*x^2)\"\n@test rf_2pt_collocation_2(1//3, a, b) <= eps()\n@test rf_2pt_collocation_2(2//3, a, b) <= eps()\n", "meta": {"hexsha": "c5b6e95a314819fba6f1fdd31e9cc6e59184457c", "size": 1315, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/ch07/WRM/Ex.7.15b.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/NumericalMethodsforEngineers.jl-00e1d38a-71a9-5665-8612-32ae585a75a3", "max_stars_repo_head_hexsha": "e230c3045d98da0cf789e4a6acdccfbfb21ef49e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-23T18:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-25T03:32:45.000Z", "max_issues_repo_path": "examples/ch07/WRM/Ex.7.15b.jl", "max_issues_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_issues_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-23T21:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-27T23:14:46.000Z", "max_forks_repo_path": "examples/ch07/WRM/Ex.7.15b.jl", "max_forks_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_forks_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2018-10-27T14:13:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-20T18:54:06.000Z", "avg_line_length": 30.5813953488, "max_line_length": 71, "alphanum_fraction": 0.5376425856, "num_tokens": 601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9449947132556619, "lm_q2_score": 0.8947894618940992, "lm_q1q2_score": 0.8455713109668023}} {"text": "# # Portfolio Optimization\n#\n# In this problem, we will find the portfolio allocation that minimizes risk while achieving a given expected return $R_\\text{target}$.\n#\n# Suppose that we know the mean returns $\\mu \\in \\mathbf{R}^n$ and the covariance $\\Sigma \\in \\mathbf{R}^{n \\times n}$ of the $n$ assets. We would like to find a portfolio allocation $w \\in \\mathbf{R}^n$, $\\sum_i w_i = 1$, minimizing the *risk* of the portfolio, which we measure as the variance $w^T \\Sigma w$ of the portfolio. The requirement that the portfolio allocation achieve the target expected return can be expressed as $w^T \\mu >= R_\\text{target}$. We suppose further that our portfolio allocation must comply with some lower and upper bounds on the allocation, $w_\\text{lower} \\leq w \\leq w_\\text{upper}$.\n#\n# This problem can be written as\n#\n# $$\n# \\begin{array}{ll}\n# \\text{minimize} & w^T \\Sigma w \\\\\n# \\text{subject to} & w^T \\mu >= R_\\text{target} \\\\\n# & \\sum_i w_i = 1 \\\\\n# & w_\\text{lower} \\leq w \\leq w_\\text{upper}\n# \\end{array}\n# $$\n#\n# where $w \\in \\mathbf{R}^n$ is our optimization variable.\n\nusing Convex, SCS\n\n## generate problem data\nμ = [11.5; 9.5; 6]/100 #expected returns\nΣ = [166 34 58; #covariance matrix\n 34 64 4;\n 58 4 100]/100^2\n\nn = length(μ) #number of assets \n\nR_target = 0.1\nw_lower = 0\nw_upper = 0.5;\n\n# If you want to try the optimization with more assets, uncomment and run the next cell. It creates a vector or average returns and a variance-covariance matrix that have scales similar to the numbers above.\n\n#=\nusing Random\nRandom.seed!(123)\n\nn = 15 #number of assets, CHANGE IT?\n\nμ = (6 .+ (11.5-6)*rand(n))/100 #mean\nA = randn(n,n)\nΣ = (A * A' + diagm(0=>rand(n)))/500; #covariance matrix\n=#\n\n#-\n\nw = Variable(n)\nret = dot(w,μ)\nrisk = quadform(w,Σ)\n\np = minimize( risk, \n ret >= R_target, \n sum(w) == 1, \n w_lower <= w, \n w <= w_upper )\n\nsolve!(p, SCS.Optimizer()) #use SCS.Optimizer(verbose = false) to suppress printing\n\n#-\n\n# Optimal portfolio weights:\nw.value\n\n#-\n\nsum(w.value)\n", "meta": {"hexsha": "ca15785a37d9512829205667101c2a88c5064791", "size": 2224, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "docs/examples_literate/portfolio_optimization/portfolio_optimization.jl", "max_stars_repo_name": "danspielman/Convex.jl", "max_stars_repo_head_hexsha": "fb7098c983553458a2e636e105f1fed97c72a7f2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/examples_literate/portfolio_optimization/portfolio_optimization.jl", "max_issues_repo_name": "danspielman/Convex.jl", "max_issues_repo_head_hexsha": "fb7098c983553458a2e636e105f1fed97c72a7f2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/examples_literate/portfolio_optimization/portfolio_optimization.jl", "max_forks_repo_name": "danspielman/Convex.jl", "max_forks_repo_head_hexsha": "fb7098c983553458a2e636e105f1fed97c72a7f2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.231884058, "max_line_length": 617, "alphanum_fraction": 0.6092625899, "num_tokens": 668, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.949669363129097, "lm_q2_score": 0.8902942363098473, "lm_q1q2_score": 0.8454851603938784}} {"text": "# # [Implicit Nonlinear Dynamics : Autoregulation](@id autoregulation)\n#\n# The following is another example on how to use the [`ImplicitOptimizer`](@ref) describing a biological autoregulation process using\n# two coupled implicit equations. \n\nusing DataDrivenDiffEq\nusing ModelingToolkit\nusing OrdinaryDiffEq\nusing LinearAlgebra\n#md using Plots\n#md gr()\n\n@parameters begin\n t\n α = 1.0\n β = 1.3\n γ = 2.0\n δ = 0.5\nend\n\n@variables begin\n x[1:2](t) = [20.0; 12.0]\nend\n\nx = collect(x)\nD = Differential(t)\n\neqs = [\n D(x[1]) ~ α/(1+x[2])-β*x[1];\n D(x[2]) ~ γ/(1+x[1])-δ*x[2];\n]\n\nsys = ODESystem(eqs, t, x, [α, β, γ, δ], name = :Autoregulation)\n\nx0 = [x[1] => 20.0; x[2] => 12.0] \n\ntspan = (0.0, 5.0) \n\nde_problem = ODEProblem(sys, x0, tspan) \nde_solution = solve(de_problem, Tsit5(), saveat = 0.005);\n#md plot(de_solution)\n\n# As always, we start by defining a [DataDrivenProblem](@ref problem) and a sufficient basis for sparse regression.\n\ndd_prob = ContinuousDataDrivenProblem(de_solution)\n\neqs = [\n polynomial_basis(x, 4); D.(x); x .* D(x[1]); x .* D(x[2])\n ]\n \nbasis = Basis(eqs, x, independent_variable = t, implicits = D.(x))\n\n#md plot(dd_prob)\n\n# Next to varying over different sparsity penalties, we want also to batch our data using the [DataSampler](@ref).\n# We define a train-test split of 80-20 for our data and batch the resulting training data into 10 minibatches, allowing \n# shuffled and repeated values. Our goal is to find the model with the minimal error.\n\nsampler = DataSampler(\n Split(ratio = 0.8), Batcher(n = 10, shuffle = true, repeated = true, batchsize_min = 30)\n)\n\nres = solve(dd_prob, basis, ImplicitOptimizer(STLSQ(1e-1:1e-1:9e-1)), by = :min, sampler = sampler, digits = 1)\nprint(res) #hide\nprint(result(res)) #hide\n\n# We have recovered the correct equations of motion! \n# Another visual check using the problem and the result yields \n\nsystem = result(res)\n@named ode = ODESystem(equations(system), t, x, parameters(system));\node_prob = ODEProblem(ode, x0, tspan, parameter_map(res));\n\nprediction = solve(ode_prob, Tsit5(), saveat = 0.2);\n#md plot(de_solution, label = [\"Groundtruth\" nothing]) \n#md scatter!(prediction, label = [\"Prediction\" nothing]) \n\n\n#md # ## [Copy-Pasteable Code](@id autoregulation_copy_paste)\n#md #\n#md # ```julia\n#md # @__CODE__\n#md # ```\n\nfor r_ in [res] #src\n @test all(l2error(r_) .< 0.5) #src\n @test all(aic(r_) .> 1e3) #src\n @test all(determination(r_) .>= 0.9) #src\nend #src\n\n\n", "meta": {"hexsha": "8ac771d487cdc9bdd2169bc7150266aa5da01983", "size": 2476, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "docs/examples/6_autoregulation.jl", "max_stars_repo_name": "JuliaDiffEq/DataDrivenDiffEq.jl", "max_stars_repo_head_hexsha": "27f5537d2d04719b80f5785531b2b02e27a8208f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2019-11-22T17:20:54.000Z", "max_stars_repo_stars_event_max_datetime": "2020-03-07T17:10:27.000Z", "max_issues_repo_path": "docs/examples/6_autoregulation.jl", "max_issues_repo_name": "JuliaDiffEq/DataDrivenDiffEq.jl", "max_issues_repo_head_hexsha": "27f5537d2d04719b80f5785531b2b02e27a8208f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 53, "max_issues_repo_issues_event_min_datetime": "2019-11-22T17:20:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-03-22T15:11:55.000Z", "max_forks_repo_path": "docs/examples/6_autoregulation.jl", "max_forks_repo_name": "AlCap23/DataDrivenDiffEq", "max_forks_repo_head_hexsha": "3be031060d5a3cad08473506be60108e572ac0fa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2019-11-22T18:36:01.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T11:36:14.000Z", "avg_line_length": 26.9130434783, "max_line_length": 133, "alphanum_fraction": 0.6744749596, "num_tokens": 803, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9433475810629193, "lm_q2_score": 0.8962513800615312, "lm_q1q2_score": 0.8454765714053485}} {"text": "module RandomNumbers\n# Note:\n# Unless mentioned otherwise, the single variable continuous functions return\n# Float64 and the multiple variable continuous functions return Array{Float64, 1}.\n# Similarly, the single variable discrete functions return Int64 and the multiple\n# variable discrete functions return Array{Int64, 1}.\nexport rn_unif, rn_exp, rn_standard_cauchy, rn_standard_normal, rn_normal,\n\t rn_chi_sq, rn_bernoulli, rn_binom, rn_geo, rn_gamma, rn_beta\n\n# Generate a single uniform random variable between a and b.\nfunction rn_unif(a, b)\n\treturn a + rand() * (b - a)\nend\n\n# Generate n random uniform variables between a and b.\nfunction rn_unif(a, b, n::Int64)\n\treturn [rn_unif(a,b) for x in 1:n]\nend\n\n# Generate a single exponential random variable with rate parameter theta.\nfunction rn_exp(theta)\n\t# Don't allow negative theta values.\n\tif theta <= 0.0\n\t\tthrow(ArgumentError(\"Theta cannot be less than zero.\"))\n\tend\n\n\treturn -log(rn_unif(0, 1)) / theta\nend\n\n# Generate n exponential random variables with rate parameter theta.\nfunction rn_exp(theta, n::Int64)\n\treturn [rn_exp(theta) for x in 1:n]\nend\n\n# Generate a single standard Cauchy random variable.\nfunction rn_standard_cauchy()\n\treturn tan(pi * (rn_unif(0, 1) - .5))\nend\n\n# Generate n standard Cauchy random variables.\nfunction rn_standard_cauchy(n::Int64)\n\treturn [rn_standard_cauchy() for x in 1:n]\nend\n\n# Generate a single standard normal random variable.\nfunction rn_standard_normal()\n\tu1 = rand()\n\tu2 = rand()\n\n\treturn sqrt(-2 * log(u1)) * cos(2 * pi * u2)\nend\n\n# Generate n standard normal random variables.\nfunction rn_standard_normal(n::Int64)\n\treturn [rn_standard_normal() for x in 1:n]\nend\n\n# Generate a single normal random variable with mean mu and\n# standard deviation sigma.\nfunction rn_normal(mu, sigma)\n\t# Don't allow negative sigma.\n\tif sigma <= 0.0\n\t\tthrow(ArgumentError(\"Standard deviation cannot be negative.\"))\n\tend\n\n\treturn (sigma * rn_standard_normal()) + mu\nend\n\n# Generate n normal random variables with mean mu and standard deviation sigma.\nfunction rn_normal(mu, sigma, n::Int64)\n\treturn [rn_normal(mu, sigma) for x in 1:n]\nend\n\n# Generate a single chi-square random normal variable with k degrees of freedom.\n# Is specifying Integer in argument necessary?\nfunction rn_chi_sq(k::Integer)\n\t# Only accept positive natural numbers as input.\n\tif k < 1\n\t\tthrow(ArgumentError(\"Degrees of freedom must be at least 1.\"))\n\tend\n\n\t# Generate k squared random normal variables.\n\tnormals = rn_standard_normal(k).^2\n\n\t# Return the sum of the squared random normal variables.\n\treturn sum(normals)\nend\n\n# Generate n chi-square random normal variables with k degrees of freedom.\nfunction rn_chi_sq(k::Integer, n::Int64)\n\treturn [rn_chi_sq(k) for x in 1:n]\nend\n\n# Generate a single Bernoulli random variable with success probability\n# p as an Int64.\nfunction rn_bernoulli(p)\n\t# Only allow success probabilities in between 0 and 1.\n\tif !(0 < p < 1)\n\t\tthrow(ArgumentError(\"Success probability must be between 0 and 1.\"))\n\tend\n\n\t# Generate a single uniform random variable.\n\tunif = rn_unif(0, 1)\n\n\t# Return whether the random variable is below p.\n\treturn unif < p ? 1 : 0\nend\n\n# Generate n Bernoulli random variables with success probability p as\n# an Array{Int64, 1}.\nfunction rn_bernoulli(p, n::Int64)\n\treturn [rn_bernoulli(p) for x in 1:n]\nend\n\n# Generate a single binomial random variable with success probability p.\n# Returns Int64.\nfunction rn_binom(trials::Int64, p)\n\t# Only allow success probabilities in between 0 and 1.\n\tif !(0 < p < 1)\n\t\tthrow(ArgumentError(\"Success probability must be between 0 and 1.\"))\n\tend\n\n\t# Only allow positive numbers of trials.\n\tif trials < 1\n\t\tthrow(ArgumentError(\"Number of trials must be at least 1.\"))\n\tend\n\n\t# Generate Bernoulli random variable for each trial.\n\tberns = rn_bernoulli(p, trials)\n\n\t# Return the sum of successful trials.\n\treturn sum(berns)\nend\n\n# Generate n binomial random variables with success probability p as an\n# Array{Int64, 1}.\nfunction rn_binom(trials, p, n::Int64)\n\treturn [rn_binom(trials, p) for x in 1:n]\nend\n\n# Generate a single geometric random variable with success probability p.\nfunction rn_geo(p)\n\t# Only allow success probabilities in between 0 and 1.\n\tif !(0 < p < 1)\n\t\tthrow(ArgumentError(\"Success probability must be between 0 and 1.\"))\n\tend\n\n\tlambda = -log(1 - p)\n\n\texp = rn_exp(lambda)\n\n\treturn ifloor(exp) + 1\nend\n\n# Generate n geometric random variables with success probability p.\nfunction rn_geo(p, n::Int64)\n\treturn [rn_geo(p) for x in 1:n]\nend\n\n# Generate a single gamma random variable with shape parameter alpha and rate\n# parameter beta. This function implements Marsaglia and Tsang's method for \n# generating gamma random variables.\nfunction rn_gamma(alpha, beta)\n\tif alpha <= 0 || beta <= 0\n\t\tthrow(ArgumentError(\"Alpha and beta must be greater than zero.\"))\n\tend\n\n\tif (alpha >= 1)\n\t\t# Set two constants used in the algorithm.\n\t\td = alpha - (1.0 / 3.0)\n\t\tc = 1.0 / sqrt(9.0 * d)\n\n\t\t# We need to loop until we generate appropriate random numbers\n\t\twhile true\n\t\t\t# Generate Z~N(0,1).\n\t\t\tz = rn_standard_normal()\n\n\t\t\t# We need a z that will never result in a negative v.\n\t\t\tif z > -1.0 / c\n\t\t\t\tv = (1.0 + c * z)^3\n\n\t\t\t\t# Generate U~U(0,1).\n\t\t\t\tu = rn_unif(0, 1)\n\n\t\t\t\t# Compute the upper bound for log(u).\n\t\t\t\tu_bound = (.5 * z^2) + d - (d * v) + (d * log(v))\n\n\t\t\t\t# If this condition holds we've found a suitable result.\n\t\t\t\tif log(u) < u_bound\n\t\t\t\t\treturn (d * v) / beta\n\t\t\t\tend\n\t\t\tend\n\t\tend\n\telse\n\t\t# When 0 < alpha < 1 we can return a transformed answer.\n\t\tx = rn_gamma(alpha + 1, beta)\n\t\tu = rn_unif(0, 1)\n\t\treturn x * u ^ (1 / alpha)\n\tend\nend\n\n# Generate n gamma random variables with shape parameter alpha and rate\n# parameter beta.\nfunction rn_gamma(alpha, beta, n::Int64)\n\treturn [rn_gamma(alpha, beta) for x in 1:n]\nend\n\n# Generate a beta random variable with shape parameters alpha and beta.\n# Note that this does not use the rejection method.\nfunction rn_beta(alpha, beta)\n\tif alpha <= 0 || beta <= 0\n\t\tthrow(ArgumentError(\"Alpha and beta must be greater than zero.\"))\n\tend\n\n\t# Generate two independent gamma variables\n\tx = rn_gamma(alpha, 1)\n\ty = rn_gamma(beta, 1)\n\n\treturn x / (x + y)\nend\n\nfunction rn_beta(alpha, beta, n::Int64)\n\treturn [rn_beta(alpha, beta) for x in 1:n]\nend\n\nend", "meta": {"hexsha": "2bdb1fcda507e4d43dfa372d2796f82b9defdf75", "size": 6262, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "RandomNumbers.jl", "max_stars_repo_name": "andrewjlm/StatComputing", "max_stars_repo_head_hexsha": "b18167f6ecd3e1689316a3f6077e50bde7dfaa03", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "RandomNumbers.jl", "max_issues_repo_name": "andrewjlm/StatComputing", "max_issues_repo_head_hexsha": "b18167f6ecd3e1689316a3f6077e50bde7dfaa03", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "RandomNumbers.jl", "max_forks_repo_name": "andrewjlm/StatComputing", "max_forks_repo_head_hexsha": "b18167f6ecd3e1689316a3f6077e50bde7dfaa03", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.7079646018, "max_line_length": 82, "alphanum_fraction": 0.7226125838, "num_tokens": 1734, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9511422241476943, "lm_q2_score": 0.8887587875995482, "lm_q1q2_score": 0.8453360099682425}} {"text": "# # Consumption Savings Problem\n# In this case study, a household endowed with ``B_0`` dollars of wealth\n# must decide how much to consume and save \n# to maximize its utility over its finite lifecycle. \n\n# ## Formulation\n\n# The corresponding dynamic optimization problem is expressed:\n# ```math\n# \\begin{aligned}\n# \t&V(B_0,0) = &&\\underset{c(t), B(t)}{\\text{max}} \\int_{t = 0}^{t=T} e^{-\\rho t} u(c(t)) dt \\\\\n# \t&\\text{s.t.} \\\\\n# \t&&& \\frac{dB}{dt} = r \\times B(t) - c(t), && t \\in [0,T] \\\\\n# &&& B(0) = B_0 \\\\\n# &&& B(T) = 0\n# \\end{aligned}\n# ```\n# where \n# the household lives during time ``t \\in [0,T]``,\n# the state variable ``B(t)`` is the household's stock of wealth at time ``t``,\n# the choice variable ``c(t)`` is the household's consumption at time ``t``,\n# ``r`` is the interest rate,\n# ``B_0`` is the household's wealth endowment (initial condition),\n# ``B(T) = 0`` is the terminal condition,\n# and ``T`` is the time\n# horizon.\n\n# ## Model Definition\n\n# Let's implement this in `InfiniteOpt` and first import the packages we need:\nusing InfiniteOpt, Ipopt\n\n# We set the preference and constraint parameters:\nρ = 0.025 # discount rate \nk = 100.0 # utility bliss point \nT = 10.0 # life horizon \nr = 0.05 # interest rate\nB0 = 100.0 # endowment\nu(c; k=k) = -(c - k)^2 # utility function \ndiscount(t; ρ=ρ) = exp(-ρ*t) # discount function \nBC(B, c; r=r) = r*B - c # budget constraint \n\n# We set the hyperparameters:\nopt = Ipopt.Optimizer # desired solver\nns = 1_000; # number of points in the time grid\n\n# We initialize the infinite model and choose the Ipopt solver:\nm = InfiniteModel(opt)\n\n# Let's specify our infinite parameter which is time ``t \\in [0, T]``:\n@infinite_parameter(m, t in [0, T], num_supports = ns)\n\n# Now let's specify the variables:\n@variable(m, B, Infinite(t)) ## state variables \n@variable(m, c, Infinite(t)) ## control variables \n\n# Specify the objective:\n@objective(m, Max, integral(u(c), t, weight_func = discount)) \n\n# Set the initial/terminal conditions:\n@constraint(m, B == B0, DomainRestrictions(t => 0)) \n@constraint(m, B == 0, DomainRestrictions(t => T)) \n\n# Set the budget constraint:\n@constraint(m, c1, deriv(B, t) == BC(B, c; r=r))\n\n# ## Problem Solution\n\n# Optimize the model:\noptimize!(m)\ntermination_status(m)\n\n# Extract the results:\nc_opt = value(c)\nB_opt = value(B)\nts = supports(t)\nopt_obj = objective_value(m) # V(B0, 0)\n\n# Plot the results:\nusing Plots\nix = 2:(length(ts)-1) # index for plotting \nplot(ts[ix], B_opt[ix], lab = \"B: wealth balance\")\nplot!(ts[ix], c_opt[ix], lab = \"c: consumption\")\n\n# That's it, now we have our optimal trajectory!\n\n# This very simple problem has a closed form solution: \nλ1 = exp((r)T)\nλ2 = exp(-(r-ρ)T)\nden = (λ1-λ2)r\nΩ1 = (k + (r*B0-k)λ2)/den \nΩ2 = (k + (r*B0-k)λ1)/den \nc0 = r*B0 + (r)Ω1 + (r-ρ)Ω2\nBB(t; k=k,r=r,ρ=ρ,Ω1=Ω1,Ω2=Ω2) = (k/r) - Ω1*exp((r)t) + Ω2*exp(-(r-ρ)t)\ncc(t; k=k,r=r,ρ=ρ,c0=c0) = k + (c0-k)*exp(-(r-ρ)t)\n\n# Compare the solution given by `InfiniteOpt` with the closed form:\nplot(legend=:topright);\nplot!(ts[ix], c_opt[ix], color = 1, lab = \"c: consumption, InfiniteOpt\");\nplot!(ts[ix], cc, color = 1, linestyle=:dash, lab = \"c: consumption, closed form\");\nplot!(ts[ix], B_opt[ix], color = 4, lab = \"B: wealth balance, InfiniteOpt\");\nplot!(ts[ix], BB, color = 4, linestyle=:dash, lab = \"B: wealth balance, closed form\")\n\n# Not bad!\n\n\n\n# ### Maintenance Tests\n# These are here to ensure this example stays up to date. \nusing Test\n@test termination_status(m) == MOI.LOCALLY_SOLVED\n@test has_values(m)\n@test B_opt isa Vector{<:Real}\n@test c_opt isa Vector{<:Real}\n", "meta": {"hexsha": "3e8c657cb43797c57a6a4eb913ecfb2a0b12af1a", "size": 3691, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "docs/src/examples/Optimal Control/consumption_savings.jl", "max_stars_repo_name": "azev77/InfiniteOpt.jl", "max_stars_repo_head_hexsha": "db734856e6d89fd105f7bdb4fb5b8e16a72bd7fd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/src/examples/Optimal Control/consumption_savings.jl", "max_issues_repo_name": "azev77/InfiniteOpt.jl", "max_issues_repo_head_hexsha": "db734856e6d89fd105f7bdb4fb5b8e16a72bd7fd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/src/examples/Optimal Control/consumption_savings.jl", "max_forks_repo_name": "azev77/InfiniteOpt.jl", "max_forks_repo_head_hexsha": "db734856e6d89fd105f7bdb4fb5b8e16a72bd7fd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.0956521739, "max_line_length": 95, "alphanum_fraction": 0.627743159, "num_tokens": 1186, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037282594921, "lm_q2_score": 0.9124361664296878, "lm_q1q2_score": 0.8451930227626183}} {"text": "\"\"\"\n# chebyshevfun\n普通递归给出第一类切比雪夫函数的表达式\n```math\nT_n(x)=2xT_{n-1}(x)-T_{n-2}(x)\n```\n\"\"\"\nfunction chebyshevT(n::Int64,x)\n if n<0\n error(\"chebyshev(n,x), n>=0 is violated\")\n elseif n==0\n return 1\n elseif n==1\n return x\n elseif n==2\n return 2*x^2-1\n elseif n==3\n return 4*x^3-3*x\n elseif n==4\n return 8*x^4-8*x^2+1\n elseif n==5\n return 16*x^5-20*x^3+5*x\n elseif n==6\n return 16*x^5-20*x^3+5*x\n else\n return 2*x*chebyshevT(n-1,x)-chebyshevT(n-2,x)\n end\nend\n\n\"\"\"\n# chebyshevfun\n普通递归给出第一类切比雪夫函数的表达式\n```math\nU_n(x)=2xU_{n-1}(x)-U_{n-2}(x)\n```\n\"\"\"\nfunction chebyshevU(n::Int64,x)\n if n<0\n error(\"chebyshev(n,x), n>=0 is violated\")\n elseif n==0\n return 1\n elseif n==1\n return 2*x\n elseif n==2\n return -1 + 4*x^2\n elseif n==3\n return -4*x + 8*x^3\n elseif n==4\n return 1 - 12*x^2 + 16*x^4\n elseif n==5\n return 6*x - 32*x^3 + 32*x^5\n elseif n==6\n return -1 + 24*x^2 - 80*x^4 + 64*x^6\n else\n return 2*x*chebyshevU(n-1,x)-chebyshevU(n-2,x)\n end\nend\n\n# \"\"\"\n# # chebyshevexpand\n# 切比雪夫展开函数\n# 输入一个函数，返回在指定点处切比雪夫展开后的版本\n# \"\"\"\n\"\"\"\n为切比雪夫T而写的宏\n\"\"\"\nmacro chebyT(n,x)\n return esc(:(2*$x*chebyshevT($n-1,$x)-chebyshevT($n-2,$x)))\nend\n\n\"\"\"\n# chebyshevTmeta\n为切比雪夫展开的多项式\n用元编程进行操作\n\"\"\"\nfunction chebyshevTmeta(n::Int64,x)\n if n==0\n return 1\n elseif n==1\n return x\n elseif n>=2\n return @chebyT(n,x)\n end\nend\n\n# \"\"\"\n# 为切比雪夫U而写的宏\n# \"\"\"\n# macro chebyU(n,x)\n# return esc(:(2*x*chebyshevT($n-1,$x)-chebyshevT($n-2,$x)))\n# end\n\n# \"\"\"\n# # chebyshevTmeta\n# 为切比雪夫展开的多项式\n# 用元编程进行操作\n# \"\"\"\n# function chebyshevUmeta(n::Int64,x)\n# if n==0\n# return 1\n# elseif n==1\n# return 2*x\n# elseif n>=2\n# return @chebyU(n,x)\n# end\n# end", "meta": {"hexsha": "3ec5d252a62103055526c4e264c2e3fac1919e63", "size": 1843, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/chebyshev.jl", "max_stars_repo_name": "sesameman/ChebyshevFun.jl", "max_stars_repo_head_hexsha": "638c6b7d05b1e43b3a076c7631371968df95cd1f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/chebyshev.jl", "max_issues_repo_name": "sesameman/ChebyshevFun.jl", "max_issues_repo_head_hexsha": "638c6b7d05b1e43b3a076c7631371968df95cd1f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/chebyshev.jl", "max_forks_repo_name": "sesameman/ChebyshevFun.jl", "max_forks_repo_head_hexsha": "638c6b7d05b1e43b3a076c7631371968df95cd1f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.3867924528, "max_line_length": 64, "alphanum_fraction": 0.5355398806, "num_tokens": 872, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9669140225647108, "lm_q2_score": 0.8740772450055545, "lm_q1q2_score": 0.845157545000601}} {"text": "\"\"\"\n euler_totient(n::Int64)\n\nCompute the positive integers up to a given integer n that are relatively prime to n; also\nknown as Euler's totient. For more information see: [https://en.wikipedia.org/wiki/Euler%27s_totient_function](https://en.wikipedia.org/wiki/Euler%27s_totient_function)\n\n\n# Arguments\n- `n::Int64`: Number of elements\n\n\n# Examples\n```julia-repl\njulia> import ClassicAlgorithmsCollections\njulia> ClassicAlgorithmsCollections.euler_totient(20)\n[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8]\n```\n\"\"\"\nfunction euler_totient(n::Int64)\n\n result = collect(Int64, 1:n)\n\n for i in 1:n\n m = i\n p = 2\n while m >= p^2\n if (m % p == 0)\n while (m % p == 0)\n m = convert(Int64, m / p)\n end\n result[i] -= convert(Int64, result[i] / p)\n end\n p += 1\n end\n if m > 1\n result[i] -= convert(Int64, result[i] / m)\n end\n end\n return result\nend\n", "meta": {"hexsha": "9e97b2bee6121134e4169dd048980e7a4169a90b", "size": 1022, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/NumberTheory/EulerTotient.jl", "max_stars_repo_name": "Anselmoo/ClassicAlgorithmsCollections", "max_stars_repo_head_hexsha": "9f802c4f317492e19b0b8bb6d9020d8450e00772", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/NumberTheory/EulerTotient.jl", "max_issues_repo_name": "Anselmoo/ClassicAlgorithmsCollections", "max_issues_repo_head_hexsha": "9f802c4f317492e19b0b8bb6d9020d8450e00772", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 21, "max_issues_repo_issues_event_min_datetime": "2020-09-03T06:47:45.000Z", "max_issues_repo_issues_event_max_datetime": "2020-11-20T06:58:58.000Z", "max_forks_repo_path": "src/NumberTheory/EulerTotient.jl", "max_forks_repo_name": "Anselmoo/ClassicAlgorithmsCollections", "max_forks_repo_head_hexsha": "9f802c4f317492e19b0b8bb6d9020d8450e00772", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.9268292683, "max_line_length": 168, "alphanum_fraction": 0.560665362, "num_tokens": 323, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.966914017797636, "lm_q2_score": 0.8740772236840656, "lm_q1q2_score": 0.8451575202177628}} {"text": "using Revise\nusing DynamicalSystems\nusing Basins\nusing Plots\nusing BenchmarkTools\n\n\n# FRACTAL BASIN BOUNDARIES\n# Steven W. MCDONALD, Celso GREBOGY Edward OTT and James A. YORKE\n# Physica D 17 1985\nfunction chaotic_map(dz,z, p, n)\n\n xn = z[1]; yn = z[2]\n λ₁=p[1];λ₂=p[2];\n dz[1]=mod(λ₁*xn,1)\n dz[2]=λ₂*yn+cos(2*π*xn)\n return\nend\n\n# dummy function to keep the initializator happy\nfunction chaotic_map_J(J,z0, p, n)\n return\nend\nds = DiscreteDynamicalSystem(chaotic_map,[1., 0.], [1.1, 1.] , chaotic_map_J)\ninteg = integrator(ds)\n\n# This map has two escapes to ±∞\nfunction escape_function(y)\n if y > 10000\n return 1\n elseif y < -10000\n return 2\n end\n return 0\nend\n\nfunction get_color(integ,x,y)\n reinit!(integ,[x,y])\n while escape_function(integ.u[2]) == 0\n step!(integ)\n integ.t > 2000 && break\n end\n return escape_function(integ.u[2]);\nend\n\n\nxg=range(0.,1.,length=600)\nyg=range(-2.,2.,length=600)\ninteg.p[1] = 3\ninteg.p[2] = 1.5\nbasin_t = zeros(length(xg),length(yg))\n\n# Compute escape basin\nfor (k,x) in enumerate(xg), (n,y) in enumerate(yg)\n basin_t[k,n]=get_color(integ,x,y);\nend\n\n@show bd = box_counting_dim(xg, yg, basin_t)\n\n\n# Estimation using the method of the paper.\n# The results do not match exactly. But the paper talks about\n# approximate results\nN=[]\neps_r = 10 .^range(-8,-1,length=10)\n\nfor eps in eps_r\n @show eps\n N2=0\n N1=0\n\n for k in 1:8192*20\n\n k1 = rand(1:length(xg))\n k2 = rand(1:length(yg))\n\n c1 = get_color(integ, xg[k1],yg[k2])\n c2 = get_color(integ, xg[k1]+eps,yg[k2])\n c3 = get_color(integ, xg[k1]-eps, yg[k2])\n\n if length(unique([c1,Int(c2),Int(c3)]))>1\n N1 += 1\n end\n N2 += 1\n end\n @show N1/N2\n push!(N,N1/N2)\nend\n\nusing LsqFit\n# get exponent\n@. model(x, p) = p[1]*x+p[2]\nfit = curve_fit(model, vec(log10.(eps_r)), vec(log10.(N)), [2., 2.])\nD = coef(fit)\n@show 2-D[1]\n\nplot(xg,yg,basin_t',seriestype=:heatmap)\n", "meta": {"hexsha": "94c1cc4049c49058d9b6ec18a30d28532616ddf7", "size": 1999, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/examples/tst_map_mcdonald.jl", "max_stars_repo_name": "awage/Basins.jl", "max_stars_repo_head_hexsha": "765a232b0568a257a845ee95a14c0cf2776046a4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2021-04-27T14:46:36.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-10T08:39:36.000Z", "max_issues_repo_path": "src/examples/tst_map_mcdonald.jl", "max_issues_repo_name": "awage/Basins.jl", "max_issues_repo_head_hexsha": "765a232b0568a257a845ee95a14c0cf2776046a4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-08-12T15:47:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-14T13:47:35.000Z", "max_forks_repo_path": "src/examples/tst_map_mcdonald.jl", "max_forks_repo_name": "awage/Basins.jl", "max_forks_repo_head_hexsha": "765a232b0568a257a845ee95a14c0cf2776046a4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.3979591837, "max_line_length": 77, "alphanum_fraction": 0.6193096548, "num_tokens": 741, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632876167045, "lm_q2_score": 0.9111797015700341, "lm_q1q2_score": 0.8449946036575945}} {"text": "```\r\nReg(args...) -->Plot\r\n\r\nEstimate Linear, Exponential and Power regression type\r\n``Linear\\quad \\Rightarrow y=mx+b\\\\ Exponential\\quad \\Rightarrow y=b{ e }^{ mx }\\\\ Power\\quad \\Rightarrow y=b{ x }^{ m }``\r\n\r\nThe next table show the list of variables allowed:\r\n\r\n|PropertyName|Args|Default|Input|Description\r\n|---|---|---|---|---|\r\n|x|Mandatory|--|Array{Number,1}| x Data|\r\n|y|Mandatory|--|Array{Number,1}| y Data|\r\n|TypeReg|Optional|TypeReg=\"Linear\"|TypeReg= \"linear\" or \"Exponential\" or \"Power\"|Type of regression|\r\n|pr|Optional|pr=false|pr=Bool|Print the results of regression|\r\n```\r\nfunction Reg(x,y; TypeReg=\"Linear\", pr=false)\r\n\r\n if TypeReg==\"Linear\"\r\n m=(mean(x)*mean(y)-mean(x.*y))/(mean(x)^2-mean(x.^2))\r\n b=mean(y)-m*mean(x)\r\n SELine=sum(map((X,Y)->((X*m+b)-Y)^2,x,y))\r\n SEYmean=sum(map(Y->(Y-mean(y))^2,y))\r\n R2=1-(SELine/SEYmean)\r\n\r\n pr ? println(\" **Linear Regression \\n form y = mx+b \\n m=$m \\n b=$b \\n r2=$R2\") : :none\r\n return m,b,R2\r\n elseif TypeReg==\"Exponential\"\r\n m=(mean(x)*mean(log.(y))-mean(x.*log.(y) ) )/(mean(x)^2-mean(x.^2))\r\n Logb=mean(log.(y))-m*mean(x)\r\n b=exp(Logb)\r\n SELine=sum(map((X,Y)->((X*m+Logb)-Y)^2,x,log.(y)))\r\n SEYmean=sum(map(Y->(Y-mean(log.(y)))^2,log.(y)))\r\n R2=1-(SELine/SEYmean)\r\n\r\n pr ? println(\" **Exponential Regression \\n form y = b*exp(mx) \\n m=$m \\n b=$b \\n r2=$R2\") : :none\r\n return m, b,R2\r\n elseif TypeReg==\"Power\"\r\n m=(mean(log.(x))*mean(log.(y))-mean(log.(x).*log.(y) ) )/(mean(log.(x))^2-mean(log.(x).^2))\r\n Logb=mean(log.(y))-m*mean(log.(x))\r\n b=exp(Logb)\r\n SELine=sum(map((X,Y)->((X*m+Logb)-Y)^2,log.(x),log.(y)))\r\n SEYmean=sum(map(Y->(Y-mean(log.(y)))^2,log.(y)))\r\n R2=1-(SELine/SEYmean)\r\n\r\n pr ? println(\" **Power Regression \\n form y = b x^(m) \\n m=$m \\n b=$b \\n r2=$R2\") : :none\r\n return m, b,R2\r\n end\r\n\r\nend\r\n", "meta": {"hexsha": "e4a210f2c4cf9394fc625668adbc8fb823d07a59", "size": 1951, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Reg.jl", "max_stars_repo_name": "scuervo91/WellLogs.jl", "max_stars_repo_head_hexsha": "bc691fdd0500414b4f78172f905ca81930364f8c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-05-23T12:19:51.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-11T21:34:04.000Z", "max_issues_repo_path": "src/Reg.jl", "max_issues_repo_name": "scuervo91/WellLogs.jl", "max_issues_repo_head_hexsha": "bc691fdd0500414b4f78172f905ca81930364f8c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-06-21T23:04:55.000Z", "max_issues_repo_issues_event_max_datetime": "2019-06-25T15:37:20.000Z", "max_forks_repo_path": "src/Reg.jl", "max_forks_repo_name": "scuervo91/WellLogs.jl", "max_forks_repo_head_hexsha": "bc691fdd0500414b4f78172f905ca81930364f8c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2019-06-21T22:50:14.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-09T18:28:16.000Z", "avg_line_length": 39.02, "max_line_length": 122, "alphanum_fraction": 0.5422860072, "num_tokens": 672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9539660949832346, "lm_q2_score": 0.8856314647623016, "lm_q1q2_score": 0.844862390033575}} {"text": "### A Pluto.jl notebook ###\n# v0.18.4\n\nusing Markdown\nusing InteractiveUtils\n\n# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).\nmacro bind(def, element)\n quote\n local iv = try Base.loaded_modules[Base.PkgId(Base.UUID(\"6e696c72-6542-2067-7265-42206c756150\"), \"AbstractPlutoDingetjes\")].Bonds.initial_value catch; b -> missing; end\n local el = $(esc(element))\n global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)\n el\n end\nend\n\n# ╔═╡ cf07621a-9a65-11ec-1fc0-51fbadbb086f\nusing Pkg; Pkg.activate()\n\n# ╔═╡ 345a90f7-f8ac-43b2-b5bd-0282d9447556\nbegin\n\tusing DifferentialEquations\n\tusing ModelingToolkit\n\tusing DynamicalSystems\n\tusing ForwardDiff\n\tusing LinearAlgebra\n\tusing Graphs\n\tusing Plots\n\tusing PlutoUI\n\tusing LaTeXStrings\nend\n\n# ╔═╡ eaf9dda2-d898-4771-9f86-ca4be174712c\nmd\"\"\"\n# Dynamical Systems in Julia\nIn this tutorial, we'll looking at specifying, simulating and analyzing dynamical systems in Julia. This is one area where Julia especially shines, thanks in no small part to the `DifferentialEquations` package, which pretty much provides the largest and fastest suite of ODE solvers on the market today.\n\"\"\"\n\n# ╔═╡ eb6aed5a-0f95-4a76-8a99-b8719984bbe5\nmd\"## Defining systems symbolically\"\n\n# ╔═╡ 0e28ede5-0af6-49a7-b95d-7094e320eee4\nmd\"\"\"\nWe'll start by using ModelingToolkit to define the famous Lorenz system. The process starts by a symbolic description of the problem.\n\"\"\"\n\n# ╔═╡ 6ccfc87f-40e7-4ee4-baae-ee41b81efe67\nbegin\n\t@parameters t σ ρ β\n\t@variables x(t) y(t) z(t)\n\tD = Differential(t)\n\n\teqs = [\n\t\tD(x) ~ σ*(y-x),\n \tD(y) ~ x*(ρ-z)-y,\n \tD(z) ~ x*y - β*z\n\t]\n\n\t@named lorenz = ODESystem(eqs)\nend\n\n# ╔═╡ 3e4ec586-482f-4c1b-a8e7-4ae88e48fa38\nmd\"\"\"\nSo far, the `lorenz` is purely symbolic, which has the added benefit of being able to render the equations in LaTeX as above! The next step before simulating it is defining values for our initial condition and parameters.\n\"\"\"\n\n# ╔═╡ ce0b65b5-e57a-4def-acb6-892c49f182d9\nbegin\n\tu0 = [\n\t\tx => 0.0,\n \ty => 0.5,\n \tz => 0.5\n\t]\n\n\tp = [\n\t\tρ => 28.0,\n \tσ => 10.0,\n \tβ => 8/3\n\t]\nend\n\n# ╔═╡ 1a7e8273-8069-42a8-ab7b-62f5fe1ade67\nmd\"\"\"\nSimulating our system is a two step process. After defining our timespan of integration, we first create an `ODEProblem`, we describes an ODE + initial conditions and parameters, and then pass it to the `solve` function to compute the solution of our problem. The second argument to solve is the solver we want to use (`Tsit5` is a 4-5 order Runge-Kutta scheme which is generally a good default choice for non-stiff problems)\n\"\"\"\n\n# ╔═╡ 1d77a2be-773c-4edd-9d68-1ed84fb5dadf\nbegin\n\ttspan = (0.0,100.0)\n\tprob = ODEProblem(lorenz,u0,tspan,p,jac=true)\n\tsol = solve(prob,Tsit5())\nend\n\n# ╔═╡ 7c712f73-3931-4cf4-82d8-9d9e2263df14\nplot(sol, vars=(x,y,z))\n\n# ╔═╡ 432fa70e-9225-4e76-aec3-c27d31887fb9\nmd\"\"\"\nNB. There are also packages for modeling specific classes of dynamical systems, such as `Catalyst` for chemical reaction networks, or `PowerDynamics` for power grids.\n\"\"\"\n\n# ╔═╡ 60063cc3-e71f-461b-be0a-491b1b566718\nmd\"## Defining Systems by hand\"\n\n# ╔═╡ ea2eeac2-5514-4fe8-8921-6235cdb0d1ad\nmd\"\"\"\nWhile defining systems symbolically is cool, this can be tedious for large systems, and sometimes you just want full control over how your ODE function is defined. To that end, we will now define another famous system, the Brusselator by hand and then build a network reaction diffusion system with it.\n\nThe Brusselator equations are as follows,\n```math\n\\begin{align*}\n\\dot{x} &= a + x^2y -bx -x \\\\\n\\dot{y} &= bx - x^2y\n\\end{align*}\n```\n\"\"\"\n\n# ╔═╡ 2bf13604-71c6-474f-90de-5b270c5c0708\nmd\"\"\"\nWe can construct ODEProblems by providing a function that computes the derivative, just like we do in Matlab. The functions we can use come in two flavours. The first type just takes as input the state `u`, parameters stored in `p` and time `t` (for non autonomous systems), and returns the new state.\n\"\"\"\n\n# ╔═╡ 743b31cd-6b2a-472c-8a70-a3955a2c26b6\nfunction brusselator(u, p, t)\n\ta, b = p\n\t[\n\t\ta + u[1]^2 * u[2] - b*u[1] - u[1],\n\t\tb*u[1] - u[1]^2 * u[2]\n\t]\nend\n\n# ╔═╡ 03bd6ff6-107f-4b04-9cb8-fe5c28570e58\nmd\"\"\"\nHowever, it is always recommended to use the second form, which also takes as input a preallocated array for the output and modifies it. Doing this is more efficient in terms of memory allocation, which is actually an important source of slowdowns.\n\"\"\"\n\n# ╔═╡ 08fc4616-d2af-471b-adc0-195bb566334f\nfunction brusselator!(du, u, p, t)\n\ta, b = p\n\tdu[1] = a + u[1]^2 * u[2] - b*u[1] - u[1]\n\tdu[2] = b*u[1] - u[1]^2 * u[2]\nend\n\n# ╔═╡ d0c1e91e-d1df-4a12-a0e2-8513eeee7ba6\nbegin\n\tu0_b = [1.0, 1.0]\n\tp_b = [1.0, 3.0]\nend\n\n# ╔═╡ 8623c9b7-5724-4678-9b53-4312389a276d\nbegin\n\tprob_b = ODEProblem(brusselator!, u0_b, (0.0, 50.0), p_b)\n\tsol_b = solve(prob_b, Tsit5())\nend\n\n# ╔═╡ da0220ea-99aa-4a0d-8826-f0e461fd503e\nplot(sol_b)\n\n# ╔═╡ 60d1ea9b-b935-4e88-9e92-07497f06aeaa\nmd\"\"\"\nNow that we have our brusselator function, let's use it on a network to make a reaction diffusion system. Such a system is of the form\n\n```math\n\\dot{u_i} = f(u_i) + D \\sum_{j=1}^N A_{ij}(u_j - u_i),\n```\nwhere ``u_i\\in\\mathbb{R}^m`` is the state of node ``i``, subject to local dynamics ``f:\\mathbb{R}^m\\rightarrow \\mathbb{R}^m``, ``A\\in\\mathbb{R}^{N\\times N}`` is the adjacency matrix of the graph (i.e. ``A_{ij}=1`` iff there is a link between nodes ``i`` and ``j``), and ``D`` is a diagonal matrix of diffusion coefficients.\n\nWe can rewrite this system in vector form (glossing over some details) as\n\n```math\n\\dot{\\mathbf{u}} = F(\\mathbf{u}) + \\mathbf{D} L \\mathbf{u},\n```\nwhere ``L`` is the *Laplacian matrix* of the graph.\n\nReaction-Diffusion systems can take more general forms, but we'll stick to this one for today.\n\"\"\"\n\n# ╔═╡ efbb46be-c442-4ab3-85a0-b10cbc3a2a9e\nmd\"\"\"\nNow, we need to create a function for our reaction-diffusion system.\n\"\"\"\n\n# ╔═╡ 8b2ec98a-b37f-4bd0-9fc8-9f3dcf9550c9\nfunction brusselator_rd(u, p, t)\n\ta, b, L, D = p\n\tdu = similar(u) # creates an array of same shape and element type as u\n\tfor i in 1:size(du, 1)\n\t\tdu[i,:] .= brusselator(u[i,:], [a, b], t)\n\tend\n\tdu[:,1] .-= D[1] .* (L*u[:,1])\n\tdu[:,2] .-= D[2] .* (L*u[:,2])\n\treturn du\nend\n\n# ╔═╡ 25c22a1f-6ae4-4b72-8e4c-de844e2699d1\nfunction brusselator_rd!(du, u, p, t)\n\t# NB. the commented out parts should be equivalent to what is used, but for some reason it is not\n\ta, b, L, D = p\n\tdu .= 0.0\n\tfor i in 1:size(du, 1)\n\t\t#brusselator!(du[i,:], u[i,:], [a, b], t)\n\t\tdu[i,:] .+= brusselator(u[i,:], [a, b], t)\n\tend\n\t#mul!(du[:,1], L, u[:,1], -D[1], 0.0) # du[:,1] .+= D[1]*L*u[:,1]\n\t#mul!(du[:,2], L, u[:,2], -D[2], 0.0)\n\tdu[:,1] .-= D[1] .* (L*u[:,1])\n\tdu[:,2] .-= D[2] .* (L*u[:,2])\nend\n\n# ╔═╡ 582b74c7-aaf6-4a8e-901f-489199ccc0d6\nmd\"\"\"\nNow, we need a network to run our system on. To do this, we can use the `Graphs` package.\n\"\"\"\n\n# ╔═╡ 3e575a45-50d6-4297-84e5-6699e36d00dd\ng = erdos_renyi(10, 0.4)\n\n# ╔═╡ ed1de34a-a42e-474d-ba36-411163a343f8\nmd\"\"\"\nWe can use the function `laplacian_matrix` to get the Laplacian matrix of the graph. The nice thing about this function is that it returns a sparse matrix, which uses optimized methods for matrix multiplications.\n\"\"\"\n\n# ╔═╡ 947c169c-a900-4728-ad29-5200b52f9599\nlaplacian_matrix(g)\n\n# ╔═╡ 04bbc594-17c9-4b41-a599-d41e2e541c9f\nbegin\n\tu_rd_0 = 5*rand(nv(g), 2)\nend\n\n# ╔═╡ 29a0f8c8-5529-496c-99f1-b436d15f2517\nmd\"\"\"\nHere's a nice feature of Pluto. It provides allows interactively modifying variables with html elements (sliders, buttons, ...) via the `PlutoUI` package.\n\nLet's use it to make sliders for the parameters of our system.\n\"\"\"\n\n# ╔═╡ f346a0b3-4d1d-49ad-94ec-cc42ee75a05b\nmd\"\"\"\na: $(@bind a Slider(0.0:0.01:5.0, default=1.0, show_value=true))\n\nb: $(@bind b Slider(0.0:0.01:5.0, default=3.0, show_value=true))\n\nD₁: $(@bind D_1 Slider(0.0:0.01:10.0, default=1.0, show_value=true))\n\nD₂ $(@bind D_2 Slider(0.0:0.01:10.0, default=1.0, show_value=true))\n\"\"\"\n\n# ╔═╡ 8a1c55e5-1008-44ee-b853-c53c2dbf3bb2\np_rd = [a, b, laplacian_matrix(g), [D_1, D_2]];\n\n# ╔═╡ 2ee789c5-ae38-4e85-95e1-920a580df058\nbegin\n\tprob_rd = ODEProblem(brusselator_rd!, u_rd_0, (0.0, 50.0), p_rd)\n\tsol_rd = solve(prob_rd, Tsit5())\n\tnothing\nend\n\n# ╔═╡ 9d1f3408-e0f5-4093-bb6b-862485e99476\nbegin\n\tlocal p1 = plot(sol_rd, vars=(0,1:nv(g)), label=\"\", ylabel=L\"x_i\")\n\tlocal p2 = plot(sol_rd, vars=(0,(nv(g)+1:2nv(g))), label=\"\", ylabel=L\"y_i\")\n\tplot(p1, p2)\nend\n\n# ╔═╡ a4d4a26f-ff25-4bf0-b8ab-2db56131ee3e\nmd\"\"\"\n### Exercise\nImplement your favourite dynamical system and play around with it's parameters.\n\"\"\"\n\n# ╔═╡ 91f1ce37-0d68-4ed2-9359-dbcfd06f6cdd\nmd\"\"\"\n## Intermezzo: using `DynamicalSystems` to study chaotic Systems\n\nBefore moving to the last part of this tutorial, let us make a detour to look at the `DynamicalSystems` package, which provides a library of famous dynamical systems, as well as generic tools for ananlyzing them, particularly chaotic systems. \n\"\"\"\n\n# ╔═╡ 4328e814-82a8-4e7d-8b8a-44a9d5ba17c5\nds = Systems.logistic()\n\n# ╔═╡ 7b11ac9b-a470-4059-bbad-ba6bc443c68b\nmd\"\"\"\nHere we load the well known logistic map from the library of dynamical systems, and generate its orbit diagram for varying values of the `r` parameter. The magic function is `orbitdiagram`, which should work for pretty much any discrete map.\n\"\"\"\n\n# ╔═╡ a5629c28-53bf-4b19-a49f-ab26df1d1b46\nbegin\n\ti = 1\n\tpvalues = 1:0.005:4\n ics = [rand() for m in 1:10]\n n = 2000\n Ttr = 2000\n p_index = 1\n\t\n output = orbitdiagram(ds, i, p_index, pvalues; n = n, Ttr = Ttr)\nend\n\n# ╔═╡ 74adb98a-1ad5-4181-a3be-ec5fe2565a35\nbegin\n\tlocal L = length(pvalues)\n\tlocal x = Vector{Float64}(undef, n*L)\n\tlocal y = copy(x)\n\tfor j in 1:L\n \tx[(1 + (j-1)*n):j*n] .= pvalues[j]\n \t\ty[(1 + (j-1)*n):j*n] .= output[j]\n\tend\n\tscatter(x,y,\n\t\txlabel=\"r\",\n\t\tylabel=\"x\",\n\t\tlabel=\"\",\n\t\tmarkersize=0.8,\n\t\tcolor=:black,\n\t\tmarkeralpha=0.05,\n\t\tmarkerstrokealpha=0.0\n\t)\nend\n\n# ╔═╡ c38e202f-a2d2-4449-8308-df22774b7af1\nmd\"\"\"\nNB. While `DynamicalSystems` provides a bunch of cool tools, a lot of them seem to inexplicably break when used outside of the examples. For example, try changing the above code to use values of `r` above 4.\n\"\"\"\n\n# ╔═╡ d5ab9847-ace4-4a5b-8c81-d2eb8b56c896\nmd\"\"\"\n## Analyzing the stability of our system\n\nReaction-Diffusion systems like the one we just saw are interesting because of their asymptotic behaviour. Because the laplacian always has a zero eigenvalue associated to an eigenvector whose entries are all the same, a reaction-diffusion system always admits a homogeneous solution where the nodes are all synchronised.\nBy tuning the diffusion coefficients, it's possible to (de)synchronize the states of the nodes.\n\nThe standard tool for analysing this phenomenon is the Master Stability Function [Pecora-Caroll. 1998].\n\nThe main idea consists in expressing the system in the basis of eigenvectors of the laplacian matrix. By considering a small perturbation around the synchronized solution, and expressing its linearized equations in the eigenbasis of ``L``, one can obtain a set of decoupled equations involving the laplacian eigenvalues and the jacobian of the local dynamics.\n\nSkipping over the details, we can check that the synchronized solution is linearly stable by checking that the following ODE is stable for every eigenvalue ``\\lambda`` of the laplacian matrix\n\n```math\n\\dot{\\xi} = [\\mathcal{J}f(u_*) - \\lambda D] \\xi\n```\nwhere ``\\mathcal{J}f`` denotes the jacobian of ``f``.\n\nThis can be done, for example, by computing the Lyapunov exponent of this equation, thereby yielding a function ``MSF(\\lambda)`` called the *Master Stability Function* of the system.\n\"\"\"\n\n# ╔═╡ 575e3106-704e-4bea-8155-7ee6e6c9da5d\nmd\"First, let's compute the eigenvalues of the laplacian matrix. We don't actually need to compute the eigenvectors.\"\n\n# ╔═╡ bff34b78-f7ad-452e-9b95-a498f3e377f5\nλ = laplacian_spectrum(g)\n\n# ╔═╡ bba5eaa1-16f2-401d-8cf3-f05e9761583d\nmd\"\"\"\nNext, let's start by writing a function that takes the local dynamics as input, and generates a function for the MSF ODE.\n\nNotice the use, of `ForwardDiff.jacobian` to automagically compute the Jacobian matrix using Automatic Differentiation (this is different from Finite Differences or Symbolic methods)\n\"\"\"\n\n# ╔═╡ ddbb84d1-945a-4184-850b-31ee3540e960\nfunction msf_ode_fun(f)\n\tfunction msf_ode!(du, u, p, t)\n\t\tp_f, D, λ = p\n\t\tdu[:,1] .= f(u[:,1], p_f, t)\n\t\tJ = ForwardDiff.jacobian(x -> f(x,p_f,t), u[:,1])\n\t\tdu[:,2] .= J*u[:,2] - λ*D*u[:,2]\n\tend\n\treturn msf_ode!\nend\n\n# ╔═╡ f0215a78-146d-4009-ae73-8a2f35072db9\nmd\"\"\"\nNow, we can move on to computing the MSF proper. To do this, we solve the MSF ODE, then estimating the Lyapunov exponent from the time series.\n\nNB. Recall that the Lyapunov exponent for a dynamical sytem is defined by\n```math\n\\lambda = \\lim_{t\\rightarrow \\infty} \\lim_{\\xi_0 \\rightarrow 0} \\frac{1}{t} \\ln \\frac{\\|\\xi(t)\\|}{\\|\\xi_0\\|},\n```\nwhere ``\\xi(0) = \\xi_0`` is a small perturbation applied to the orbit of interest (here, the periodic solution).\n\"\"\"\n\n# ╔═╡ 84d86795-e2c4-4c2c-888b-ae999bcda2dd\nfunction master_stability_function(f, u_0, p_f, D, t; δ=0.1, k=50, Δt=0.1)\n\tmsf_ode! = msf_ode_fun(f)\n\tn = length(u_0)\n\tu_msf_0 = hcat(u_0, δ*randn(n))\n\tfunction msf(λ; u_msf_0=u_msf_0, p_f=p_f, D=D, t=t, Δt=Δt)\n\t\tp_msf = [p_f, D, λ]\n\t\tprob = ODEProblem(msf_ode!, u_msf_0, (0.0,t), p_msf)\n\t\tsol = solve(prob, Tsit5(), saveat=0.0:Δt:t, abstol=1e-10, reltol=1e-10)\n\t\tξ_0 = norm(sol.u[1][:,2])\n\t\tl = length(sol.u)\n\t\tξ_t = norm([sol.u[k][:,2] for k in l-k:l])\n\t\treturn log(ξ_t/ξ_0)/sol.t[end]\n\tend\n\treturn msf\nend\n\n# ╔═╡ 8c592351-4da7-4bed-a99c-595156c1ed8b\nbegin\n\tu0_msf = sol_b.u[end]\n\tp_f = [1.0, 3.0]\n\tD_msf = Diagonal([0.2,5.0])\nend\n\n# ╔═╡ e3411cdb-acf7-40da-9f5e-6852c1370797\nmsf = master_stability_function(brusselator, u0_msf, p_f, D_msf, 200.0)\n\n# ╔═╡ 5fbfe4aa-f2e5-4845-92e7-9de00063059f\nmsf(0.0)\n\n# ╔═╡ ffecadd1-95b5-412f-86e4-e4dee733cab1\nmd\"\"\"\nNote that this implementation is not super fast, as it takes about a minute to run the next cell. There's lots of room for improvements though, and in the mean time, we still have a generic funcion for computing the MSF.\n\"\"\"\n\n# ╔═╡ bad93338-72e3-46fc-888f-e507c62cefc6\nbegin\n\tlocal λs = 0.0:0.01:7.0\n\tplot(λs, msf.(λs), \n\t\tlabel=\"\",\n\t\txlabel=\"-Re λ\"\n\t)\n\t#plot!(λs, msf_ds.(λs), label=\"ds\")\n\tplot!([0.0,7.0],[0.0,0.0], linestyle=:dash, label=\"\", color=:black)\n\tscatter!(λ, msf.(λ), label=\"\")\nend\n\n# ╔═╡ 63c14c9e-808e-464b-b5cc-25921aa34f56\nmd\"\"\"\nBelow, we examine the trajectory of the MSF equations for λ = 0. Note that this is quite ill-behaved (notice the log scale), so it's no wonder this is tricky to get right.\n\"\"\"\n\n# ╔═╡ 8f8a9dec-4de2-4fc7-9d87-a3638ec47ee3\nbegin\n\tf! = msf_ode_fun(brusselator)\n\tu0_ = hcat(u0_msf, 0.1*randn(2))\n\tp_msf = [p_f, D_msf, 0.0]\n\tprob_msf = ODEProblem(f!, u0_, (0.0,100.0), p_msf)\n\tsol_msf = solve(prob_msf, Rodas5(), abstol=1e-10, reltol=1e-10)\nend\n\n# ╔═╡ e85b9b6d-736d-446a-acb1-b6ed0ccdd907\nbegin\n\tlocal foo = (t,u,v) -> (t,norm([u;v]))\n\tplot(sol_msf, vars=(foo,0,3,4), \n\t\tyscale=:log10, \n\t\tlabel=\"\", \n\t\tylabel=L\"\\ln \\|\\xi\\|\"\n\t)\nend\n\n# ╔═╡ Cell order:\n# ╟─eaf9dda2-d898-4771-9f86-ca4be174712c\n# ╠═cf07621a-9a65-11ec-1fc0-51fbadbb086f\n# ╠═345a90f7-f8ac-43b2-b5bd-0282d9447556\n# ╟─eb6aed5a-0f95-4a76-8a99-b8719984bbe5\n# ╟─0e28ede5-0af6-49a7-b95d-7094e320eee4\n# ╠═6ccfc87f-40e7-4ee4-baae-ee41b81efe67\n# ╟─3e4ec586-482f-4c1b-a8e7-4ae88e48fa38\n# ╠═ce0b65b5-e57a-4def-acb6-892c49f182d9\n# ╟─1a7e8273-8069-42a8-ab7b-62f5fe1ade67\n# ╠═1d77a2be-773c-4edd-9d68-1ed84fb5dadf\n# ╠═7c712f73-3931-4cf4-82d8-9d9e2263df14\n# ╟─432fa70e-9225-4e76-aec3-c27d31887fb9\n# ╟─60063cc3-e71f-461b-be0a-491b1b566718\n# ╟─ea2eeac2-5514-4fe8-8921-6235cdb0d1ad\n# ╟─2bf13604-71c6-474f-90de-5b270c5c0708\n# ╠═743b31cd-6b2a-472c-8a70-a3955a2c26b6\n# ╟─03bd6ff6-107f-4b04-9cb8-fe5c28570e58\n# ╠═08fc4616-d2af-471b-adc0-195bb566334f\n# ╠═d0c1e91e-d1df-4a12-a0e2-8513eeee7ba6\n# ╠═8623c9b7-5724-4678-9b53-4312389a276d\n# ╠═da0220ea-99aa-4a0d-8826-f0e461fd503e\n# ╟─60d1ea9b-b935-4e88-9e92-07497f06aeaa\n# ╟─efbb46be-c442-4ab3-85a0-b10cbc3a2a9e\n# ╠═8b2ec98a-b37f-4bd0-9fc8-9f3dcf9550c9\n# ╠═25c22a1f-6ae4-4b72-8e4c-de844e2699d1\n# ╟─582b74c7-aaf6-4a8e-901f-489199ccc0d6\n# ╠═3e575a45-50d6-4297-84e5-6699e36d00dd\n# ╟─ed1de34a-a42e-474d-ba36-411163a343f8\n# ╠═947c169c-a900-4728-ad29-5200b52f9599\n# ╠═04bbc594-17c9-4b41-a599-d41e2e541c9f\n# ╟─29a0f8c8-5529-496c-99f1-b436d15f2517\n# ╟─f346a0b3-4d1d-49ad-94ec-cc42ee75a05b\n# ╠═8a1c55e5-1008-44ee-b853-c53c2dbf3bb2\n# ╠═2ee789c5-ae38-4e85-95e1-920a580df058\n# ╠═9d1f3408-e0f5-4093-bb6b-862485e99476\n# ╟─a4d4a26f-ff25-4bf0-b8ab-2db56131ee3e\n# ╟─91f1ce37-0d68-4ed2-9359-dbcfd06f6cdd\n# ╠═4328e814-82a8-4e7d-8b8a-44a9d5ba17c5\n# ╟─7b11ac9b-a470-4059-bbad-ba6bc443c68b\n# ╠═a5629c28-53bf-4b19-a49f-ab26df1d1b46\n# ╠═74adb98a-1ad5-4181-a3be-ec5fe2565a35\n# ╟─c38e202f-a2d2-4449-8308-df22774b7af1\n# ╟─d5ab9847-ace4-4a5b-8c81-d2eb8b56c896\n# ╟─575e3106-704e-4bea-8155-7ee6e6c9da5d\n# ╠═bff34b78-f7ad-452e-9b95-a498f3e377f5\n# ╟─bba5eaa1-16f2-401d-8cf3-f05e9761583d\n# ╠═ddbb84d1-945a-4184-850b-31ee3540e960\n# ╟─f0215a78-146d-4009-ae73-8a2f35072db9\n# ╠═84d86795-e2c4-4c2c-888b-ae999bcda2dd\n# ╠═8c592351-4da7-4bed-a99c-595156c1ed8b\n# ╠═e3411cdb-acf7-40da-9f5e-6852c1370797\n# ╠═5fbfe4aa-f2e5-4845-92e7-9de00063059f\n# ╟─ffecadd1-95b5-412f-86e4-e4dee733cab1\n# ╠═bad93338-72e3-46fc-888f-e507c62cefc6\n# ╟─63c14c9e-808e-464b-b5cc-25921aa34f56\n# ╠═8f8a9dec-4de2-4fc7-9d87-a3638ec47ee3\n# ╠═e85b9b6d-736d-446a-acb1-b6ed0ccdd907\n", "meta": {"hexsha": "b31533b1790e50dcaf7adb3fc1498860712b5b42", "size": 17529, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "notebook.jl", "max_stars_repo_name": "csimal/Julia-for-Dynamical-Systems", "max_stars_repo_head_hexsha": "6fe31fdfc023c607037a173bacdb6f46b4536ccc", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebook.jl", "max_issues_repo_name": "csimal/Julia-for-Dynamical-Systems", "max_issues_repo_head_hexsha": "6fe31fdfc023c607037a173bacdb6f46b4536ccc", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebook.jl", "max_forks_repo_name": "csimal/Julia-for-Dynamical-Systems", "max_forks_repo_head_hexsha": "6fe31fdfc023c607037a173bacdb6f46b4536ccc", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.4381139489, "max_line_length": 425, "alphanum_fraction": 0.7072850705, "num_tokens": 7426, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9489172630429475, "lm_q2_score": 0.8902942261220292, "lm_q1q2_score": 0.844815560354655}} {"text": "module Donquicote\n\nusing PrettyTables, Statistics, StatsBase, ForwardDiff\n\nD(f) = x -> ForwardDiff.derivative(f,float(x))\n\nfunction newton(f, ∂f, x0; xatol = 1e-7)\n\n diff = 1\n while diff >= xatol\n x1 = x0 - f(x0) / ∂f(x0);\n diff = abs(x1 - x0);\n x0 = x1;\n end\n return x0\nend\n\nfunction haley(f, ∂f, ∂2f, x0; xatol = 1e-7)\n diff = 1\n while diff >= xatol\n x1 = x0 - (2*f(x0)*∂f(x0)) / (2*∂f(x0)^2 - f(x0)*∂2f(x0));\n diff = abs(x1 - x0);\n x0 = x1;\n end\n return x0\n\nend\n\nfunction schroeder(f, ∂f, ∂2f, x0; xatol = 1e-7)\n diff = 1\n while diff >= xatol\n x1 = x0 - f(x0) / ∂f(x0) - (∂2f(x0)*f(x0)^2)/(2*∂f(x0)^3);\n diff = abs(x1 - x0);\n x0 = x1;\n end\n return x0\n\nend\n\nfunction weightedmean(x, w)\n wxsum = wsum = 0\n for (x,w) in zip(x,w)\n wx = w*x\n if !ismissing(wx) | isfinite(wx)\n wxsum += wx\n wsum += w\n end\n end\n return wxsum / wsum\nend\n\nfunction weightedstd(x, w)\n μ = weightedmean(x, w)\n wxsum = wsum = 0\n for (x,w) in zip(x,w)\n wx = w*(x - μ)^2\n if !ismissing(wx) | isfinite(wx)\n wxsum += wx\n wsum += w\n end\n end\n σ = sqrt((1/wsum)*wxsum)\n return σ\nend\n\nfunction gini(x::Vector{Float64})\n n = length(x)\n sarray = sort(x)\n return 2*(sum(collect(1:n).*sarray))/(n*sum(sarray))-1\nend\n\nlogm(x) = x > 0.0 ? log(x) : missing\n\nlvar(x) = var(logm.(x))\n\nnanmean(x) = mean(filter(!isnan,x))\n\nformat(x) = round(x, digits = 3)\n\nfunction savehtml(filename, data)\n open(\"$filename.html\", \"w\") do f\n pretty_table(f, data, backend = :html)\n end\nend\n\nexport newton, haley, schroeder\nexport gini, nanmean, weightedmean, weightedstd, format, logm, lvar\nexport ones, zeros\nexport savehtml\n\nend", "meta": {"hexsha": "85844cf5869d44d39f64cb39d0bedcab4288a7c1", "size": 1809, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Donquicote.jl", "max_stars_repo_name": "Donquicote/donquicote.jl", "max_stars_repo_head_hexsha": "6b1d7b18f8676a80341288fd43d6805ecfcd0678", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Donquicote.jl", "max_issues_repo_name": "Donquicote/donquicote.jl", "max_issues_repo_head_hexsha": "6b1d7b18f8676a80341288fd43d6805ecfcd0678", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Donquicote.jl", "max_forks_repo_name": "Donquicote/donquicote.jl", "max_forks_repo_head_hexsha": "6b1d7b18f8676a80341288fd43d6805ecfcd0678", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.8791208791, "max_line_length": 67, "alphanum_fraction": 0.535655058, "num_tokens": 683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9381240142763573, "lm_q2_score": 0.9005297807787537, "lm_q1q2_score": 0.8448086129195724}} {"text": "using NumericalMethodsforEngineers\n\n#=\n@sym begin\n LagrangePolynomial(xi_, yi_) := Module([N, sum, num, den],\n begin\n N = Length(xi)\n sum = 0\n For( i=1, i <= N, Increment(i),\n begin\n num = 1\n den = 1\n For( j=1, j <= N, Increment(j),\n begin\n If(j != i, num *= (x-xi[j]))\n If(j != i, den *= (xi[i]-xi[j]))\n end\n )\n sum += yi[i] * num/den\n end\n ),\n Return(Simplify(sum))\n end\n )\nend\n=#\n\nprintln(\"\\n\\n--------1------------\\n\")\n\n@sym begin\n ClearAll(xi, yi, Y, R, sol, ytilde)\n xi = [0, 1//2, 1]\n yi = [0, a, 1]\n Y(x_) := LagrangePolynomial(xi, yi)\n Println(Y(x))\nend\n\nprintln(\"\\n---------2-----------\\n\")\n\n@sym begin\n ClearAll(xi, yi, Y1, R, sol, ytilde)\n Y1(x_) := x^3 -3x^2 +6x - 9\n R(x_) = Simplify(D(Y1(x), x, 2))\n Println(R(x))\nend\n\nprintln(\"\\n----------3----------\\n\")\n\n@sym begin\n ClearAll(xi, yi, Y, R, sol, ytilde)\n xi = [0, 1//2, 1]\n yi = [0, a, 1]\n Y(x_) := LagrangePolynomial(xi, yi)\n R(x_) = Simplify(D(Y(x), x, 2) - 3*x - 4*Y(x))\n R(x_) = Simplify(R(x) ./ (a => -C1/4))\n Println(R(x))\n SetJ(r, ToString(Simplify(R(x))))\nend\n\nprintln(\"\\n----------4----------\\n\")\n\nprintln(r)\n\nprintln(\"\\n----------5----------\\n\")\n\nprintln(\"\\n\\nExample 7.14: y'' = 3x + 4y, y(0)=0, y(1)=1\")\nprintln(\"Residual for Weighted Residual Method using 2 point Lagragian Polynomial\")\n@sym Println(\"\\nY(x): \", Expand(Y(x)), \"\\n\")\nprintln(\"( Example 7.14 gives: ytilde = 1/2*x^3*(27a-27b+9)-x^2*(45a-36b+9)+x*(18a-9b+2) )\")\n@sym Println(\"\\nR(x) = \", R(x), \"\\n\")\n", "meta": {"hexsha": "1c0d40c13e3d755f5376d5fd6d14dda554aa540f", "size": 1599, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test/wrm_with_using_nmfe.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/NumericalMethodsforEngineers.jl-00e1d38a-71a9-5665-8612-32ae585a75a3", "max_stars_repo_head_hexsha": "e230c3045d98da0cf789e4a6acdccfbfb21ef49e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-23T18:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-25T03:32:45.000Z", "max_issues_repo_path": "test/wrm_with_using_nmfe.jl", "max_issues_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_issues_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-23T21:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-27T23:14:46.000Z", "max_forks_repo_path": "test/wrm_with_using_nmfe.jl", "max_forks_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_forks_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2018-10-27T14:13:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-20T18:54:06.000Z", "avg_line_length": 22.5211267606, "max_line_length": 92, "alphanum_fraction": 0.4752970607, "num_tokens": 608, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533069832973, "lm_q2_score": 0.9059898153067649, "lm_q1q2_score": 0.8447931993759797}} {"text": "### A Pluto.jl notebook ###\n# v0.12.10\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ e077ab16-22e5-11eb-15ed-5f0654bf7328\nusing Pkg, DrWatson\n\n# ╔═╡ e75a9a06-22e5-11eb-2e36-8d4f62b830ed\nbegin\n\t@quickactivate \"StatisticsWithJuliaPlutoNotebooks\"\n\tusing Random, Distributions, StatsPlots, Plots, Measures\n\tRandom.seed!(0)\n end\n\n# ╔═╡ ca5fbdc8-22e5-11eb-0a60-cb0a127a3ca5\nmd\"## Listing4.21\"\n\n# ╔═╡ fa68607e-22e5-11eb-0558-c9a4d9f77426\nbegin\n\tb1, b2 = 0.5 , 2\n\tdist1, dist2, = Beta(b1,b1), Beta(b2,b2)\n\n\tn = 2000\n\tdata1 = rand(dist1,n)\n\tdata2 = rand(dist2,n)\nend\n\n# ╔═╡ 65067674-250d-11eb-12ab-f7aeb1fb1a16\nbegin\n\tstephist(data1, bins=15, label = \"beta($b1,$b1)\", c = :red, normed = true)\n\tp1 = stephist!(data2, bins=15, label = \"beta($b2,$b2)\",\n\t\t\tc = :blue, xlabel=\"x\", ylabel=\"Density\",normed = true)\n\n\tp2 = qqplot(data1, data2, c=:black, ms=1, msw =0,\n\t\t\txlabel=\"Quantiles for beta($b1,$b1) sample\",\n\t\t\tylabel=\"Quantiles for beta($b2,$b2) sample\",\n\t\t\tlegend=false)\n\n\tplot(p1, p2, size=(800,400), margin = 5mm)\nend\n\n# ╔═╡ 475ff888-22e6-11eb-2354-09f8f40a8e12\nmd\"## End of listing4.21\"\n\n# ╔═╡ Cell order:\n# ╟─ca5fbdc8-22e5-11eb-0a60-cb0a127a3ca5\n# ╠═e077ab16-22e5-11eb-15ed-5f0654bf7328\n# ╠═e75a9a06-22e5-11eb-2e36-8d4f62b830ed\n# ╠═fa68607e-22e5-11eb-0558-c9a4d9f77426\n# ╠═65067674-250d-11eb-12ab-f7aeb1fb1a16\n# ╟─475ff888-22e6-11eb-2354-09f8f40a8e12\n", "meta": {"hexsha": "07cda7ee7cd111f73af1ac6b697fd8db201fc777", "size": 1348, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "notebooks/04/listing4.21.jl", "max_stars_repo_name": "pitmonticone/StatisticsWithJuliaPlutoNotebooks.jl", "max_stars_repo_head_hexsha": "47a10ee915d87dd3241d2863e47b5f9630a03409", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 68, "max_stars_repo_stars_event_min_datetime": "2020-11-08T07:32:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-22T16:58:01.000Z", "max_issues_repo_path": "notebooks/04/listing4.21.jl", "max_issues_repo_name": "pitmonticone/StatisticsWithJuliaPlutoNotebooks.jl", "max_issues_repo_head_hexsha": "47a10ee915d87dd3241d2863e47b5f9630a03409", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-12-07T08:07:30.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-20T16:16:06.000Z", "max_forks_repo_path": "notebooks/04/listing4.21.jl", "max_forks_repo_name": "pitmonticone/StatisticsWithJuliaPlutoNotebooks.jl", "max_forks_repo_head_hexsha": "47a10ee915d87dd3241d2863e47b5f9630a03409", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 14, "max_forks_repo_forks_event_min_datetime": "2020-11-14T05:07:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-16T21:05:35.000Z", "avg_line_length": 24.962962963, "max_line_length": 75, "alphanum_fraction": 0.6958456973, "num_tokens": 668, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.942506716354847, "lm_q2_score": 0.8962513745192026, "lm_q1q2_score": 0.8447229400266117}} {"text": "import Pkg; Pkg.add(Pkg.PackageSpec(url=\"https://github.com/JuliaComputing/JuliaAcademyData.jl\"))\nusing JuliaAcademyData; activate(\"Foundations of machine learning\")\n\n## Run this cell to load the graphics packages\nusing Plots; gr()\n\n#-\n\n## Run this cell to use a couple of definitions from the previous Tools notebook\n\nσ(x) = 1 / (1 + exp(-x))\nf(x, w) = σ(w * x)\n\n# ## Multiple function parameters\n#\n# In notebook 6, we saw how we could adjust a parameter to make a curve fit a single data point. What if there is more data to fit?\n#\n# We'll see that as we acquire more data to fit, it can sometimes be useful to add complexity to our model via additional parameters.\n\n#-\n\n# ## Adding more data\n\n#-\n\n# Suppose there are now two data points to fit, the previous $(x_0, y_0)$ used in notebook 6 and also $(x_1, y_1) = (-3, 0.3)$.\n\n#-\n\n# #### Exercise 1\n#\n# Make an interactive plot of the function $f_w$ together with the two data points. Can you make the graph of $f_w$ pass through *both* data points at the *same* time?\n\nx0, y0 = 2, 0.8\nx1, y1 = -3, 0.3\n\nw = 5.0 # try manipulating w between 0 and 10\nplot(x->f(x, w), -5, 5, ylim=(0,1))\nscatter!([x0, x1], [y0, y1])\n\n# You should have found that it's actually *impossible* to fit both data points at the same time! The best we could do is to *minimise* how far away the function is from the data. To do so, we need to somehow balance the distance from each of the two data points.\n\n#-\n\n# #### Exercise 2\n#\n# Play with the slider to find the value $w^*$ of $w$ that you think has the \"least error\" in an intuitive sense.\n\nw=0.45\nplot(x->f(x, w), -5, 5, ylim=(0,1))\nscatter!([x0, x1], [y0, y1])\n\n# ## Defining a loss function\n\n#-\n\n# How can we *quantify* some kind of overall measure of the distance from *all* of the data? Well, we just need to define a new loss function! One way to do so would be to sum up the loss functions for each data point, i.e. the sum of the squared vertical distances from the graph to each data point:\n#\n# $$L(w) = [y_0 - f_w(x_0)]^2 + [y_1 - f_w(x_1)]^2.$$\n#\n# Since the two pieces that we are adding have the same mathematical \"shape\" or structure, we can abbreviate this by writing it as a sum:\n#\n# $$L(w) = \\sum_{i=0}^1 [y_i - f_w(x_i)]^2.$$\n\n#-\n\n# So now we want to find the value $w^*$ of $w$ that minimizes this new function $L$!\n\n#-\n\n# #### Exercise 3\n#\n# Make a visualization to show the function $f_w$ and to visualize the distance from each of the data points.\n\n#-\n\n# #### Solution\n\nxs = [2, -3]\nys = [0.8, 0.3]\n\nL(w) = sum( (ys .- f.(xs, w)) .^ 2 )\n\nw = 0.0 # Try manipulating w between -2 and 2\n\n plot(x->f(x, w), -5, 5, ylims=(0, 1), lw = 3, label=\"f_w\")\n\n scatter!(xs, ys, label=\"data\")\n\n for i in 1:2\n plot!([xs[i], xs[i]], [ys[i], f(xs[i], w)], c=:green)\n end\n\n\n title!(\"L(w) = $(round(L(w); sigdigits = 5))\")\n\nend\n\n# #### Exercise 4\n#\n# After playing with this for a while, it is intuitively obvious that we cannot make the function pass through both data points for any value of $w$. In other words, our loss function, `L(w)`, is never zero.\n#\n# What is the minimum value of `L(w)` that you can find by altering `w`? What is the corresponding value of `w`?\n\nw = 0.42\nplot(x->f(x, w), -5, 5, ylims=(0, 1), lw = 3, label=\"f_w\")\n\nscatter!(xs, ys, label=\"data\")\n\nfor i in 1:2\n plot!([xs[i], xs[i]], [ys[i], f(xs[i], w)], c=:green)\nend\n\n\ntitle!(\"L(w) = $(round(L(w); sigdigits = 5))\")\n\n# ### Sums in Julia\n\n#-\n\n# To generate the above plot we used the `sum` function. `sum` can add together all the elements of a collection or range, or it can add together the outputs of applying a function to all the elements of a collection or range.\n#\n# Look up the docs for `sum` via\n#\n# ```julia\n# ?sum\n# ```\n# if you need more information.\n\n#-\n\n# #### Exercise 5\n#\n# Use `sum` to add together all integers in the range 1 to 16, inclusive. What is the result?\n\n#-\n\n# #### Solution\n\nsum(1:16)\n\n# #### Exercise 6\n#\n# What is the sum of the absolute values of all integers between -3 and 3? Use `sum` and the `abs` function.\n\n#-\n\n# #### Solution\n\nsum(abs, -3:3)\n\n# ## What does the loss function $L$ look like?\n\n#-\n\n# In our last attempt to minimize `L(w)` by varying `w`, we saw that `L(w)` always seemed to be greater than 0. Is that true? Let's plot it to find out!\n\n#-\n\n# #### Exercise 7\n#\n# Plot the new loss function $L(w)$ as a function of $w$.\n\n#-\n\n# #### Solution\n\nplot(L, -2, 2, xlabel=\"w\", ylabel=\"L(w)\", ylims=(0, 1.2))\n\n# ### Features of the loss function\n\n#-\n\n# The first thing to notice is that $L$ is always positive. Since it is the sum of squares, and squares cannot be negative, the sum cannot be negative either!\n#\n# However, we also see that although $L$ dips close to $0$ for a single, special value $w^* \\simeq 0.4$, it never actually *reaches* 0! Again we could zoom in on that region of the graph to estimate it more precisely.\n\n#-\n\n# We might start suspecting that there should be a better way of using the computer to minimize $L$ to find the location $w^*$ of the minimum, rather than still doing everything by eye. Indeed there is, as we will see in the next two notebooks!\n\n#-\n\n# ## Adding more parameters to the model\n\n#-\n\n# If we add more parameters to a function, we may be able to improve how it fits to data. For example, we could define a new function $g$ with another parameter, a shift or **bias**:\n#\n# $$g(x; w, b) := \\sigma(w \\, x + b).$$\n\ng(x, w, b) = σ(w*x + b)\n\n# #### Exercise 8\n#\n# Make an interactive visualization that allows you to change $w$ and $b$. Play with the sliders to try and fit *both* data points at once!\n\n#-\n\n# #### Solution\n\nxs = [2, -3]\nys = [0.8, 0.3]\n\nL2D(w, b) = sum( (ys .- g.(xs, w, b)) .^ 2)\n\nw = 0.0 # Try manipulating w between -2 and 2\nb = 0.0 # Try manipulating b between -2 and 2\nplot(x -> g(x, w, b), -5, 5, ylims=(0, 1), lw=3)\n\nscatter!(xs, ys)\n\nfor i in 1:2\n plot!([xs[i], xs[i]], [ys[i], g(xs[i], w, b)])\nend\n\n\ntitle!(\"L2D(w, b) = $(L2D(w, b))\")\n\n# You should be able to convince yourself that we can now make the curve pass through both points simultaneously.\n\n#-\n\n# #### Exercise 9\n#\n# For what values of `w` and `b` does the line pass through both points?\n\nw = 0.45\nb = 0.48\nplot(x -> g(x, w, b), -5, 5, ylims=(0, 1), lw=3)\n\nscatter!(xs, ys)\n\nfor i in 1:2\n plot!([xs[i], xs[i]], [ys[i], g(xs[i], w, b)])\nend\n\n\ntitle!(\"L2D(w, b) = $(L2D(w, b))\")\n\n# ## Fitting both data points: a loss function\n\n#-\n\n# Following the procedure that we used when we had a single parameter, we can think of the fitting procedure once again as minimizing a suitable *loss function*. The expression for the loss function is almost the same, except that now $f$ has two parameters $w$ and $b$, so the loss function $L_2$ is itself a function of $w$ *and* $b$:\n\n#-\n\n# $$L_2(w, b) = \\sum_i [y_i - f(x_i; w, b)]^2.$$\n\n#-\n\n# So we want to minimize this loss function over *both* of the parameters $w$ and $b$! Let's plot it.\n\n#-\n\n# #### Exercise 10\n#\n# Define the function `L2` in Julia.\n\n#-\n\n# #### Solution\n\nL2(w, b) = sum( (ys .- g.(xs, w, b)) .^ 2 )\n\n# To plot the loss as a function of *two* variables (the weights *and* the bias), we will make use of the `surface` function. To get a nice interactive 3D plot, we will use the PlotlyJS \"backend\" (plotting engine):\n\ngr() # load the backend\n\n#-\n\nws = -2:0.05:2\nbs = -2:0.05:2\n\ncontour(ws, bs, L2, levels=0.001:0.025:1.25, xlabel=\"value of w\", ylabel=\"value of b\")\n\n# We can see that indeed there is a unique point $(w^*, b^*)$ where the function $L_2$ attains its minimum. You can try different ranges for the $x$, $y$ and $z$ coordinates to get a better view.\n\n#-\n\n# ## More data\n\n#-\n\n# If we add more data, however, we will again not be able to fit all of the data; we will only be able to attain a \"best fit\".\n#\n# Let's create `xs` and `ys` with some more data:\n\nxs = [2, -3, -1, 1]\nys = [0.8, 0.3, 0.4, 0.4]\n\n# #### Exercise 11\n#\n# a) Make an interactive visualization of the function $f$ and the data. Try to find the values of $w$ and $b$ that give the best fit.\n#\n# b) Define the loss function and plot it.\n\n#-\n\n# #### Solution\n\ngr()\n\n#-\n\nL2(w, b) = sum( (ys .- g.(xs, w, b)) .^ 2)\n\nw = 0.0 # Try manipulating w between -2 and 2\nb = 0.0 # Try manipulating b between -2 and 2\nplot(x->g(x, w, b), -5, 5, ylims=(-0.2, 1))\n\nscatter!(xs, ys)\n\nfor i in 1:length(xs)\n plot!([xs[i], xs[i]], [ys[i], g(xs[i], w, b)])\nend\n\n\ntitle!(\"L2(w, b) = $(round(L2(w, b); sigdigits = 5))\")\n\n# Let's define the loss function that we're using above so that we can plot it as a function of the parameters `w` and `b`.\n\n#-\n\n# #### Exercise 12\n#\n# We've seen the loss function, $$L_{2D}(w, b) = \\sum_i[\\mathrm{ys}_i - g(\\mathrm{xs}_i, w, b)]^2,$$ written in Julia as\n#\n# ```julia\n# L2(w, b) = sum( (ys .- g.(xs, w, b)) .^ 2 )\n# ```\n#\n# a few times now. To ensure you understand what this function is doing, implement your own loss function using the commented code below. Do this without using `sum`, `abs2`, or broadcasting dot syntax (for example, `.-`). Hint: you'll want to use a `for` loop to do this.\n\nfunction myL2(w, b)\n loss = 0.0\n for (x, y) in zip(xs, ys)\n loss += (y - g(x, w, b))^2\n end\n return loss\nend\n\n# Now that you've defined `L2`, we can plot it using either `surface` or `contour` to see how the loss changes as `w` and `b` change!\n\n## plotlyjs() # Use Plotly for 3d surface plots (if possible)\n\nws = -2:0.05:2\nbs = -2:0.05:2\n\n## surface(ws, bs, L2, alpha=0.8, zlims=(0,3)) # alpha gives transparency\ncontour(ws, bs, L2, levels=0.055:0.05:1.5, xlabel=\"value of w\", ylabel=\"value of b\")\n\n", "meta": {"hexsha": "0f314110d166d3d5693b3d3ef8a76976ba8b4807", "size": 9533, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Courses/Foundations of machine learning/0700.Model-complexity-quiz.jl", "max_stars_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_stars_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 45, "max_stars_repo_stars_event_min_datetime": "2020-02-13T00:50:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-05T07:57:22.000Z", "max_issues_repo_path": "Courses/Foundations of machine learning/0700.Model-complexity-quiz.jl", "max_issues_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_issues_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 52, "max_issues_repo_issues_event_min_datetime": "2019-10-30T16:22:28.000Z", "max_issues_repo_issues_event_max_datetime": "2020-01-26T20:02:43.000Z", "max_forks_repo_path": "Courses/Foundations of machine learning/0700.Model-complexity-quiz.jl", "max_forks_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_forks_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 27, "max_forks_repo_forks_event_min_datetime": "2020-02-26T11:33:28.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-25T22:34:53.000Z", "avg_line_length": 26.9293785311, "max_line_length": 336, "alphanum_fraction": 0.6375747404, "num_tokens": 3131, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391727723469, "lm_q2_score": 0.9149009549929797, "lm_q1q2_score": 0.8444894206653503}} {"text": "#Partitions\n\nexport\n integer_partitions,\n ncpartitions,\n partitions,\n prevprod\n #nextprod,\n\n\n#integer partitions\n\nstruct IntegerPartitions\n n::Int\nend\n\nBase.length(p::IntegerPartitions) = npartitions(p.n)\n\nfunction Base.iterate(p::IntegerPartitions, xs = Int[])\n length(xs) == p.n && return\n xs = nextpartition(p.n,xs)\n (xs, xs)\nend\n\n\"\"\"\n partitions(n)\n\nGenerate all integer arrays that sum to `n`. Because the number of partitions can be very\nlarge, this function returns an iterator object. Use `collect(partitions(n))` to get an\narray of all partitions. The number of partitions to generate can be efficiently computed\nusing `length(partitions(n))`.\n\"\"\"\npartitions(n::Integer) = IntegerPartitions(n)\n\n\n\nfunction nextpartition(n, as)\n isempty(as) && return Int[n]\n\n xs = similar(as, 0)\n sizehint!(xs, length(as) + 1)\n\n for i = 1:length(as)-1\n if as[i+1] == 1\n x = as[i]-1\n push!(xs, x)\n n -= x\n while n > x\n push!(xs, x)\n n -= x\n end\n push!(xs, n)\n\n return xs\n end\n push!(xs, as[i])\n n -= as[i]\n end\n push!(xs, as[end]-1)\n push!(xs, 1)\n\n xs\nend\n\nlet _npartitions = Dict{Int,Int}()\n global npartitions\n function npartitions(n::Int)\n if n < 0\n 0\n elseif n < 2\n 1\n elseif (np = get(_npartitions, n, 0)) > 0\n np\n else\n np = 0\n sgn = 1\n for k = 1:n\n np += sgn * (npartitions(n - (k*(3k-1)) >> 1) + npartitions(n - (k*(3k+1)) >> 1))\n sgn = -sgn\n end\n _npartitions[n] = np\n end\n end\nend\n\n# Algorithm H from TAoCP 7.2.1.4\n# Partition n into m parts\n# in colex order (lexicographic by reflected sequence)\n\nstruct FixedPartitions\n n::Int\n m::Int\nend\n\nBase.length(f::FixedPartitions) = npartitions(f.n,f.m)\n\n\"\"\"\n partitions(n, m)\n\nGenerate all arrays of `m` integers that sum to `n`. Because the number of partitions can\nbe very large, this function returns an iterator object. Use `collect(partitions(n, m))` to\nget an array of all partitions. The number of partitions to generate can be efficiently\ncomputed using `length(partitions(n, m))`.\n\"\"\"\npartitions(n::Integer, m::Integer) =\n n >= 1 && m >= 1 ?\n FixedPartitions(n, m) :\n throw(DomainError((n, m), \"n and m must be positive\"))\n\nfunction Base.iterate(f::FixedPartitions, s::Vector{Int} = Int[])\n f.m <= f.n || return\n if !isempty(s)\n (f.m == 1 || s[1]-1 <= s[end]) && return\n end\n\n xs = nextfixedpartition(f.n,f.m,s)\n (xs, xs)\nend\n\nfunction nextfixedpartition(n, m, bs)\n as = copy(bs)\n if isempty(as)\n # First iteration\n as = [n-m+1; ones(Int, m-1)]\n elseif as[2] < as[1]-1\n # Most common iteration\n as[1] -= 1\n as[2] += 1\n else\n # Iterate\n j = 0\n s = as[1]+as[2]-1\n for jj = 3:m # TODO: use `for outer j = ...` on 0.7\n j = jj\n as[jj] < as[1]-1 && break\n s += as[jj]\n end\n x = as[j] += 1\n for k = j-1:-1:2\n as[k] = x\n s -= x\n end\n as[1] = s\n end\n\n return as\nend\n\nlet _nipartitions = Dict{Tuple{Int,Int},Int}()\n global npartitions\n function npartitions(n::Int, m::Int)\n if n < m || m == 0\n 0\n elseif n == m\n 1\n elseif (np = get(_nipartitions, (n,m), 0)) > 0\n np\n else\n _nipartitions[(n, m)] = npartitions(n-1, m-1) + npartitions(n-m, m)\n end\n end\nend\n\n# Algorithm H from TAoCP 7.2.1.5\n# Set partitions\n\nstruct SetPartitions{T<:AbstractVector}\n s::T\nend\n\nBase.length(p::SetPartitions) = nsetpartitions(length(p.s))\n\n\"\"\"\n partitions(s::AbstractVector)\n\nGenerate all set partitions of the elements of an array `s`, represented as arrays of\narrays. Because the number of partitions can be very large, this function returns an\niterator object. Use `collect(partitions(s))` to get an array of all partitions. The\nnumber of partitions to generate can be efficiently computed using\n`length(partitions(s))`.\n\"\"\"\npartitions(s::AbstractVector) = SetPartitions(s)\n\nfunction Base.iterate(p::SetPartitions)\n n = length(p.s)\n iterate(p, (zeros(Int32, n), ones(Int32, n-1), n, 1))\nend\nfunction Base.iterate(p::SetPartitions, s)\n s[1][1] > 0 && return\n nextsetpartition(p.s, s...)\nend\n\nfunction nextsetpartition(s::AbstractVector, a, b, n, m)\n function makeparts(s, a, m)\n temp = [similar(s, 0) for k = 0:m]\n for i = 1:n\n push!(temp[a[i]+1], s[i])\n end\n filter!(!isempty, temp)\n end\n\n isempty(s) && return ([s], ([1], Int[], n, 1))\n\n part = makeparts(s,a,m)\n\n if a[end] != m\n a[end] += 1\n else\n j = 0\n for jj = n-1:-1:1\n j = jj\n a[jj] == b[jj] || break\n end\n a[j] += 1\n m = b[j] + (a[j] == b[j])\n for k = j+1:n-1\n a[k] = 0\n b[k] = m\n end\n a[end] = 0\n end\n\n return (part, (a,b,n,m))\nend\n\nlet _nsetpartitions = Dict{Int,Int}()\n global nsetpartitions\n function nsetpartitions(n::Int)\n if n < 0\n 0\n elseif n < 2\n 1\n elseif (wn = get(_nsetpartitions, n, 0)) > 0\n wn\n else\n wn = 0\n for k = 0:n-1\n wn += binomial(n-1,k)*nsetpartitions(n-1-k)\n end\n _nsetpartitions[n] = wn\n end\n end\nend\n\nstruct FixedSetPartitions{T<:AbstractVector}\n s::T\n m::Int\nend\n\nBase.length(p::FixedSetPartitions) = nfixedsetpartitions(length(p.s),p.m)\n\n\"\"\"\n partitions(s::AbstractVector, m::Int)\n\nGenerate all set partitions of the elements of an array `s` into exactly `m` subsets,\nrepresented as arrays of arrays. Because the number of partitions can be very large,\nthis function returns an iterator object. Use `collect(partitions(s, m))` to get\nan array of all partitions. The number of partitions into `m` subsets is equal to the\nStirling number of the second kind, and can be efficiently computed using\n`length(partitions(s, m))`.\n\"\"\"\npartitions(s::AbstractVector, m::Int) =\n length(s) >= 1 && m >= 1 ?\n FixedSetPartitions(s, m) :\n throw(DomainError((length(s), m), \"length(s) and m must be positive\"))\n\nfunction Base.iterate(p::FixedSetPartitions)\n n = length(p.s)\n m = p.m\n state = m <= n ? (vcat(ones(Int, n-m),1:m), vcat(1,n-m+2:n), n) : (Int[], Int[], n)\n # state consists of:\n # vector a of length n describing to which partition every element of s belongs\n # vector b of length n describing the first index b[i] that belongs to partition i\n # integer n\n\n iterate(p, state)\nend\n\nfunction Base.iterate(p::FixedSetPartitions, s)\n (isempty(s[1]) || s[1][1] > 1) && return\n nextfixedsetpartition(p.s,p.m, s...)\nend\n\nfunction nextfixedsetpartition(s::AbstractVector, m, a, b, n)\n function makeparts(s, a)\n part = [similar(s, 0) for k = 1:m]\n for i = 1:n\n push!(part[a[i]], s[i])\n end\n return part\n end\n\n part = makeparts(s,a)\n\n if m == 1\n a[1] = 2\n return (part, (a, b, n))\n end\n\n if a[end] != m\n a[end] += 1\n else\n j = k = 0\n for jj = n-1:-1:1\n j = jj\n if a[j] < m && b[a[j]+1] < j\n break\n end\n end\n if j > 1\n a[j] += 1\n for p = j+1:n\n if b[a[p]] != p\n a[p] = 1\n end\n end\n else\n for kk = m:-1:2\n k = kk\n if b[k-1] < b[k] - 1\n break\n end\n end\n b[k] -= 1\n b[k+1:m] = n-m+k+1:n\n a[1:n] .= 1\n a[b] = 1:m\n end\n end\n\n return (part, (a,b,n))\nend\n\nfunction nfixedsetpartitions(n::Int, m::Int)\n numpart = 0\n for k = 0:m\n numpart += (-1)^(m-k) * binomial(m, k) * (k^n)\n end\n numpart = div(numpart, factorial(m))\n return numpart\nend\n\n# TODO: Base.DSP is no longer a thing in Julia 0.7\n#This function is still defined in Base because it is being used by Base.DSP\n#\"\"\"\n#Next integer not less than `n` that can be written as $\\prod k_i^{p_i}$ for integers $p_1$, $p_2$, etc.\n#\n#For a list of integers i1, i2, i3, find the smallest\n# i1^n1 * i2^n2 * i3^n3 >= x\n#for integer n1, n2, n3\n#\"\"\"\n#function nextprod(a::Vector{Int}, x)\n# if x > typemax(Int)\n# throw(ArgumentError(\"unsafe for x > typemax(Int), got $x\"))\n# end\n# k = length(a)\n# v = ones(Int, k) # current value of each counter\n# mx = [nextpow(ai,x) for ai in a] # maximum value of each counter\n# v[1] = mx[1] # start at first case that is >= x\n# p::widen(Int) = mx[1] # initial value of product in this case\n# best = p\n# icarry = 1\n#\n# while v[end] < mx[end]\n# if p >= x\n# best = p < best ? p : best # keep the best found yet\n# carrytest = true\n# while carrytest\n# p = div(p, v[icarry])\n# v[icarry] = 1\n# icarry += 1\n# p *= a[icarry]\n# v[icarry] *= a[icarry]\n# carrytest = v[icarry] > mx[icarry] && icarry < k\n# end\n# if p < x\n# icarry = 1\n# end\n# else\n# while p < x\n# p *= a[1]\n# v[1] *= a[1]\n# end\n# end\n# end\n# best = mx[end] < best ? mx[end] : best\n# return Int(best) # could overflow, but best to have predictable return type\n#end\n\n\"\"\"\n prevprod(a::Vector{Int}, x)\n\nPrevious integer not greater than `x` that can be written as ``\\\\prod k_i^{p_i}`` for\nintegers ``p_1``, ``p_2``, etc.\n\nFor integers ``i_1``, ``i_2``, ``i_3``, this is equivalent to finding the largest ``x``\nsuch that\n```math\ni_1^n_1 i_2^n_2 i_3^n_3 \\\\leq x\n```\nfor integers ``n_1``, ``n_2``, ``n_3``.\n\"\"\"\nfunction prevprod(a::Vector{Int}, x)\n if x > typemax(Int)\n throw(ArgumentError(\"unsafe for x > typemax(Int), got $x\"))\n end\n k = length(a)\n v = ones(Int, k) # current value of each counter\n mx = [nextpow(ai,x) for ai in a] # allow each counter to exceed p (sentinel)\n first = Int(prevpow(a[1], x)) # start at best case in first factor\n v[1] = first\n p::widen(Int) = first\n best = p\n icarry = 1\n\n while v[end] < mx[end]\n while p <= x\n best = p > best ? p : best\n p *= a[1]\n v[1] *= a[1]\n end\n if p > x\n carrytest = true\n while carrytest\n p = div(p, v[icarry])\n v[icarry] = 1\n icarry += 1\n p *= a[icarry]\n v[icarry] *= a[icarry]\n carrytest = v[icarry] > mx[icarry] && icarry < k\n end\n if p <= x\n icarry = 1\n end\n end\n end\n best = x >= p > best ? p : best\n return Int(best)\nend\n\n\n\"\"\"\n integer_partitions(n)\n\nList the partitions of the integer `n`.\n\n!!! note\n The order of the resulting array is consistent with that produced by the computational\n discrete algebra software GAP.\n\"\"\"\nfunction integer_partitions(n::Integer)\n if n < 0\n throw(DomainError(n, \"n must be nonnegative\"))\n elseif n == 0\n return Vector{Int}[]\n elseif n == 1\n return Vector{Int}[[1]]\n end\n\n list = Vector{Int}[]\n\n for p in integer_partitions(n-1)\n push!(list, [p; 1])\n if length(p) == 1 || p[end] < p[end-1]\n push!(list, [p[1:end-1]; p[end]+1])\n end\n end\n\n list\nend\n\n\n\n#Noncrossing partitions\n\nif VERSION >= v\"0.7.0-DEV.3251\"\n const _cmp = cmp\nelse\n const _cmp = lexcmp\nend\n\n#Produces noncrossing partitions of length n\nfunction ncpartitions(n::Int)\n partitions = Vector{Vector{Int}}[]\n _ncpart!(1,n,n,Vector{Int}[], partitions)\n partitions\nend\n\nfunction _ncpart!(a::Int, b::Int, nn::Int, x::Vector, partitions::Vector)\n n = b - a + 1\n for k = 1:n, root in CoolLexCombinations(n, k)\n root .+= a - 1\n #Abort if construction is out of lex order\n !isempty(x) && _cmp(x[end], root) == 1 && return\n\n #Save if we've filled all the holes\n sofar = Vector{Int}[x..., root]\n ssofaru = sort!(union(sofar...))\n if length(ssofaru) == nn && ssofaru == collect(1:nn)\n push!(partitions, sofar)\n return\n end\n\n #otherwise patch all remaining holes\n blob = [ssofaru; nn + 1]\n for l = 1:length(blob)-1\n ap, bp = blob[l] + 1, blob[l+1] - 1\n ap <= bp && _ncpart!(ap, bp, nn, sofar, partitions)\n end\n end\nend\n", "meta": {"hexsha": "f84ead100623257f2c38621e47ffd4f0a92f9c1b", "size": 12908, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/partitions.jl", "max_stars_repo_name": "mschauer/Combinatorics.jl", "max_stars_repo_head_hexsha": "0e0edc6b204f3cbd726074a9d3f8d4288b3ab16b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/partitions.jl", "max_issues_repo_name": "mschauer/Combinatorics.jl", "max_issues_repo_head_hexsha": "0e0edc6b204f3cbd726074a9d3f8d4288b3ab16b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/partitions.jl", "max_forks_repo_name": "mschauer/Combinatorics.jl", "max_forks_repo_head_hexsha": "0e0edc6b204f3cbd726074a9d3f8d4288b3ab16b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.2109375, "max_line_length": 104, "alphanum_fraction": 0.5257979548, "num_tokens": 3970, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391664210672, "lm_q2_score": 0.9149009544128984, "lm_q1q2_score": 0.8444894143191206}} {"text": "x = -11:11\ny = sin.(x)\nx = x[y .> 0] # Only keep values of x where y=sin x is positive\n\n# Example: Consider the matrix\nA = rand(-10:10, 3, 10)\n\n# Compute a boolean array with `true` if the column sums are >=0\npick = sum(A, dims=1) .≥ 0\n\n# Create a new vector with only the columns of A given by the `pick` variable\n# Note that since `pick` is a 2D vector (row vector), we use [:] before indexing\nB = A[:, pick[:]]\n", "meta": {"hexsha": "862ab79dbab8ba75871fdff082a2d45086f3ac6a", "size": 423, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "textbook/_build/jupyter_execute/content/Vectorization/Logical_Indexing.jl", "max_stars_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_stars_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "textbook/_build/jupyter_execute/content/Vectorization/Logical_Indexing.jl", "max_issues_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_issues_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "textbook/_build/jupyter_execute/content/Vectorization/Logical_Indexing.jl", "max_forks_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_forks_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.2142857143, "max_line_length": 80, "alphanum_fraction": 0.6359338061, "num_tokens": 140, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418241572635, "lm_q2_score": 0.9136765163620469, "lm_q1q2_score": 0.8443666825204759}} {"text": "# This file illustrates the calculation of power spectral density and \n# total average power of a sinusoidal signal. \n\nusing FFTW \nusing Plots \n\n# Construct the signal\nf = 20\nfs = 100 \nl = 2^10\nts = 1 / fs \nt = (0 : l -1) * ts \nx = cos.(2π * f * t)\n\n# Compute fft\nxf = fft(x) * ts \nff = 0 : fs / (l - 1) : fs \n\n# Compute power \npsd = abs.(xf).^2 / (l - 1) / ts \nΔf = fs / (l - 1) \nPf = sum(psd) * Δf # Average power calculated in time \nPt = sum(abs.(x).^2) / (l - 1) # Average power calculated in frequency\n@show Pf, Pt\n\n# Plots \nplt = plot(layout=1)\nplot!(ff, psd, subplot=1)\nxlabel!(\"Frequency [Hz]\") \nylabel!(\"Ψ(f)\")\ntitle!(\"Power Spectal Density\")\ndisplay(plt)", "meta": {"hexsha": "86ab6237eefae9be3fd991a2cde2e0e708189e83", "size": 678, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "example/dsp/sinusoidal.jl", "max_stars_repo_name": "zekeriyasari/DigitalCommunications.jl", "max_stars_repo_head_hexsha": "7ac2e6afe42aa996d94d211c1ea590ac3c0beb15", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-12-03T20:02:21.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-13T06:43:38.000Z", "max_issues_repo_path": "example/dsp/sinusoidal.jl", "max_issues_repo_name": "zekeriyasari/DigitalCommunications.jl", "max_issues_repo_head_hexsha": "7ac2e6afe42aa996d94d211c1ea590ac3c0beb15", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2020-11-26T21:56:29.000Z", "max_issues_repo_issues_event_max_datetime": "2021-01-03T19:54:59.000Z", "max_forks_repo_path": "example/dsp/sinusoidal.jl", "max_forks_repo_name": "zekeriyasari/DigitalCommunications.jl", "max_forks_repo_head_hexsha": "7ac2e6afe42aa996d94d211c1ea590ac3c0beb15", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-20T12:53:43.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-20T12:53:43.000Z", "avg_line_length": 21.1875, "max_line_length": 71, "alphanum_fraction": 0.598820059, "num_tokens": 239, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.967899289579129, "lm_q2_score": 0.8723473663814338, "lm_q1q2_score": 0.8443443961868139}} {"text": "\"Square the sum of the first `n` positive integers\"\nfunction square_of_sum(n)\n tmp=0\n for i in 1:n\n tmp+=i\n end\n return tmp^2\nend\n\n\"Sum the squares of the first `n` positive integers\"\nfunction sum_of_squares(n)\n tmp=0\n for i in 1:n\n tmp+=i^2\n end\n return tmp\nend\n\n\"Subtract the sum of squares from square of the sum of the first `n` positive ints\"\nfunction difference(n)\n return abs(sum_of_squares(n) - square_of_sum(n))\nend\n", "meta": {"hexsha": "5735899c9df90a9ee3e2957b185ebc2204eafb09", "size": 466, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "difference-of-squares/difference-of-squares.jl", "max_stars_repo_name": "stepinski/julia-in", "max_stars_repo_head_hexsha": "08c82e2de236ead3b1eb356d60a04c61e275d4fa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-01T11:27:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-01T11:27:13.000Z", "max_issues_repo_path": "difference-of-squares/difference-of-squares.jl", "max_issues_repo_name": "stepinski/julia-init", "max_issues_repo_head_hexsha": "e70b08c72c4db3677483480e72a9f79827a41fff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "difference-of-squares/difference-of-squares.jl", "max_forks_repo_name": "stepinski/julia-init", "max_forks_repo_head_hexsha": "e70b08c72c4db3677483480e72a9f79827a41fff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.2608695652, "max_line_length": 83, "alphanum_fraction": 0.669527897, "num_tokens": 139, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9465966671870766, "lm_q2_score": 0.8918110454379296, "lm_q1q2_score": 0.8441853633721667}} {"text": "function absolute_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(abs(y_true .- y_pred))\nend\n\nfunction percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return((y_true .- y_pred) ./ y_true)\nend\n\nfunction log_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(log(y_true - y_pred))\nend\n\nfunction squared_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return((y_true - y_pred) .^ 2)\nend\n\nfunction squared_log_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(log_error(y_true, y_pred) .^ 2)\nend\n\nfunction absolute_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(abs(percent_error(y_true, y_pred)))\nend\n\nfunction mean_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(y_true - y_pred))\nend\n\nfunction mean_absolute_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(absolute_error(y_true, y_pred)))\nend\n\nfunction median_absolute_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(median(absolute_error(y_true, y_pred)))\nend\n\nfunction mean_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(percent_error(y_true, y_pred)))\nend\n\nfunction median_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(median(percent_error(y_true, y_pred)))\nend\n\nfunction mean_squared_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(squared_error(y_true, y_pred)))\nend\n\nfunction median_squared_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(median(squared_error(y_true, y_pred)))\nend\n\nfunction sum_squared_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(sum(squared_error(y_true, y_pred)))\nend\n\nfunction mean_squared_log_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(squared_log_error(y_true, y_pred)))\nend\n\nfunction mean_absolute_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(absolute_percent_error(y_true, y_pred)))\nend\n\nfunction median_absolute_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(median(absolute_percent_error(y_true, y_pred)))\nend\n\nfunction symmetric_mean_absolute_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(mean(abs(y_pred - y_true) / ((abs(y_true) + abs(y_pred)) / 2)))\nend\n\nfunction symmetric_median_absolute_percent_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(median(abs(y_pred - y_true) / ((abs(y_true) + abs(y_pred)) / 2)))\nend\n\nfunction mean_absolute_scaled_error(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n n = max(length(y_true), length(y_pred))\n numerator = sum(abs(y_true - y_pred))\n denominator = (n / (n - 1)) * sum(abs(y_true[2:n] - y_pred[1:(n-1)]))\n return(numerator / denominator)\nend\n\nfunction total_variance_score(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(sum((y_true - mean(y_true)) .^ 2))\nend\n\nfunction explained_variance_score(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(sum((y_pred - mean(y_true)) .^ 2))\nend\n\nfunction unexplained_variance_score(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(sum((y_true - y_pred) .^ 2))\nend\n\nfunction r2_score(y_true::Vector, y_pred::Vector)\n check_args(y_true, y_pred)\n return(explained_variance_score(y_true, y_pred) / total_variance_score(y_true, y_pred))\nend\n", "meta": {"hexsha": "e5bf0b98b739cf8af0ee83b74afc1442fa966723", "size": 3710, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/regression.jl", "max_stars_repo_name": "JuliaPackageMirrors/MachineLearningMetrics.jl", "max_stars_repo_head_hexsha": "6ca8b9c26bab6ef5e08ea9e2278b1eb74e69aae7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2016-04-06T13:02:18.000Z", "max_stars_repo_stars_event_max_datetime": "2018-12-11T09:35:51.000Z", "max_issues_repo_path": "src/regression.jl", "max_issues_repo_name": "JuliaPackageMirrors/MachineLearningMetrics.jl", "max_issues_repo_head_hexsha": "6ca8b9c26bab6ef5e08ea9e2278b1eb74e69aae7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2016-03-24T18:23:32.000Z", "max_issues_repo_issues_event_max_datetime": "2019-10-09T08:26:32.000Z", "max_forks_repo_path": "src/regression.jl", "max_forks_repo_name": "JuliaPackageMirrors/MachineLearningMetrics.jl", "max_forks_repo_head_hexsha": "6ca8b9c26bab6ef5e08ea9e2278b1eb74e69aae7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2016-07-12T02:14:56.000Z", "max_forks_repo_forks_event_max_datetime": "2018-02-26T18:19:25.000Z", "avg_line_length": 30.162601626, "max_line_length": 91, "alphanum_fraction": 0.7447439353, "num_tokens": 994, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9566341987633821, "lm_q2_score": 0.8824278741843883, "lm_q1q2_score": 0.8441606823868568}} {"text": "# 2520 is the smallest number that can be divided by each of the numbers from 1 to 10\n# without any remainder.\n# What is the smallest positive number that is evenly\n# divisible by all of the numbers from 1 to 20?\n\n# (1,2) (1,3) Means 1 * 2 * 1 * 3\n# (2,2) means 2 * 2\ninclude(\"./primes.jl\")\nusing Primes\n\n\nfunction pop(arr, value)\n inx = findfirst(x -> x == value, arr)\n if inx != 0\n deleteat!(arr, inx)\n return value\n end\n return 0\nend\n\n\nfunction merge(factor_set_1, factor_set_2)\n results = []\n for factor in factor_set_1\n pop(factor_set_2, factor)\n push!(results, factor)\n end\n\n for factor in factor_set_2\n push!(results, factor)\n end\n return results\nend\n\n\n@assert Primes.prime_factors(2) == [2]\n@assert Primes.prime_factors(4) == [2,2]\n@assert Primes.prime_factors(8) == [2,2,2]\n@assert Primes.prime_factors(21) == [3,7]\n\n@assert merge([2,2], [2,2,3]) == [2,2,3]\n@assert merge([2,7], [2,2,3]) == [2,7,2,3]\n@assert merge([7], [2,3,3]) == [7,2,3,3]\n\nfunction min_common_factor(nb)\n factors = map(x -> Primes.prime_factors(x), 2:nb)\n min_factors = reduce(merge, [], factors)\n return prod(min_factors)\nend\n\n@assert min_common_factor(10) == 2520\nprintln(min_common_factor(20))\n", "meta": {"hexsha": "655bce9731ddec31b2ca0083ddecb304e326d95d", "size": 1250, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "5.jl", "max_stars_repo_name": "ddugue/project-euler", "max_stars_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "5.jl", "max_issues_repo_name": "ddugue/project-euler", "max_issues_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "5.jl", "max_forks_repo_name": "ddugue/project-euler", "max_forks_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.5849056604, "max_line_length": 85, "alphanum_fraction": 0.6424, "num_tokens": 420, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9566342049451596, "lm_q2_score": 0.8824278618165526, "lm_q1q2_score": 0.844160676010335}} {"text": "# ------------------------------------------------------------------------------------------\n# ## How can we calculate $\\pi$?\n#\n# Given a square of length $2r$, the square's area is\n#\n# $$A_{square} = (2r)^2 = 4r^2$$\n#\n# whereas the area of a circle with radius $r$ is\n# $$A_{circle} = \\pi r^2$$\n#\n#
![\"Drawing\"](\"images/area_ratio.png\")
\n#\n# Therefore the ratio of the area of the circle to that of the square above is\n#\n# $$\\frac{A_{circle}}{A_{square}} = \\frac{\\pi r^2}{4r^2} = \\frac{\\pi}{4}$$\n#\n# and we can define $\\pi$ as\n#\n# $$\\pi = 4\\frac{A_{circle}}{A_{square}}$$\n#\n# This suggests a way to calculate $\\pi$: if we have a square and the largest circle that\n# fits inside that square, we can determine the ratio of areas of a circle and a square. We\n# can calculate this ratio using a monte carlo simulation. We select random points inside a\n# square, and we keep track of how often those points also fall inside the circle that fits\n# perfectly inside that square.\n#\n# Given a large enough sampling points, $\\frac{A_{circle}}{A_{square}}$ will be equal to the\n# fraction of randomly chosen points inside the square that also fall inside the circle.\n# Then we can figure out $\\pi$!\n#\n# #### Pseudo-code\n#\n# Given the above, our algorithm for determining $\\pi$ looks like this:\n#\n# 1. For each of $N$ iterations,\n# 1. Select a random point inside a square of area $4r^2$ as Cartesian, $(x, y)$,\n# coordinates.\n# 1. Determine if the point also falls inside the circle embedded within this square of\n# area $\\pi r^2$.\n# 1. Keep track of whether or not this point fell inside the circle. At the end of $N$\n# iterations, you want to know $M$ -- the number of the $N$ random points that fell inside\n# the circle!\n# 1. Calculate $\\pi$ as $4\\frac{M}{N}$\n#\n# #### Exercise\n#\n# Write a function that calculates $\\pi$ using Julia.\n#\n# The algorithm above should work for any value of $r$ that you choose to use. Make sure you\n# make $N$ big enough that the value of $\\pi$ is correct to at least a couple numbers after\n# the decimal point!\n#\n# *Hint*:\n#\n# This will probably be easier if you center your circle and square at the coordinate (0, 0)\n# and use a radius of 1. For example, to choose random coordinates within your square at\n# position (x, y), you may want to choose x and y so that they are each a value between -1\n# and +1. Then any point within a distance of 1 from (0, 0) will fall inside the circle!\n#\n# ![\"Drawing\"](\"images/hint.png\")
\n#\n#\n# #### Solution\n#\n# In what follows are two functions, `calculate_pi` and `calculate_pi_2`. They use the same\n# algorithm to determine `pi` with a couple minor differences in execution. Each of these\n# functions is then called to show that as the number of samples, `N`, increases, our value\n# for `pi` becomes more precise.\n# ------------------------------------------------------------------------------------------\n\n\"\"\"\ncalculate_pi(N = 1000)\n\nReturn the value of pi, calculated using a Monte Carlo simulation with N samples. N defaults to 1000.\nThe radius of the circle, r, defaults to 1.\n\"\"\"\nfunction calculate_pi(N = 1000)\n how_often_in_circle = 0\n for i in 1:N\n # Generate a random point (x, y) inside the square centered at (0, 0) \n # that has corners at (-1, 1), (-1, 1), (1, -1), and (1, 1)\n x, y = rand([-1, 1])*rand(), rand([-1, 1])*rand()\n # Check if the distance to (x, y) from (0, 0) is less than the radius of 1\n if sqrt(x^2 + y^2) < 1\n how_often_in_circle += 1\n end\n end\n return 4 * how_often_in_circle / N\nend\n\ncalculate_pi.([10, 100, 1000, 10000, 100_000, 1_000_000_000])\n\n\"\"\"\ncalculate_pi_2(N = 1000)\n\nReturn the value of pi, calculated using a Monte Carlo simulation with N samples. N defaults to 1000.\nThe radius of the circle, r, defaults to 1.\n\"\"\"\nfunction calculate_pi_2(N = 1000)\n how_often_in_circle = 0\n for i in 1:N\n # Generate a random point (x, y) inside the square centered at (0, 0) \n # that has corners at (-1, 1), (-1, 1), (1, -1), and (1, 1)\n x = 1-2*rand()\n y = 1-2*rand()\n # Check if the distance to (x, y) from (0, 0) is less than the radius of 1\n if x^2 + y^2 < 1\n how_often_in_circle += 1\n end\n end\n return 4 * how_often_in_circle / N\nend\n\ncalculate_pi.([10, 100, 1000, 10000, 100_000, 1_000_000_000])\n", "meta": {"hexsha": "8801da2b4ca801c0f820c3fd006c00f433337cd2", "size": 4408, "ext": "jl", "lang": "Julia", "max_stars_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/calculate_pi_solution.jl", "max_stars_repo_name": "grenkoca/JuliaTutorials", "max_stars_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 535, "max_stars_repo_stars_event_min_datetime": "2020-07-15T14:56:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T12:50:32.000Z", "max_issues_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/calculate_pi_solution.jl", "max_issues_repo_name": "grenkoca/JuliaTutorials", "max_issues_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 42, "max_issues_repo_issues_event_min_datetime": "2018-02-25T22:53:47.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-14T02:15:50.000Z", "max_forks_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/calculate_pi_solution.jl", "max_forks_repo_name": "grenkoca/JuliaTutorials", "max_forks_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 394, "max_forks_repo_forks_event_min_datetime": "2020-07-14T23:22:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T20:12:57.000Z", "avg_line_length": 38.3304347826, "max_line_length": 101, "alphanum_fraction": 0.6349818512, "num_tokens": 1301, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.94815453335022, "lm_q2_score": 0.8902942283051332, "lm_q1q2_score": 0.8441365085830478}}
{"text": "@doc raw\"\"\"\n RationalQuadraticKernel([α=1 [,β=1]])\n\nThe rational-quadratic kernel is given by:\n\n```math\n\\kappa(\\mathbf{x},\\mathbf{y}) \n= \\left(1 +\\alpha ||\\mathbf{x},\\mathbf{y}||^2\\right)^{-\\beta} \n\\qquad \\alpha > 0, \\; \\beta > 0\n```\n\nwhere ``\\alpha`` is a scaling parameter and ``\\beta`` is a shape parameter. This kernel can \nbe seen as an infinite sum of Gaussian kernels. If one sets ``\\alpha = \\alpha_0 / \\beta``, \nthen taking the limit ``\\beta \\rightarrow \\infty`` results in the Gaussian kernel with \nscaling parameter ``\\alpha_0``.\n\"\"\"\nstruct RationalQuadraticKernel{T<:AbstractFloat} <: MercerKernel{T}\n alpha::HyperParameter{T}\n beta::HyperParameter{T}\n RationalQuadraticKernel{T}(α::Real, β::Real) where {T<:AbstractFloat} = new{T}(\n HyperParameter(convert(T,α), interval(OpenBound(zero(T)), nothing)),\n HyperParameter(convert(T,β), interval(OpenBound(zero(T)), nothing))\n )\nend\nfunction RationalQuadraticKernel(α::T1 = 1.0, β::T2 = one(T1)) where {T1<:Real,T2<:Real}\n RationalQuadraticKernel{floattype(T1,T2)}(α, β)\nend\n\n@inline rationalquadratickernel(z::T, α::T, β::T) where {T<:AbstractFloat} = (1 + α*z)^(-β)\n\n@inline basefunction(::RationalQuadraticKernel) = SquaredEuclidean()\n@inline function kappa(κ::RationalQuadraticKernel{T}, z::T) where {T}\n rationalquadratickernel(z, getvalue(κ.alpha), getvalue(κ.beta))\nend", "meta": {"hexsha": "47690d6760e07065c1aec3b76f2f446d704d4ad3", "size": 1368, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/kernelfunctions/mercer/rationalquadratic.jl", "max_stars_repo_name": "holtri/MLKernels.jl", "max_stars_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/kernelfunctions/mercer/rationalquadratic.jl", "max_issues_repo_name": "holtri/MLKernels.jl", "max_issues_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/kernelfunctions/mercer/rationalquadratic.jl", "max_forks_repo_name": "holtri/MLKernels.jl", "max_forks_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.2352941176, "max_line_length": 92, "alphanum_fraction": 0.6885964912, "num_tokens": 425, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545392102522, "lm_q2_score": 0.8902942159342104, "lm_q1q2_score": 0.844136502070654}}
{"text": "function rectfunc(x::T, x_c::T)::T where T\n if abs(x) <= x_c\n return one(T)\n end\n\n return zero(T)\nend\n\n\nfunction raisedcosinefunction(x::T, β, 𝑇)::T where T\n return raisedcosinefunction(x,convert(T,β),convert(T,𝑇))\nend\n\n\"\"\"\n raisedcosinefunc(x::T, β::T, 𝑇::T)::T\nLet low := (1-β)/(2𝑇),\n |x| := abs(x),\n high := (1+β)/(2𝑇).\n\nReturns 1 if |x| ≤ low,\nReturns 0.5*(1+cos(π*𝑇/β *(|x|-low))) if low ≤ |x| ≤ high.\nReturns 0 otherwise.\n\"\"\"\nfunction raisedcosinefunction(x::T, β::T, 𝑇::T)::T where T\n two = one(T) + one(T)\n low = (one(T)-β)/(two*𝑇)\n high = (one(T)+β)/(two*𝑇)\n\n if abs(x) <= low\n return one(T)\n elseif low < abs(x) <= high\n return one(T)/two*( one(T) + cos(π*𝑇/β *(abs(x)-low)) )\n end\n\n return zero(T)\nend\n\n\n# transition band is cosine. This allows satisfies the power complementary condition.\n# cos( (yp-y)/(yp-ys) * π/2 ) for a falling edge. yp is pass band, ys is stop band.\nfunction cosinetransitionlowpassfunction(y::T, yp::T, ys::T)::T where T\n abs_y = abs(y)\n\n if abs_y <= yp\n return one(T)\n elseif yp < abs_y <= ys\n #return cos( (yp-y)/(ys-yp)*π/2 )\n return cos( (yp-abs_y)/(ys-yp)*π/2 )\n end\n\n return zero(T)\nend\n\nfunction cosinetransitionhighpassfunction(x::T, xp::T, xs::T)::T where T\n abs_x = abs(x)\n\n if abs_x <= xs\n return zero(T)\n elseif xs < abs_x <= xp\n #return cos( (xp-x)/(xp-xs)*π/2 )\n return cos( (xp-abs_x)/(xp-xs)*π/2 )\n end\n\n return one(T)\nend\n\n\n# rs is rising edge's stopband.\n# rp is rising edge's passband.\n# fp is falling edge's passband.\n# fs is falling edge's stopband.\nfunction cosinetransitionbandpassfunction(y::T, rs, rp, fp, fs)::T where T\n abs_y = abs(y)\n\n # rising edge.\n if abs_y <= rs\n return zero(T)\n elseif rs < abs_y <= rp\n #return cos( (rp-y)/(rp-rs)*π/2 )\n return cos( (rp-abs_y)/(rp-rs)*π/2 )\n end\n\n # falling edge.\n if rp <= abs_y <= fp\n return one(T)\n elseif fp < abs_y <= fs\n #return cos( (fp-y)/(fs-fp)*π/2 )\n return cos( (fp-abs_y)/(fs-fp)*π/2 )\n end\n\n return zero(T)\nend\n", "meta": {"hexsha": "8f7304ff0b40beaa193a9f537f9180b04ce2a9d1", "size": 2136, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/misc/template_signals.jl", "max_stars_repo_name": "RoyCCWang/FIRRemez", "max_stars_repo_head_hexsha": "41a3ad4e8bd851d51a62c7fcd35ac3abf6fdd84c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/misc/template_signals.jl", "max_issues_repo_name": "RoyCCWang/FIRRemez", "max_issues_repo_head_hexsha": "41a3ad4e8bd851d51a62c7fcd35ac3abf6fdd84c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/misc/template_signals.jl", "max_forks_repo_name": "RoyCCWang/FIRRemez", "max_forks_repo_head_hexsha": "41a3ad4e8bd851d51a62c7fcd35ac3abf6fdd84c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.9677419355, "max_line_length": 85, "alphanum_fraction": 0.5627340824, "num_tokens": 748, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9433475778774728, "lm_q2_score": 0.8947894632969136, "lm_q1q2_score": 0.8440974729114272}}
{"text": "\r\n# =========================================================================== #\r\n# pdf functions\r\n# =========================================================================== #\r\n\r\nfunction pdf(d::UniformDistribution, x)\r\n h = 1 / (d.b - d.a)\r\n if d.a <= x <= d.b\r\n y = h\r\n else\r\n y = 0.0\r\n end\r\n return y\r\nend\r\n\r\nfunction pdf(d::NormalDistribution, x)\r\n y = 1 ./ (sqrt(2*pi) .* d.sigma) .* exp.( -0.5 .* (x .- d.mu).^2 ./ d.sigma.^2 )\r\n return y\r\nend\r\n\r\nfunction pdf(d::CauchyDistribution, x)\r\n return 1 ./ ( (pi * d.c) .* (1 .+ ((x .- d.mu) ./ d.c).^2 ) )\r\nend\r\n\r\n\r\nfunction pdf(d::MultiNormalDistribution, x)\r\n n_dim = length(d.mu)\r\n det_sigma = prod( diag(d.L) )\r\n dx = x .- d.mu\r\n arg = -0.5 .* dot(dx, d.inv_sigma * dx )\r\n nom = exp.(arg)\r\n den = sqrt( (2*pi)^n_dim * det_sigma )\r\n return nom ./ den\r\nend", "meta": {"hexsha": "6a3f27767ac3e0dd68e191299b5ce401a0c42498", "size": 912, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "__lib__/math/common/distributions/src/pdf.jl", "max_stars_repo_name": "HomoModelicus/julia", "max_stars_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "__lib__/math/common/distributions/src/pdf.jl", "max_issues_repo_name": "HomoModelicus/julia", "max_issues_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "__lib__/math/common/distributions/src/pdf.jl", "max_forks_repo_name": "HomoModelicus/julia", "max_forks_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.8235294118, "max_line_length": 85, "alphanum_fraction": 0.3837719298, "num_tokens": 258, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9693242000616578, "lm_q2_score": 0.870597270087091, "lm_q1q2_score": 0.8438910024030325}}
{"text": "\"\"\"\n mandelbrot( c )\n\nComputes the number of evaluations of the following equation before the resulting complex number `z` beginning with `z=c`.\n- Input: `c::Number`\n- Output: `itrMax::??`\n\n### Questions\n- What do you think is the output type of this function?\n- Which of the part of the code determines the type?\n\nSometimes forcing the type of output can be an advantage.\n\"\"\"\nfunction mandelbrot( c )\n z = c\n itrMax = 80\n for n in 1:itrMax\n if abs(z) > 2\n return n-1\n end\n z = z^2 + c\n end\n return itrMax\nend\n\n", "meta": {"hexsha": "b82463f35e09ab8fe346768184ffbb1f2433b5eb", "size": 561, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "01-HPC/Mandelbrot.jl", "max_stars_repo_name": "jybantang/phys215-202122-1", "max_stars_repo_head_hexsha": "187d0d36b9a8b2e48024242a5fca4d38accfc805", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "01-HPC/Mandelbrot.jl", "max_issues_repo_name": "jybantang/phys215-202122-1", "max_issues_repo_head_hexsha": "187d0d36b9a8b2e48024242a5fca4d38accfc805", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "01-HPC/Mandelbrot.jl", "max_forks_repo_name": "jybantang/phys215-202122-1", "max_forks_repo_head_hexsha": "187d0d36b9a8b2e48024242a5fca4d38accfc805", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.5769230769, "max_line_length": 122, "alphanum_fraction": 0.632798574, "num_tokens": 157, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.951142225532629, "lm_q2_score": 0.8872045832787205, "lm_q1q2_score": 0.8438577418424709}}
{"text": "using NumericalMethodsforEngineers, Test\n\na = [10. 1. -5.; -20. 3. 20.; 5. 3. 5.]\nb = [1., 2., 6.]\nn = size(A, 1)\nx = 0.0\n\n# Convert to upper triang;e\nfunction to_upper(a::Matrix, b::Vector)\n at = deepcopy(a)\n bt = deepcopy(b) \n r, c = size(at)\n\n for k in 1:r-1\n if abs(at[k, k]) > 1.0e-6\n for i in k+1:r\n x = at[i, k] / at[k, k]\n at[i, k] = 0.0\n for j in k+1:r\n at[i, j] -= at[k, j] * x\n end\n bt[i] -= bt[k] * x\n end\n else\n md\"Zero pivot found in row $k\"\n end\n end\n (at, bt)\nend\n\nfunction direct_solve(a::Matrix, b::Vector)\n at = deepcopy(a)\n bt = deepcopy(b)\n r, c = size(at)\n\n for i in r:-1:1\n x = bt[i]\n if i < r\n for j in i+1:r\n x -= at[i, j] * bt[j]\n end\n end\n bt[i] = x / at[i, i]\n end\n bt\nend\n\nprintln(\"Modified lhs matrix and rhs\")\nat, bt = to_upper(a, b)\nat |> display\nprintln()\nbt |> display\nprintln()\n\nprintln(\"Solution vector:\")\nc_nmfe = direct_solve(at, bt)\nc_nmfe |> display\nprintln()\n\nc = a \\ [1., 2., 6.]\nc |> display\n\nprintln()\n@test ≈(c, c_nmfe, atol=1e-10)\n@test round.(a * c; digits=14) == b\n", "meta": {"hexsha": "fb6bae9feb75f3c1c94731c6892ee4cb60012410", "size": 1097, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/ch02/f-02-01.jl", "max_stars_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_stars_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-23T18:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-25T03:32:45.000Z", "max_issues_repo_path": "examples/ch02/f-02-01.jl", "max_issues_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_issues_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-23T21:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-27T23:14:46.000Z", "max_forks_repo_path": "examples/ch02/f-02-01.jl", "max_forks_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_forks_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2018-10-27T14:13:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-20T18:54:06.000Z", "avg_line_length": 16.6212121212, "max_line_length": 43, "alphanum_fraction": 0.5278030994, "num_tokens": 442, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9511422199928905, "lm_q2_score": 0.8872045862611166, "lm_q1q2_score": 0.8438577397642724}}
{"text": "\"\"\"\n ChirpSignal\nChirp Swept-frequency sine generator.\n\"\"\"\nmodule ChirpSignal\n\nexport chirp\n\n\n\"\"\"\n chirp(T, fs, fl, fh; method=\"linear\", phase=0.0) -> Array{Float64,1}\n\nChirp Swept-frequency sine generator.\n# Arguments\n- `T` : signal's length in second\n- `fs` : sampling rate\n- `fl` : lower frequency at the ending time step, fl <= fs\n- `fh ` : higher frequency at the ending time step, fh <= fs\n- `phase` : phase of the chirp sigal, like sin(wt + phase)\n- `method`: available methods are 'linear','quadratic','exponential' and 'logarithmic'; the default is 'linear'\n\"\"\"\nfunction chirp(T, fs, fl, fh; method=\"linear\", phase=0.0)\n method == \"linear\" && return chirpLinear(T, fs, fl, fh; phase=phase)\n method == \"quadratic\" && return chirpQuadratic(T, fs, fl, fh; phase=phase)\n method == \"logarithmic\" && return chirpLogarithmic(T, fs, fl, fh; phase=phase)\n method == \"exponential\" && return chirpExponential(T, fs, fl, fh; phase=phase)\nend\n\n\nfunction chirpLinear(T, fs, fl, fh; phase=0.0)\n Δt = 1.0/fs;\n fl = min(fs/2,max(0,fl));\n fh = min(fs/2,max(0,fh));\n k = (fh - fl)/T/2;\n t = 0:Δt:T-Δt;\n y = @. sin(6.2831853*(k*t + fl)*t + phase);\nend\n\n\nfunction chirpQuadratic(T, fs, fl, fh; phase=0.0)\n # f(t) = k * t^2 + fl, k = (fh-fl)/T^2\n # F(t) = k/3 * t^3 + fl*t\n Δt = 1.0/fs;\n fl = min(fs/2,max(0,fl));\n fh = min(fs/2,max(0,fh));\n t = 0:Δt:T-Δt;\n y = @. sin(6.2831853*( (fh-fl)/(3*T^2) * t^3 + fl * t) + phase)\nend\n\n\nfunction chirpLogarithmic(T, fs, fl, fh; phase=0.0)\n # f(t) = k*log(t+1) + fl, k = (fh-fl)/log(T+1)\n # F(t) = k*(t+1)*log(t+1) + (fl-k)*t\n (fl <= fh) || error(\"fl <= fh not met\")\n Δt = 1.0/fs;\n fl = min(fs/2,max(0,fl));\n fh = min(fs/2,max(0,fh));\n t = 0:Δt:T-Δt;\n k = (fh-fl)/log(T+1);\n y = @. sin(6.2831853*( k*(t+1)*log(t+1) + (fl-k)*t ) + phase);\nend\n\n\nfunction chirpExponential(T, fs, fl, fh; phase=0.0)\n # f(t) = exp(k * t) + (fl-1), k = 1/T*log(fh-fl+1)\n # F(t) = 1/k * exp(k * t) + (fl-1)*t\n (fl <= fh) || error(\"fl <= fh not met\")\n Δt = 1.0/fs;\n fl = min(fs/2,max(0,fl));\n fh = min(fs/2,max(0,fh));\n t = 0:Δt:T-Δt;\n k = 1/T*log(fh-fl+1);\n y = @. sin(6.2831853*( (1/k) * exp(k*t) + (fl-1)*t ) + phase);\nend\n\n\n\"\"\"\n chirp(T, fs, f::Function; phase=0.0) -> Array{Float64,1}\n\nFunction customizable chirp swept-frequency sine generator.\n# Arguments\n- `T` : signal's length in second\n- `fs` : sampling rate\n- `f` : function that defines how the frequency changes vs time\n- `phase` : phase of the chirp sigal, like sin(wt + phase)\n\"\"\"\nfunction chirp(T, fs, f::Function; phase=0.0)\n Δt = 1.0/fs;\n τ = 0:Δt:T-Δt;\n ϕ = zeros(length(τ),1);\n x = zeros(length(τ),1);\n ϕ[1] = f(0)*Δt;\n x[1] = sin(6.2831853*ϕ[1] + phase);\n for t = 2:length(τ)\n ϕ[t] = ϕ[t-1] + f(τ[t-1])*Δt;\n x[t] = sin(6.2831853*ϕ[t] + phase);\n end\n return x\nend\n\nend # module\n", "meta": {"hexsha": "7e07c1ce3f04537a3ac0a228a70eafafb8a97f96", "size": 2945, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/ChirpSignal.jl", "max_stars_repo_name": "sonosole/ChirpSignal", "max_stars_repo_head_hexsha": "18752dfddf43bd45e6e27c3a615bca7342a7ba5b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-08-13T16:27:54.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-08T17:40:01.000Z", "max_issues_repo_path": "src/ChirpSignal.jl", "max_issues_repo_name": "sonosole/ChirpSignal", "max_issues_repo_head_hexsha": "18752dfddf43bd45e6e27c3a615bca7342a7ba5b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/ChirpSignal.jl", "max_forks_repo_name": "sonosole/ChirpSignal", "max_forks_repo_head_hexsha": "18752dfddf43bd45e6e27c3a615bca7342a7ba5b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-08T17:40:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-08T17:40:05.000Z", "avg_line_length": 28.8725490196, "max_line_length": 111, "alphanum_fraction": 0.5487266553, "num_tokens": 1195, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632956467157, "lm_q2_score": 0.9099070035949656, "lm_q1q2_score": 0.8438143575858553}}
{"text": "# ------------------------------------------------------------------------------------------\n# Often used in machine learning, a \"one hot\" vector is a vector of all zeros, except for a\n# single `1` entry.\n# Representing it as a standard vector is memory-inefficient, so it cries out for a special\n# implementation.\n# ------------------------------------------------------------------------------------------\n\nstruct OneHotVector <: AbstractVector{Int}\n idx::Int\n len::Int\nend\n\nBase.size(v::OneHotVector) = (v.len,)\n\nBase.getindex(v::OneHotVector, i::Integer) = Int(i == v.idx)\n\nOneHotVector(3, 10)\n\nA = rand(5,5)\n\nA * OneHotVector(3, 5)\n\nVector(OneHotVector(3,5))\n\n# ------------------------------------------------------------------------------------------\n# ## Exercise\n#\n# Generalize it to any element type.\n# ------------------------------------------------------------------------------------------\n\n\n", "meta": {"hexsha": "5819b200ab9d6d69e11b4773dd4ba0eae9ef5d4c", "size": 912, "ext": "jl", "lang": "Julia", "max_stars_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/long-version/130 OneHot Vector.jl", "max_stars_repo_name": "grenkoca/JuliaTutorials", "max_stars_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 535, "max_stars_repo_stars_event_min_datetime": "2020-07-15T14:56:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T12:50:32.000Z", "max_issues_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/long-version/130 OneHot Vector.jl", "max_issues_repo_name": "grenkoca/JuliaTutorials", "max_issues_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 42, "max_issues_repo_issues_event_min_datetime": "2018-02-25T22:53:47.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-14T02:15:50.000Z", "max_forks_repo_path": ".nbexports/introductory-tutorials/intro-to-julia/long-version/130 OneHot Vector.jl", "max_forks_repo_name": "grenkoca/JuliaTutorials", "max_forks_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 394, "max_forks_repo_forks_event_min_datetime": "2020-07-14T23:22:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T20:12:57.000Z", "avg_line_length": 28.5, "max_line_length": 92, "alphanum_fraction": 0.4122807018, "num_tokens": 178, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133531922388, "lm_q2_score": 0.8976952996340946, "lm_q1q2_score": 0.8437557992239934}}
{"text": "using Base.MathConstants\n\nfunction getHarmonicNumber(n::Integer)::Float64\n H = 0\n for i = 1:n\n H = H + (1 / i)\n end\n return H\nend\n\nfunction getHarmonicNumberWithOrder(n::Integer, order::Integer)::Float64\n H = 0\n for i = 1:n\n H = H + (1 / i^order)\n end\n return H\nend\n\neulermascheroni = convert(Float64, γ)", "meta": {"hexsha": "76bfd4daf87d6835ab93278cc4ca991eef51c79b", "size": 342, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/harmonic.jl", "max_stars_repo_name": "EllianCarlos/number-theory-but-in-julia", "max_stars_repo_head_hexsha": "74368fee8002bc53031b8ef6026e72f806efe84d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/harmonic.jl", "max_issues_repo_name": "EllianCarlos/number-theory-but-in-julia", "max_issues_repo_head_hexsha": "74368fee8002bc53031b8ef6026e72f806efe84d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/harmonic.jl", "max_forks_repo_name": "EllianCarlos/number-theory-but-in-julia", "max_forks_repo_head_hexsha": "74368fee8002bc53031b8ef6026e72f806efe84d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.0, "max_line_length": 72, "alphanum_fraction": 0.6081871345, "num_tokens": 116, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9615338101862455, "lm_q2_score": 0.8774767746654976, "lm_q1q2_score": 0.8437235864940534}}
{"text": "using DrWatson\n@quickactivate \"StatReth\"\n\n# %%\nusing Plots\nusing Distributions\n\n# %%\nlen = 100\np = range(0, 1, length = len)\n\n# %%\nprior = ones(len)\nprior ./= sum(prior)\n\nlikelihood(w, l) = @. pdf(Binomial(w + l, p), w)\n\nfunction posterior(prior, likelih)\n post = prior .* likelih\n return post ./ sum(post)\nend\n\nposterior_a = posterior(prior, likelihood(3, 0)) # www\nposterior_b = posterior(prior, likelihood(3, 1)) # wwwl\nposterior_c = posterior(prior, likelihood(5, 2)) # lwwlwww\n\n# %%\nplot(xlabel = \"p\", ylabel = \"probability\", dpi = 200, legend = :topleft)\nplot!(p, posterior_a, label = \"a\")\nplot!(p, posterior_b, label = \"b\")\nplot!(p, posterior_c, label = \"c\")\n\n\n# %%\nprior = ones(len)\nprior[p .< 0.5] .= 0.0\nprior ./= sum(prior)\n\nposterior_a = posterior(prior, likelihood(3, 0)) # www\nposterior_b = posterior(prior, likelihood(3, 1)) # wwwl\nposterior_c = posterior(prior, likelihood(5, 2)) # lwwlwww\n\n# %%\nplot(xlabel = \"p\", ylabel = \"probability\", dpi = 200, legend = :topleft)\nplot!(p, posterior_a, label = \"a\")\nplot!(p, posterior_b, label = \"b\")\nplot!(p, posterior_c, label = \"c\")\n", "meta": {"hexsha": "aa496fbb722a3d8c4dfff065e8a13698d1bc3cf5", "size": 1102, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "exercises/ch02/2M1.jl", "max_stars_repo_name": "karajan9/statisticalrethinking", "max_stars_repo_head_hexsha": "e7516c468ca182c4b1c9cae0cfd0bb1feef7ed3c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2020-06-03T14:18:11.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-05T16:52:26.000Z", "max_issues_repo_path": "exercises/ch02/2M1.jl", "max_issues_repo_name": "karajan9/statisticalrethinking", "max_issues_repo_head_hexsha": "e7516c468ca182c4b1c9cae0cfd0bb1feef7ed3c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2020-06-13T05:56:35.000Z", "max_issues_repo_issues_event_max_datetime": "2020-08-12T14:05:57.000Z", "max_forks_repo_path": "exercises/ch02/2M1.jl", "max_forks_repo_name": "karajan9/statisticalrethinking", "max_forks_repo_head_hexsha": "e7516c468ca182c4b1c9cae0cfd0bb1feef7ed3c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-07-01T13:00:14.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-03T23:40:53.000Z", "avg_line_length": 22.9583333333, "max_line_length": 72, "alphanum_fraction": 0.6470054446, "num_tokens": 377, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9632305360354471, "lm_q2_score": 0.875787001374006, "lm_q1q2_score": 0.8435847827863606}}
{"text": "# ------------------------------------------------------------------\n# Licensed under the MIT License. See LICENSE in the project root.\n# ------------------------------------------------------------------\n\n\"\"\"\n signarea(p₁, p₂, p₃)\n\nCompute signed area of triangle formed\nby points `p₁`, `p₂` and `p₃`.\n\"\"\"\nfunction signarea(p₁::Point{2}, p₂::Point{2}, p₃::Point{2})\n a = coordinates(p₁)\n b = coordinates(p₂)\n c = coordinates(p₃)\n ((b[1]-a[1])*(c[2]-a[2]) - (b[2]-a[2])*(c[1]-a[1])) / 2\nend\n\n\"\"\"\n signarea(triangle)\n\nCompute the signed area of `triangle`.\n\"\"\"\nfunction signarea(t::Triangle{2})\n vs = vertices(t)\n signarea(vs[1], vs[2], vs[3])\nend\n\n\"\"\"\n sideof(point, segment)\n\nDetermines on which side of the oriented `segment`\nthe `point` lies. Possible results are `:LEFT`,\n`:RIGHT` or `:ON` the segment.\n\"\"\"\nfunction sideof(p::Point{2,T}, s::Segment{2,T}) where {T}\n a, b = vertices(s)\n area = signarea(p, a, b)\n ifelse(area > atol(T), :LEFT, ifelse(area < -atol(T), :RIGHT, :ON))\nend\n\n\"\"\"\n sideof(point, chain)\n\nDetermines on which side of the closed `chain` the\n`point` lies. Possible results are `:INSIDE` or\n`:OUTSIDE` the chain.\n\"\"\"\nfunction sideof(p::Point{2,T}, c::Chain{2,T}) where {T}\n w = windingnumber(p, c)\n ifelse(isapprox(w, zero(T), atol=atol(T)), :OUTSIDE, :INSIDE)\nend\n", "meta": {"hexsha": "e3532de295e17b06e8eaf414fa79e0dc79d96f59", "size": 1312, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/utils.jl", "max_stars_repo_name": "hyrodium/Meshes.jl", "max_stars_repo_head_hexsha": "02fb99be21c6d1f567e0886afe4997e75166f7e0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-10T17:32:33.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-10T17:32:33.000Z", "max_issues_repo_path": "src/utils.jl", "max_issues_repo_name": "hyrodium/Meshes.jl", "max_issues_repo_head_hexsha": "02fb99be21c6d1f567e0886afe4997e75166f7e0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/utils.jl", "max_forks_repo_name": "hyrodium/Meshes.jl", "max_forks_repo_head_hexsha": "02fb99be21c6d1f567e0886afe4997e75166f7e0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.2307692308, "max_line_length": 69, "alphanum_fraction": 0.5670731707, "num_tokens": 409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9632305297023093, "lm_q2_score": 0.8757869997529962, "lm_q1q2_score": 0.8435847756784748}}
{"text": "# Julia program to Find the Sum of Digits of a Number \n\n# Extracting Each digits\nfunction sum_of_digits(num)\n sum = 0\n while(num != 0)\n sum = sum + (num % 10)\n num = num ÷ 10\n end\n return sum\nend\n\nprint(\"Enter the number: \")\nnum = readline()\nnum = parse(Int, num)\nsum = sum_of_digits(abs(num))\nprintln(\"The sum of digits of the given number is $sum.\")\n\n\"\"\"\nTime Complexity: O(log(num)), where 'num' is the length of the given number\nSpace Complexity: O(1)\n\nSAMPLE INPUT AND OUTPUT\n\nSAMPLE 1\nEnter the number: -12\nThe sum of digits of the given number is 3.\n\nSAMPLE 2\nEnter the number: 43258\nThe sum of digits of the given number is 22.\n\n\"\"\"\n", "meta": {"hexsha": "984988d655390fa220e9abe9c79a7a841ea5bea9", "size": 667, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Julia/math/sum_of_digits.jl", "max_stars_repo_name": "zhcet19/NeoAlgo-1", "max_stars_repo_head_hexsha": "c534a23307109280bda0e4867d6e8e490002a4ee", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 897, "max_stars_repo_stars_event_min_datetime": "2020-06-25T00:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-24T00:49:31.000Z", "max_issues_repo_path": "Julia/math/sum_of_digits.jl", "max_issues_repo_name": "zhcet19/NeoAlgo-1", "max_issues_repo_head_hexsha": "c534a23307109280bda0e4867d6e8e490002a4ee", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5707, "max_issues_repo_issues_event_min_datetime": "2020-06-24T17:53:28.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-22T05:03:15.000Z", "max_forks_repo_path": "Julia/math/sum_of_digits.jl", "max_forks_repo_name": "zhcet19/NeoAlgo-1", "max_forks_repo_head_hexsha": "c534a23307109280bda0e4867d6e8e490002a4ee", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1817, "max_forks_repo_forks_event_min_datetime": "2020-06-25T03:51:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-29T05:14:07.000Z", "avg_line_length": 19.6176470588, "max_line_length": 75, "alphanum_fraction": 0.6731634183, "num_tokens": 188, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533126145178, "lm_q2_score": 0.9046505325302034, "lm_q1q2_score": 0.8435443858162758}}
{"text": "using Plots\r\nusing PrettyTables\r\n\r\ninclude(\"rfiod.jl\")\r\nusing Main.RFIOD\r\n\r\nio = stdout\r\n# Uncomment below to save to file instead\r\n# io = open(\"output.txt\", \"w\")\r\n\r\nF₁(x) = 2x - 3sin(x) + 5\r\nF₁′(x) = 2 - 3cos(x)\r\nF₁root = -2.8832368725582835\r\n\r\nF₂(x) = x^3 - 8.5x^2 + 20x - 8\r\nF₂′(x) = 3x^2 - 17x + 20\r\nF₂root = 4\r\n\r\n# -- Question 1 -- #\r\n\r\nq1plot = plot(F₁, -5:0.1:5, framestyle=:origin, legend = false, title = \"F₁(x) = 2x - 3sin(x) + 5 has a single real root in [-5,5]\")\r\n# F₁′(x) > 0 outside of this interval with F₁(-5) < 0, F₁(5) > 0, so no roots outside\r\n\r\n# -- Binary Search Programming Task -- #\r\nprintln(io, \"-- Binary Search Programming Task --\")\r\n\r\n# Initial interval [-5, 5]\r\nBSresult1 = binarysearch(F₁, -5, 5)\r\n# Initial interval [-5, 5], but with inputs swapped\r\nBSresult2 = binarysearch(F₁, 5, -5)\r\n# Initial interval [-2.88323688, -2.88323687]\r\nBSresult3 = binarysearch(F₁, -2.88323688, -2.88323687)\r\n# Initial interval [-100, 100]\r\nBSresult4 = binarysearch(F₁, -100, 100)\r\n\r\nprintln(io, \"Binary search with initial interval [a,b]\")\r\n# Table created for display\r\ntabledata = hcat(\r\n [-5, 5, -2.88323688, -100],\r\n [5, -5, -2.88323687, 100],\r\n Plots.unzip([BSresult1, BSresult2, BSresult3, BSresult4])...\r\n)\r\npretty_table(io, tabledata, header = [\"a\", \"b\", \"root\", \"iterations\"])\r\n\r\n# Initial interval [-1, 1], which does not contain a root\r\nBSresult5 = try binarysearch(F₁, -1, 1) catch ErrorException \"Error, root not found\" end\r\nprintln(io, \"Binary search with the invalid initial interval [-1, 1] gives:\")\r\nshow(io, BSresult5)\r\n\r\n# -- Question 3 -- #\r\nprintln(io, \"\\n\\n-- Question 3 --\")\r\n\r\nf(k) = x -> x - F₁(x)/(2+k)\r\n\r\n\"\"\"\r\n q3FPI(k, x₀, Nₘₐₓ, [ε])\r\n\r\nRun the fixed point iteration algorithm used in Question 3, with parameter `k` and starting value `x₀`.\r\n\r\nTerminate if consecutive iterations differ by less than `ε`, or after `Nₘₐₓ` iterations.\r\n\r\nOutputs a table to the REPL with the results, and returns a staircase/cobweb diagram.\r\n\r\n# Examples\r\n\r\n```julia-repl\r\njulia> q3FPI(3,-2,10)\r\nFixed point iteration with k = 3, Nₘₐₓ = 10\r\n┌─────┬──────────┬────────────┬───────────┐\r\n│ n │ xₙ │ ϵₙ │ ϵₙ/ϵₙ₋₁ │\r\n├─────┼──────────┼────────────┼───────────┤\r\n│ 0.0 │ -2.0 │ 0.883237 │ 0.0 │\r\n│ 1.0 │ -2.74558 │ 0.137658 │ 0.155857 │\r\n│ 2.0 │ -2.87879 │ 0.00444334 │ 0.032278 │\r\n│ 3.0 │ -2.88315 │ 9.00029e-5 │ 0.0202557 │\r\n│ 4.0 │ -2.88324 │ 1.79286e-6 │ 0.0199201 │\r\n│ 5.0 │ -2.88324 │ 3.57019e-8 │ 0.0199133 │\r\n└─────┴──────────┴────────────┴───────────┘\r\n```\r\n\r\n\"\"\"\r\nfunction q3FPI(k, x₀, Nₘₐₓ, ε = 1e-5)\r\n\r\n result = fixedpointiteration(f(k), x₀, Nₘₐₓ, ε)\r\n n = length(result) - 1\r\n\r\n errors = result .- F₁root\r\n errorratios = [0, (errors[2:(n+1)] ./ errors[1:n])...]\r\n tabledata = hcat(\r\n collect(0:n),\r\n result,\r\n errors,\r\n errorratios\r\n )\r\n println(io, \"Fixed point iteration with k = $k, Nₘₐₓ = $Nₘₐₓ\")\r\n pretty_table(io, tabledata, header = [\"n\", \"xₙ\", \"ϵₙ\", \"ϵₙ/ϵₙ₋₁\"])\r\n\r\n FPIplot = staircasediagram(f(k), result)\r\n plot!(FPIplot, title = \"k = $k\")\r\n return FPIplot\r\n\r\nend\r\n\r\n# k = 0, ε = 10⁻⁵, x₀ = -2, Nₘₐₓ = 10\r\nq3plot1 = q3FPI(0, -2, 10)\r\n\r\n# k = 4 gives monotonic convergence as f(8)'(x*) > 0\r\nq3plot2 = q3FPI(4, -2, 20)\r\n\r\n# k = 2 gives oscillatory convergence as f(1)'(x*) < 0\r\nq3plot3 = q3FPI(2, -2, 20)\r\n\r\n# k = 16, Nₘₐₓ = 50\r\nq3FPI(16, -2, 50)\r\n\r\n# -- Question 4 -- #\r\nprintln(io, \"\\n-- Question 4 --\")\r\n\r\ng(x) = x - F₂(x)/20\r\n\r\nq4result = fixedpointiteration(g, 5, 1000, 1e-5)\r\n# n and xₙ take the final values given below\r\nn = length(q4result) - 1\r\nxₙ = q4result[end]\r\nprintln(io, \"Fixed point iteration to find the double root of\r\nF₂(x) = x³ - 8.5x² + 20x - 8 results in n = $n iterations\r\nwith estimated root xₙ = $xₙ\")\r\n\r\n# -- Question 5 -- #\r\nprintln(io, \"\\n-- Question 5 --\")\r\n\r\n\"\"\"\r\n q5NR(f, x₀, Nₘₐₓ, root, order)\r\n\r\nRun the Newton-Raphson algorithm used in Question 5 on the function `f(x) = x - F(x)/F'(x)` starting at `x₀`\r\n\r\nTerminate the fixed point iteration after at most Nₘₐₓ iterations.\r\n\r\nOutputs a table with errors to `root` and error ratios of order `order`, and returns the vector of iterations.\r\n\r\n# Examples\r\n\r\n```julia-repl\r\njulia> q5NR(x -> x - sin(x)/cos(x), 3, 10, π, 2)\r\n┌─────┬─────────┬──────────────┬──────────────┐\r\n│ n │ xₙ │ ϵₙ │ ϵₙ/ϵₙ₋₁² │\r\n├─────┼─────────┼──────────────┼──────────────┤\r\n│ 0.0 │ 3.0 │ -0.141593 │ 0.0 │\r\n│ 1.0 │ 3.14255 │ 0.000953889 │ 0.0475791 │\r\n│ 2.0 │ 3.14159 │ -2.89316e-10 │ -0.000317963 │\r\n│ 3.0 │ 3.14159 │ 0.0 │ 0.0 │\r\n│ 4.0 │ 3.14159 │ 0.0 │ NaN │\r\n└─────┴─────────┴──────────────┴──────────────┘\r\n5-element Vector{Float64}:\r\n 3.0\r\n 3.142546543074278\r\n 3.141592653300477\r\n 3.141592653589793\r\n 3.141592653589793\r\n```\r\n\"\"\"\r\nfunction q5NR(f, x₀, Nₘₐₓ, root, order)\r\n result = fixedpointiteration(f, x₀, Nₘₐₓ)\r\n n = length(result) - 1\r\n errors = result .- root\r\n errorratios = [0, (errors[2:(n+1)] ./ errors[1:n].^order)...]\r\n tabledata = hcat(\r\n collect(0:n),\r\n result,\r\n errors,\r\n errorratios\r\n )\r\n pretty_table(io, tabledata, header = [\"n\", \"xₙ\", \"ϵₙ\", \"ϵₙ/ϵₙ₋₁\" * (order == 2 ? \"²\" : \"\")])\r\n return result\r\nend\r\n\r\n\"\"\"\r\n q5NRgraph(F, result, nlines)\r\nCreate a graph from the first `nvalues` iterations of `result`, coming from the Newton-Raphson algorithm on `F`.\r\n\r\n# Examples\r\n\r\n```julia-repl\r\njulia> q5NRgraph(sin, q5NR(x -> x - sin(x)/cos(x), 3, 10, π, 2), 4)\r\n``` \r\n\"\"\"\r\nfunction q5NRgraph(F, result, nlines)\r\n NRplot = scatter((result[1],F(result[1])), markercolor = :black, framestyle = :origin,\r\n title = \"Newton-Raphson with x₀ = $(result[1])\", legend = false)\r\n for n = 1:nlines\r\n plot!(NRplot, [result[n],result[n+1],result[n+1]],\r\n [F(result[n]), 0, F(result[n+1])], linecolor = :black)\r\n end\r\n plot!(NRplot, F, linecolor = :orange)\r\n return NRplot\r\nend\r\n\r\nf₁(x) = x - F₁(x)/F₁′(x)\r\nf₂(x) = x - F₂(x)/F₂′(x)\r\n\r\n# x₀ = 0 for F₁ gives divergence\r\nprintln(io, \"Newton-Raphson method for \\nF₁(x) = 2x - 3sin(x) + 5 with x₀ = 0\")\r\nq5result1 = q5NR(f₁, 0, 10, F₁root, 2)\r\nq5plot1 = q5NRgraph(F₁, q5result1, 4)\r\n\r\n# x₀ = -2 for F₁ gives convergence\r\nprintln(io, \"Newton-Raphson method for \\nF₁(x) = 2x - 3sin(x) + 5 with x₀ = -2\")\r\nq5result2 = q5NR(f₁, -2, 10, F₁root, 2)\r\nq5plot2 = q5NRgraph(F₁, q5result2, 2)\r\n\r\n# x₀ = 5 for F₂ gives convergence\r\nprintln(io, \"Newton-Raphson method for \\nF₂(x) = x³ - 8.5x² + 20x - 8 with x₀ = 5\")\r\nq5result3 = q5NR(f₂, 5, 50, F₂root, 1)\r\nq5plot3 = q5NRgraph(F₂, q5result3, 10)\r\n\r\n# Saves images automatically, uncomment to use\r\n#=\r\n\r\n# File extension e.g. \".png\", \".pdf\"\r\next = \".pdf\"\r\n\r\n# Makes a new directory for the images (or ignores the error if one exists already)\r\ntry\r\n mkdir(\"images\")\r\ncatch IOError\r\n nothing\r\nend\r\n\r\n# Saves the figures created above\r\ntry\r\n savefig(q1plot,\"images\\\\q1-singleroot$ext\")\r\n savefig(q3plot1,\"images\\\\q3-k0-graph$ext\")\r\n savefig(q3plot2,\"images\\\\q3-k4-graph$ext\")\r\n savefig(q3plot3,\"images\\\\q3-k2-graph$ext\")\r\n savefig(q5plot1,\"images\\\\q5-f1divergence-graph$ext\")\r\n savefig(q5plot2,\"images\\\\q5-f1convergence-graph$ext\")\r\n savefig(q5plot3,\"images\\\\q5-f2convergence-graph$ext\")\r\ncatch IOError\r\n println(\"Failed to save images\")\r\nend\r\n\r\n=#\r\n\r\nio == stdout || close(io)\r\nnothing", "meta": {"hexsha": "e72b5d16c1b9d7f32368fddb4e2f6c66a2abb10f", "size": 7375, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "introductoryproject/rfiod-run.jl", "max_stars_repo_name": "sje30/catam-julia", "max_stars_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2021-07-13T12:55:37.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T14:40:23.000Z", "max_issues_repo_path": "introductoryproject/rfiod-run.jl", "max_issues_repo_name": "sje30/catam-julia", "max_issues_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2021-07-11T21:35:47.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-25T12:10:58.000Z", "max_forks_repo_path": "introductoryproject/rfiod-run.jl", "max_forks_repo_name": "sje30/catam-julia", "max_forks_repo_head_hexsha": "8778e5d08888c6d98c41261d9640d2e2c21f4629", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-10-13T21:00:47.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-13T21:00:47.000Z", "avg_line_length": 30.1020408163, "max_line_length": 133, "alphanum_fraction": 0.5732881356, "num_tokens": 2875, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896715436482, "lm_q2_score": 0.9161096175976053, "lm_q1q2_score": 0.8435442738856762}}
{"text": "#md # ---\n\nusing DifferentialEquations, Plots\n\ng = 9.79 # Gravitational constants\nL = 1.00 # Length of the pendulum\n\n#Initial Conditions\nu₀ = [0, π / 60] # Initial speed and initial angle\ntspan = (0.0, 6.3) # time domain\n\n#Define the problem\nfunction simplependulum(du, u, p, t)\n θ = u[1]\n dθ = u[2]\n du[1] = dθ\n du[2] = -(g/L)*θ\nend\n\n#Pass to solvers\nprob = ODEProblem(simplependulum, u₀, tspan)\nsol = solve(prob, Tsit5(), reltol = 1e-6)\n#md nothing # hide\n\n#md # ---\n\n# Analytic solution\nu = u₀[2] .* cos.(sqrt(g / L) .* sol.t)\n\nplot(sol.t, getindex.(sol.u, 2), label = \"Numerical\")\nplot!(sol.t, u, label = \"Analytic\")\n#md savefig(\"pendulum1.svg\"); nothing # hide \n\n#md # \n\n#md # ---\n\n# [Numbers with Uncertainties](http://tutorials.juliadiffeq.org/html/type_handling/02-uncertainties.html)\n\nusing Measurements\n\ng = 9.79 ± 0.02; # Gravitational constants\nL = 1.00 ± 0.01; # Length of the pendulum\n\n#Initial Conditions\nu₀ = [0 ± 0, π / 60 ± 0.01] # Initial speed and initial angle\n\n#Define the problem\nfunction simplependulum(du, u, p, t)\n θ = u[1]\n dθ = u[2]\n du[1] = dθ\n du[2] = -(g/L)*θ\nend\n\n#Pass to solvers\nprob = ODEProblem(simplependulum, u₀, tspan)\nsol = solve(prob, Tsit5(), reltol = 1e-6);\nnothing # hide\n\n#md # ---\n\n# Analytic solution\nu = u₀[2] .* cos.(sqrt(g / L) .* sol.t)\n\nplot(sol.t, getindex.(sol.u, 2), label = \"Numerical\")\nplot!(sol.t, u, label = \"Analytic\")\n#md savefig(\"pendulum2.svg\"); nothing # hide \n\n#md # \n\n", "meta": {"hexsha": "ec81fb6c0d9089d7aa58b4e6e8f0aac5b078443c", "size": 1494, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/01.Measurements.jl", "max_stars_repo_name": "pnavaro/DiffEq", "max_stars_repo_head_hexsha": "42214888a98165b43caaa3bada1e0c8429821896", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/01.Measurements.jl", "max_issues_repo_name": "pnavaro/DiffEq", "max_issues_repo_head_hexsha": "42214888a98165b43caaa3bada1e0c8429821896", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/01.Measurements.jl", "max_forks_repo_name": "pnavaro/DiffEq", "max_forks_repo_head_hexsha": "42214888a98165b43caaa3bada1e0c8429821896", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.75, "max_line_length": 105, "alphanum_fraction": 0.6305220884, "num_tokens": 555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9381240160063031, "lm_q2_score": 0.8991213691605412, "lm_q1q2_score": 0.8434873497139728}}
{"text": "using Statistics\n\n\"\"\"\n linear_regression(X::Matrix, Y::Matrix, alpha::Float64, iterations::Int64)\n\n Linear Regression Algorithm using Gradient Descent \n to find the best fit line for the data.\n \n Arguments :\n - X : Training data set of shape [m x n]\n - Y : Labels corresponding to the training data set of shape [m x 1]\n - alpha : Learning rate\n - iterations : Number of iterations to train\n Returns :\n - W : Weight vector \n - b : Bias\n - J_history : Cost Function History\n\n \n\n gradient_descent(W::Matrix, b, X::Matrix, Y::Matrix, alpha::Float64, num_iters::Int64)\n \n Performs gradient descent to learn W and b\n\n Arguments\n - `W`: matrix of weights\n - `b`: bias vector\n - `X`: matrix of features\n - `Y`: vector of labels\n - `alpha`: learning rate ( generally ranges from 0.01 to 0.00001 )\n - `num_iters`: number of iterations to run gradient descent\n\n Returns:\n - `W`: matrix of weights\n - `b`: bias vector\n - `J_history`: vector of cost values after each iteration\n\n\n \n # Contributors:- [Navaneeth Sharma](https://github.com/Navaneeth-Sharma) \n\"\"\"\n\nfunction gradient_descent(\n W::Matrix,\n b,\n X::Matrix,\n Y::Matrix,\n alpha::Float64,\n num_iters::Int64,\n)\n m = length(Y)\n J_history = []\n for i in 1:num_iters\n W = W - alpha .* (1 / m) * (X * (X'W .+ b - Y'))\n b = b - alpha .* (1 / m) * sum((X'W .+ b - Y'))\n push!(J_history, 1 / 2m * sum((X'W .+ b - Y') .^ 2))\n end\n return W, b, J_history\nend\n\nfunction linear_regression(\n X::Matrix,\n Y::Matrix,\n alpha::Float64,\n num_iters::Int64,\n)\n W = zeros(1, size(X, 1))\n b = 0\n W, b, J_history = gradient_descent(W, b, X, Y, alpha, num_iters)\n return W, b, J_history\nend\n\n# Function to predict the output of a given data vector\npredict(X::Matrix, W::Matrix, b::Float64) = X'W .+ b;\n\n# Example usage\nX = [1, 3, 2, 5, 7, 8, 8, 9, 10, 12]\nY = [2, 4, 3, 6, 8, 9, 9, 10, 11, 13]\n\nX = reshape(X, 1, length(X));\nY = reshape(Y, 1, length(Y));\n\nW, b, J_history = linear_regression(X, Y, 0.01, 1000);\n\noutput = predict(X, W, b)\nprintln(\"The predicted is: $output\")\n", "meta": {"hexsha": "9fe9018b22663f969585755ab22341d47c3bb60c", "size": 2226, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/machine_learning/linear_regression.jl", "max_stars_repo_name": "Whiteshark-314/Julia", "max_stars_repo_head_hexsha": "3285d8d6b7585cc1075831c2c210b891151da0c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-10-14T21:48:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-14T21:48:50.000Z", "max_issues_repo_path": "src/machine_learning/linear_regression.jl", "max_issues_repo_name": "AugustoCL/Julia", "max_issues_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/machine_learning/linear_regression.jl", "max_forks_repo_name": "AugustoCL/Julia", "max_forks_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.5862068966, "max_line_length": 90, "alphanum_fraction": 0.5795148248, "num_tokens": 693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9728307716151472, "lm_q2_score": 0.8670357494949105, "lm_q1q2_score": 0.8434790571990513}}
{"text": "\"A module to plot Ulam spirals\"\nmodule UlamSpiral\n\nimport PyPlot\nusing Primes\nimport Base: position\n \nexport position, text_spiral, point_spiral, plot_quadratic\n \n\"\"\" \n position(n[, start_n=1])\n\nCompute (x,y) location in the lattice spiral of the integer n, assuming the spiral\nstarts at the integer start_n (which takes default value 1).\n\n# Arguments\n- `n::Integer`: the integer to find the position of, N.B. n >= start_n\n- `start_n::Integer=1`: the starting integer of the spiral\n\n# Examples\n```jldoctest\njulia> UlamSpiral.position(23, 1)\n(0, -2)\n```\n\"\"\"\nfunction position( n::Integer, start_n::Integer=1 )\n if n![\"Drawing\"](\"https://raw.githubusercontent.com/JuliaComputing/JuliaAcademyData.jl/master/courses/Foundations%20of%20machine%20learning/data/data_flow.png\")
\n#\n#\n# ![\"Drawing\"](\"https://raw.githubusercontent.com/JuliaComputing/JuliaAcademyData.jl/master/courses/Foundations%20of%20machine%20learning/data/what_is_model.png\")
\n\n#-\n\n# A common function for performing this kind of **classification** is the sigmoid that we saw in the last notebook, and that we will now extend by adding two **parameters**, $w$ and $b$:\n#\n# $$\\sigma(x; w, b) := \\frac{1}{1 + \\exp(-wx + b)}$$\n#\n# $$ x = \\mathrm{data} $$\n#\n# \\begin{align}\n# \\sigma(x;w,b) &\\approx 0 \\implies \\mathrm{apple} \\\\\n# \\sigma(x;w,b) &\\approx 1 \\implies \\mathrm{banana}\n# \\end{align}\n\n#-\n\n# In our mathematical notation above, the `;` in the function differentiates between the **data** and the **parameters**. `x` is the data and is determined from the image. The parameters, `w` and `b`, are numbers which we choose to make our function match the results it should be modeling.\n#\n# Note that in the code below, we don't distinguish between data and parameters - both are just inputs to our function, σ!\n\nusing Images, Statistics\n\napple = load(datapath(\"data/10_100.jpg\"))\nbanana = load(datapath(\"data/104_100.jpg\"))\n\napple_green_amount = mean(Float64.(green.(apple)))\nbanana_green_amount = mean(Float64.(green.(banana)))\n\nprintln(\"Average green for apple = $apple_green_amount\")\nprintln(\"Average green for banana = $banana_green_amount\")\n\n#-\n\nσ(x, w, b) = 1 / (1 + exp(-w * x + b))\n\n# What we want is that when we give σ as input the average green for the apple, roughly `x = 0.3385`, it should return as output something close to 0, meaning \"apple\". And when we give σ the input `x = 0.8808`, it should output something close to 1, meaning \"banana\".\n#\n# By changing the parameters of the function, we can change the shape of the function, and hence make it represent, or **fit**, the data better!\n\n#-\n\n# ## Data fitting by varying parameters\n\n#-\n\n# We can understand how our choice of `w` and `b` affects our model by seeing how our values for `w` and `b` change the plot of the $\\sigma$ function.\n\nusing Plots; gr() # GR works better for interactive manipulations\n\n# Run the code in the next cell. You should see two \"sliders\" appear, one for `w` and one for `b`.\n#\n# **Game**:\n# Change w and b around until the blue curve, labeled \"model\", which is the graph of the `\\sigma` function, passes through *both* of the data points at the same time.\n\nw = 10.0 # try manipulating w between -10 and 30\nb = 10.0 # try manipulating b between 0 and 20\n\nplot(x -> σ(x, w, b), xlim=(-0,1), ylim=(-0.1,1.1), label=\"model\", legend=:topleft, lw=3)\n\nscatter!([apple_green_amount], [0.0], label=\"apple\", ms=5) # marker size = 5\nscatter!([banana_green_amount], [1.0], label=\"banana\", ms=5)\n\n# Notice that the two parameters do two very different things. The **weight**, `w`, determines *how fast* the transition between 0 and 1 occurs. It encodes how trustworthy we think our data actually is, and in what range we should be putting points between 0 and 1 and thus calling them \"unsure\". The **bias**, `b`, encodes *where* on the $x$-axis the switch should take place. It can be seen as shifting the function left-right. We'll come to understand these *parameters* more in notebook 6.\n#\n# Here are some parameter choices that work well:\n\nw = 25.58; b = 15.6\n\nplot(x -> σ(x, w, b), xlim=(0,1), ylim=(-0.1,1.1), label=\"model\", legend=:topleft, lw=3)\n\nscatter!([apple_green_amount], [0.0], label=\"apple\")\nscatter!([banana_green_amount],[1.0], label=\"banana\")\n\n# (Note that in this problem there are many combinations of `w` and `b` that fit the data well.)\n\n#-\n\n# Once we have a model, we have a computational representation for how to choose between \"apple\" and \"banana\". So let's pull in some new images and see what our model says about them!\n\napple2 = load(datapath(\"data/107_100.jpg\"))\n\n#-\n\ngreen_amount = mean(Float64.(green.(apple2)))\n@show green_amount\n\nscatter!([green_amount], [0.0], label=\"new apple\")\n\n# Our model successfully says that our new image is an apple! Pat yourself on the back: you've actually just trained your first neural network!\n\n#-\n\n# #### Exercise 1\n#\n# Load the image of a banana in `data/8_100.jpg` as `mybanana`. Edit the code below to calculate the amount of green in `mybanana` and to overlay data for this image with the existing model and data points.\n\n#-\n\n# #### Solution\n\n#-\n\n# To get the desired overlay, the code we need is\n\nmybanana = load(datapath(\"data/8_100.jpg\"))\nmybanana_green_amount = mean(Float64.(green.(banana)))\nscatter!([mybanana_green_amount], [1.0], label=\"my banana\")\n\n# ## Closing remarks: bigger models, more data, more accuracy\n#\n# That last apple should start making you think: not all apples are red; some are yellow. \"Redness\" is one attribute of being an apple, but isn't the whole thing. What we need to do is incorporate more ideas into our model by allowing more inputs. However, more inputs would mean more parameters to play with. Also, we would like to have the computer start \"learning\" on its own, instead of modifying the parameters ourselves until we think it \"looks right\". How do we take the next step?\n#\n# The first thing to think about is, if you wanted to incorporate more data into the model, how would you change the sigmoid function? Play around with some ideas. But also, start thinking about how you chose parameters. What process did you do to finally end up at good parameters? These two problems (working with models with more data and automatically choosing parameters) are the last remaining step to understanding deep learning.\n\n", "meta": {"hexsha": "f10ba6504aa4d3823a14d24ed23647b0352392ca", "size": 6122, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Courses/Foundations of machine learning/0500.Building-models.jl", "max_stars_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_stars_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 45, "max_stars_repo_stars_event_min_datetime": "2020-02-13T00:50:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-05T07:57:22.000Z", "max_issues_repo_path": "Courses/Foundations of machine learning/0500.Building-models.jl", "max_issues_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_issues_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 52, "max_issues_repo_issues_event_min_datetime": "2019-10-30T16:22:28.000Z", "max_issues_repo_issues_event_max_datetime": "2020-01-26T20:02:43.000Z", "max_forks_repo_path": "Courses/Foundations of machine learning/0500.Building-models.jl", "max_forks_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_forks_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 27, "max_forks_repo_forks_event_min_datetime": "2020-02-26T11:33:28.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-25T22:34:53.000Z", "avg_line_length": 47.0923076923, "max_line_length": 494, "alphanum_fraction": 0.7252531852, "num_tokens": 1632, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9539660962919971, "lm_q2_score": 0.8791467564270272, "lm_q1q2_score": 0.8386761992964624}}
{"text": "# Linear Regression\n\nusing CSV\nusing Statistics\nusing Missings\nusing Plots\nusing StatsBase\nusing HypothesisTests\n\n# Read data\ndataset = CSV.read(\"C:/Users/Andrew.Bannerman/Desktop/Machine Learning/Machine-Learning/Machine-Learning/linear_regression/Salary_Data.csv\", header=true, types=[Float64, Float64])\ndataset[:index] = collect(1:1:size(dataset,1)) # index the data\ndataset = dataset[sample(axes(dataset, 1), size(dataset,1); replace = false, ordered = false), :] # Random shuffle the dataframe\n\n# Feature Scaling\ndataset[:Salary] = standardization(Float64.(dataset[:Salary]))\ndataset[:YearsExperience] = standardization(Float64.(dataset[:YearsExperience]))\n\n# Split data to train and test sets\nsplit_ratio = 2/3\nlen = size(dataset,1)\nsplit_no = Int64.(round(split_ratio * len))\ntrain_set = dataset[1:split_no,:]\ntest_set = dataset[split_no+1:len,:]\n\n# Visualize the data\ngr() # We will continue onward using the GR backend\nplot(train_set[:YearsExperience],train_set[:Salary],seriestype=:scatter,title=\"My Scatter Plot (Training Data)\")\n\n# Find the regression formula\nx = Float64.(train_set[:YearsExperience])\ny = Float64.(train_set[:Salary])\n\n# Regression coefficients (y = b0 .+ (b1 .* x))\nb1 = cov(x,y) / var(x) # b (slope)\nb0 = mean(y) - b1 * mean(x) # a (intercept)\n\n# Apply the regression formula to the training data\ny_hat_train = fill(0.0,size(train_set,1)) # initialize array\ny_hat_train = b0 .+ (b1 .* x)\nplot!(train_set[:YearsExperience],y_hat_train, smooth=true)\nplot_annotation = \"y = \",round(b0,digits=2),\" + (\",round(b1,digits=2),\" * x)\"\nannotate!(-.5, 1.0, text(plot_annotation, :red, :left, 10))\ntraining_set_residuals = y .- y_hat_train\nr_2 = cor(x,y) ^2\nn = size(train_set,1)\nstandard_error = sqrt((sum(y_hat_train .- y)^2) / (n-2))\n\n# Using the same regression coefficients found on training data - plot the regression line on unseen test data\nx = Float64.(test_set[:YearsExperience])\ny = Float64.(test_set[:Salary])\ny_hat_test = fill(0.0,size(test_set,1))\ny_hat_test = b0 .+ (b1 .* x)\nplot(test_set[:YearsExperience],test_set[:Salary],seriestype=:scatter,title=\"My Scatter Plot (Test Data)\")\nplot!(test_set[:YearsExperience],y_hat_test, smooth=true)\nplot_annotation = \"y = \",round(b0,digits=2),\" + (\",round(b1,digits=2),\" * x)\"\nannotate!(-.5, 1.0, text(plot_annotation, :red, :left, 10))\ntest_set_residuals = y .- y_hat_test\nr_2 = cor(x,y) ^2\nn = size(test_set,1)\nstandard_error = sqrt((sum(y_hat_test .- y)^2) / (n-2))\n\n\"\"\"\n```\nlin_reg_ols(x_train::Array{T},y_train::Array{T},x_test::Array{T},y_test::Array{T})\n```\nTrain the regression model on training data - derive coefficients b0 and b1\nApply training coefficients on test set data:: y_hat_test = b0 .+ (b1 .* x_test)\n\"\"\"\nfunction lin_reg_ols(x_train::Array{T},y_train::Array{T},x_test::Array{T},y_test::Array{T})::Array{T} where {T<:Real}\n let y_hat_train = size(x_train,1), y_hat_test = size(x_test,1)\n b1 = cov(x_train,y_train) / var(x_train) # b (slope)\n b0 = mean(y_train) - b1 * mean(x_train) # a (intercept)\n coefficients = b1,b0\n y_hat_train = b0 .+ (b1 .* x_train)\n # Test set\n y_hat_test = b0 .+ (b1 .* x_test)\n return y_hat_test\nend\nend\n\n# Test function\nx_train = Float64.(train_set[:YearsExperience])\ny_train= Float64.(train_set[:Salary])\nx_test = Float64.(test_set[:YearsExperience])\ny_test= Float64.(test_set[:Salary])\n\ntest_predictions = lin_reg_ols(x_train,y_train,x_test,y_test)\ntest_set_residuals = y_test .- test_predictions\nr_2 = cor(x_test,y_test) ^2\n\n# How to calculate the p-value\n# https://www.wikihow.com/Calculate-P-Value\n", "meta": {"hexsha": "4bbb885dee2c65234d5bf2820778037bb85f18d8", "size": 3552, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "linear_regression/linear_regression.jl", "max_stars_repo_name": "JuliaTagBot/Machine-Learning", "max_stars_repo_head_hexsha": "a4fc7e1b4222a77d2182953a1edaf44f2c8ab35d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "linear_regression/linear_regression.jl", "max_issues_repo_name": "JuliaTagBot/Machine-Learning", "max_issues_repo_head_hexsha": "a4fc7e1b4222a77d2182953a1edaf44f2c8ab35d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "linear_regression/linear_regression.jl", "max_forks_repo_name": "JuliaTagBot/Machine-Learning", "max_forks_repo_head_hexsha": "a4fc7e1b4222a77d2182953a1edaf44f2c8ab35d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-02-08T10:43:29.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T10:43:29.000Z", "avg_line_length": 37.7872340426, "max_line_length": 179, "alphanum_fraction": 0.7179054054, "num_tokens": 1067, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109812297141, "lm_q2_score": 0.8824278664544911, "lm_q1q2_score": 0.8386691344214561}}
{"text": "using LinearAlgebra\r\n\r\n#Givens from Problem Statement\r\n#Elastic Modulus(psi),Area(in^2),length(in),Force(lb)\r\nE = 1.9*10^6\r\nA = 8\r\nl = 3*12\r\nl2 = 3*sqrt(2)*12\r\nF = 500\r\n\r\n# Torsional Stiffness Function\r\nk(A,E,l) = (A*E)/l\r\n\r\n# k_local(Elastic Modulus,Area,length,theta,number of nodes,\r\n# row/column one,row/column two, ==1 left fixed/ ==2 right fixed)\r\nfunction k_local(E,A,l,theta,i,n1,n2,fixed=0)\r\n #k_local = zeros(4,4)\r\n p1 = (n1*2)-1\r\n p2 = n1*2\r\n p3 = (n2*2)-1\r\n p4 = n2*2\r\n \r\n k_global = zeros(2*i,2*i)\r\n kin = k(A,E,l)\r\n s = (sind(theta))^2\r\n\tc = (cosd(theta))^2\r\n\tsc = sind(theta)*cosd(theta)\r\n\tk_quad = [c sc;\r\n sc s]\r\n k_global[p1:p2,p1:p2] = k*k_quad\r\n k_global[p1:p2,p3:p4] = -k*k_quad\r\n k_global[p3:p4,p1:p2] = -k*k_quad\r\n k_global[p3:p4,p3:p4] = k*k_quad\r\n # Need to remove rows and columns; check: push!, pop! \r\n if fixed == 1\r\n adj = zeros(1,i)\r\n #adj[n1] = 1\r\n k_global[p1,:] = adj \r\n k_global[:,p1] = adj \r\n k_global[p2,:] = adj \r\n k_global[:,p2] = adj \r\n elseif fixed == 2\r\n adj = zeros(1,i)\r\n #adj[n2] = 1 \r\n k_global[p3,:] = adj \r\n k_global[:,p3] = adj \r\n k_global[p4,:] = adj \r\n k_global[:,p4] = adj \r\n end\r\n return k_global\r\nend\r\n\r\n# Unadjusted Global \r\nk_1 = k_local(E,A,l,0,5,1,2,1)\r\nk_2 = k_local(E,A,l,135,5,2,3)\r\nk_3 = k_local(E,A,l,0,5,3,4,1)\r\nk_4 = k_local(E,A,l,90,5,2,4)\r\nk_5 = k_local(E,A,l,45,5,2,5)\r\nk_6 = k_local(E,A,l,0,5,4,5)\r\nkG = k_1+k_2+k_3+k_4+k_5+k_6\r\n\r\n#Adjusted Global \r\nkG_adj = kG\r\n\r\n# Load Vector\r\np = Float64[0 0 0 -F 0 -F]\r\nP = transpose(p)\r\n\r\n# Angular Displacement Vector\r\nu = kG_adj\\P\r\nprintln(\"Displacement Vector (in)\")\r\ndisplay(u)\r\n\r\n# Moment Reaction Vector\r\nP_abs = kG*u_adj\r\nprintln(\"Load Reaction Vector (lb):\")\r\ndisplay(P_abs)\r\n", "meta": {"hexsha": "0819a410a14f4a771ce789855a448b041edfb9a3", "size": 1785, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "trejo_nicholas_IC4.jl", "max_stars_repo_name": "UltraHeckerNick/MechanicalPrograms_small", "max_stars_repo_head_hexsha": "1059fb6c0d391be5ef75c4ba165e4f48819bfb2e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "trejo_nicholas_IC4.jl", "max_issues_repo_name": "UltraHeckerNick/MechanicalPrograms_small", "max_issues_repo_head_hexsha": "1059fb6c0d391be5ef75c4ba165e4f48819bfb2e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "trejo_nicholas_IC4.jl", "max_forks_repo_name": "UltraHeckerNick/MechanicalPrograms_small", "max_forks_repo_head_hexsha": "1059fb6c0d391be5ef75c4ba165e4f48819bfb2e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.8846153846, "max_line_length": 66, "alphanum_fraction": 0.5770308123, "num_tokens": 709, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9732407168145568, "lm_q2_score": 0.8615381987656672, "lm_q1q2_score": 0.8384840541298201}}
{"text": "function count_histogram(nbr_outcomes, sequence)\n count = zeros(nbr_outcomes)\n for x in sequence\n count[x] += 1\n end\n count\nend\n\nusing PyPlot\n\nfunction simulate_die(ntrials)\n outcomes = collect(1:6) # Simulate a fair die\n x = rand(outcomes, ntrials);\n bar(outcomes, count_histogram(6, x) / ntrials)\n xlabel(\"Die outcome\")\n ylabel(\"Probability\")\nend\n\nsimulate_die(1000);\n\nfunction simulate_sum_of_n_dice(ntrials, ndice)\n outcomes = collect(1:6) # Simulate fair dice\n x = zeros(Int64, ntrials)\n for i = 1:ndice\n x .+= rand(outcomes, ntrials)\n end\n outcomesn = collect(1:6ndice)\n bar(outcomesn, count_histogram(6ndice, x) / ntrials)\n xlabel(\"Sum of n dice outcome\")\n ylabel(\"Probability\")\nend\n\nsimulate_sum_of_n_dice(1000, 2); # Two dice\n\nsimulate_sum_of_n_dice(10000, 50);\n\nx = randn(10000)\nnbins = 50\n\nplt.hist(x, nbins)\nxlabel(\"Random variable x\")\nylabel(\"Count\");\n", "meta": {"hexsha": "3b0bb6a7e60f22adc15075e69c4c82559c8de709", "size": 941, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "textbook/_build/jupyter_execute/content/Random_Numbers/Histograms.jl", "max_stars_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_stars_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "textbook/_build/jupyter_execute/content/Random_Numbers/Histograms.jl", "max_issues_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_issues_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "textbook/_build/jupyter_execute/content/Random_Numbers/Histograms.jl", "max_forks_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_forks_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.8837209302, "max_line_length": 56, "alphanum_fraction": 0.6737513284, "num_tokens": 289, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9433475746920261, "lm_q2_score": 0.8887588008585925, "lm_q1q2_score": 0.8384084592761466}}
{"text": "using LinearAlgebra\nusing Plots\n\n\nfunction newtons_method(∇f, H, x, ϵ, k_max)\n k, Δ = 1, Inf\n while abs(Δ) > ϵ && k ≤ k_max\n Δ = ∇f(x)/abs(H(x))\n x -= Δ\n k += 1\n end\n return x\nend\n\nf = x->sin(10x)+cos(3x)\ndf(x) = 10*cos(10*x)-3*sin(3*x)\nddf(x) = -100*sin(10x)-9*cos(3x)\n\n#converge para o ponto de minimo\ninitial_point = 5.1 #ponto inicial\nconvergence = 1e-8 #convergencia\nmaximum_iterations = 10 #numero maximo de iteracoes\nxs = newtons_method(df,ddf,initial_point,convergence,maximum_iterations)\nxs,f(xs),ddf(xs),df(xs)\n\n\n#converge para o ponto de maximo\n#initial_point = 4.45 #ponto inicial\n#convergence = 1e-8 #convergencia\n#maximum_iterations = 10 #numero maximo de iteracoes\n#xs = newtons_method(df,ddf,initial_point,convergence,maximum_iterations)\n#xs,f(xs),ddf(xs),df(xs)\n\nplot(f,0.,8.,lw=2,draw_arrow=\"true\")\n#scatter!([a,c],[f(a),f(c)])\nscatter!([initial_point],[f(initial_point)])\nscatter!([xs],[f(xs)])", "meta": {"hexsha": "15398b0e1ab737fb1f9d203bac0f19b5b449ae4e", "size": 970, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Newtons_Method.jl", "max_stars_repo_name": "GilbertoLucas/Structural_Optimization", "max_stars_repo_head_hexsha": "e12ed13bbb97db275d79a0e5bfcad6438a643441", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Newtons_Method.jl", "max_issues_repo_name": "GilbertoLucas/Structural_Optimization", "max_issues_repo_head_hexsha": "e12ed13bbb97db275d79a0e5bfcad6438a643441", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Newtons_Method.jl", "max_forks_repo_name": "GilbertoLucas/Structural_Optimization", "max_forks_repo_head_hexsha": "e12ed13bbb97db275d79a0e5bfcad6438a643441", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.2162162162, "max_line_length": 73, "alphanum_fraction": 0.6567010309, "num_tokens": 348, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9591542864252022, "lm_q2_score": 0.8740772384450967, "lm_q1q2_score": 0.838374929921318}}
{"text": "using RDatasets, DataFrames, GLM, Plots, Measures; pyplot()\n\ndf = dataset(\"datasets\", \"anscombe\")\n\nmodel1 = lm(@formula(Y1 ~ X1), df)\nmodel2 = lm(@formula(Y2 ~ X2), df)\nmodel3 = lm(@formula(Y3 ~ X3), df)\nmodel4 = lm(@formula(Y4 ~ X4), df)\n\nprintln(\"Model 1. Coefficients: \", coef(model1),\"\\t R squared: \",r2(model1))\nprintln(\"Model 2. Coefficients: \", coef(model2),\"\\t R squared: \",r2(model2))\nprintln(\"Model 3. Coefficients: \", coef(model3),\"\\t R squared: \",r2(model3))\nprintln(\"Model 4. Coefficients: \", coef(model4),\"\\t R squared: \",r2(model4))\n\nyHat(model, X) = coef(model)' * [ 1 , X ]\nxlims = [0, 20]\n\np1 = scatter(df.X1, df.Y1, c=:blue, msw=0, ms=8)\np1 = plot!(xlims, [yHat(model1, x) for x in xlims], c=:red, xlims=(xlims))\n\np2 = scatter(df.X2, df.Y2, c=:blue, msw=0, ms=8)\np2 = plot!(xlims, [yHat(model2, x) for x in xlims], c=:red, xlims=(xlims))\n\np3 = scatter(df.X3, df.Y3, c=:blue, msw=0, ms=8)\np3 = plot!(xlims, [yHat(model3, x) for x in xlims], c=:red, xlims=(xlims))\n\np4 = scatter(df.X4, df.Y4, c=:blue, msw=0, ms=8)\np4 = plot!(xlims, [yHat(model4, x) for x in xlims], c=:red, msw=0, xlims=(xlims))\n\nplot(p1, p2, p3, p4, layout = (2,2), xlims=(0,20), ylims=(0,14), \n\tlegend=:none, xlabel = \"x\", ylabel = \"y\",\n \tsize=(1200, 800), margin = 10mm)", "meta": {"hexsha": "ebf1d8d36bade35a2683bbc26c373cf7a56af31f", "size": 1261, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "8_chapter/anscombeQuartet.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "8_chapter/anscombeQuartet.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "8_chapter/anscombeQuartet.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 39.40625, "max_line_length": 81, "alphanum_fraction": 0.6193497224, "num_tokens": 511, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9465966686936262, "lm_q2_score": 0.8856314753275017, "lm_q1q2_score": 0.8383358042352346}}
{"text": "using Plots\n\n#Useful functions\n\nfunction hat(i, N, x)\n @assert 0 <= i < N\n h = 1 / (N-1) #Domain then goes from 0 to 1\n xim1 = h * (i-1)\n xi = h * i\n xip1 = h * (i+1)\n if x <= xim1\n return 0.\n elseif x <= xi\n return (x - xim1) / h\n elseif x <= xip1\n return -(x - xip1) / h\n else\n return 0.\n end\nend\n\nfunction sample(f, N)\n h = 1 / (N-1)\n return coeffs = Float64[f(h*i) for i in 0:N-1]\nend\n\nfunction evaluate(coeffs, x)\n N = length(coeffs)\n sum(coeffs[i+1] * hat(i, N, x) for i in 0:N-1)\nend\n\nfunction deriv(f)\n N = length(f)\n h = 1 / (N-1)\n deriv_coeffs = similar(f)\n for i in 1:N\n if i == 1\n deriv_coeffs[i] = (f[2] - f[1]) / h\n elseif i == N\n deriv_coeffs[i] = (f[N] - f[N-1]) / h\n else\n deriv_coeffs[i] = (f[i+1] - f[i-1]) / (2*h)\n end\n end\n return deriv_coeffs\nend\n\nfunction deriv2(f)\n N = length(f)\n h = 1 / (N-1)\n deriv2_coeffs = similar(f)\n for i in 1:N\n if i == 1\n deriv2_coeffs[i] = (f[3] - 2*f[2] + f[1]) / h^2\n elseif i == N\n deriv2_coeffs[i] = (f[N] - 2*f[N-1] + f[N-2]) / h^2\n else\n deriv2_coeffs[i] = (f[i+1] - 2*f[i] + f[i-1]) / h^2\n end\n end\n return deriv2_coeffs\nend\n\nfunction rk2step(f, y0, h)\n k0 = f(y0)\n y1 = y0 + (h/2)*k0\n k1 = f(y1)\n y2 = y0 + h*k1\n y2\nend\n\nfunction rk2integrate(f, y0, h, n)\n res = [y0]\n for i in 1:n\n y1 = rk2step(f, y0, h)\n push!(res, y1)\n y0 = y1\n end\n res\nend\n\n#Solving now the wave equation in a box.\n#My state vector will be an array where the first index\n#indicates position and the second index indicates time step\n\nL=100\nT=100\n\n#Create the array\n\nu = zeros(L,T)\n\n#Boundary Conditions\n\nfor n=1:T\n u[1,n]=0.\n u[L,n]=0.\nend\n\n#Initial Conditions\n\nf(x)=sin(2*pi*x)\n\n#Discretization Parameter.\n#It has a big effect on stability\n\ns = 1\n\n#Finite difference scheme\n\nfi=sample(f,L)\nu[:,1]=fi\n\nfor j=2:L-1\n u[j,2]=u[j,1]-1/2*s*(u[j+1,1]-2*u[j,1]+u[j-1,1])\nend\n\nfor n=2:T-1\n for j=2:L-1\n u[j,n+1]=s*(u[j+1,n]+u[j-1,n]) + 2*(1.0-s)*u[j,n]-u[j,n-1]\n end\nend\n\n#Plot the solution\n\ng(x,j)=evaluate(u[:,j],x)\n\nu5(x)=g(x,5)\nu10(x)=g(x,10)\nu25(x)=g(x,25)\nu27(x)=g(x,27)\nu30(x)=g(x,30)\nu50(x)=g(x,50)\n\nlabels = [\"t=5\" \"t=10\" \"t=25\" \"t=27\" \"t=30\" \"t=50\"]\n\nplot([u5,u10,u25,u27,u30,u50],0,1,xlabel=\"x\",ylabel=\"y\",plot_title=`Hello`,label=labels)\n\n#Energy of the solution (Conserved)- Using to check for stability\n\nud=similar(u)\nudt=similar(u)\n\nfor n=1:T\n ud[:,n]=deriv(u[:,n])\nend\n\nfor i=1:L\n udt[i,:]=deriv(u[i,:])\nend\n\nh(x,j)=evaluate(ud[:,j],x)\nk(x,j)=evaluate(udt[:,j],x)\n\nE=[]\nind=[]\n\nfor j=1:T\n\n p(x)=(h(x,j)^2+k(x,j)^2)\n\n Energy=rk2integrate(p,0.001,0.001,9000)[9000]\n\n push!(E,Energy)\n\n push!(ind,j)\n\nend\n\nplot(ind,E,ylabel=\"Energy\",xlabel=\"Time step\",label=\"Energy\")\n\n## Check for convergence\n\nE=[]\nind=[]\n\nfor i=1:10\n\n T=10*i\n\n L=10*i\n\n u = zeros(L,T)\n\n for n=1:T\n u[1,n]=0.\n u[L,n]=0.\n end\n\n fi=sample(f,L)\n u[:,1]=fi\n\n for j=2:L-1\n u[j,2]=u[j,1]-1/2*s*(u[j+1,1]-2*u[j,1]+u[j-1,1])\n end\n\n for n=2:T-1\n for j=2:L-1\n u[j,n+1]=s*(u[j+1,n]+u[j-1,n]) + 2*(1.0-s)*u[j,n]-u[j,n-1]\n end\n end\n\n ud=similar(u)\n udt=similar(u)\n\n for n=1:T\n ud[:,n]=deriv(u[:,n])\n end\n\n for i=1:L\n udt[i,:]=deriv(u[i,:])\n end\n\n #Let's take a particular j, that is, a particular time step\n\n j=10\n\n g(x,j)=evaluate(u[:,j],x)\n h(x,j)=evaluate(ud[:,j],x)\n k(x,j)=evaluate(udt[:,j],x)\n\n p(x)=(h(x,j)^2+k(x,j)^2)\n\n Energy=rk2integrate(p,0.001,0.001,9000)[9000]\n\n push!(E,Energy)\n\n push!(ind,L)\n\nend\n\nplot(ind,ylabel=\"Energy\",xlabel=\"Resolution\",label=\"Energy\")\n", "meta": {"hexsha": "2c3dd84459e8b9330cf578624e5a5eb67d2e9794", "size": 3810, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "wave.jl", "max_stars_repo_name": "JuliMaristany/WaveEquationBox", "max_stars_repo_head_hexsha": "c4df74ace6aa62e8e525c7577f2431f07730eba6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "wave.jl", "max_issues_repo_name": "JuliMaristany/WaveEquationBox", "max_issues_repo_head_hexsha": "c4df74ace6aa62e8e525c7577f2431f07730eba6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-03-02T20:25:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-03-02T20:50:26.000Z", "max_forks_repo_path": "wave.jl", "max_forks_repo_name": "JuliMaristany/WaveEquationBox", "max_forks_repo_head_hexsha": "c4df74ace6aa62e8e525c7577f2431f07730eba6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 16.3519313305, "max_line_length": 88, "alphanum_fraction": 0.5141732283, "num_tokens": 1545, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# # Example\n# Lets try to find a sparse representation for a matrix, so that the Frobeniusnorm of the difference \n# is as small as possible. \n# Lets consider a symmetric matrix $M$:\nusing SubspacePreservingSparsification\nusing SparseArrays\nusing LinearAlgebra\nM = [17.05 16.98 0.3 6.99 7; 16.98 0.2 7.1 6.9 0; 0.3 7.1 -12 0.01 17; 6.99 6.9 0.01 -11.97 0; 7 0 17 0 -0.1]\n\n# First we calculate a sparsity pattern:\nM_id = sparsity_pattern(M, 0.6, 2)\n# For every column and every row of the Matrix the function solves the optimization problem\n# ```math\n# max_{Z}\\, ||x - x\\circ Z(x)||_{0}\\,\\, s.t. \\\\\n# ```\n# ```math\n# \\begin{align*}\n# (Z(x))_{i} &= 0 \\, \\text{or}\\, 1 \\\\\n# ||Z(x)||_{0} &\\ge N \\\\\n# ||x - x\\circ Z(x)||_{p} \t\\,&\\le\\, (1-q)||x||_{p},\n# \\end{align*}\n# ```\n# where $Z$ is the sparsity pattern for the column or row. $N$ is a maximum number for non zero entries, \n# $q$ is a factor that controls how sparse the the row or column should be. All the individual row and column\n# patter are overlayed, if either row or column or both return a one for an entry the entry is one.\n# This ensures that the pattern will preserve different Subspaces of the Matrix. \n\n# Than we can take the sparsity pattern and modify it to a binning pattern:\nbin_sparse_matrix!(M, M_id, 200)\n# The function looks at the maximum and minimum values of the matrix and then finds a partition \n# of the negative and positive area according to the specified maximum number of bins.\n# For each entry of the matrix it is checked to which bin it belongs to. The binning controls the \n# number of unknowns and so the computation cost to solve the optimization problem. It also typically \n# improves the conditioning of the optimization problem.\n\n# With the binning pattern we can find the Matrix so that the norm of the difference \n# is as small as possible and so that the binning constrains are fullfilled:\nbinned_minimization(M_id, zeros(5, 5), Matrix{Float64}(I, 5, 5), M)\n\n# The function solves the optimization problem\n# ```math\n# min ||M - X||_{F} \\,\\, s.t.\n# ```\n# ```math\n# X\\, \\text{has the specified sparsity pattern}\\, B(M) = M_{id}\n# ```\n# Of course our example optimization problem is not too interesting, because without the binning\n# only some entries are set to zero, while the others remain the same. However, \n# more sophisticated optimization problems can be used here.\n\n# We can also use the sparsify function on $M$:\nsparsify(M, 0.6, 2, 200)\n# This function calculates the sparsity and binning pattern like we did and uses as optimization problem\n\n# ```math\n# \\min\\limits_{X} \\frac{1}{2} \\sum\\limits_{i=1}^r \\frac{1}{\\sigma_i^2} ||\\left( X - M\\right) v_i||_2^2 \n# + \\frac{1}{2} \\sum\\limits_{i=1}^r \\frac{1}{\\sigma_i^2} ||\\left( X^* - M^*\\right) u_i||_2^2,\n# ```\n\n# where $r$ is the rank, $\\sigma_i$ are the $r$ biggest singular values, $v_i$ the corresponding right singular vectors and $u_i$ \n# the corresponding left singular vectors of $M$. $M^*$ denotes the conjugate transpose of $M$. The function compares the action of \n# the unknown matrix $X$ with the action of $M$ on the singular vectors of $M$ and penalizes the differences in near null-space with \n# larger weights. It can be formulatet in the form\n\n# ```math\n# X M^+(M^+)^* + (M^+)^*M^+X = MM^+(M^+)^* (M^+)^*M^+M,\n# ```\n\n# where $M^+$ is the pseudo inverse of $M$.\n# We can also set `impose_null_spaces` true:\nsparsify(M, 0.6, 2, 200, true)\n# Then after the described optimization problem was solved the function also solves an additional optimization problem that\n# ensures that the left and right null-spaces are preserved exactly. Because our matrix has full rank this makes no difference in our case.\n# \n", "meta": {"hexsha": "d3cf82a5db08b5d223bd4db495373bbab388361e", "size": 3668, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "docs/literate/example.jl", "max_stars_repo_name": "Matt1h/SSA.jl", "max_stars_repo_head_hexsha": "c7cc678e287f14c2d24561442bdd7439b6f9502c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2022-03-29T09:47:54.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T12:02:56.000Z", "max_issues_repo_path": "docs/literate/example.jl", "max_issues_repo_name": "Matt1h/SSA.jl", "max_issues_repo_head_hexsha": "c7cc678e287f14c2d24561442bdd7439b6f9502c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2022-03-25T15:21:18.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-28T17:20:56.000Z", "max_forks_repo_path": "docs/literate/example.jl", "max_forks_repo_name": "Matt1h/SSA.jl", "max_forks_repo_head_hexsha": "c7cc678e287f14c2d24561442bdd7439b6f9502c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.9066666667, "max_line_length": 139, "alphanum_fraction": 0.7020174482, "num_tokens": 1116, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "import RowEchelon\n\n#=\nHomework 2\nSection 1.4\nProblem 11\n=#\n\n# Create the first Matrix\nA = [1 3 -4 ; \n\t 1 5 2 ; \n\t -3 -7 6]\n\n# Create the second matrix\nB = [-2 ; 4 ; 12]\n\n# Create an augmented matrix\nAugment = [1 3 -4 -2 ;\n\t\t 1 5 2 4 ;\n\t\t -3 -7 6 12]\n\n# Solve the matrix\nAugment_rref = RowEchelon.rref(Augment)\n\n# Print the results\nprintln()\nprintln(\"Original Matrix, A:\")\nprintln(A)\nprintln(\"Original Matrix, B\")\nprintln(B)\nprintln(\"Augmented Matrix:\")\nprintln(Augment)\nprintln(\"Row Reduced Echelon Matrix:\")\nprintln(Augment_rref)\nprintln(\"Solution as a vector:\")\nprintln(Augment_rref[:,4])\nprintln()\n\n", "meta": {"hexsha": "7fe42f04d6de941fe06375cf05ff52c9cfe0a326", "size": 606, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "HW2_S1.4_P11.jl", "max_stars_repo_name": "tleecsm/J4LA", "max_stars_repo_head_hexsha": "0ff51fa1771e9b5fc0b8a684a65cb36cebb8eefb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "HW2_S1.4_P11.jl", "max_issues_repo_name": "tleecsm/J4LA", "max_issues_repo_head_hexsha": "0ff51fa1771e9b5fc0b8a684a65cb36cebb8eefb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "HW2_S1.4_P11.jl", "max_forks_repo_name": "tleecsm/J4LA", "max_forks_repo_head_hexsha": "0ff51fa1771e9b5fc0b8a684a65cb36cebb8eefb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 15.5384615385, "max_line_length": 39, "alphanum_fraction": 0.6716171617, "num_tokens": 205, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9553191271831558, "lm_q2_score": 0.8774767906859265, "lm_q1q2_score": 0.838270361801556}}
{"text": "\n\"\"\"\n SqExponential(lnℓ::AbstractFloat) <: RadialKernel{SqEuclidean}\n\nThe squared exponential kernel (aka the Gaussian kernel) with length scale \n``\\\\exp(2 \\\\ln\\\\ell) = \\\\ell^2 > 0``. \n\n```math\nk_{\\\\ell}(x, y) = k_{\\\\ell}(\\\\lVert x - y\\\\rVert_2^2) = \n\\\\exp\\\\bigg\\\\lbrace -\\\\frac{\\\\lVert x - y\\\\rVert_2^2}{2\\\\ell^2}\\\\bigg\\\\rbrace\n```\n\nExternal links\n* [RBF Kernel on Wikipedia](https://en.wikipedia.org/wiki/Radial_basis_function_kernel)\n\n\"\"\"\nmutable struct SqExponential{F<:AbstractFloat} <: RadialKernel{SqEuclidean}\n dist::SqEuclidean\n lnℓ::F\n twoℓ²::F\n function SqExponential(dist::SqEuclidean, lnℓ::AbstractFloat)\n new{typeof(lnℓ)}(dist, lnℓ, exp(2 * lnℓ + float(logtwo)))\n end\nend\n\nSqExponential(lnℓ::AbstractFloat) = SqExponential(SqEuclidean(), lnℓ)\n\n@inline (k::SqExponential)(τ::AbstractFloat) = exp(-τ / k.twoℓ²)\n\nGaussian = SqExponential\n\nnumparams(::SqExponential) = (1,)\nparamtypes(::SqExponential{F}) where F = (F,)\nparams(k::SqExponential) = (lnℓ = k.lnℓ,)\nfunction setparams!(k::SqExponential{F}, lnℓ::F) where F\n k.lnℓ = lnℓ\n k.twoℓ² = exp(2 * lnℓ + float(logtwo))\nend\n", "meta": {"hexsha": "2d107871859cda47ad9a7aa89eae7c932b2dbb4a", "size": 1117, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/kernels/radial/SqExponential.jl", "max_stars_repo_name": "cameton/SKI.jl", "max_stars_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/kernels/radial/SqExponential.jl", "max_issues_repo_name": "cameton/SKI.jl", "max_issues_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/kernels/radial/SqExponential.jl", "max_forks_repo_name": "cameton/SKI.jl", "max_forks_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.641025641, "max_line_length": 87, "alphanum_fraction": 0.6723366159, "num_tokens": 400, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133464597458, "lm_q2_score": 0.8918110382493035, "lm_q1q2_score": 0.8382250973706432}}
{"text": "# Exercise 1\n\nusing LinearAlgebra, Statistics\n\nfunction lyapunov(A,Σ; tolerance = 1.0E-8, maxiter = 500, S0 = Σ*Σ')\n\n #initialise loop\n normdiff = Inf\n iter = 1\n S_old = S0\n Q = Σ * Σ'\n\n while normdiff > tolerance && iter ≤ maxiter\n\n S_new = A * S_old * A' + Q\n\n #continue looping\n normdiff = norm(S_old - S_new)\n S_old = S_new\n iter = iter +1\n end\n return (S_old,iter)\nend\n\n# test function\nA = [0.8 -0.2; -0.1 0.7]\nΣ = [0.5 0.4; 0.4 0.6]\n\n@show maximum(abs, eigvals(A))\n\nour_solution, iter = lyapunov(A, Σ)\n\n###############\n# Exercise 2\n###############\nusing Plots\n\nΘ = [0.8, 0.9, 0.98] #parameters\nγ = 1\nσ = 1\nT = 150 #horizon\np = plot() #naming plot to add\ny0 = 1\nτ = 5 # moving average - interval\n\nfor θ in Θ\n y = zeros(T+1)\n y[1] = y0\n for i ∈ 1:T\n y[i+1] = γ + θ * y[i] + σ * randn()\n end\n plot!(p, y, label = \"θ = $θ\")\n\n # moving average\n x = zeros(T+1)\n for i ∈ τ:(T+1)\n x[i]= sum(y[(i-τ+1):i])/τ\n end\n\n plot!(p, x, label = \"rolling mean θ = $θ\" )\nend\np\n\n# simulate \"nsamp\" times\n\nfunction sim_randomwalk_drift(Θ,γ,σ,T)\n\n y_sim = zeros(nsamp,length(Θ))\n for j in 1:nsamp\n\n y = zeros(T+1,length(Θ))\n y[1,:] = ones(eltype(Θ), length(Θ))\n\n for i in 1:T\n y[i+1,:] = γ .+ Θ .*y[i,:] + σ.*randn(3)\n end\n\n y_sim[j,:] = y[T+1,:]\n end\n return y_sim\nend\n\ny_sim = sim_randomwalk_drift(Θ,γ,σ,T)\n\n# find mean and variance of ensemble averages\n\n@show (mean_θ1, mean_θ2, mean_θ3) = (mean(y_sim[:,1]),mean(y_sim[:,2]),mean(y_sim[:,3]))\n@show (var_θ1, var_θ2, var_θ3) = (var(y_sim[:,1]),var(y_sim[:,2]),var(y_sim[:,3]))\n\n# plot histogram of T+1 values\n\nh=histogram()\nfor (i,θ) in enumerate(Θ)\n histogram!(h, y_sim[:,i],α =0.5, label = \"θ = $θ\",bins =20)\nend\nh\n\n##############\n# Exercise 3\n##############\n\n\n# simulate observations\n\nN = 50 # number of observations\n\nx_1, x_2 = (randn(N), randn(N))\n\na, b, c, d, σ = (0.1, 0.2, 0.5, 1.0, 0.1) #parameters\n\n# simulate data\n\nfunction simdata(a, b, c, d, σ, N; M = 20)\n y = zeros(N,M)\n for i in 1:N\n y[i,:] = a * x_1[i] + b * x_1[i]^2 + c * x_2[i] + d .+ σ.*randn(M)\n end\n return y\nend\n\nY = simdata(a, b, c, d, σ, N)\n\n# perform OLS\n\nX = [x_1 x_1.^2 x_2 ones(N)]\nβ = zeros(M,size(X)[2])\n\nfor m in 1:M\n β[m,:] = inv(X'*X)*X'*Y[:,m]\nend\nβ\n\n\n# plot histograms\nΘ = [\"a=0.1\" \"b=0.2\" \"c=0.5\" \"d=1.0\"]\nh = histogram()\n\nfor (i,θ) in enumerate(Θ)\n histogram!(h, β[:,i],α = 0.8, label = θ, bins = 10)\nend\nh\n\n############################\n# Exercise 4 - use NL solve\n############################\n\nusing NLsolve\n\n\nA = [0.8 -0.2; -0.1 0.7]\nΣ = [0.5 0.4; 0.4 0.6]\nQ = Σ * Σ'\nS0 = Σ * Σ'\n\nf(S) = A * S * A' + Q\n\nf(S0)\nsol = fixedpoint(f,S0)\nsol\n", "meta": {"hexsha": "53f647a3d8df638eea544a4366f1c0f3adab56ef", "size": 2755, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "GettingStarted_ArraysTuplesRanges.jl", "max_stars_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_stars_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "GettingStarted_ArraysTuplesRanges.jl", "max_issues_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_issues_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "GettingStarted_ArraysTuplesRanges.jl", "max_forks_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_forks_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.21875, "max_line_length": 88, "alphanum_fraction": 0.5085299456, "num_tokens": 1149, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545348152282, "lm_q2_score": 0.8840392710530071, "lm_q1q2_score": 0.8382058438036573}}
{"text": "# Simple Quadratic Regression\n# Originally written by Keegan Go for lsqpy\n# Translated into LinearLeastSquares.jl by Karanveer Mohan and David Zeng\n\nusing LinearLeastSquares\nusing Gadfly\n# Set the random seed to get consistent data\nsrand(1)\n\n# Number of examples to use\nn = 100\n\n# Specify the true value of the variable\ntrue_coeffs = [2; -2; 0.5]\n\n# Generate data\nx_data = rand(n, 1) * 5\nx_data_expanded = hcat([x_data .^ i for i in 1 : 3]...)\ny_data = x_data_expanded * true_coeffs + 0.5 * rand(n, 1)\n\nquadratic_coeff = Variable()\nslope = Variable()\noffset = Variable()\nquadratic = offset + x_data * slope + quadratic_coeff * x_data .^ 2\nresiduals = quadratic - y_data\nfit_error = sum_squares(residuals)\noptval = minimize!(fit_error)\n\n# Create some evenly spaced points for plotting, again replicate powers\nt = reshape([0 : 0.1 : 5], length([0 : 0.1 : 5]), 1)\nt_squared = t .^ 2\n\n# Plot our regressed function\np = plot(\n layer(x=x_data, y=y_data, Geom.point),\n layer(x=t, y=evaluate(offset) + t * evaluate(slope) + t_squared * evaluate(quadratic_coeff), Geom.line),\n Theme(panel_fill=color(\"white\"))\n)\n# draw(PNG(\"quadratic_regression.png\", 16cm, 12cm), p)\n", "meta": {"hexsha": "ef743554ee714757df38617c723e565547cc3ee6", "size": 1161, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/regression/quadratic_regression.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/LinearLeastSquares.jl-a8002f64-6652-5b53-92ba-d4864fedfa0a", "max_stars_repo_head_hexsha": "f50c74231e7204bb63a9f49ea606ba28dddd3fa5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2015-06-25T15:56:13.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-04T16:36:46.000Z", "max_issues_repo_path": "examples/regression/quadratic_regression.jl", "max_issues_repo_name": "UnofficialJuliaMirrorSnapshots/LinearLeastSquares.jl-a8002f64-6652-5b53-92ba-d4864fedfa0a", "max_issues_repo_head_hexsha": "f50c74231e7204bb63a9f49ea606ba28dddd3fa5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2016-07-12T03:00:37.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-06T12:25:56.000Z", "max_forks_repo_path": "examples/regression/quadratic_regression.jl", "max_forks_repo_name": "UnofficialJuliaMirrorSnapshots/LinearLeastSquares.jl-a8002f64-6652-5b53-92ba-d4864fedfa0a", "max_forks_repo_head_hexsha": "f50c74231e7204bb63a9f49ea606ba28dddd3fa5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2015-02-14T02:30:05.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-01T18:11:37.000Z", "avg_line_length": 29.025, "max_line_length": 106, "alphanum_fraction": 0.7166236003, "num_tokens": 358, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545304202039, "lm_q2_score": 0.8840392725805823, "lm_q1q2_score": 0.8382058413666608}}
{"text": "# Astronomy example from Hairer et al 1992 p.245. As Hairer writes, \"a\n# celestial mechanics problem( which we call 'the Pleiades'): seven stars in\n# the plan with coordinates x_i,y_i and masses m_i = i\". Time span from\n#t =0 to t= 3. This is a non-stiff problem.\n#\n# Reference solution and a more in depth description can be found in\n# https://archimede.dm.uniba.it/~testset/report/plei.pdf\n\n\nexport plei\n\n# create an instance of TestCaseExplicit by filling in the ...:\nplei = let\n tcname = :plei ::Symbol # name should be same as variable name (except for upper/lower case)\n\n T = Float64 # the datatype used, probably Float64\n Tarr = [Matrix, SparseMatrixCSC][2] # the container datatype used\n # for the Jacobian and mass\n # matrix, probably Matrix\n # (default) or\n # SparseMatrixCSC.\n const N = 7\n const dof = 4N::Int # degrees of freedom\n dae = 0 ::Int # index of DAE, ==0 for ODE\n # stiffness of system, one of the three constants\n stiffness = [nonstiff, mildlystiff, stiff][1]\n\n ## the problem function\n function fn!(t::T,y::Vector{T},dydt::Vector{T})\n # The ode function dydt = f(t,y) modifying dydt in-place.\n # Note that all indices of dydt need to be written!\n\n # Problem is of the form z = (x,y) and z'' = f(z), (z(0),z'(0)) = (z_0,z'_0)\n\n #input (z,w), where w = z'\n #output dydt = (z,w)' = (w, f(z)) , where f(z) = z''\n\n for i=1:2N\n dydt[i] = y[2N+i]\n end\n\n dydt[2N+1:end] = 0\n\n #parameters\n for i = 1:N\n for j =1:N\n if i != j\n r = ((y[i] -y[j])^2 + (y[i+N] -y[j+N])^2)^(3/2)\n #Note, for what follows, dydt[2N+i] = x″[i] and dydt[3N+i] = y″[i]\n dydt[2N+i] += j*(y[j] - y[i])/r #j factor is mass of particles,\n dydt[3N+i] += j*(y[j+N] - y[i+N])/r\n end\n end\n end\n\n return nothing\n end\n # initializes storage for y:\n fn!( ; T_::Type=T, dof_=dof) = zeros(T_,dof_)\n\n # NOTE, using keyword arguments more intuitively like:\n # fn!( ; T::Type=T, dof=dof) = zeros(T,dof)\n # does not work: https://github.com/JuliaLang/julia/issues/9948\n\n jac! = nothing\n mass! = nothing\n\n ic = T[3, 3, -1, -3, 2, -2, 2, #x(0)\n 3, -3, 2, 0, 0, -4, 4, #y(0)\n 0, 0, 0, 0, 0, 1.75, -1.5, #x'(0)\n 0, 0, 0, -1.25, 1, 0, 0] #y'(0)\n # vector of initial conditions\n tspan = T[0, 3] # integration interval\n\n refsol = T[0.3706139143970502,3.237284092057233,-3.222559032418324,0.6597091455775310, 0.3425581707156584, 1.562172101400631,-0.7003092922212495,\n -3.943437585517392,-3.271380973972550, 5.225081843456543, -2.590612434977470, 1.198213693392275, -0.2429682344935824,1.091449240428980,\n 3.417003806314313, 1.354584501625501, -2.590065597810775, 2.025053734714242, -1.155815100160448, -0.8072988170223021, 0.5952396354208710,\n -3.741244961234010, 0.3773459685750630, 0.9386858869551073, 0.3667922227200571, -0.3474046353808490, 2.344915448180937, -1.947020434263292]\n # reference solution at tspan[2]\n\n\n refsolinds = trues(dof) # if refsol does not contain all indices\n # then specify which indices of the\n # solution are to be compared to refsol\n\n scd_absinds = Int[] # set where refsol is very small or\n # zero. To avoid it dominating the\n # relative error\n\n tc = TestCaseExplicit{tcname, T, Tarr}(\n stiffness,\n dae,\n dof,\n fn!,\n jac!,\n mass!,\n ic,\n tspan,\n refsol,\n refsolinds,\n scd_absinds)\n\n # put into the right buckets\n tc_all[tcname] = tc\n tc_nonstiff[tcname] = tc\n # Note that the return-result of the last statement of the let\n # block needs to be tc!\n tc\nend\n", "meta": {"hexsha": "f21b133048bf9d211e98574b45282e3f8e3b246c", "size": 4329, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/testcases/plei.jl", "max_stars_repo_name": "pwl/IVPTestSuite.jl", "max_stars_repo_head_hexsha": "15d92b5b943c35493ce29e8aa8def14619d96760", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/testcases/plei.jl", "max_issues_repo_name": "pwl/IVPTestSuite.jl", "max_issues_repo_head_hexsha": "15d92b5b943c35493ce29e8aa8def14619d96760", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/testcases/plei.jl", "max_forks_repo_name": "pwl/IVPTestSuite.jl", "max_forks_repo_head_hexsha": "15d92b5b943c35493ce29e8aa8def14619d96760", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.3545454545, "max_line_length": 155, "alphanum_fraction": 0.5255255255, "num_tokens": 1324, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037384317887, "lm_q2_score": 0.9046505267461572, "lm_q1q2_score": 0.8379811648992522}}
{"text": "using ProgressBars, Plots, LaTeXStrings, JLD\n\nchoices = ((1,0),(0,1),(-1,0),(0,-1))\n\nfunction RecurseSAW(R, Network, N)\n if N == 0\n return 1\n end\n count_ = 0\n Network[R...] = true\n for step in choices\n Rₙ = R .+ step\n if !Network[Rₙ...]\n count_ += RecurseSAW(Rₙ, Network, N-1)\n end\n end\n Network[R...] = false\n return count_\nend\n\nfunction maincal(N)\n dim = 2N+1\n Network = falses(dim,dim)\n R = [N+1,N+1]\n return RecurseSAW(R, Network, N)\nend\n\nTotList = [maincal(n) for n ∈ ProgressBar(0:17)]\nsave(\"../../Data/Q3/Q3-SAW.jld\", \"Data\", TotList)\n\nP1 = begin\n plot(0:17,log.(TotList), c = :black, label = nothing)\n scatter!(0:17,log.(TotList), c = :steelblue,framestyle = :box,\n ylabel = L\"ln\\ (Number\\ of\\ Paths)\", xlabel = L\"N\",\n label = L\"Self-Avoiding\\ Walker\", legend = 140)\n plot!(0:17, log.(4 .^(0:17)), c = :black, label = nothing)\n scatter!(0:17, log.(4 .^(0:17)), c = :purple, label = L\"Random\\ Walker\")\nend\n\n\nP2 = begin\n plot(0:17,TotList, c = :black, label = nothing)\n scatter!(0:17,TotList, c = :steelblue,framestyle = :box,\n ylabel = L\"Number\\ of\\ Paths\", xlabel = L\"N\",\n label = L\"Self-Avoiding\\ Walker\", legend = 140)\n plot!(0:13,4 .^(0:13), c = :black, label = nothing)\n scatter!(0:13, 4 .^(0:13), c = :purple, label = L\"Random\\ Walker\")\nend\n\nsavefig(P1,\"../../Figs/Q3/Q3-1.pdf\")\nsavefig(P2,\"../../Figs/Q3/Q3-2.pdf\")\n\nP3 = begin\n plot(0:17, TotList./ 4 .^(0:17), c = :black, label = nothing)\n scatter!(0:17, TotList./ 4 .^(0:17), c = :steelblue,framestyle = :box,\n ylabel = L\"Ratio\\ of\\ Number\\ of\\ Paths\", xlabel = L\"N\", legend = nothing,\n title = L\"Ratio\\ of\\ Number\\ of\\ Paths\\ of\\ SAW\\ and\\ RW\")\nend\n\nsavefig(P3,\"../../Figs/Q3/Q3-3.pdf\")\n", "meta": {"hexsha": "3bfc7476eefed92219798a3465c091b8363f316a", "size": 1821, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "ProblemSet5/Codes/Q3/Q3-SAW.jl", "max_stars_repo_name": "shahmari/ComputationalPhysics-Fall2021", "max_stars_repo_head_hexsha": "f1681e32258c55697d11009e1702eb86d5f119d4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ProblemSet5/Codes/Q3/Q3-SAW.jl", "max_issues_repo_name": "shahmari/ComputationalPhysics-Fall2021", "max_issues_repo_head_hexsha": "f1681e32258c55697d11009e1702eb86d5f119d4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ProblemSet5/Codes/Q3/Q3-SAW.jl", "max_forks_repo_name": "shahmari/ComputationalPhysics-Fall2021", "max_forks_repo_head_hexsha": "f1681e32258c55697d11009e1702eb86d5f119d4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-10-21T11:07:08.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-21T11:07:08.000Z", "avg_line_length": 29.8524590164, "max_line_length": 82, "alphanum_fraction": 0.5579352004, "num_tokens": 651, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425223682086, "lm_q2_score": 0.9111797015700343, "lm_q1q2_score": 0.8379595990825779}}
{"text": "import Pkg; Pkg.add(Pkg.PackageSpec(url=\"https://github.com/JuliaComputing/JuliaAcademyData.jl\"))\nusing JuliaAcademyData; activate(\"Parallel_Computing\")\n\n# # Parallel Algorithms: Thinking in Parallel\n#\n# Now that we're starting to see the challenges of parallelism, it's worth taking\n# a step back and examining how we might go about designing parallel algorithms.\n#\n# This is adapted from a [workshop paper](http://jiahao.github.io/parallel-prefix/) by Jiahao Chen and\n# Alan Edelman entitled \"Parallel Prefix Polymorphism Permits Parallelization, Presentation & Proof\" and\n# will appear in the proceedings of the [First Workshop for High Performance Technical Computing in Dynamic\n# Languages](http://jiahao.github.io/hptcdl-sc14/), held in conjunction with [SC14: The International Conference on High Performance Computing, Networking, Storage and Analysis](http://sc14.supercomputing.org/)\n\n#-\n\nusing Compose, Gadfly\n\n# # `reduce()`\n#\n# Reduction applies a binary operator to a vector repeatedly to return a scalar. Thus + becomes sum, and * becomes prod.\n#\n# It is considered a basic parallel computing primitive.\n\nreduce(+, 1:8), sum(1:8) # triangular numbers\n\n#-\n\nreduce(*, 1:8), prod(1:8) # factorials\n\n#-\n\nboring(a,b)=a\n@show reduce(boring, 1:8)\nboring2(a,b)=b\n@show reduce(boring2, 1:8)\n\n# You can also use reduce to compute Fibonacci numbers using their recurrences.\n#\n# $$\\begin{pmatrix} f_2 \\\\f_1 \\end{pmatrix} = \\begin{pmatrix} f_1 + f_0 \\\\ f_1 \\end{pmatrix}\n# = \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} f_1 \\\\ f_0 \\end{pmatrix} $$\n#\n# $$\\begin{pmatrix} f_{n+1} \\\\ f_n \\end{pmatrix} = \\dots\n# = \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n \\begin{pmatrix} f_1 \\\\ f_0 \\end{pmatrix} $$\n#\n# From this, you can show that\n#\n# $$\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} f_{n+1} & f_n \\\\ f_n & f_{n-1} \\end{pmatrix} $$\n#\n# (this applies reduce to the same argument over and over again -- there are of course other ways)\n\nM=[1 1; 1 0]\nreduce(*,fill(M,3))\nprod(fill(M,3))\n\n#-\n\nn= 50 # Try manipulating n to pick different values (try between 0-100)\nprod(fill(big.(M),n))\n\n#-\n\nfib(j)=reduce(*, fill(M,j))\nfib.([4,7])\n\n# You can solve recurrences of any complexity using `reduce`. For example, `reduce` can compute a Hadamard matrix from its definition in terms of its submatrices:\n#\n# $$H_2 = \\begin{pmatrix} H_1 & H_1 \\\\ H_1 & -H_1 \\end{pmatrix} = \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \\otimes H_1$$\n#\n# and so on.\n#\n# (Note: this is just using reduce to compute a matrix power.\n# One can think of alternative ways for sure.)\n\n## If A is m x n\n## If B is p x q\n## then kron(A,B) is mp x nq and has all the elements of A times all of the elements of B\n\n#-\n\nA=[1 2;3 4]\nB=[10 100; 1 -10]\n⊗(A,B)=kron(A,B)\n\nM=[ 1 1;1 -1]\nH=⊗(⊗(⊗(M,M),M),M)\n\n#-\n\nH'H\n\n#-\n\nHadamard(n)=reduce(⊗, fill(M,n))\nH=Hadamard(4)\n\n#-\n\nusing LinearAlgebra\ncumsum(1:8) # It is useful to know that cumsum is a linear operator\n## You can use power method! Below is the underlying matrix\nA=tril(ones(Int,8,8))\n\n#-\n\n[A*(1:8),cumsum(1:8)]\n\n# # `prefix`\n\n#-\n\n# Having discussed `reduce`, we are now ready for the idea behind prefix sum.\n# Prefix or scan is long considered an important parallel\n# primitive as well.\n#\n# Suppose you wanted to compute the partial sums of a vector, i.e. given\n# `y[1:n]`, we want to overwrite the vector `y` with the vector of partial sums\n#\n# ```julia\n# new_y[1] = y[1]\n# new_y[2] = y[1] + y[2]\n# new_y[3] = y[1] + y[2] + y[3]\n# ...\n# ```\n#\n# At first blush, it seems impossible to parallelize this, since\n#\n# ```julia\n# new_y[1] = y[1]\n# new_y[2] = new_y[1] + y[2]\n# new_y[3] = new_y[2] + y[3]\n# ...\n# ```\n#\n# which appears to be an intrinsically serial process. As written with a `+`\n# operator, this is `cumsum` — but note that it can generalize to any operation.\n\nfunction prefix_serial!(y, ⊕)\n for i=2:length(y)\n y[i] = y[i-1] ⊕ y[i]\n end\n y\nend\n\n#-\n\nprefix_serial!([1:8;],+)\n\n#-\n\ncumsum(1:8)\n\n#-\n\nprefix_serial!([1:8;], *)\n\n#-\n\ncumprod(1:8)\n\n# However, it turns out that because these operations are associative, we can regroup the _order_ of how these sums or products are carried out. (This of course extends to other associative operations, too.) Another ordering of 8 associative operations is provided by `prefix8!`:\n\n## Magic :)\nfunction prefix8!(y, ⊕)\n length(y)==8 || error(\"length 8 only\")\n for i in (2,4,6,8); y[i] = y[i-1] ⊕ y[i]; end\n for i in ( 4, 8); y[i] = y[i-2] ⊕ y[i]; end\n for i in ( 8); y[i] = y[i-4] ⊕ y[i]; end\n for i in ( 6 ); y[i] = y[i-2] ⊕ y[i]; end\n for i in ( 3,5,7 ); y[i] = y[i-1] ⊕ y[i]; end\n y\nend\n\n#-\n\nprefix8!([1:8;], +) == cumsum(1:8)\n\n# In fact, this can generalize beyond just length-8 arrays:\n\n## More magic\nfunction prefix!(y, ⊕)\n l=length(y)\n k=ceil(Int, log2(l))\n @inbounds for j=1:k, i=2^j:2^j:min(l, 2^k) #\"reduce\"\n y[i] = y[i-2^(j-1)] ⊕ y[i]\n end\n @inbounds for j=(k-1):-1:1, i=3*2^(j-1):2^j:min(l, 2^k) #\"expand\"\n y[i] = y[i-2^(j-1)] ⊕ y[i]\n end\n y\nend\n\n# -\n\nA = rand(0:9, 123)\nprefix!(copy(A), *) == cumprod(A)\n\n# ## What is this magic?\n\n#-\n\n# We can visualize the operations with a little bit of trickery. In Julia, arrays are simply types that expose the array protocol. In particular, they need to implement methods for the generic functions `length`, `getindex` and `setindex!`. The last two are used in indexing operations, since statements\n#\n# y = A[1]\n# A[3] = y\n#\n# get desugared to\n#\n# y = getindex(A, 1)\n# setindex!(A, y, 3)\n#\n# respectively.\n#\n# We can trace through the iterable by introduce a dummy array type, `AccessArray`, which records every access to `getindex` and `setindex!`.\n#\n# Specifically:\n#\n# - `length(A::AccessArray)` returns the length of the array it wraps\n# - `getindex(A::AccessArray, i)` records read access to the index `i` in the `A.read` field and then actually retuns the value in the array it wraps.\n# - `setindex!(A::AccessArray, x, i)` records write access to the index `i`. The `A.history` field is appended with a new tuple consisting of the current `A.read` field and the index `i`, and then it performs the assignment.\n#\n# The way `AccessArray` works, it assumes an association between a single `setindex!` call and and all the preceding `getindex` calls since the previous `setindex!` call, which is sufficient for the purposes of tracing through prefix calls.\n\nmutable struct AccessArray{T,N,A}\n data :: A\n read :: Vector{Int}\n history :: Vector{Tuple{Vector{Int},Int}}\nend\nAccessArray(A) = AccessArray{eltype(A), ndims(A), typeof(A)}(A, Int[], Int[])\n\nBase.length(A::AccessArray) = length(A.data)\n\nfunction Base.getindex(A::AccessArray, i::Int)\n push!(A.read, i)\n A.data[i]\nend\n\nfunction Base.setindex!(A::AccessArray, x, i::Int)\n push!(A.history, (A.read, i))\n A.read = Int[]\n A.data[i] = x\nend\n\n#-\n\nM = AccessArray(rand(8))\n\n#-\n\nM[7] = M[3] + M[2]\n\n#-\n\nM.history\n\n# So now we can trace the access pattern when calling `prefix8`!\n\nA=prefix8!(AccessArray(rand(8)),+)\n\n#-\n\nA.history\n\n# Now let's visualize this! Each entry in `A.history` is rendered by a gate object:\n\nusing Compose: circle, mm\n\n#-\n\nstruct Gate{I,O}\n ins :: I\n outs :: O\nend\n\nimport Gadfly.render\n\nfunction render(G::Gate, x₁, y₁, y₀; rᵢ=0.1, rₒ=0.25)\n ipoints = [(i, y₀+rᵢ) for i in G.ins]\n opoints = [(i, y₀+0.5) for i in G.outs]\n igates = [circle(i..., rᵢ) for i in ipoints]\n ogates = [circle(i..., rₒ) for i in opoints]\n lines = [line([i, j]) for i in ipoints, j in opoints]\n compose(context(units=UnitBox(0.5,0,x₁,y₁+1)),\n compose(context(), stroke(colorant\"black\"), fill(colorant\"white\"),\n igates..., ogates...),\n compose(context(), linewidth(0.3mm), stroke(colorant\"black\"),\n lines...))\nend\n\nA=Gate([1,2],2)\nrender(A,2,0,0)\n\n# Now we render the whole algorithm. We have to scan through the trace twice; the first time merely calculates the maximum depth that needs to be drawn and the second time actually generates the objects.\n\nfunction render(A::AccessArray)\n #Scan to find maximum depth\n olast = depth = 0\n for y in A.history\n (any(y[1] .≤ olast)) && (depth += 1)\n olast = maximum(y[2])\n end\n maxdepth = depth\n\n olast = depth = 0\n C = []\n for y in A.history\n (any(y[1] .≤ olast)) && (depth += 1)\n push!(C, render(Gate(y...), length(A), maxdepth, depth))\n olast = maximum(y[2])\n end\n\n push!(C, compose(context(units=UnitBox(0.5,0,length(A),1)),\n [line([(i,0), (i,1)]) for i=1:length(A)]...,\n linewidth(0.1mm), stroke(colorant\"grey\")))\n compose(context(), C...)\nend\n\n#-\n\nrender(prefix!(AccessArray(zeros(8)), +))\n\n# Now we can see that `prefix!` rearranges the operations to form two spanning trees:\n\nrender(prefix!(AccessArray(zeros(120)),+))\n\n# as contrasted with the serial code:\n\nrender(prefix_serial!(AccessArray(zeros(8)),+))\n\n#-\n\nnpp=90 # Try manipulating npp to choose different values between 1 and 180\nrender(prefix!(AccessArray(zeros(npp)),+))\n\n#-\n\nnpp=90 # And manipulate it again, this time with the serial algorithm\nrender(prefix_serial!(AccessArray(zeros(npp)),+))\n\n# # Now exploit the parallelism in the _algorithm_ to use a parallel _implementation_\n\nusing .Threads\nfunction prefix_threads!(y, ⊕)\n l=length(y)\n k=ceil(Int, log2(l))\n for j=1:k\n @threads for i=2^j:2^j:min(l, 2^k) #\"reduce\"\n @inbounds y[i] = y[i-2^(j-1)] ⊕ y[i]\n end\n end\n for j=(k-1):-1:1\n @threads for i=3*2^(j-1):2^j:min(l, 2^k) #\"expand\"\n @inbounds y[i] = y[i-2^(j-1)] ⊕ y[i]\n end\n end\n y\nend\n\nA = rand(500_000);\n\nusing BenchmarkTools\n@btime prefix_serial!($(copy(A)), +);\n@btime prefix!($(copy(A)), +);\n@btime prefix_threads!($(copy(A)), +);\n\nprefix_threads!(copy(A), +) == prefix!(copy(A), +) ≈ cumsum(A)\n\n# # Thinking in parallel\n#\n# Notice how we didn't need to contort ourselves in making our algorithm\n# work with `@threads`. We really did _just_ take a `@threads` on it and it\n# just worked. It was both accurate _and_ fast.\n#\n# Coming up with rearrangements that make your particular algorithm parallel\n# friendly isn't always easy, but when possible it makes everything else\n# just fall out naturally.\n#\n# Finally, note that there can be clever ways to visualize algorithms as sanity checks.\n\n", "meta": {"hexsha": "79100a5579ccd7234a83e2886e1c4fe0184ad490", "size": 10360, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Courses/Parallel_Computing/050 Parallel Algorithms.jl", "max_stars_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_stars_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 45, "max_stars_repo_stars_event_min_datetime": "2020-02-13T00:50:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-05T07:57:22.000Z", "max_issues_repo_path": "Courses/Parallel_Computing/050 Parallel Algorithms.jl", "max_issues_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_issues_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 52, "max_issues_repo_issues_event_min_datetime": "2019-10-30T16:22:28.000Z", "max_issues_repo_issues_event_max_datetime": "2020-01-26T20:02:43.000Z", "max_forks_repo_path": "Courses/Parallel_Computing/050 Parallel Algorithms.jl", "max_forks_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_forks_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 27, "max_forks_repo_forks_event_min_datetime": "2020-02-26T11:33:28.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-25T22:34:53.000Z", "avg_line_length": 27.3350923483, "max_line_length": 304, "alphanum_fraction": 0.6466216216, "num_tokens": 3367, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425245706048, "lm_q2_score": 0.9111796991580949, "lm_q1q2_score": 0.8379595988712346}}
{"text": "## matern.jl : implementation of Mat\\'ern covariance function?\n\n## Matern ##\nstruct Matern{T} <: IsotropicCovarianceStructure{T}\n λ::T\n ν::T\n σ::T\n p::T\n\n function Matern{T}(λ::T, ν::T, σ::T, p::T) where T\n λ > 0 || throw(DomainError(λ, \"correlation length λ of Mat\\u00E9rn covariance cannot be negative or zero\"))\n ν > 0 || throw(DomainError(ν, \"smoothness ν of Mat\\u00E9rn covariance cannot be negative or zero\"))\n σ > 0 || throw(DomainError(σ, \"marginal standard deviation σ of Mat\\u00E9rn covariance cannot be negative or zero\"))\n p >= 1 || throw(DomainError(p, \"in p-norm, p must be greater than or equal to 1\"))\n \tisinf(p) && throw(DomainError(p, \"in p-norm, p cannot be infinity\"))\n\n new{T}(λ, ν, σ, p)\n end\nend\n\n\"\"\"\n Matern(λ, ν, [σ = 1], [p = 2])\n\nMat\\u00E9rn covariance structure with correlation length `λ`, smoothness `ν`, (optional) marginal standard deviation `σ` and (optional) `p`-norm, defined as\n\n``C(x, y) = σ \\\\displaystyle\\\\frac{2^{1 - ν}}{Γ(ν)} \\\\left(\\\\frac{\\sqrt{2\\nu}ρ}{λ}\\\\right)^ν K_ν\\\\left(\\\\frac{\\sqrt{2\\nu}ρ}{λ}\\\\right)``\n\nwith ``ρ = ||x - y||_p``.\n\n# Examples\n```jldoctest\njulia> Matern(0.1, 1.0)\nMatérn (λ=0.1, ν=1.0, σ=1.0, p=2.0)\n\njulia> Matern(1, 1, σ=2.0)\nMatérn (λ=1.0, ν=1.0, σ=2.0, p=2.0)\n\n```\nSee also: [`Exponential`](@ref), [`Linear`](@ref), [`Spherical`](@ref), [`Whittle`](@ref), [`Gaussian`](@ref), [`SquaredExponential`](@ref)\n\"\"\"\nMatern(λ::Real, ν::Real; σ::Real=1.0, p::Real=2) =\n Matern{promote_type(typeof(λ),typeof(ν),typeof(σ),typeof(p))}(promote(λ, ν, σ, p)...)\n\n# TODO : Evaluation of the besselk function is extremely expensive; maybe try to catch special cases for ν = 1/2, 3/2 etc.\n# evaluate Matern covariance\nfunction apply(m::Matern, x::Real)\n if iszero(x)\n float(one(x))\n else\n 2^(1 - m.ν) / gamma(m.ν) * (sqrt(2 * m.ν) * x / m.λ)^m.ν * besselk(m.ν, sqrt(2 * m.ν) * x / m.λ)\n end\nend\n\n# short name\nshortname(::Matern) = \"Mat\\u00E9rn\"\n", "meta": {"hexsha": "687cb3471a5b2b8998de3cdcb82c285128f66f65", "size": 1980, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/covariance_functions/matern.jl", "max_stars_repo_name": "gerrygralton/GaussianRandomFields.jl", "max_stars_repo_head_hexsha": "be2740877a18a82e015e1a6d149036312ce79504", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/covariance_functions/matern.jl", "max_issues_repo_name": "gerrygralton/GaussianRandomFields.jl", "max_issues_repo_head_hexsha": "be2740877a18a82e015e1a6d149036312ce79504", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/covariance_functions/matern.jl", "max_forks_repo_name": "gerrygralton/GaussianRandomFields.jl", "max_forks_repo_head_hexsha": "be2740877a18a82e015e1a6d149036312ce79504", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.3571428571, "max_line_length": 156, "alphanum_fraction": 0.6090909091, "num_tokens": 740, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9334308128813471, "lm_q2_score": 0.8976952941600963, "lm_q1q2_score": 0.8379364481476187}}
{"text": "# Computing deviation in a variety of ways\n\n# squared L2 distance\nfunction sqL2dist{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T})\n n = length(a)\n length(b) == n || throw(DimensionMismatch(\"Input dimension mismatch\"))\n r = 0.0\n for i = 1:n\n @inbounds r += abs2(a[i] - b[i]) \n end\n return r\nend\n\n# L2 distance\nL2dist{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T}) = sqrt(sqL2dist(a, b))\n\n# L1 distance\nfunction L1dist{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T})\n n = length(a)\n length(b) == n || throw(DimensionMismatch(\"Input dimension mismatch\"))\n r = 0.0\n for i = 1:n\n @inbounds r += abs(a[i] - b[i]) \n end\n return r \nend\n\n# Linf distance\nfunction Linfdist{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T})\n n = length(a)\n length(b) == n || throw(DimensionMismatch(\"Input dimension mismatch\"))\n r = 0.0\n for i = 1:n\n @inbounds v = abs(a[i] - b[i]) \n if r < v\n r = v\n end\n end\n return r \nend\n\n# Generalized KL-divergence\nfunction gkldiv{T<:FloatingPoint}(a::AbstractArray{T}, b::AbstractArray{T})\n n = length(a)\n r = 0.0\n for i = 1:n\n @inbounds ai = a[i]\n @inbounds bi = b[i]\n if ai > 0\n r += (ai * log(ai / bi) - ai + bi)\n else\n r += bi\n end\n end\n return r::Float64\nend\n\n# MeanAD: mean absolute deviation\nmeanad{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T}) = L1dist(a, b) / length(a)\n\n# MaxAD: maximum absolute deviation\nmaxad{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T}) = Linfdist(a, b)\n\n# MSD: mean squared deviation\nmsd{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T}) = sqL2dist(a, b) / length(a)\n\n# RMSD: root mean squared deviation\nfunction rmsd{T<:Number}(a::AbstractArray{T}, b::AbstractArray{T}; normalize::Bool=false) \n\tv = sqrt(msd(a, b))\n\tif normalize\n\t\tamin, amax = extrema(a)\n\t\tv /= (amax - amin)\n\tend\n\treturn v\nend\n\n# PSNR: peak signal-to-noise ratio\nfunction psnr{T<:Real}(a::AbstractArray{T}, b::AbstractArray{T}, maxv::Real)\n 20. * log10(maxv) - 10. * log10(msd(a, b))\nend\n", "meta": {"hexsha": "b15c80f93d4e56e3e626763bbf6c136b53837644", "size": 2120, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/deviation.jl", "max_stars_repo_name": "wildart/StatsBase.jl", "max_stars_repo_head_hexsha": "3885542a93d2de71a1102da030c798cd51749481", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/deviation.jl", "max_issues_repo_name": "wildart/StatsBase.jl", "max_issues_repo_head_hexsha": "3885542a93d2de71a1102da030c798cd51749481", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/deviation.jl", "max_forks_repo_name": "wildart/StatsBase.jl", "max_forks_repo_head_hexsha": "3885542a93d2de71a1102da030c798cd51749481", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.1728395062, "max_line_length": 90, "alphanum_fraction": 0.6033018868, "num_tokens": 690, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582612793112, "lm_q2_score": 0.9005297921244243, "lm_q1q2_score": 0.8379053846103114}}
{"text": "\"\"\"\n Difference array\n\n# Brief:\n For any array A with N numbers, we can create difference array D with N numbers also.\n Every k-th element of D is equal to difference between A[k] and A[k-1]\n D[k] = A[k-1] - A[k] \n\n# Complexity of some operations\n - add some value x to the m consecutive elements of array - O(1)\n - print array after any numbers of changes - O(N)\n\n# Functions\n - create_diff_arr(original::Array{T}) - Create difference array for array 'original'\n - calculate_arr(diff_arr::Array{T}) - Create a original array from the given difference array\n - add_to_arr(diff_arr::Array{T}, l::Int, r::Int, x::Number) - Add x to all elements with index from [l, r]\n\n\n# Contributed by: [Nikola Mircic](https://github.com/Nikola-Mircic)\n\"\"\"\nmodule DifferenceArray\n# Create difference array for array 'original'\n# Parameters:\n# - original - an array which is used for calculating the difference array\nfunction create_diff_arr(original::Array{T}) where {T<:Number}\n n = length(original)\n\n diff_arr = Array(original)\n\n for i in 2:n\n diff_arr[i] = original[i] - original[i-1]\n end\n\n return diff_arr\nend\n\n# Create a original array from the given difference array\n# Parameters:\n# - diff_arr - an difference array which is used for calculating the original array\nfunction calculate_arr(diff_arr::Array{T}) where {T<:Number}\n n = length(diff_arr)\n\n arr = Array(diff_arr)\n\n for i in 2:n\n arr[i] = diff_arr[i] + arr[i-1]\n end\n\n return arr\nend\n\n# Add x to all elements with index from [l, r]\n# Parameters:\n# - dif_arr - a difference array of the array you want to change\n# - l - leftmost index of the affected range\n# - r - rightmost index of the affected range\n# - x - a value to be added to all elements from a given range\nfunction add_to_arr(\n diff_arr::Array{T},\n l::Int,\n r::Int,\n x::Number,\n) where {T<:Number}\n diff_arr[l] += x\n if r < length(diff_arr)\n diff_arr[r+1] -= x\n end\nend\nend\n", "meta": {"hexsha": "005ff439f0816d18ac9b5be26c722fe7fb33cbcd", "size": 2011, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/basic/difference_arr.jl", "max_stars_repo_name": "Whiteshark-314/Julia", "max_stars_repo_head_hexsha": "3285d8d6b7585cc1075831c2c210b891151da0c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-10-14T21:48:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-14T21:48:50.000Z", "max_issues_repo_path": "src/basic/difference_arr.jl", "max_issues_repo_name": "AugustoCL/Julia", "max_issues_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/basic/difference_arr.jl", "max_forks_repo_name": "AugustoCL/Julia", "max_forks_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.7285714286, "max_line_length": 110, "alphanum_fraction": 0.6663351566, "num_tokens": 541, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582593509315, "lm_q2_score": 0.9005297861178929, "lm_q1q2_score": 0.8379053772849213}}
{"text": "# ------------------------------------------------------------------------------------------\n# # An Intro to DifferentialEquations.jl\n#\n# ## Basic Introduction Via Ordinary Differential Equations\n#\n# This notebook will get you started with DifferentialEquations.jl by introducing you to the\n# functionality for solving ordinary differential equations (ODEs). The corresponding\n# documentation page is the [ODE\n# tutorial](http://docs.juliadiffeq.org/latest/tutorials/ode_example.html). While some of\n# the syntax may be different for other types of equations, the same general principles hold\n# in each case. Our goal is to give a gentle and thorough introduction that highlights these\n# principles in a way that will help you generalize what you have learned.\n#\n# ### Background\n#\n# If you are new to the study of differential equations, it can be helpful to do a quick\n# background read on [the definition of ordinary differential\n# equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation). We define an\n# ordinary differential equation as an equation which describes the way that a variable $u$\n# changes, that is\n#\n# $$ u' = f(u,p,t) $$\n#\n# where $p$ are the parameters of the model, $t$ is the time variable, and $f$ is the\n# nonlinear model of how $u$ changes. The initial value problem also includes the\n# information about the starting value:\n#\n# $$ u(t_0) = u_0 $$\n#\n# Together, if you know the starting value and you know how the value will change with time,\n# then you know what the value will be at any time point in the future. This is the\n# intuitive definition of a differential equation.\n# ------------------------------------------------------------------------------------------\n\n# ------------------------------------------------------------------------------------------\n# ### First Model: Exponential Growth\n#\n# Our first model will be the canonical exponential growth model. This model says that the\n# rate of change is proportional to the current value, and is this:\n#\n# $$ u' = au $$\n#\n# where we have a starting value $u(0)=u_0$. Let's say we put 1 dollar into Bitcoin which is\n# increasing at a rate of $98\\%$ per year. Then calling now $t=0$ and measuring time in\n# years, our model is:\n#\n# $$ u' = 0.98u $$\n#\n# and $u(0) = 1.0$. We encode this into Julia by noticing that, in this setup, we match the\n# general form when\n#\n# $$ f(u,p,t) = 0.98u $$\n#\n# with $ u_0 = 1.0 $. If we want to solve this model on a time span from `t=0.0` to `t=1.0`,\n# then we define an `ODEProblem` by specifying this function `f`, this initial condition\n# `u0`, and this time span as follows:\n# ------------------------------------------------------------------------------------------\n\nusing DifferentialEquations\nf(u,p,t) = 0.98u\nu0 = 1.0\ntspan = (0.0,1.0)\nprob = ODEProblem(f,u0,tspan)\n\n# ------------------------------------------------------------------------------------------\n# To solve our `ODEProblem` we use the command `solve`.\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob)\n\n# ------------------------------------------------------------------------------------------\n# and that's it: we have succesfully solved our first ODE!\n#\n# #### Analyzing the Solution\n#\n# Of course, the solution type is not interesting in and of itself. We want to understand\n# the solution! The documentation page which explains in detail the functions for analyzing\n# the solution is the [Solution\n# Handling](http://docs.juliadiffeq.org/latest/basics/solution.html) page. Here we will\n# describe some of the basics. You can plot the solution using the plot recipe provided by\n# [Plots.jl](http://docs.juliaplots.org/latest/):\n# ------------------------------------------------------------------------------------------\n\nusing Plots; gr()\nplot(sol)\n\n# ------------------------------------------------------------------------------------------\n# From the picture we see that the solution is an exponential curve, which matches our\n# intuition. As a plot recipe, we can annotate the result using any of the [Plots.jl\n# attributes](http://docs.juliaplots.org/latest/attributes/). For example:\n# ------------------------------------------------------------------------------------------\n\nplot(sol,linewidth=5,title=\"Solution to the linear ODE with a thick line\",\n xaxis=\"Time (t)\",yaxis=\"u(t) (in μm)\",label=\"My Thick Line!\") # legend=false\n\n# ------------------------------------------------------------------------------------------\n# Using the mutating `plot!` command we can add other pieces to our plot. For this ODE we\n# know that the true solution is $u(t) = u_0 exp(at)$, so let's add some of the true\n# solution to our plot:\n# ------------------------------------------------------------------------------------------\n\nplot!(sol.t, t->1.0*exp(0.98t),lw=3,ls=:dash,label=\"True Solution!\")\n\n# ------------------------------------------------------------------------------------------\n# In the previous command I demonstrated `sol.t`, which grabs the array of time points that\n# the solution was saved at:\n# ------------------------------------------------------------------------------------------\n\nsol.t\n\n# ------------------------------------------------------------------------------------------\n# We can get the array of solution values using `sol.u`:\n# ------------------------------------------------------------------------------------------\n\nsol.u\n\n# ------------------------------------------------------------------------------------------\n# `sol.u[i]` is the value of the solution at time `sol.t[i]`. We can compute arrays of\n# functions of the solution values using standard comprehensions, like:\n# ------------------------------------------------------------------------------------------\n\n[t+u for (u,t) in tuples(sol)]\n\n# ------------------------------------------------------------------------------------------\n# However, one interesting feature is that, by default, the solution is a continuous\n# function. If we check the print out again:\n# ------------------------------------------------------------------------------------------\n\nsol\n\n# ------------------------------------------------------------------------------------------\n# you see that it says that the solution has a 4th order interpolation, meaning that it is a\n# continuous function of 4th order accuracy. We can call the solution as a function of time\n# `sol(t)`. For example, to get the value at `t=0.45`, we can use the command:\n# ------------------------------------------------------------------------------------------\n\nsol(0.45)\n\n# ------------------------------------------------------------------------------------------\n# #### Controlling the Solver\n#\n# DifferentialEquations.jl has a common set of solver controls among its algorithms which\n# can be found [at the Common Solver\n# Options](http://docs.juliadiffeq.org/latest/basics/common_solver_opts.html) page. We will\n# detail some of the most widely used options.\n#\n# The most useful options are the tolerances `abstol` and `reltol`. These tell the internal\n# adaptive time stepping engine how precise of a solution you want. Generally, `reltol` is\n# the relative accuracy while `abstol` is the accuracy when `u` is near zero. These\n# tolerances are local tolerances and thus are not global guarantees. However, a good rule\n# of thumb is that the total solution accuracy is 1-2 digits less than the relative\n# tolerances. Thus for the defaults `abstol=1e-6` and `reltol=1e-3`, you can expect a global\n# accuracy of about 1-2 digits. If we want to get around 6 digits of accuracy, we can use\n# the commands:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,abstol=1e-8,reltol=1e-8)\n\n# ------------------------------------------------------------------------------------------\n# Now we can see no visible difference against the true solution:\n# ------------------------------------------------------------------------------------------\n\nplot(sol)\nplot!(sol.t, t->1.0*exp(0.98t),lw=3,ls=:dash,label=\"True Solution!\")\n\n# ------------------------------------------------------------------------------------------\n# Notice that by decreasing the tolerance, the number of steps the solver had to take was\n# `9` instead of the previous `5`. There is a trade off between accuracy and speed, and it\n# is up to you to determine what is the right balance for your problem.\n#\n# Another common option is to use `saveat` to make the solver save at specific time points.\n# For example, if we want the solution at an even grid of `t=0.1k` for integers `k`, we\n# would use the command:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,saveat=0.1)\n\n# ------------------------------------------------------------------------------------------\n# Notice that when `saveat` is used the continuous output variables are no longer saved and\n# thus `sol(t)`, the interpolation, is only first order. We can save at an uneven grid of\n# points by passing a collection of values to `saveat`. For example:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,saveat=[0.2,0.7,0.9])\n\n# ------------------------------------------------------------------------------------------\n# By default it always saves the first and last values, but we can turn this off as well:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,saveat=[0.2,0.7,0.9],save_start = false, save_end = false)\n\n# ------------------------------------------------------------------------------------------\n# If we need to reduce the amount of saving, we can also turn off the continuous output\n# directly via `dense=false`:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,dense=false)\n\n# ------------------------------------------------------------------------------------------\n# and to turn off all intermediate saving we can use `save_everystep=false`:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,save_everystep=false)\n\n# ------------------------------------------------------------------------------------------\n# More advanced saving behaviors, such as saving functionals of the solution, are handled\n# via the `SavingCallback` in the [Callback Library](http://docs.juliadiffeq.org/latest/feat\n# ures/callback_library.html#SavingCallback-1) which will be addressed later in the\n# tutorial.\n# ------------------------------------------------------------------------------------------\n\n# ------------------------------------------------------------------------------------------\n# #### Choosing Solver Algorithms\n#\n# There is no best algorithm for numerically solving a differential equation. When you call\n# `solve(prob)`, DifferentialEquations.jl makes a guess at a good algorithm for your\n# problem, given the properties that you ask for (the tolerances, the saving information,\n# etc.). However, in many cases you may want more direct control. A later notebook will help\n# introduce the various *algorithms* in DifferentialEquations.jl, but for now let's\n# introduce the *syntax*.\n#\n# The most crucial determining factor in choosing a numerical method is the stiffness of the\n# model. Stiffness is roughly characterized by a Jacobian `f` with large eigenvalues. That's\n# quite mathematical, and we can think of it more intuitively: if you have big numbers in\n# `f` (like parameters of order `1e5`), then it's probably stiff. Or, as the creator of the\n# MATLAB ODE Suite, Lawrence Shampine, likes to define it, if the standard algorithms are\n# slow, then it's stiff. We will go into more depth about diagnosing stiffness in a later\n# tutorial, but for now note that if you believe your model may be stiff, you can hint this\n# to the algorithm chooser via `alg_hints = [:stiff]`.\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,alg_hints=[:stiff])\n\n# ------------------------------------------------------------------------------------------\n# Stiff algorithms have to solve implicit equations and linear systems at each step so they\n# should only be used when required.\n#\n# If we want to choose an algorithm directly, you can pass the algorithm type after the\n# problem as `solve(prob,alg)`. For example, let's solve this problem using the `Tsit5()`\n# algorithm, and just for show let's change the relative tolerance to `1e-6` at the same\n# time:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob,Tsit5(),reltol=1e-6)\n\n# ------------------------------------------------------------------------------------------\n# ### Systems of ODEs: The Lorenz Equation\n#\n# Now let's move to a system of ODEs. The [Lorenz\n# equation](https://en.wikipedia.org/wiki/Lorenz_system) is the famous \"butterfly attractor\"\n# that spawned chaos theory. It is defined by the system of ODEs:\n#\n# $$ \\frac{dx}{dt} = \\sigma (y - x) $$\n# $$ \\frac{dy}{dt} = x (\\rho - z) -y $$\n# $$ \\frac{dz}{dt} = xy - \\beta z $$\n#\n# To define a system of differential equations in DifferentialEquations.jl, we define our\n# `f` as a vector function with a vector initial condition. Thus, for the vector `u =\n# [x,y,z]'`, we have the derivative function:\n# ------------------------------------------------------------------------------------------\n\nfunction lorenz!(du,u,p,t)\n σ,ρ,β = p\n du[1] = σ*(u[2]-u[1])\n du[2] = u[1]*(ρ-u[3]) - u[2]\n du[3] = u[1]*u[2] - β*u[3] \nend\n\n# ------------------------------------------------------------------------------------------\n# Notice here we used the in-place format which writes the output to the preallocated vector\n# `du`. For systems of equations the in-place format is faster. We use the initial condition\n# $u_0 = [1.0,0.0,0.0]$ as follows:\n# ------------------------------------------------------------------------------------------\n\nu0 = [1.0,0.0,0.0]\n\n# ------------------------------------------------------------------------------------------\n# Lastly, for this model we made use of the parameters `p`. We need to set this value in the\n# `ODEProblem` as well. For our model we want to solve using the parameters $\\sigma = 10$,\n# $\\rho = 28$, and $\\beta = 8/3$, and thus we build the parameter collection:\n# ------------------------------------------------------------------------------------------\n\np = (10,28,8/3) # we could also make this an array, or any other type!\n\n# ------------------------------------------------------------------------------------------\n# Now we generate the `ODEProblem` type. In this case, since we have parameters, we add the\n# parameter values to the end of the constructor call. Let's solve this on a time span of\n# `t=0` to `t=100`:\n# ------------------------------------------------------------------------------------------\n\ntspan = (0.0,100.0)\nprob = ODEProblem(lorenz!,u0,tspan,p)\n\n# ------------------------------------------------------------------------------------------\n# Now, just as before, we solve the problem:\n# ------------------------------------------------------------------------------------------\n\nsol = solve(prob)\n\n# ------------------------------------------------------------------------------------------\n# The same solution handling features apply to this case. Thus `sol.t` stores the time\n# points and `sol.u` is an array storing the solution at the corresponding time points.\n#\n# However, there are a few extra features which are good to know when dealing with systems\n# of equations. First of all, `sol` also acts like an array. `sol[i]` returns the solution\n# at the `i`th time point.\n# ------------------------------------------------------------------------------------------\n\nsol.t[10],sol[10]\n\n# ------------------------------------------------------------------------------------------\n# Additionally, the solution acts like a matrix where `sol[j,i]` is the value of the `j`th\n# variable at time `i`:\n# ------------------------------------------------------------------------------------------\n\nsol[2,10]\n\n# ------------------------------------------------------------------------------------------\n# We can get a real matrix by performing a conversion:\n# ------------------------------------------------------------------------------------------\n\nA = convert(Array,sol)\n\n# ------------------------------------------------------------------------------------------\n# This is the same as sol, i.e. `sol[i,j] = A[i,j]`, but now it's a true matrix. Plotting\n# will by default show the time series for each variable:\n# ------------------------------------------------------------------------------------------\n\nplot(sol)\n\n# ------------------------------------------------------------------------------------------\n# If we instead want to plot values against each other, we can use the `vars` command. Let's\n# plot variable `1` against variable `2` against variable `3`:\n# ------------------------------------------------------------------------------------------\n\nplot(sol,vars=(1,2,3))\n\n# ------------------------------------------------------------------------------------------\n# This is the classic Lorenz attractor plot, where the `x` axis is `u[1]`, the `y` axis is\n# `u[2]`, and the `z` axis is `u[3]`. Note that the plot recipe by default uses the\n# interpolation, but we can turn this off:\n# ------------------------------------------------------------------------------------------\n\nplot(sol,vars=(1,2,3),denseplot=false)\n\n# ------------------------------------------------------------------------------------------\n# Yikes! This shows how calculating the continuous solution has saved a lot of computational\n# effort by computing only a sparse solution and filling in the values! Note that in vars,\n# `0=time`, and thus we can plot the time series of a single component like:\n# ------------------------------------------------------------------------------------------\n\nplot(sol,vars=(0,2))\n\n# ------------------------------------------------------------------------------------------\n# ### A DSL for Parameterized Functions\n#\n# In many cases you may be defining a lot of functions with parameters. There exists the\n# domain-specific language (DSL) defined by the `@ode_def` macro for helping with this\n# common problem. For example, we can define the Lotka-Volterra equation:\n#\n# $$ \\frac{dx}{dt} = ax - bxy $$\n# $$ \\frac{dy}{dt} = -cy + dxy $$\n#\n# as follows:\n# ------------------------------------------------------------------------------------------\n\nfunction lotka_volterra!(du,u,p,t)\n du[1] = p[1]*u[1] - p[2]*u[1]*u[2]\n du[2] = -p[3]*u[2] + p[4]*u[1]*u[2]\nend\n\n# ------------------------------------------------------------------------------------------\n# However, that can be hard to follow since there's a lot of \"programming\" getting in the\n# way. Instead, you can use the `@ode_def` macro:\n# ------------------------------------------------------------------------------------------\n\nlv! = @ode_def LotkaVolterra begin\n dx = a*x - b*x*y\n dy = -c*y + d*x*y\nend a b c d\n\n# ------------------------------------------------------------------------------------------\n# We can then use the result just like an ODE function from before:\n# ------------------------------------------------------------------------------------------\n\nu0 = [1.0,1.0]\np = (1.5,1.0,3.0,1.0)\ntspan = (0.0,10.0)\nprob = ODEProblem(lv!,u0,tspan,p)\nsol = solve(prob)\nplot(sol)\n\n# ------------------------------------------------------------------------------------------\n# Not only is the DSL convenient syntax, but it does some magic behind the scenes. For\n# example, further parts of the tutorial will describe how solvers for stiff differential\n# equations have to make use of the Jacobian in calculations. Here, the DSL uses symbolic\n# differentiation to automatically derive that function:\n# ------------------------------------------------------------------------------------------\n\nlv!.Jex\n\n# ------------------------------------------------------------------------------------------\n# The DSL can derive many other functions; this ability is used to speed up the solvers. An\n# extension to DifferentialEquations.jl, [Latexify.jl](https://korsbo.github.io/Latexify.jl/\n# latest/tutorials/parameterizedfunctions.html), allows you to extract these pieces as LaTeX\n# expressions.\n# ------------------------------------------------------------------------------------------\n\n# ------------------------------------------------------------------------------------------\n# ## Internal Types\n#\n# The last basic user-interface feature to explore is the choice of types.\n# DifferentialEquations.jl respects your input types to determine the internal types that\n# are used. Thus since in the previous cases, when we used `Float64` values for the initial\n# condition, this meant that the internal values would be solved using `Float64`. We made\n# sure that time was specified via `Float64` values, meaning that time steps would utilize\n# 64-bit floats as well. But, by simply changing these types we can change what is used\n# internally.\n#\n# As a quick example, let's say we want to solve an ODE defined by a matrix. To do this, we\n# can simply use a matrix as input.\n# ------------------------------------------------------------------------------------------\n\nA = [1. 0 0 -5\n 4 -2 4 -3\n -4 0 0 1\n 5 -2 2 3]\nu0 = rand(4,2)\ntspan = (0.0,1.0)\nf(u,p,t) = A*u\nprob = ODEProblem(f,u0,tspan)\nsol = solve(prob)\n\n# ------------------------------------------------------------------------------------------\n# There is no real difference from what we did before, but now in this case `u0` is a `4x2`\n# matrix. Because of that, the solution at each time point is matrix:\n# ------------------------------------------------------------------------------------------\n\nsol[3]\n\n# ------------------------------------------------------------------------------------------\n# In DifferentialEquations.jl, you can use any type that defines `+`, `-`, `*`, `/`, and has\n# an appropriate `norm`. For example, if we want arbitrary precision floating point numbers,\n# we can change the input to be a matrix of `BigFloat`:\n# ------------------------------------------------------------------------------------------\n\nbig_u0 = big.(u0)\n\n# ------------------------------------------------------------------------------------------\n# and we can solve the `ODEProblem` with arbitrary precision numbers by using that initial\n# condition:\n# ------------------------------------------------------------------------------------------\n\nprob = ODEProblem(f,big_u0,tspan)\nsol = solve(prob)\n\nsol[1,3]\n\n# ------------------------------------------------------------------------------------------\n# To really make use of this, we would want to change `abstol` and `reltol` to be small!\n# Notice that the type for \"time\" is different than the type for the dependent variables,\n# and this can be used to optimize the algorithm via keeping multiple precisions. We can\n# convert time to be arbitrary precision as well by defining our time span with `BigFloat`\n# variables:\n# \n# ------------------------------------------------------------------------------------------\n\nprob = ODEProblem(f,big_u0,big.(tspan))\nsol = solve(prob)\n\n# ------------------------------------------------------------------------------------------\n# Let's end by showing a more complicated use of types. For small arrays, it's usually\n# faster to do operations on static arrays via the package\n# [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl). The syntax is similar\n# to that of normal arrays, but for these special arrays we utilize the `@SMatrix` macro to\n# indicate we want to create a static array.\n# ------------------------------------------------------------------------------------------\n\nusing StaticArrays\nA = @SMatrix [ 1.0 0.0 0.0 -5.0\n 4.0 -2.0 4.0 -3.0\n -4.0 0.0 0.0 1.0\n 5.0 -2.0 2.0 3.0]\nu0 = @SMatrix rand(4,2)\ntspan = (0.0,1.0)\nf(u,p,t) = A*u\nprob = ODEProblem(f,u0,tspan)\nsol = solve(prob)\n\nsol[3]\n\n# ------------------------------------------------------------------------------------------\n# ## Conclusion\n#\n# These are the basic controls in DifferentialEquations.jl. All equations are defined via a\n# problem type, and the `solve` command is used with an algorithm choice (or the default) to\n# get a solution. Every solution acts the same, like an array `sol[i]` with `sol.t[i]`, and\n# also like a continuous function `sol(t)` with a nice plot command `plot(sol)`. The Common\n# Solver Options can be used to control the solver for any equation type. Lastly, the types\n# used in the numerical solving are determined by the input types, and this can be used to\n# solve with arbitrary precision and add additional optimizations (this can be used to solve\n# via GPUs for example!). While this was shown on ODEs, these techniques generalize to other\n# types of equations as well.\n# ------------------------------------------------------------------------------------------\n", "meta": {"hexsha": "3623c073acdd1e52d683a579fa4322590072b9ba", "size": 25247, "ext": "jl", "lang": "Julia", "max_stars_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/intro-to-solving-diffeq-in-julia/1.ODEIntroduction.jl", "max_stars_repo_name": "grenkoca/JuliaTutorials", "max_stars_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 535, "max_stars_repo_stars_event_min_datetime": "2020-07-15T14:56:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T12:50:32.000Z", "max_issues_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/intro-to-solving-diffeq-in-julia/1.ODEIntroduction.jl", "max_issues_repo_name": "grenkoca/JuliaTutorials", "max_issues_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 42, "max_issues_repo_issues_event_min_datetime": "2018-02-25T22:53:47.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-14T02:15:50.000Z", "max_forks_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/intro-to-solving-diffeq-in-julia/1.ODEIntroduction.jl", "max_forks_repo_name": "grenkoca/JuliaTutorials", "max_forks_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 394, "max_forks_repo_forks_event_min_datetime": "2020-07-14T23:22:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T20:12:57.000Z", "avg_line_length": 49.9940594059, "max_line_length": 92, "alphanum_fraction": 0.4825127738, "num_tokens": 4990, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "import Pkg; Pkg.add(Pkg.PackageSpec(url=\"https://github.com/JuliaComputing/JuliaAcademyData.jl\"))\nusing JuliaAcademyData; activate(\"Foundations of machine learning\")\n\n# # Minimizing functions - how a computer learns\n\n#-\n\n# In the previous notebooks, we've seen that by changing **parameters** in a function, we could find a \"best fit\" of that function to some data. We use a **loss function** to quantify the \"goodness\" of a set of parameters and we look for those parameters that **optimize**, in fact **minimize**, the loss function. If we teach a machine how to minimize the loss function on its own, we say that machine is able to **learn** how to model our data.\n#\n# In the last notebook, we threw around terms like **derivative**, **gradient**, and **gradient descent** to give you a rough sense of how we minimize a function on a computer. In this notebook, we will step through these concepts more carefully, with the aim of being able to implement them using Julia.\n\n#-\n\n# ## Minimizing a 1D function using calculus\n\n#-\n\n# Let's draw the 1D loss function $L_1$ from a previous notebook again:\n\nσ(x) = 1 / (1 + exp(-x))\nf(x, w) = σ(w * x)\n\nx1 = 2\ny1 = 0.8\n\nL1(w) = (y1 - f(x1, w))^2\n\n#-\n\nusing Plots; gr()\n\n#-\n\nplot(L1, -2, 1.5, xlabel=\"w\", ylabel=\"L1(w)\", leg=false)\n\n# By eye, we can see that the minimum is around $w=0.6$. But how can we get the computer to work this out on its own?\n\n#-\n\n# In the previous notebook, we thought of this function plot as a hill, viewed from the side. We could find the minimum by making the hill sticky, and letting a ball roll down it. The ball will find and settle in the minimum of the function. Now let's see how to teach a computer to do this.\n#\n# We need to find the downhill direction along the hill, which is related to its *slope* (how steep it is). Calculus provides us with tools to calculate that slope!\n\n#-\n\n# Namely, the slope of a curve $L_1(w)$ at $w$ is given by its **derivative** $L_1'(w)$; geometrically, this is the slope of the **tangent line** to the curve at that point, i.e. the straight line which touches the curve at that point.\n\n#-\n\n# Calculus provides us with some rules to calculate an analytical formula for the derivative, and we will see later how to apply these rules, albeit indirectly, for machine learning.\n# To gain understanding, however, we will see here how to get the computer to help us out by calculating the derivatives numerically instead!\n\n#-\n\n# ## Approximating derivatives\n\n#-\n\n# Let's recall that the derivative $L_1'(w)$ of a function is defined as\n#\n# $$L_1'(w) \\simeq \\frac{L_1(w+h) - L_1(w)}{h},$$\n#\n# for a small step size $h$. (Strictly speaking, we must take the limit when $h$ tends to $0$ to obtain the exact value of the derivative.)\n\n#-\n\n# #### Exercise 1\n#\n# Write a function to calculate the derivative of a function at a given point. Note that in Julia, we can easily pass functions as arguments to other functions!\n# The function should take the function, the point $w$ at which to calculate the derivative, and the value of $h$, which should have a default value of 0.001.\n#\n# *Note*: A function input argument can have a *default value* if we set the input argument equal to that default value when we define the function. For example,\n#\n# ```julia\n# f(x, a = 3) = a * x^2\n# ```\n#\n# The function `f` will square the input we give it and multiply by `a`. However, if we choose to call `f(x)` *without* passing it an `a` input, it will assume `a` is `3` and return `3*x^2`.\n\n#-\n\n# #### Solution\n#\n# We could define the `derivative` function as\n\nfunction derivative(f, w, h=0.001)\n return ( f(w + h) - f(w) ) / h\nend\n## Or the short form with:\n## derivative(f, w, h=0.001) = ( f(w + h) - f(w) ) / h\n\n# **Tests**\n\n@assert derivative(x->x^2, 0.2) ≈ 0.401 # check default h works\n@assert derivative(x->x^2, 0.2, 0.5) ≈ .9 # check we can specify our own h\n@assert derivative(x->x^3, 0.2) ≈ 0.120601\n@assert derivative(x->x^3, -0.2) ≈ 0.119401\n\n# #### Exercise 2\n#\n# Write an interactive visualization of the tangent line to the graph of $L_1$, so that we can visualize the tangent at any point on $L_1$. Include the current value of the derivative in the title.\n#\n# *Hint*: Recall that the straight line through the point $(x_0, y_0)$ with slope $m$ is given by\n#\n# $$ \\frac{y - y_0}{x - x_0} = m,$$\n#\n# so\n#\n# $$y = y_0 + m*(x - x_0).$$\n\n#-\n\n# #### Solution\n\nwmin = -2\nwmax = 1\n\nw = 0.5 # Try manipulating w between wmin and wmax (-2 to 1)\nplot(L1, wmin, wmax, label=\"L_1\")\n\nx0 = w\ny0 = L1(w)\n\nscatter!([x0], [y0], label=\"\")\n\nm = derivative(L1, w)\n\nplot!(x->y0 + m*(x - x0), ls=:dash, label=\"tangent line\")\n\nylims!(-0.1, 0.6)\nxlims!(wmin, wmax)\n\ntitle!(\"Derivative = $m\")\nxlabel!(\"w\")\nylabel!(\"L1(w)\")\n\n# #### Exercise 3\n#\n# What is the value of the derivative (slope of the tangent line) at a minimum? Can this happen anywhere else?\n\n#-\n\n# #### Solution\n\n#-\n\n# The derivative is `0` at a minimum. This happens at any minimum, maximum or \"point of inflection\" of a smooth function.\n\n#-\n\n# ## Minimization by gradient descent\n\n#-\n\n# The tangent line at a point is a good approximation of the function near that point. Therefore the derivative tells us in which direction, and how fast, a function grows or shrinks when we move a small distance away from that point.\n#\n# As we saw in the previous notebook, we can think of the function as being a hill, and having a ball that we want to move down the hill. Gravity will automatically pull the ball in the direction *down* the hill; we want to emulate this in the computer!\n\n#-\n\n# #### Exercise 4\n#\n# When the derivative $L_1'(w)$ is positive, that means that $L_1$ increases from left to right at point $w$.\n#\n# If the derivative $L_1'(w)$ is positive (> 0), in which direction should we move $w$ to *decrease* $L_1$?\n\n#-\n\n# #### Solution\n#\n# We should move $w$ left, i.e. decrease it.\n\n#-\n\n# #### Exercise 5\n#\n# If the derivative $L_1'(w)$ is negative (< 0), in which direction should we move $w$ to *decrease* $L_1$?\n\n#-\n\n# We can use this information to tell the computer which way to take a step. This constitutes the numerical algorithm called **gradient descent**; it is called this since we are descending (moving downwards) along the graph of the function by using information about its gradient (slope).\n\n#-\n\n# #### Exercise 6\n#\n# Implement gradient descent by following this prescription for an **iterative (repetitive) algorithm**:\n#\n# 1. Start at an initial guess $w_0$ for $w$.\n#\n# 2. At each step, calculate the derivative, $L_1'(w_n)$ at the current value of $w_n$ using the function that you created above.\n#\n# 3. Modify the value of $w$ by a small multiple (for example, $\\eta=0.01$) of the value of the derivative you just created, via $w_{n+1} = w_n - \\eta L_1'(w_n)$.\n#\n# For this problem, start with $w_0 = -2.0$. Repeat steps 2 and 3 a total of `2000` times.\n#\n# Package this code into a function called `gradient_descent` that takes two inputs, a function and a range for values of $w$, and returns the final value of $w$ and $L_1(w)$.\n#\n# Using `L1` and `-2:0.01:1.5` as your inputs to `gradient_descent`, for what value of $w$ is $L_1$ at a minimum?\n\n#-\n\n# #### Solution\n\nfunction gradient_descent(f, wrange)\n derivative(f, w, h=0.001) = ( f(w + h) - f(w) ) / h\n w = -2.0\n for i in 1:2000\n fderiv = derivative(f, w)\n step = .01 * fderiv\n w = w - step\n end\n return w, f(w)\nend\n\ngradient_descent(L1, -2:0.01:1.5)\n\n# **Tests**\n\n@assert 0.6383 < gradient_descent(L1, -2:0.01:1.5)[1] < 0.6385\n@assert 0.00032 < gradient_descent(L1, -2:0.01:1.5)[2] < 0.00033\n\n# #### Exercise 7\n#\n# Modify your code for gradient descent to return the result once it has found an answer within some *tolerance*, rather than taking a set number of steps. The new prescription for this algorithm is:\n#\n# 1. Start at an initial guess $w_0$ for $w$.\n#\n# 2. At each step, calculate the derivative, $L_1'(w_n)$ at the current value of $w_n$, using the function that you created above.\n#\n# 3. Modify the value of $w$ by a small multiple (for example, $\\eta=0.01$) of the value of the derivative you just created, via $w_{n+1} = w_n - \\eta L_1'(w_n)$.\n#\n# 4. Check how different $w_{n+1}$ is from $w_n$. If you're satisfied that $L_1(w_{n+1})$ is minimized, return $w_{n+1}$ and $L_1(w_{n+1})$. Otherwise, go to step (2) and continue.\n#\n# Edit `gradient_descent` so that it takes three inputs: a function, a range for values of $w$, and a tolerance that tells you how close $w_{n+1}$ must be to $w_n$ before you can stop iterating.\n#\n# Using `L1`, `-2:0.01:1.5`, and `.000001` as your inputs to `gradient_descent`, for what value of $w$ is $L_1$ at a minimum?\n\n#-\n\n# #### Solution\n#\n#\n# Your code should look something like this\n#\n# $L_1$ is at a minimum when `gradient_descent(L1, -2:0.01:1.5, .000001)[1]`, or $w$, is around `0.69216`.\n\nfunction gradient_descent(f, wrange, tol)\n w = -2.0\n while true\n fderiv = derivative(f, w)\n step = .01 * fderiv\n w = w - step\n if abs(step) < tol\n return w, f(w)\n end\n end\nend\n\ngradient_descent(L1, -2:0.01:1.5, .000001)\n\n# **Tests**\n\n@assert 0.69215 < gradient_descent(L1, -2:0.01:1.5, .000001)[1] < 0.69217\n\n# After you've completed this step, you will have found a good approximation to the value $w^*$ where $L_1$ reaches its minimum, and the minimum value $L_1(w^*)$, which in this case is (almost) $0$.\n\n#-\n\n# #### Exercise 8\n#\n# Alter the function `gradient_descent` so that it stores the results `(w, L1(w))` at each step of the algorithm as an array and returns this array. How many steps does the algorithm take for input parameters `(L1, -2:0.01:1.5, .000001)` before terminating? You should count your starting $w_0$ as your first step.\n\n#-\n\n# #### Solution\n\n#-\n\n# Your altered code should look like this, getting the number of steps with `length`:\n\nfunction gradient_descent(f, wrange, tol)\n w = -2.0\n store_w = [(w, f(w))]\n while true\n fderiv = derivative(f, w)\n step = .01 * fderiv\n w = w - step\n push!(store_w, (w, f(w)))\n if abs(step) < tol\n return store_w\n end\n end\nend\n\nlength(gradient_descent(L1, -2:0.01:1.5, .000001))\n\n# **Tests**\n\n@assert length(gradient_descent(L1, -2:0.01:1.5, .000001)) == 4255\n\n# #### Exercise 9\n#\n# Overlay the steps taken in the last exercise with a plot of $L_1(w)$ vs. $w$.\n#\n# Where does our algorithm take the largest steps? (Where does the ball move fastest down the hill?)\n#\n# A) Between w = -2:-1\n# B) Between w = -1:0
\n# C) Between w = 0:.6\n#\n# *Hint*: It may be easier to see what's going on if you only plot, for example, every 15th step.\n\n#-\n\n# #### Solution\n#\n# B) The following code generates the desired figure:\n\nwsteps = gradient_descent(L1, -2:0.01:1.5, .000001)\nfig1 = plot(L1, -2, 1.5, xlabel=\"w\", ylabel=\"C1(w)\", leg=false)\nfor step in wsteps[1:15:end]\n scatter!([step[1]], [step[2]])\nend\nfig1\n\n# ## Functions of 2 variables and their derivatives\n\n#-\n\n# So far, we have been looking at the minimizing a loss function $L_1(w)$ that depends on a single parameter, $w$. Now let's turn to the cost function $L_2(w, b)$ from a previous notebook, that is a function of *two* parameters, $w$ and $b$, and try to minimize it.\n#\n# As we've seen, we get a **surface**, instead of a curve, when we graph $L_2$ as a function of both of its parameters.\n\n#-\n\n# **Exercise 10**\n#\n# Draw a surface plot of $L_2$, given by\n#\n# $$L_{2}(w, b) = \\sum_i(y_i - g(x_i, w, b))^2$$\n#\n# using the `surface` function from `Plots.jl`. For this plot, use the values of `xs` and `ys` from notebook 5:\n#\n# We can get a nice interactive 3D plot by using the Plotly backend of `Plots.jl` by executing the command\n#\n# plotly()\n\nxs = [2, -3, -1, 1]\nys = [0.8, 0.3, 0.4, 0.4]\n\n# #### Solution\n\n#-\n\n# The following code will generate the desired graphic:\n\nplotly()\nσ(x) = 1 / (1 + exp(-x))\ng(x, w, b) = σ(w*x) + b\n\nxs = [2, -3, -1, 1]\nys = [0.8, 0.3, 0.4, 0.4]\n\nL2(w, b) = sum(abs2, ys .- g.(xs, w, b))\n\nsurface(-2:0.02:2, -2:0.02:2, L2, alpha=0.6, xlabel = \"w\", ylabel = \"b\", zlabel = \"L2(w, b)\")\n\n# ### Finding the minimum\n\n#-\n\n# We can just about see, by rotating the graph, that $L_2$ has a single minimum. We want to find the values of $w$ and $b$ where this minimum value is reached.\n\n#-\n\n# Following what we did for the function $L_1$ above, we expect that we will need to calculate derivatives of $L_2$. Since the function is more complicated, though, the derivatives are too!\n#\n# It turns out that the right concept is that of the [**gradient**](https://en.wikipedia.org/wiki/Gradient) of $L_2$, denoted $\\nabla L_2(w, b)$. This is a **vector** consisting of $2$ numbers if there are $2$ parameters [or, in general, $n$ numbers if there are $n$ parameters].\n#\n# The numbers that form the gradient $\\nabla L_2(w, b)$ are called the **partial derivatives** of $L_2$ with respect to $w$ and $b$, written as\n#\n# $$\\frac{\\partial L_2}{\\partial w} \\quad \\text{and} \\quad \\frac{\\partial L_2}{\\partial b}.$$\n#\n# Although this notation might look complicated, all it means is that we calculate derivatives just like we did before, except that we fix the value of the other variable.\n# For example, to calculate $\\frac{\\partial L_2}{\\partial w}$, the partial derivative of $L_2$ with respect to $w$, we fix the value of $b$ and think of the resulting function as a function of the single variable $w$; then we use the formula for derivatives of functions of a single variable.\n#\n# [Note that $\\frac{\\partial L_2}{\\partial w}$ is itself a function of $w$ and $b$; we could write $\\frac{\\partial L_2}{\\partial w}(w, b)$.]\n\n#-\n\n# #### Exercise 11\n#\n# Write functions that will allow you to calculate the partial derivatives of a function $f$ of two variables.\n#\n# In particular, declare functions called `partial_w` and `partial_b`. Each should take four inputs - a function $f$ of two variables, the first input argument to $f$, the second input argument to $f$, and a step size `h` with default value `0.001`. `partial_w` should return the partial derivative of $f$ with respect to its first input argument and `partial_b` should return the partial derivative of $f$ with respect to its second input argument.\n\n#-\n\n# #### Solution\n#\n# The partial derivatives are\n\npartial_w(f, w, b, h = 0.001) = ( f(w + h, b) - f(w, b) ) / h\npartial_b(f, w, b, h = 0.001) = ( f(w, b + h) - f(w, b) ) / h\n\n# **Tests**\n\n## Test that default parameters work:\n@assert partial_w((x, y) -> 3 * x * y, 0.3, 0.3) ≈ 0.9\n@assert partial_b((x, y) -> 3 * x * y, 0.3, 0.3) ≈ 0.9\n## Test that you can specify your own step sizes:\n@assert partial_w((x, y) -> 3 * x^2 * y^3, 2.3, 10.2, 1) ≈ 17828.2944\n@assert partial_b((x, y) -> 3 * x^2 * y^3, 2.3, 10.2, 1) ≈ 5454.8364\n## Test that x and y are treated separately:\n@assert partial_w((x, y) -> 3 * x * y^2, 0.3, 0.2) ≈ 0.12\n@assert partial_b((x, y) -> 3 * x * y^2, 0.3, 0.2) ≈ 0.3609\n@assert partial_w((x, y) -> 3 * x^2 * y, 0.3, 0.2, 0.1) ≈ 0.42\n@assert partial_b((x, y) -> 3 * x^2 * y, 0.3, 0.2, 0.1) ≈ 0.27\n\n# #### Exercise 12\n#\n# Use `partial_b` from the last exercise to find the partial derivative of $L_2$ with respect to $w$ at b = 0.3, $\\frac{\\partial L_2}{\\partial w}|_{b = 0.3}$ for `w = -2:0.01:1`\n\n#-\n\n# #### Solution\n\nwrange = -2:0.01:1\npartialw = partial_w.(L2, wrange, 0.3 )\n\n# **Tests**\n\n@assert isapprox(partialw[3], -0.212573, rtol=2e-6)\n\n# #### Exercise 13\n#\n# Plot the cross section of the surface of $L_2(w, b)$ at $b = 0.3$. Make this plot interactive to show that the function `partial_w` gives the slope of the tangent to this cross section for any point `w` in the range `-2:0.01:1`.\n#\n# For what value of $w$ in this range is the slope of the cross section closest to -1?\n\n#-\n\n# #### Solution\n\n#-\n\n# The slope is closest to -1.0 at w = -0.55. You can see this by running the following code:\n\nL2_fixedb = L2.(wrange, 0.3)\n\nw = 0.0 # Try manipulating w to pick a value within `wrange`\nplot(wrange, L2_fixedb)\nscatter!([w], [L2(w, 0.3)])\nslope = partial_w(L2, w, 0.3)\ntangent_points = slope .* wrange .- slope * w .+ L2.(w, 0.3)\nplot!(wrange, tangent_points)\nylims!(0.3, 2.0)\ntitle!(\"Slope = $slope @ $(w)\")\n\n# ## ***Optional**: Functions with $n$ inputs\n\n#-\n\n# If a function $f$ takes $n$ input arguments, we can write them as $p_1, \\ldots, p_n$, where $p_i$ means the \"$i$th parameter\". In Julia, we can wrap them up into a single **vector**. Now we can calculate the partial derivative $\\frac{\\partial L_2}{\\partial p_i}$ with respect to the $i$th variable.\n\n#-\n\n# #### Exercise 14\n#\n# For the next exercise, you will need to use the splat command, `...`. You can use this command to \"open up\" a collection and pass all the elements of that collection as inputs to a function.\n#\n# For example, say you have an array, `numbers`,\n#\n# ```julia\n# numbers = [4, 3, 2]\n# ```\n#\n# and you want to use `numbers` to create a $4\\times3\\times3$ randomly populated array via `rand`. `rand(numbers)` will not do what you want. You could index into `numbers` to grab the values you want and pass them to `rand`, as in\n#\n# ```julia\n# rand(numbers[1], numbers[2], numbers[3])\n# ```\n#\n# or you could use a splat:\n#\n# ```julia\n# rand(numbers...)\n# ```\n#\n# Use `...` to pass the contents of `inputs`\n#\n# ```julia\n# inputs = [30, 12, \"cats\"]\n# ```\n#\n# to the function `dreams`\n#\n# ```julia\n# dreams(i, j, perfect_mammal) = \"I wish I had $(i + j) $perfect_mammal.\"\n# ```\n\n#-\n\n# #### Solution\n\ninputs = [30, 12, \"cats\"]\ndreams(i, j, perfect_mammal) = \"I wish I had $(i + j) $perfect_mammal.\"\ndreams(inputs...)\n\n# **Tests**\n\n@assert dreams(inputs...) == \"I wish I had 42 cats.\"\n\n# #### Exercise 15:\n#\n# Write a function, `partial`, to calculate the $i$th partial derivative of a function. This function should have four inputs\n#\n# * a function, $f$, for which you want to compute the partial derivative\n# * an array, *p*, specifying the values of all input arguments to $f$ at the point where you want $\\frac{\\partial f}{\\partial p_i}$ computed\n# * the index, $i$, of the variable with respect to which you want to calculate the partial derivative of $f$\n# * a step size with default value 0.001\n#\n# Hint: you will need to `copy` and modify `p` within `partial`.\n\n#-\n\n# #### Solution\n\nfunction partial(f, p, i, h = 0.001)\n p_next = copy(p)\n p_next[i] += h\n return ( f(p_next...) - f(p...) ) / h\nend\n\n# **Tests**\n\n@assert partial((x, y, z) -> x^2 * y * z, [1.0, 2.0, 1.0], 1, 0.0001) ≈ 4.0002\n@assert partial((x, y, z) -> x^2 * y * z, [1.0, 2.0, 1.0], 1, 1) ≈ 6\n\n@assert partial((w, x, y, z) -> w * x^2 * y^3 * z^4, [1.1, 2.2, 3.3, 4.4], 1) ≈ 65192.53776074038\n@assert partial((w, x, y, z) -> w * x^2 * y^3 * z^4, [1.1, 2.2, 3.3, 4.4], 2) ≈ 65207.35424659506\n@assert partial((w, x, y, z) -> w * x^2 * y^3 * z^4, [1.1, 2.2, 3.3, 4.4], 3) ≈ 65212.29507069802\n@assert partial((w, x, y, z) -> w * x^2 * y^3 * z^4, [1.1, 2.2, 3.3, 4.4], 4) ≈ 65214.7658571339\n\n# ## Gradient descent in 2 dimensions\n\n#-\n\n# It turns out that the gradient vector of the function $L_2(w, b)$ gives the direction in the plane $(w, b)$ in which the function $L_2$ **increases fastest**.\n#\n# In other words, if you start at the position $(w_0, b_0)$ and take a small step of length $\\eta$ to new values $(w', b')$, the value of the function will change to a new value $L_2(w', b')$. How do we get to the minimum of $L_2(w, b)$ as fast as possible? We want to step in the direction where $L_2$ decreases fastest!\n#\n# In multivariable calculus courses, it is shown that $L_2$ will *increase* fastest if you take a step **in the direction of the gradient $\\nabla L_2(w, b)$**! To decrease $L_2$ the fastest, we should take a step in the *opposite* direction, $-\\nabla L_2(w, b)$.\n#\n# Let's now generalize the gradient descent algorithm that we wrote previously to work our 2-dimensional function.\n\n#-\n\n# #### Exercise 16\n#\n# Extend your 1D implementation of `gradient_descent` so that it will minimize the function $L_2$.\n#\n# Requirements:\n#\n# * Your new method for `gradient_descent` will take four input arguments: the function $f$ for which you seek the minimum, the range of values for $f$'s first input argument that you will consider, the range of values for $f$'s second input argument that you will consider, and a tolerance that will specify the maximum allowable step size, $\\sum_i \\eta \\frac{\\partial f}{\\partial p_i}$\n#\n# * Use $\\eta = .01$. For example, for a function $f(w, b)$, update $w$ such that $w_{n+1} = w_n - 0.01 * \\frac{\\partial f}{\\partial w_n}$\n#\n# * Seed `gradient_descent` with the starting coordinates [-2.0, -2.0], i.e. $w_0 = -2.0$ and $b_0 = -2.0$.\n#\n# * Return all steps (their coordinates) taken during gradient descent and the values of the loss function at these coordinates.\n#\n# Once you have done this, execute\n#\n# ```julia\n# gradient_descent(L2, -2:0.02:2, -2:0.02:2, .001)\n# ```\n#\n# How many steps were taken by gradient descent?\n#\n# Hint: Do not count your starting coordinates `[-2.0, -2.0]` as a step.\n\n#-\n\n# #### Solution\n#\n#\n# The new code for `gradient_descent` should look something like\n\nfunction gradient_descent(f, wrange, brange, tol)\n w, b = -2.0, -2\n store_steps = [[w, b]]\n store_C2_vals = [f(w, b)]\n while true\n # Calculate partial derivatives\n fpartial1 = partial(f, [w, b], 1)\n fpartial2 = partial(f, [w, b], 2)\n # Update parameters\n w = w - .01 * fpartial1\n b = b - .01 * fpartial2\n # Store parameters and resulting function\n push!(store_steps, [w, b])\n push!(store_C2_vals, f(w, b))\n\n if abs(w + b) < tol\n return store_steps, store_C2_vals\n end\n end\nend\n\n# **Tests**\n\nsteps, values = gradient_descent(L2, -2:0.02:2, -2:0.02:2, .001 )\n@assert length(values) == 396\n\n# #### Exercise 17\n#\n# Use the `surface` and `scatter!` commands to illustrate the path taken by `gradient_descent` from [-2.0, -2.0] to wherever the algorithm terminates.\n#\n# Where do the scattered points representing the steps of gradient descent appear the most dense?\n#\n# A) Near the starting point at [-2.0, -2.0]
\n# B) Near the point [-1.8, 0] where $C_2$ appears nearly flat
\n# C) Near the minimum of $C_2$\n\n#-\n\n# #### Solution\n#\n# B) The following code will plot the function `C2` with an overlay of the steps taken in gradient descent\n#\n# You can change the frequency with which steps are plotted by changing the iterator in the `for` loop from `1:length(values)` to `1:n:length(values)` for varying values of `n`. This will allow you to see more clearly where steps are densely or sparsely packed.\n\nfig = surface(-2:0.02:2, -2:0.02:2, L2, alpha=0.6, xlabel = \"w\", ylabel = \"b\", zlabel = \"C2(w, b)\")\nfor i in 1:length(values)\n scatter!([steps[i][1]], [steps[i][2]], [values[i]])\nend\nfig\n\n# ## What we have learnt\n\n#-\n\n# To recap, in this notebook, we have seen that the computer can calculate approximate derivatives, and that we can use those inside a relatively simple algorithm, gradient descent, to minimize functions of 1 and 2 variables!\n#\n# Now that we've used gradient descent to **learn** the best values of `b` and `w` for a cost function, we can imagine how we could do our classification of apples vs. bananas by allowing the machine to choose our parameters! To do this, we need to define a loss function for apple vs. banana classification, using images that have already been categorised as apples or bananas.\n\n", "meta": {"hexsha": "3c19d7c4a370ed67c9e10cd6dac7af29be846af1", "size": 23353, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Courses/Foundations of machine learning/0900.What-is-learning-quiz.jl", "max_stars_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_stars_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 45, "max_stars_repo_stars_event_min_datetime": "2020-02-13T00:50:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-05T07:57:22.000Z", "max_issues_repo_path": "Courses/Foundations of machine learning/0900.What-is-learning-quiz.jl", "max_issues_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_issues_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 52, "max_issues_repo_issues_event_min_datetime": "2019-10-30T16:22:28.000Z", "max_issues_repo_issues_event_max_datetime": "2020-01-26T20:02:43.000Z", "max_forks_repo_path": "Courses/Foundations of machine learning/0900.What-is-learning-quiz.jl", "max_forks_repo_name": "fercarozzi/JuliaAcademyMaterials", "max_forks_repo_head_hexsha": "4c7501d42e698379050fd6e6d469f3f84428cdcd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 27, "max_forks_repo_forks_event_min_datetime": "2020-02-26T11:33:28.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-25T22:34:53.000Z", "avg_line_length": 35.9830508475, "max_line_length": 449, "alphanum_fraction": 0.6626129405, "num_tokens": 7444, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896824119663, "lm_q2_score": 0.9099070115349837, "lm_q1q2_score": 0.837832988175719}} {"text": "# # Investment Science - Module 1\n#\n# \\tableofcontents\n#\n# ## Problem 4\n# The IRR is generally calculated using an iterative procedure.\n# Suppose that we define $f(\\lambda) = -a_{1} + a_{2}\\lambda^{2} + \\ldots + a_{n}\\lambda^{n}$,\n# where all $a_{i}$'s are positive and $n > 1$.\n# Here is an iterative technique that generates a sequence\n# $\\lambda_{0}, \\lambda_{1}, \\lambda_{2}, \\ldots \\lambda_{k}, \\ldots$\n# of estimates that converges to the root $\\overline{\\lambda} > 0$, solving $f(\\overline{\\lambda}) = 0$.\n# Start with any $\\lambda_{0} > 0$ close to the solution.\n# Assuming $\\lambda_{k}$ has been calculated, evaluate\n#\n# $$\n# f'(\\lambda_{k}) = a_{1} + 2a_{2}\\lambda_{k} + 3a_{3}\\lambda_{k}^{2} + \\ldots + na_{n}\\lambda_{k}^{n-1}\n# $$\n#\n# and define\n#\n# $$\n# \\lambda_{k+1} = \\lambda_{k} - \\frac{f(\\lambda_{k})}{f'(\\lambda_{k})}\n# $$\n#\n# This is Newton's method.\n# It is based on approximating the function $f$ by a line tangent to its graph at $\\lambda_{k}$.\n# Try the procedure on $f(\\lambda) = -1 + \\lambda + \\lambda^{2}$.\n# Start with $\\lambda_{0} = 1$ and compute four additional estimates.\n#\n# ### Solution\n#\n# We know\n#\n# $$\n# \\begin{aligned}\n# \\lambda_{0} &= 1 \\\\\n# \\lambda_{k+1} &= \\lambda_{k} - \\frac{f(\\lambda_{k})}{f'(\\lambda_{k})} \\\\\n# f(\\lambda) &= -1 + \\lambda + \\lambda^{2} \\\\\n# f'(\\lambda) &= 1 + 2\\lambda \\\\\n# \\end{aligned}\n# $$\n#\n# Using all these equations we can compute four additional estimates.\n\n## define function f\nf(λ) = -1 + λ + λ^2\n## define derivative of f\nḟ(λ) = 1 + 2*λ\n## define function to calculate the next λ\nnext(λ) = λ - f(λ)/ḟ(λ)\n## create a list of λs starting with 1\nλs = [1.0]\n## calculate next four\nfor n ∈ 1:4\n push!(λs, next(λs[n]))\nend\n## display results\n@show λs\n\n# ## Problem 5\n#\n# Suppose that you have the opportunity to plant trees that alter can be sold for lumber. This project requires an initial outlay of money in order to purchase and plant the seedlings. No other cash flow occurs until trees are harvested. However, you have a choice as to when to harvest. If you harvest after 1 year, you get your return quickly; but if you wait, the trees will have additional growth and the revenue generated from the sale of the trees will be greater. Assume that the case flow streams associated with these alternatives are:\n#\n# - Wait $1$ year: $(-1, 2)$\n# - Wait $2$ year: $(-1, ,0, 3)$\n# - Wait $3$ year: $(-1, 0, 0, 4)$\n# - $\\vdots$\n# - Wait $n$ year: $(-1, 0, 0, \\ldots, n+1)$\n#\n# The prevailing interest rate is 10%. When is the best time to cut the trees?\n#\n# ### Solution\n#\n# Let’s first define a function for calculating the Net Present Value, say on a cash flow stream that looks like $(x_{0}, \\ldots, x_{n})$ where $r$ is the prevailing interest rate and $k$ is the number of years waited to harvest:\n#\n# $$\n# NPV = -x_{0} + \\frac{x_{k}}{(1+r)^{k}}\n# $$\n#\n# We can create a function to calculate the **_NPV_** given a _cash flow stream_ and _interest rate_. Let's create a table of the first several values to determine the optimal time to harvest.\n\n## define function to calculate net present value\nfunction net_present_value(investment, revenue, year, rate)\n -investment + revenue/(1+rate)^year\nend\n## define empty array for values\nvalues = []\n## populate some values\nfor i ∈ 1:20\n append!(values, net_present_value(1, i+1, i, 0.10))\nend\n## display some values\n@show values\n\n# To make reading easier, we can create a table or `dataframe`.\n\nusing DataFrames\ndf = DataFrame(Dict(:year => 1:20, :revenue => values))\nshow(df)\n\n# A quick glance reveals the maximum revenue.\n\nmaximum(values)\n#+\nusing PyPlot;\nfigure(figsize=(8, 6));\nplot(df.year, df.revenue);\n\nxlabel(\"Year\", fontsize=14);\nylabel(\"Revenue\", fontsize=14);\n\nplt.savefig(joinpath(@OUTPUT, \"invsci-01-time-to-harvest.svg\")); # hide\n\n# \\figalt{Time to Harvest}{invsci-01-time-to-harvest.svg}\n#\n# We can see that the graph levels off at the top, and upon inspection of the table we confirm that **years 9 and 10** will yield optimal revenue.\n", "meta": {"hexsha": "cf68a568156495ef3f7997535fcdec83b6c4ae12", "size": 3976, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "_literate/invsci-01.jl", "max_stars_repo_name": "jvaverka/lvaverka.github.io", "max_stars_repo_head_hexsha": "1cb38637671e50439fb873b766e023aa450f3a42", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_literate/invsci-01.jl", "max_issues_repo_name": "jvaverka/lvaverka.github.io", "max_issues_repo_head_hexsha": "1cb38637671e50439fb873b766e023aa450f3a42", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2022-02-09T17:10:24.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-11T17:26:04.000Z", "max_forks_repo_path": "_literate/invsci-01.jl", "max_forks_repo_name": "jvaverka/XranklinTest", "max_forks_repo_head_hexsha": "1cb38637671e50439fb873b766e023aa450f3a42", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.9829059829, "max_line_length": 544, "alphanum_fraction": 0.6680080483, "num_tokens": 1249, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070133672955, "lm_q2_score": 0.9207896693699845, "lm_q1q2_score": 0.8378329779959022}} {"text": "# Question 1\nimport Pkg; Pkg.add(\"Distributions\"); Pkg.add(\"QuantEcon\")\nusing Distributions, QuantEcon\n\nfunction profit_max_q(a, c, mu, sigma, method, n)\n b = rand(LogNormal(mu, sigma), n)\n if method == \"mc\"\n q = Float64((a-c)/(2*mean(b)))\n return q\n elseif method == \"quad\"\n nodes_weights = qnwlogn(n, mu, sigma) #http://quantecon.github.io/QuantEcon.jl/latest/api/QuantEcon.html\n nodes = nodes_weights[1]\n weights = nodes_weights[2]\n mean_b = sum(nodes .* weights)\n q = Float64((a-c)/(2*mean_b))\n end\n return q\n end\n\nprofit_max_q(3,1,1,1,\"mc\",10)\n\n# we dont have to worry about code_warntype\n@code_warntype profit_max_q(3,1,1,1,\"mc\",10)\n\n\n# Question 2\nusing Distributions\nfunction estimate_pi(lower,upper,num_draws)\n n_land_in_circle = 0\n x = rand(Uniform(lower, upper), num_draws)\n y = rand(Uniform(lower, upper), num_draws)\n for i in 1:num_draws\n r_sq = x[i]^2 + y[i]^2\n if r_sq < 1\n n_land_in_circle += 1\n end\n end\n return n_land_in_circle / num_draws * 4\n end\n\nestimate_pi(0,1,1000000)\n", "meta": {"hexsha": "0f48ab5ec0fe5e52717894febc1d282e8920333d", "size": 1177, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "hw1_jh2737.jl", "max_stars_repo_name": "jh2737/AEM7130_test", "max_stars_repo_head_hexsha": "db2f7ef1f6030c012ddc983545a891516bb0264b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "hw1_jh2737.jl", "max_issues_repo_name": "jh2737/AEM7130_test", "max_issues_repo_head_hexsha": "db2f7ef1f6030c012ddc983545a891516bb0264b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-02-24T20:52:44.000Z", "max_issues_repo_issues_event_max_datetime": "2021-02-24T20:52:44.000Z", "max_forks_repo_path": "hw1_jh2737.jl", "max_forks_repo_name": "jh2737/AEM7130_test", "max_forks_repo_head_hexsha": "db2f7ef1f6030c012ddc983545a891516bb0264b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.0238095238, "max_line_length": 116, "alphanum_fraction": 0.5964316058, "num_tokens": 356, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9362850110816423, "lm_q2_score": 0.8947894604912848, "lm_q1q2_score": 0.8377779599318194}} {"text": "\"\"\"\n jacobi(x, p::Integer, α, β)\n\nEvaluate the Legendre polynomial with parameters `α`, `β` of degree `p` at `x`\nusing the three term recursion [Karniadakis and Sherwin, Spectral/hp Element\nMethods for CFD, Appendix A].\n\"\"\"\nfunction jacobi(x, p::Integer, α, β)\n T = typeof( (2+α+β)*x / 2 )\n a = one(T)\n b = ((2+α+β)*x + α - β) / 2\n\n if p <= 0\n return a\n elseif p == 1\n return b\n end\n\n for n in 2:p\n a1 = 2n*(n+α+β)*(2n-2+α+β)\n a2 = (2n-1+α+β)*(α+β)*(α-β)\n a3 = (2n-2+α+β)*(2n-1+α+β)*(2n+α+β)\n a4 = 2*(n-1+α)*(n-1+β)*(2n+α+β)\n a, b = b, ( (a2+a3*x)*b - a4*a ) / a1\n end\n\n b\nend\n\n\"\"\"\n jacobi_and_derivative(x, p::Integer, α, β)\n\nEvaluate the Jacobi polynomial with parameters `α`, `β` of degree `p` and its\nderivative at `x` using the three term recursion [Karniadakis and Sherwin,\nSpectral/hp Element Methods for CFD, Appendix A].\n\"\"\"\nfunction jacobi_and_derivative(x, p::Integer, α, β)\n T = typeof( (2+α+β)*x / 2 )\n # Coefficients for the polynomial...\n a = one(T)\n b = ((2+α+β)*x + α - β) / 2\n # ... and for the derivative. The derivative is computed via eq. (A.1.8)\n # of Karniadakis and Sherwin as Jacobi polynomial with parameters α+1, β+1.\n aa = one(T)\n bb = ((4+α+β)*x + α - β) / 2\n\n if p <= 0\n return a, zero(T)\n elseif p == 1\n return b, (2+α+β)/2\n end\n\n for n in 2:p\n a1 = 2n*(n+α+β)*(2n-2+α+β)\n a2 = (2n-1+α+β)*(α+β)*(α-β)\n a3 = (2n-2+α+β)*(2n-1+α+β)*(2n+α+β)\n a4 = 2*(n-1+α)*(n-1+β)*(2n+α+β)\n a, b = b, ( (a2+a3*x)*b - a4*a ) / a1\n\n b1 = 2n*(n+2+α+β)*(2n+α+β)\n b2 = (2n+1+α+β)*(α+β+2)*(α-β)\n b3 = (2n+α+β)*(2n+1+α+β)*(2n+2+α+β)\n b4 = 2*(n+α)*(n+β)*(2n+2+α+β)\n aa, bb = bb, ( (b2+b3*x)*bb - b4*aa ) / b1\n end\n\n b, (α+β+p+1)*aa/2\nend\n\n\n\"\"\"\n gauss_jacobi_nodes_and_weights(p, α, β, T=Float64::Type, tol=4*eps(T), maxit=100)\n\nCompute the Gauss-Jacobi nodes and weights for polynomials of degree `p` with\nparameters `α`, `β` using the scalar type `T`, tolerance `tol` and maximal number\nof Newton iterations `maxit` [Karniadakis and Sherwin, Spectral/hp Element\nMethods for CFD, Appendix B].\n\"\"\"\nfunction gauss_jacobi_nodes_and_weights(p, α, β, T=Float64::Type, tol=4*eps(T), maxit=1000)\n T = promote_type(typeof(α), typeof(β), T)\n nodes = Vector{T}(undef, p+1)\n weights = Vector{T}(undef, p+1)\n for j in 0:p\n x = -cospi(T(2j+1)/(2p+2))\n if j > 0\n x = (x + nodes[j]) / 2\n end\n for k in 1:maxit\n s = zero(T)\n for l in 1:j\n s += 1 / (x - nodes[l])\n end\n pol, der = jacobi_and_derivative(x, p+1, α, β)\n Δ = -pol / (der - s*pol)\n x = x + Δ\n abs(Δ) <= tol*abs(x) && break\n end\n pol, der = jacobi_and_derivative(x, p+1, α, β)\n nodes[j+1] = x\n weights[j+1] = 2^(α+β+1) * gamma(α+p+2) * gamma(β+p+2) /\n (gamma(p+2) * gamma(α+β+p+2) * (1-x^2) * der^2)\n end\n\n return nodes, weights\nend\n\n\n\n\"\"\"\n jacobi_vandermonde(nodes, α, β)\n\nComputes the Vandermonde matrix with respect to the Jacobi polynomials with\nparameters `α`, `β` and the nodal basis on `nodes`.\nThe Vandermonde matrix `V` is the transformation matrix from the modal Jacobi\nbasis to the nodal Lagrange basis associated with `nodes`.\n\"\"\"\nfunction jacobi_vandermonde(nodes::AbstractVector, α, β)\n T = eltype(nodes)\n pp1 = length(nodes)\n V = Array{T}(undef, pp1, pp1)\n for j in 1:pp1, (i,x) in enumerate(nodes)\n V[i, j] = jacobi(x, j-1, α, β)\n end\n V\nend\n\njacobi_vandermonde(basis::NodalBasis{Line}, α, β) = jacobi_vandermonde(basis.nodes, α, β)\n", "meta": {"hexsha": "8a5f132ba41e292fe9cfce401918facdc04fd336", "size": 3734, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/jacobi.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/PolynomialBases.jl-c74db56a-226d-5e98-8bb0-a6049094aeea", "max_stars_repo_head_hexsha": "7a1ac70cbdae378323a4dd611c8dc1386222c71f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2019-09-11T18:12:07.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-13T21:08:56.000Z", "max_issues_repo_path": "src/jacobi.jl", "max_issues_repo_name": "UnofficialJuliaMirrorSnapshots/PolynomialBases.jl-c74db56a-226d-5e98-8bb0-a6049094aeea", "max_issues_repo_head_hexsha": "7a1ac70cbdae378323a4dd611c8dc1386222c71f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2018-02-15T06:32:55.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-11T15:59:07.000Z", "max_forks_repo_path": "src/jacobi.jl", "max_forks_repo_name": "UnofficialJuliaMirrorSnapshots/PolynomialBases.jl-c74db56a-226d-5e98-8bb0-a6049094aeea", "max_forks_repo_head_hexsha": "7a1ac70cbdae378323a4dd611c8dc1386222c71f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-02-26T18:34:02.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T11:01:42.000Z", "avg_line_length": 29.171875, "max_line_length": 91, "alphanum_fraction": 0.5431173005, "num_tokens": 1442, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9362849986365571, "lm_q2_score": 0.8947894696095783, "lm_q1q2_score": 0.8377779573334097}} {"text": "function sum_n_below(factor, max)\n # Return the sum of all the natural numbers of factor below max\n max_nb = div(max-1,factor)\n ((factor / 2 * max_nb * (max_nb + 1)))\nend\n\nfunction n_below(n)\n # Return all the natural number of 3 and 5 multipliers\n return sum_n_below(3, n) + sum_n_below(5, n) - sum_n_below(3*5, n)\nend\n\n@assert n_below(10) == 23\nprintln(n_below(1000))\n", "meta": {"hexsha": "962fb35ac990fb335d90ed4a8bf54a1852a7568a", "size": 385, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "1.jl", "max_stars_repo_name": "ddugue/project-euler", "max_stars_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "1.jl", "max_issues_repo_name": "ddugue/project-euler", "max_issues_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "1.jl", "max_forks_repo_name": "ddugue/project-euler", "max_forks_repo_head_hexsha": "ab373d896ac5051bb53a706cccfdfa3987ca9272", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.5, "max_line_length": 70, "alphanum_fraction": 0.6831168831, "num_tokens": 124, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9724147177732101, "lm_q2_score": 0.8615382058759128, "lm_q1q2_score": 0.8377724313176635}} {"text": "\"\"\"\n newton(f::Function, x0::Real; niters=10^2, tol=1e-8)\n\nCompute roots of scalar `f` by Newton's method\n\n\"\"\"\n\nfunction newton(f::Function, x0::Real; niters=10^2, tol=1e-8)\n x = xnew = x0\n for it in 1:niters\n df = ForwardDiff.derivative(f, x)\n xnew = x - f(x)/df\n abs(x - xnew) < tol && return xnew\n x, xnew = xnew, x\n end\n xnew\nend", "meta": {"hexsha": "9b8fe5ca703fd1507ee21114c0e79800e646f2b6", "size": 376, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/newton.jl", "max_stars_repo_name": "stecrotti/JuliaUtils.jl", "max_stars_repo_head_hexsha": "51e09a622a0ae96de6b8cc162ef985fc9b11c915", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/newton.jl", "max_issues_repo_name": "stecrotti/JuliaUtils.jl", "max_issues_repo_head_hexsha": "51e09a622a0ae96de6b8cc162ef985fc9b11c915", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/newton.jl", "max_forks_repo_name": "stecrotti/JuliaUtils.jl", "max_forks_repo_head_hexsha": "51e09a622a0ae96de6b8cc162ef985fc9b11c915", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.1176470588, "max_line_length": 61, "alphanum_fraction": 0.5638297872, "num_tokens": 140, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.951142221377825, "lm_q2_score": 0.8807970779778825, "lm_q1q2_score": 0.8377632893309805}} {"text": "\"\"\"\nComputes entropy of a sequence.\n\"\"\"\nfunction entropy(X :: AbstractVector)\n counts = Dict{eltype(X), Int}()\n n = length(X)\n for i = 1:length(X)\n counts[X[i]] = get(counts, X[i], 0) + 1\n end\n sum(kv -> log2(n / kv[2]) * kv[2], counts) / n\nend\n\n\"\"\"\nComputes conditional entropy `H(Y | X)`.\n\"\"\"\nfunction conditional_entropy(X :: AbstractVector, Y :: AbstractVector)\n cnts_for_pref = Dict{eltype(X), Dict{eltype(Y), Int}}()\n n = length(X)\n for i = 1:length(X)\n if !haskey(cnts_for_pref, X[i])\n cnts_for_pref[X[i]] = Dict{eltype(Y), Int}()\n end\n counts = cnts_for_pref[X[i]]\n counts[Y[i]] = get(counts, Y[i], 0) + 1\n end\n\n # this helper function computes \"unnormalized entropy\" for given counts dict\n # i.e. entropy multiplied by sum of counts\n function _prefix_helper(counts :: Dict{eltype(Y), Int})\n counts_sum = sum(kv -> kv[2], counts)\n sum(kv -> log2(counts_sum / kv[2]) * kv[2], counts)\n end\n\n sum(kv -> _prefix_helper(kv[2]), cnts_for_pref) / n\nend\n", "meta": {"hexsha": "5a757d8bf60278327afe563a8be916a4746bdf88", "size": 994, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/entropy.jl", "max_stars_repo_name": "krcz/ChaosQuantified.jl", "max_stars_repo_head_hexsha": "f6795a42f7d2bd266865f2c436d3065652fb1ba8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/entropy.jl", "max_issues_repo_name": "krcz/ChaosQuantified.jl", "max_issues_repo_head_hexsha": "f6795a42f7d2bd266865f2c436d3065652fb1ba8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/entropy.jl", "max_forks_repo_name": "krcz/ChaosQuantified.jl", "max_forks_repo_head_hexsha": "f6795a42f7d2bd266865f2c436d3065652fb1ba8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.6111111111, "max_line_length": 78, "alphanum_fraction": 0.6277665996, "num_tokens": 319, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9702399051935107, "lm_q2_score": 0.8633916099737806, "lm_q1q2_score": 0.8376969938058334}} {"text": "#simple DFT function\nfunction DFT(x)\n N = length(x)\n\n # We want two vectors here for real space (n) and frequency space (k)\n n = 0:N-1\n k = n'\n transform_matrix = exp.(-2im*pi*n*k/N)\n return transform_matrix*x\n\nend\n\n# Implementing the Cooley-Tukey Algorithm\nfunction cooley_tukey(x)\n N = length(x)\n\n if (N > 2)\n x_odd = cooley_tukey(x[1:2:N])\n x_even = cooley_tukey(x[2:2:N])\n else\n x_odd = x[1]\n x_even = x[2]\n end\n n = 0:N-1\n half = div(N,2)\n factor = exp.(-2im*pi*n/N)\n return vcat(x_odd + x_even .* factor[1:half],\n x_odd - x_even .* factor[1:half])\n\nend\n\nfunction bitreverse(a::Array)\n # First, we need to find the necessary number of bits\n digits = convert(Int,ceil(log2(length(a))))\n\n indices = [i for i = 0:length(a)-1]\n\n bit_indices = []\n for i = 1:length(indices)\n push!(bit_indices, bits(indices[i]))\n end\n\n # Now stripping the unnecessary numbers\n for i = 1:length(bit_indices)\n bit_indices[i] = bit_indices[i][end-digits:end]\n end\n\n # Flipping the bits\n for i =1:length(bit_indices)\n bit_indices[i] = reverse(bit_indices[i])\n end\n\n #replacing indices\n for i = 1:length(indices)\n indices[i] = 0\n for j = 1:digits\n indices[i] += 2^(j-1) * parse(string(bit_indices[i][end-j]))\n end\n indices[i] += 1\n end\n\n b = [float(i) for i = 1:length(a)]\n for i = 1:length(indices)\n b[i] = a[indices[i]]\n end\n\n return b\nend\n\nfunction iterative_cooley_tukey(x)\n N = length(x)\n logN = convert(Int,ceil(log2(length(x))))\n bnum = div(N,2)\n stride = 0;\n\n x = bitreverse(x)\n\n z = [Complex(x[i]) for i = 1:length(x)]\n for i = 1:logN\n stride = div(N, bnum)\n for j = 0:bnum-1\n start_index = j*stride + 1\n y = butterfly(z[start_index:start_index + stride - 1])\n for k = 1:length(y)\n z[start_index+k-1] = y[k]\n end\n end\n bnum = div(bnum,2)\n end\n\n return z\nend\n\nfunction butterfly(x)\n N = length(x)\n half = div(N,2)\n n = [i for i = 0:N-1]\n half = div(N,2)\n factor = exp.(-2im*pi*n/N)\n\n y = [0 + 0.0im for i = 1:length(x)]\n\n for i = 1:half\n y[i] = x[i] + x[half+i]*factor[i]\n y[half+i] = x[i] - x[half+i]*factor[i]\n end\n\n return y\nend\n\nfunction approx(x, y)\n val = true\n for i = 1:length(x)\n if (abs(x[i]) - abs(y[i]) > 1e-5)\n val = false\n end\n end\n println(val)\nend\n\nfunction main()\n x = rand(128)\n y = cooley_tukey(x)\n z = iterative_cooley_tukey(x)\n w = fft(x)\n approx(y, w)\n approx(z, w)\nend\n\nmain()\n", "meta": {"hexsha": "03291fb8c44586a334e520d07c0384d666fe69be", "size": 2695, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "chapters/FFT/code/julia/fft.jl", "max_stars_repo_name": "sanstorik/algorithm-archive", "max_stars_repo_head_hexsha": "74d54d09449fcf303c867834d68ea2399573b369", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-08-16T17:42:24.000Z", "max_stars_repo_stars_event_max_datetime": "2019-08-16T17:42:24.000Z", "max_issues_repo_path": "chapters/FFT/code/julia/fft.jl", "max_issues_repo_name": "sanstorik/algorithm-archive", "max_issues_repo_head_hexsha": "74d54d09449fcf303c867834d68ea2399573b369", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2018-07-12T01:07:26.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-04T19:37:47.000Z", "max_forks_repo_path": "chapters/FFT/code/julia/fft.jl", "max_forks_repo_name": "sanstorik/algorithm-archive", "max_forks_repo_head_hexsha": "74d54d09449fcf303c867834d68ea2399573b369", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-04-19T10:04:37.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-19T10:04:37.000Z", "avg_line_length": 20.572519084, "max_line_length": 73, "alphanum_fraction": 0.5395176252, "num_tokens": 873, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9441768635777511, "lm_q2_score": 0.8872046011730965, "lm_q1q2_score": 0.8376780576873638}} {"text": "\"\"\"\nbarychebinit(n)\n\nGet Chebyshev nodes of first kind and corresponding barycentric Lagrange interpolation weights. \nReference: Berrut, J.P. and Trefethen, L.N., 2004. Barycentric lagrange interpolation. SIAM review, 46(3), pp.501-517.\n\n# Arguments\n- `n`: order of the Chebyshev interpolation\n\n# Returns\n- Chebyshev nodes\n- Barycentric Lagrange interpolation weights\n\"\"\"\nfunction barychebinit(n)\n x = zeros(Float64, n)\n w = similar(x)\n for i in 1:n\n c = (2i - 1)π / (2n)\n x[n - i + 1] = cos(c)\n w[n - i + 1] = (-1)^(i - 1) * sin(c)\n end\n return x, w\nend\n\n\"\"\"\nfunction barycheb(n, x, f, wc, xc)\n\nBarycentric Lagrange interpolation at Chebyshev nodes\nReference: Berrut, J.P. and Trefethen, L.N., 2004. Barycentric lagrange interpolation. SIAM review, 46(3), pp.501-517.\n\n# Arguments\n- `n`: order of the Chebyshev interpolation\n- `x`: coordinate to interpolate\n- `f`: array of size n, function at the Chebyshev nodes\n- `wc`: array of size n, Barycentric Lagrange interpolation weights\n- `xc`: array of size n, coordinates of Chebyshev nodes\n\n# Returns\n- Interpolation result\n\"\"\"\nfunction barycheb(n, x, f, wc, xc)\n for j in 1:n\n if x == xc[j]\n return f[j]\n end \n end\n\n num, den = 0.0, 0.0\n for j in 1:n\n q = wc[j] / (x - xc[j])\n num += q * f[j]\n den += q\n end\n return num / den\nend\n\nfunction barycheb2(n, x, f, wc, xc)\n for j in 1:n\n if x == xc[j]\n return f[j, :]\n end \n end\n\n den = 0.0\n num = zeros(eltype(f), size(f)[2])\n for j in 1:n\n q = wc[j] / (x - xc[j])\n num += q .* f[j, :]\n den += q\n end\n return num ./ den\nend", "meta": {"hexsha": "9f7502c6bb8f65d1df038ef2591b680a63d19946", "size": 1691, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/old/chebyshev.jl", "max_stars_repo_name": "quantumstatistics/CompositeGrids.jl", "max_stars_repo_head_hexsha": "6bf230eb4267749adbcfb1a80ddffd6899e7317d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-08-21T03:44:39.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-29T22:58:02.000Z", "max_issues_repo_path": "src/old/chebyshev.jl", "max_issues_repo_name": "quantumstatistics/CompositeGrids.jl", "max_issues_repo_head_hexsha": "6bf230eb4267749adbcfb1a80ddffd6899e7317d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 20, "max_issues_repo_issues_event_min_datetime": "2021-08-16T19:16:41.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-17T01:55:31.000Z", "max_forks_repo_path": "src/old/chebyshev.jl", "max_forks_repo_name": "quantumstatistics/CompositeGrids.jl", "max_forks_repo_head_hexsha": "6bf230eb4267749adbcfb1a80ddffd6899e7317d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-22T21:04:22.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-22T21:04:22.000Z", "avg_line_length": 23.4861111111, "max_line_length": 118, "alphanum_fraction": 0.585452395, "num_tokens": 576, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9407897442783527, "lm_q2_score": 0.8902942239389253, "lm_q1q2_score": 0.8375796752719961}} {"text": "using Polynomials: Polynomial,fit,coeffs\nusing Plots\nusing Distributions: Normal\nusing Random\nusing Statistics: mean, std\n\np = Polynomial([1,2,3])\n@show coeffs(p)\n\nerrstd = 3\nerrNormal = Normal(0,errstd)\n\nx = 0:10\ny = 2 .+ 3 .* x .+rand(errNormal,11)\n\nscatter(x,y,yerror=errstd,label=false)\n\nfit_line = fit(x,y,1)\n@show coeffs(fit_line)\nplot!(x,fit_line.(x),label=false)\n\nb_list = []\nm_list = []\nN = 1000\nfor i in 1:N\n y = 2 .+ 3 .* x .+rand(errNormal,11)\n fit_c = coeffs(fit(x,y,1))\n push!(b_list,fit_c[1])\n push!(m_list,fit_c[2])\nend\n\n@show mean(b_list)\n@show mean(m_list)\n@show std(b_list)\n@show std(m_list)\n\nhistogram(b_list,bins=30)\nhistogram(m_list,bins=30)\n\n", "meta": {"hexsha": "9718b87f5bae9ce257b1e47293895de04d05d8da", "size": 677, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "least-squares.jl", "max_stars_repo_name": "hstrey/BME-502-2022", "max_stars_repo_head_hexsha": "01ae3787874a13adbed69cb16913f73594107f33", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "least-squares.jl", "max_issues_repo_name": "hstrey/BME-502-2022", "max_issues_repo_head_hexsha": "01ae3787874a13adbed69cb16913f73594107f33", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "least-squares.jl", "max_forks_repo_name": "hstrey/BME-502-2022", "max_forks_repo_head_hexsha": "01ae3787874a13adbed69cb16913f73594107f33", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-02-15T18:19:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-15T18:26:08.000Z", "avg_line_length": 16.925, "max_line_length": 40, "alphanum_fraction": 0.6765140325, "num_tokens": 237, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9637799441350253, "lm_q2_score": 0.8688267728417087, "lm_q1q2_score": 0.8373578185923963}} {"text": "#=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=#\r\n# Problem set 7 solutions\r\n# Written by Tyler Ransom\r\n# Commented by Giuseppe Grasso\r\n#=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=#\r\n\r\nusing Random\r\nusing LinearAlgebra\r\nusing Statistics\r\nusing Distributions\r\nusing Optim\r\nusing DataFrames\r\nusing CSV\r\nusing HTTP\r\nusing GLM\r\nusing FreqTables\r\nusing ForwardDiff\r\nusing LineSearches\r\nusing SMM\r\ncd(\"/Users/peppegrass/Documents/GitHub/fall-2020/ProblemSets/PS6-ccp/\") # GG: sets working directory\r\npwd() ## GG: prints working directory\r\nreaddir() # GG: equivalent to -ls- to see elements of working directory\r\n\r\nfunction allwrap()\r\n\r\n #=\r\n GG: In this problem set, we will practice estimating models by\r\n Generalized Method of Moments (GMM) and Simulated Method of Moments (SMM).\r\n =#\r\n\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # Question 1 - GMM for simple regression model\r\n # data & model are SAME AS PS2, QUESTION 2\r\n #\r\n # 1. Estimate the linear regression model from Question 2 of Problem Set 2 by GMM.\r\n # Write down the moment function as in slide #8 of the Lecture 9 slide deck and use Optim for estimation.\r\n # Use the N × N Identity matrix as your weighting matrix. Check your answer using the closed-form matrix formula for the OLS estimator.\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n url = \"https://raw.githubusercontent.com/OU-PhD-Econometrics/fall-2020/master/ProblemSets/PS1-julia-intro/nlsw88.csv\"\r\n df = CSV.read(HTTP.get(url).body)\r\n X = [ones(size(df,1),1) df.age df.race.==1 df.collgrad.==1]\r\n y = log.(df.wage)\r\n\r\n function ols_gmm(β, X, y) # GG: GMM function\r\n K = size(X,2) # GG: don't really needed for GMM, unless for computing RMSE in step below\r\n N = size(y,1) # GG: don't really needed for GMM, unless for computing RMSE in step below\r\n ŷ = X*β\r\n g = y .- ŷ # GG: force y and y_hat to be as close as possible\r\n J = g'*I*g # GG: GMM objective function\r\n return J\r\n end\r\n\r\n # we can also estimate the RMSE of the regression by ADDING ANOTHER MOMENT (and assuming homoskedasticity)\r\n function ols_gmm_with_σ(θ, X, y)\r\n K = size(X,2)\r\n N = size(y,1)\r\n β = θ[1:end-1]\r\n σ = θ[end]\r\n g = y .- X*β\r\n g = vcat(g,( (N-1)/(N-K) )*var(g) .- σ^2) # this builds on the assumption that MSE: SSR/(N-K) = σ² # GG: performing DOFs corrections\r\n J = g'*I*g\r\n return J\r\n end\r\n\r\n β_hat_gmm = optimize(b -> ols_gmm(b, X, y), rand(size(X,2)), LBFGS(), Optim.Options(g_tol=1e-6, iterations=100_000, show_trace=true))\r\n println(β_hat_gmm.minimizer)\r\n println(\"ans: \",X\\y) # GG: X\\y gives the least squares; cool uh!?\r\n β_hat_gmm = optimize(b -> ols_gmm_with_σ(b, X, y), rand(size(X,2)+1), LBFGS(), Optim.Options(g_tol=1e-6, iterations=100_000, show_trace=true))\r\n println(β_hat_gmm.minimizer)\r\n println(\"ans: \",vcat(X\\y,sqrt( sum( (y .- X*β_hat_gmm.minimizer[1:end-1]).^2 )/( size(y,1)-size(X,2) ) )))\r\n\r\n # GG: of course ols_gmm(b, X, y) and ols_gmm_with_σ(b, X, y) yield same results but latter with MSE\r\n\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # Question 2 - GMM for multinomial logit model\r\n # data & model are SAME AS PS2, QUESTION 5\r\n #\r\n # 2. Estimate the multinomial logit model from Question 5 of Problem Set 2 by the following means:\r\n # (a) Maximum likelihood (i.e. re-run your code [or mine] from Question 5 of Problem Set 2)\r\n # (b) GMM with the MLE estimates as starting values. Your g object should be a vector of dimension N ×J where\r\n # N is the number of rows of the X matrix and J is the dimension of the choice set. Each element, g should equal d − P,\r\n # where d and P are “stacked” vectors of dimension N×J\r\n # (c) GMM with random starting values\r\n # Compare your estimates from part (b) and (c). Is the objective function globally concave?\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n df = dropmissing(df, :occupation)\r\n df[df.occupation.==8 ,:occupation] .= 7\r\n df[df.occupation.==9 ,:occupation] .= 7\r\n df[df.occupation.==10,:occupation] .= 7\r\n df[df.occupation.==11,:occupation] .= 7\r\n df[df.occupation.==12,:occupation] .= 7\r\n df[df.occupation.==13,:occupation] .= 7\r\n\r\n #df = @transform(df, white = race.==1)\r\n df[:,:white] = df.race .== 1\r\n X = [ones(size(df,1),1) df.age df.white df.collgrad]\r\n y = df.occupation\r\n\r\n # get starting values for mlogit from series of binary logits\r\n svals = zeros(size(X,2),7)\r\n for j=1:7\r\n tempname = Symbol(string(\"occ\",j))\r\n df[:,tempname] = df.occupation.==j\r\n end\r\n # GG: this is to derive some \"reasonable starting values\" for a multinomial logit model\r\n # GG: what do we mean for \"reasonable starting values\". You don't wanna have neither a vector of 0 nor a vector of random numbers\r\n # GG: So what starting values for a Multinomial logit. Estimate a bunch of binary logits and then...\r\n svals[:,1] = coef(lm(@formula(occ1 ~ age + white + collgrad), df))\r\n svals[:,2] = coef(lm(@formula(occ2 ~ age + white + collgrad), df))\r\n svals[:,3] = coef(lm(@formula(occ3 ~ age + white + collgrad), df))\r\n svals[:,4] = coef(lm(@formula(occ4 ~ age + white + collgrad), df))\r\n svals[:,5] = coef(lm(@formula(occ5 ~ age + white + collgrad), df))\r\n svals[:,6] = coef(lm(@formula(occ6 ~ age + white + collgrad), df))\r\n svals[:,7] = coef(lm(@formula(occ7 ~ age + white + collgrad), df))\r\n svals = svals[:,1:6] .- svals[:,7] # GG: ... impose the identification normalization (i.e. you subtract for the first 6 estimates the one of the 7th)\r\n svals = svals[:] # GG: then vectorize it and use it as a starting value\r\n\r\n function mlogit_mle(α, X, y)\r\n\r\n K = size(X,2)\r\n J = length(unique(y))\r\n N = length(y)\r\n bigY = zeros(N,J)\r\n for j=1:J\r\n bigY[:,j] = y.==j\r\n end\r\n bigα = [reshape(α,K,J-1) zeros(K)]\r\n\r\n P = exp.(X*bigα) ./ sum.(eachrow(exp.(X*bigα)))\r\n\r\n loglike = -sum( bigY.*log.(P) ) # GG: MLE objective function\r\n\r\n return loglike\r\n end\r\n\r\n α_zero = zeros(6*size(X,2))\r\n α_rand = rand(6*size(X,2))\r\n # α_true = [-.5120867,-.6950393,.3165616,-1.646016,-2.300697,-.7656569]\r\n α_true = [.1910213,-.0335262,.5963968,.4165052,-.1698368,-.0359784,1.30684,-.430997,.6894727,-.0104578,.5231634,-1.492475,-2.26748,-.0053001,1.391402,-.9849661,-1.398468,-.0142969,-.0176531,-1.495123,.2454891,-.0067267,-.5382892,-3.78975]\r\n α_start = α_true.*rand(size(α_true))\r\n println(size(α_true))\r\n α_hat_optim = optimize(a -> mlogit_mle(a, X, y), svals, LBFGS(), Optim.Options(g_tol = 1e-5, iterations=100_000, show_trace=true, show_every=50))\r\n α_hat_mle = α_hat_optim.minimizer\r\n println(\"mlogit MLE estimates: \",α_hat_mle)\r\n\r\n function mlogit_gmm(α, X, y)\r\n K = size(X,2)\r\n J = length(unique(y))\r\n N = length(y)\r\n bigY = zeros(N,J)\r\n for j=1:J\r\n bigY[:,j] = y.==j\r\n end\r\n bigα = [reshape(α,K,J-1) zeros(K)]\r\n\r\n P = exp.(X*bigα) ./ sum.(eachrow(exp.(X*bigα)))\r\n\r\n g = zeros((J-1)*K)\r\n for j=1:(J-1)\r\n for k = 1:K\r\n g[(j-1)*K+k] = mean( (bigY[:,j] .- P[:,j]) .* X[:,k] ) # GG: GMM (exactly-identified) objective function\r\n # GG: (j-1)*K moment conditions; (j-1) equations and K betas; same number of elements as number non-zero elements in bigαlpha\r\n end\r\n end\r\n J = N*g⋅g #GG: exactly identified; no weighting matrix\r\n return J\r\n end\r\n\r\n function mlogit_gmm_overid(α, X, y)\r\n K = size(X,2)\r\n J = length(unique(y))\r\n N = length(y)\r\n bigY = zeros(N,J)\r\n for j=1:J\r\n bigY[:,j] = y.==j\r\n end\r\n bigα = [reshape(α,K,J-1) zeros(K)]\r\n\r\n P = exp.(X*bigα) ./ sum.(eachrow(exp.(X*bigα))) # GG: same as before P = exp.(X*bigα) ./ sum.(eachrow(exp.(X*bigα)))\r\n\r\n g = bigY[:] .- P[:] # GG: now in over-identified case; we vectorize big Y and P with the [:] operator and subtract the two\r\n # GG: this will be equivalent to NONLINEAR LEAST SQUARES\r\n\r\n J = g'*I*g # GG: now we have weighting matrix\r\n return J\r\n end\r\n #α_zero = zeros(6*size(X,2))\r\n #α_rand = rand(6*size(X,2))\r\n α_true = [.1910213,-.0335262,.5963968,.4165052,-.1698368,-.0359784,1.30684,-.430997,.6894727,-.0104578,.5231634,-1.492475,-2.26748,-.0053001,1.391402,-.9849661,-1.398468,-.0142969,-.0176531,-1.495123,.2454891,-.0067267,-.5382892,-3.78975]\r\n #α_start = α_true.*rand(size(α_true))\r\n #println(\"starting values: \",α_rand)\r\n\r\n # GG: here we're playing around with different optimization algorithms, starting values, and later using autodiff\r\n\r\n # LBFGS\r\n α_hat_optim = optimize(a -> mlogit_gmm_overid(a, X, y), α_true .+ .0001*rand(size(α_true)), LBFGS(), Optim.Options(g_tol = 1e-5, iterations=1_000, show_trace=true, show_every=100))\r\n α_hat_gmm = α_hat_optim.minimizer\r\n println(\"LBFGS GMM minimum: \",α_hat_optim.minimum)\r\n println(\"LBFGS GMM estimates: \",α_hat_gmm)\r\n\r\n # use autodiff with LBFGS\r\n td = TwiceDifferentiable(b -> mlogit_gmm_overid(b, X, y), svals; autodiff = :forward)\r\n α_hat_optim_ad = optimize(td, α_true .+ .0001*rand(size(α_true)), LBFGS(), Optim.Options(g_tol = 1e-5, iterations=10_000, show_trace=true, show_every=100))\r\n α_hat_gmm_ad = α_hat_optim_ad.minimizer\r\n println(\"LBFGS GMM autodiff minimum: \",α_hat_optim_ad.minimum)\r\n println(\"LBFGS GMM autodiff estimates: \",α_hat_gmm_ad)\r\n\r\n # use autodiff with Newton\r\n α_hat_optim_ad_newton = optimize(td, α_true .+ .0001*rand(size(α_true)), Newton(linesearch = BackTracking()), Optim.Options(g_tol = 1e-5, iterations=10_000, show_trace=true, show_every=100))\r\n α_hat_gmm_ad_newton = α_hat_optim_ad_newton.minimizer\r\n println(\"Newton GMM autodiff minimum: \",α_hat_optim_ad_newton.minimum)\r\n println(\"Newton GMM autodiff estimates: \",α_hat_gmm_ad_newton)\r\n\r\n # autodiff from bad starting values\r\n α_hat_optim_ad_fr_bad = optimize(td, α_rand.*sign.(α_true), LBFGS(), Optim.Options(g_tol = 1e-10, f_tol = 1e-10, x_tol = 1e-10, iterations=10_000, show_trace=true, show_every=50))\r\n α_hat_gmm_ad_fr_bad = α_hat_optim_ad_fr_bad.minimizer\r\n println(\"LBFGS GMM autodiff minimum: \",α_hat_optim_ad_fr_bad.minimum)\r\n println(\"LBFGS GMM autodiff estimates: \",α_hat_gmm_ad_fr_bad)\r\n\r\n # GG: putting different answers in a DataFrame and comparing them\r\n # GG: found out close answers; the one using bad starting value didn't get close to other solutions; some times GMM tolerance values needed to be played around with\r\n compare = DataFrame(mle = α_hat_mle, gmm = α_hat_gmm, gmm_ad = α_hat_gmm_ad, gmm_ad_newton = α_hat_gmm_ad_newton, gmm_ad_fr_bad = α_hat_gmm_ad_fr_bad)\r\n println(compare)\r\n # GG: objective function globally concave, but couldn't get to global maximum from random starting values, although pretty close\r\n\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # Question 3 - SIMULATE DATA from a logit model\r\n #\r\n # 3. SIMULATE A DATASET FROM A MULTINOMIAL LOGIT MODEL, and then estimate its parameter values and verify that the estimates are close to the\r\n # parameter values you set. That is, for a given sample size N, choice set dimension J and parameter vector β, write a function that outputs\r\n # data X and Y. I will let you choose N, J, β and the number of covariates in X (K), but J should be larger than 2 and K should be larger than 1.\r\n # If you haven’t done this before, you may want to follow these steps:\r\n # (a) Generate X using a random number generator—rand() or randn().\r\n # (b) Set values for β such that conformability with X and J is satisfied\r\n # (c) Generate the N×J matrix of choice probabilities P\r\n # (d) Draw the preference shocks ε as a N×1 vector of U[0,1] random numbers\r\n # (e) Generate Y as follows:\r\n # • Initialize Y as an N ×1 vector of 0s\r\n # • Update Y i = ∑ j=1 J 1 P ik > εi ∑k=j J [see text of problem set]\r\n # (f) An alternative way to generate choices would be to draw a N × J matrix of ε’s from a T1EV distribution.\r\n # This distribution is already defined in the Distributions package. Then Y i = argmax j X i β j + ε i j .\r\n # I’ll show you an example of how to do that in the solutions code for this problem set.\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n function sim_logit(N=100_000,J=4) # putting \"=\" after the arguments here sets default values for these arguments\r\n # generate X, β and P\r\n X = hcat(ones(N),randn(N),2 .+ 2 .* randn(N))\r\n if J==4\r\n β = hcat([1, -1, 0.5],[-2, 0.5, 0.3],[0, -0.5, 2],zeros(3)) # GG: numbers that square nicely with 4\r\n else\r\n β = 2 .* rand(size(X,2),J) .- 1\r\n end\r\n P = exp.(X*β) ./ sum.(eachrow(exp.(X*β))) # GG: usual one-liner to generate multinomial ogit choice probabilities; eachrow is summation\r\n # draw choices\r\n draw = rand(N)\r\n Y = zeros(N) # Initializing Y as an Nx1 vector of 0s\r\n for j=1:J\r\n Ytemp = sum(P[:,j:J];dims=2) .> draw # GG: then updating it (summing across j) based on whether choice probability is above or below draw [see formula of problem set]\r\n Y += Ytemp # GG: for each iteration of the loop adding Ytemp to Y\r\n end\r\n return Y,X\r\n end\r\n ySim,XSim = sim_logit() # GG: it will use \"default values\" for N and J set in line 244 through s\"et to the default operator =\"\"\r\n α_hat_sim_optim = optimize(a -> mlogit_mle(a, XSim, ySim), rand(9), LBFGS(), Optim.Options(g_tol = 1e-5, iterations=100_000, show_trace=true, show_every=50))\r\n α_hat_sim_mle = α_hat_sim_optim.minimizer\r\n println(\"simulated mlogit MLE estimates: \",α_hat_sim_mle) # GG: and here are our simulated estimates; values pretty close because sample size big and SEs low\r\n\r\n # now with drawing from Gumbel instead (#GG: i.e T1EV)\r\n # GG: python does not have the nice Distributions package as in Julia\r\n function sim_logit_w_gumbel(N=100_000,J=4) # putting = here sets default values for these arguments\r\n # generate X and β (No P needed)\r\n X = hcat(ones(N),randn(N),2 .+ 2 .* randn(N))\r\n if J==4\r\n β = hcat([1, -1, 0.5],[-2, 0.5, 0.3],[0, -0.5, 2],zeros(3))\r\n else\r\n β = 2 .* rand(size(X,2),J) .- 1\r\n end\r\n # draw choices\r\n ϵ = rand(Gumbel(0,1),N,J) # GG: using Julia's Distributions package; Gumbel(0,1) is \"standard\" T1EV\r\n Y = argmax.(eachrow(X*β .+ ϵ)) # GG: getting argmax of eachrow; eachrow allows avoiding to have to write a loop; tells Julia to do argmax computation row-wise and find column j maximizer\r\n # GG: that's just another way of deriving the Y; the loop in the previous function is just as fine\r\n return Y,X\r\n end\r\n ySim,XSim = sim_logit_w_gumbel()\r\n α_hat_sim_optim = optimize(a -> mlogit_mle(a, XSim, ySim), rand(9), LBFGS(), Optim.Options(g_tol = 1e-5, iterations=100_000, show_trace=true, show_every=50))\r\n α_hat_sim_mle = α_hat_sim_optim.minimizer\r\n println(\"simulated mlogit MLE estimates: \",α_hat_sim_mle)\r\n\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # Question 4 - run SMM.parallelNormal()\r\n #\r\n # 4. Use SMM.jl to run the example code on slide #21 of the Lecture 9 slide deck.\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # GG: SMM.jl has a nice count-down that Optim does not have\r\n MA = SMM.parallelNormal()\r\n dc = SMM.history(MA.chains[1])\r\n dc = dc[dc[:accepted].==true, :]\r\n println(describe(dc)) # GG: can verify you get the mean of the MVN distribution\r\n\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # Question 5 - estimate mlogit by SMM\r\n # GG: Wanna use SIMULATED METHOD OF MOMENTS (using code of Q3) to ESTIMATE RESULT OF Q2\r\n #\r\n # 5. Use your code from Question 3 to estimate the multinomial logit model from Question 2 using SMM and the code example from slide #18 of\r\n # the Lecture 9 slide deck.\r\n #:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n # GG: X and y of Q2\r\n X = [ones(size(df,1),1) df.age df.white df.collgrad]\r\n y = df.occupation\r\n # GG: chunk below was commented out by Tyler beacuse the set of moment conditions used were just not working out for him\r\n #function mlogit_smm(α, X, y, D)\r\n # K = size(X,2)\r\n # J = length(unique(y))\r\n # N = length(y)\r\n # bigY = zeros(N,J)\r\n # bigỸ = zeros(N,J)\r\n # for j=1:J\r\n # bigY[:,j] = y.==j\r\n # end\r\n # bigα = [reshape(α,K,J-1) zeros(K)]\r\n\r\n # T = promote_type(eltype(α),eltype(X))\r\n # gmodel = zeros(T,(J-1)*K,D)\r\n # gdata = zeros(T,(J-1)*K)\r\n\r\n # # data moments\r\n # for j=1:(J-1)\r\n # gdata[(j-1)*K+1] = mean( (bigY[:,j]) )\r\n # for k = 2:K\r\n # gdata[(j-1)*K+k] = cov( bigY[:,j], X[:,k] )\r\n # end\r\n # end\r\n\r\n # Random.seed!(1234)\r\n # # simulated model moments\r\n # for d=1:D\r\n # # draw choices\r\n # ε = rand(Gumbel(0,1),N,J)\r\n # ỹ = argmax.(eachrow(X*bigα .+ ε))\r\n # for j=1:J\r\n # bigỸ[:,j] = ỹ.==j\r\n # end\r\n # # data moments\r\n # for j=1:(J-1)\r\n # gmodel[(j-1)*K+1,d] = mean( (bigỸ[:,j]) )\r\n # for k = 2:K\r\n # gmodel[(j-1)*K+k,d] = cov( bigỸ[:,j], X[:,k] )\r\n # end\r\n # end\r\n # end\r\n\r\n # # minimize squared difference between data and moments\r\n # g = mean(gmodel .- gdata; dims=2)\r\n # J = N*g⋅g\r\n # return J\r\n #\r\n\r\n # GG: the function below for the SIMULATED METHOD OF MOMENTS is the one from the SLIDES OF LECTURE 9 (mind the SCROLLER; gotta use HTML and not PDF version)\r\n function mlogit_smm_overid(α, X, y, D) # GG: D is the number of draws\r\n K = size(X,2)\r\n J = length(unique(y))\r\n N = length(y)\r\n bigY = zeros(N,J)\r\n bigỸ = zeros(N,J)\r\n for j=1:J\r\n bigY[:,j] = y.==j # GG: bigY is our real data that we're trying to MATCH\r\n end\r\n bigα = [reshape(α,K,J-1) zeros(K)] # GG: bigα is our parameter vector that we're gonna try to guess value of, so that we get values so that simulated data match actual ones as closely as possible\r\n\r\n Random.seed!(1234) # GG: cardinal rule, set up the seed; don't forget that!\r\n for d=1:D\r\n # draw choices\r\n ε = rand(Gumbel(0,1),N,J) # GG: if we change the distribution of ε to a multivariate normal, we can do MNP without having to do any Monte Carlo integration. We don't need to have a closed form solution for this.\r\n # GG: and if we want to do GEV models, we need to specify nest-specific and alternative-specific component; just two errors but same principle.\r\n ỹ = argmax.(eachrow(X*bigα .+ ε)) # GG: we're goona draw ỹ through ε\r\n for j=1:J\r\n bigỸ[:,j] .+= (ỹ.==j)*(1/D) # GG: bigỹ is our SIMULATED data\r\n # GG: we set the jth column of bigỸ to a dummy if ỹ is equal to j\r\n end\r\n end\r\n\r\n g = bigY[:] .- bigỸ[:] # GG: exactly like g in Question 2; very same principle\r\n\r\n J = g'*I*g\r\n return J\r\n end\r\n println(size(X,2))\r\n println(length(unique(y))-1)\r\n println(\"size of svals: \",size(α_true .+ .0001*rand(size(α_true))))\r\n bigα = [reshape(α_true,size(X,2),length(unique(y))-1) zeros(size(X,2))]\r\n println(\"size of bigα: \",size(bigα))\r\n td = TwiceDifferentiable(th -> mlogit_smm_overid(th, X, y, 2_000), α_true .+ .0001*rand(size(α_true)); autodiff = :forward)\r\n α_hat_smm_overid = optimize(td, α_true .+ .0001*rand(size(α_true)), LBFGS(), Optim.Options(g_tol=1e-8, x_tol=1e-8, f_tol=1e-8, iterations=100_000, show_trace=true, show_every=5))\r\n println(α_hat_smm_overid.minimizer)\r\n return nothing\r\nend\r\n\r\n#:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n# Question 6\r\n#:::::::::::::::::::::::::::::::::::::::::::::::::::\r\n@time allwrap()\r\n", "meta": {"hexsha": "ecf23e19285d4a1d1aa64b1328cd83e573f46fce", "size": 20255, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "ProblemSets/PS7-smm/PS7solutions_GG.jl", "max_stars_repo_name": "peppegrass/fall-2020", "max_stars_repo_head_hexsha": "03f90548ca4d800146bbeaf9dceca917a21c1195", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-09-01T08:58:14.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-01T08:58:14.000Z", "max_issues_repo_path": "ProblemSets/PS7-smm/PS7solutions_GG.jl", "max_issues_repo_name": "peppegrass/fall-2020", "max_issues_repo_head_hexsha": "03f90548ca4d800146bbeaf9dceca917a21c1195", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ProblemSets/PS7-smm/PS7solutions_GG.jl", "max_forks_repo_name": "peppegrass/fall-2020", "max_forks_repo_head_hexsha": "03f90548ca4d800146bbeaf9dceca917a21c1195", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-10-12T17:24:05.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-12T17:24:05.000Z", "avg_line_length": 50.5112219451, "max_line_length": 243, "alphanum_fraction": 0.5925944211, "num_tokens": 6261, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9284087946129329, "lm_q2_score": 0.9019206738932334, "lm_q1q2_score": 0.8373510856857009}} {"text": "# A variety of means\n\n# Geometric mean\nfunction geomean(a::RealArray)\n s = 0.0\n n = length(a)\n for i = 1 : n\n @inbounds s += log(a[i])\n end\n return exp(s / n)\nend\n\n\n# Harmonic mean\nfunction harmmean(a::RealArray)\n s = 0.0\n n = length(a)\n for i in 1 : n\n @inbounds s += inv(a[i])\n end\n return n / s\nend\n\n\n# Trimmed mean\n\nfunction trimmean(x::RealArray, p::FloatingPoint)\n n = length(x)\n n > 0 || error(\"x can not be empty.\")\n rn = min(iround(n * p), n-1)\n if rn == n - 1\n return median(x)\n else\n sx = sort(x)\n nl = rn >> 1\n nh = (rn - nl)\n s = 0.0\n @inbounds for i = (1+nl) : (n-nh)\n s += x[i]\n end\n return s / (n - rn)\n end\nend\n\n\n# Weighted mean\nfunction wmean{T<:Number,W<:Real}(v::AbstractArray{T}, w::AbstractArray{W}; wsum::W=NaN)\n\t# wsum is the pre-computed sum of weights\n\n\tn = length(v)\n\tlength(w) == n || raise_dimerror()\n sv = zero(T)\n\n if isnan(wsum)\n \tsw = zero(W)\n \tfor i = 1 : n\n \t\t@inbounds wi = w[i]\n \t@inbounds sv += v[i] * wi\n \tsw += wi\n \tend\n else\n \tfor i = 1 : n\n \t\t@inbounds sv += v[i] * w[i]\n \tend\n \tsw = wsum\n end\n\n return sv / sw\nend\n\n# Weighted mean (fast version for contiguous arrays using BLAS)\nfunction wmean{T<:BlasReal}(v::Array{T}, w::Array{T}; wsum::T=NaN) \n\tsw = (isnan(wsum) ? sum(w) : wsum)\n\tdot(v, w) / sw\nend\n\nmean{T<:Number}(v::AbstractArray{T}, w::WeightVec) = wmean(v, values(w); wsum=sum(w))\n\n", "meta": {"hexsha": "4c14ff8a0c3dcf885d12cd3adcb472b79395d42a", "size": 1516, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/means.jl", "max_stars_repo_name": "rened/Stats.jl", "max_stars_repo_head_hexsha": "0efba512a2bf8faa21e61c9568222ae1ae96acbb", "max_stars_repo_licenses": ["Xnet", "X11"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-06-20T16:09:54.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-20T16:09:54.000Z", "max_issues_repo_path": "src/means.jl", "max_issues_repo_name": "aviks/StatsBase.jl", "max_issues_repo_head_hexsha": "2e3401eb239454d3c38b68a8ccd0c665005acf44", "max_issues_repo_licenses": ["Xnet", "X11"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/means.jl", "max_forks_repo_name": "aviks/StatsBase.jl", "max_forks_repo_head_hexsha": "2e3401eb239454d3c38b68a8ccd0c665005acf44", "max_forks_repo_licenses": ["Xnet", "X11"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.1898734177, "max_line_length": 88, "alphanum_fraction": 0.5250659631, "num_tokens": 530, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9489172673767973, "lm_q2_score": 0.8824278726384089, "lm_q1q2_score": 0.8373510455611595}} {"text": "@doc raw\"\"\"\n ExponentiatedKernel([a=1])\n\nThe exponentiated kernel is a Mercer kernel given by:\n\n```math\n\\kappa(\\mathbf{x},\\mathbf{y}) = \\exp\\left(a \\mathbf{x}^\\intercal \\mathbf{y} \\right) \n\\qquad a > 0\n```\n\"\"\"\nstruct ExponentiatedKernel{T<:AbstractFloat} <: MercerKernel{T}\n alpha::HyperParameter{T}\n ExponentiatedKernel{T}(α::Real) where {T<:AbstractFloat} = new{T}(\n HyperParameter(convert(T,α), interval(OpenBound(zero(T)), nothing))\n )\nend\nExponentiatedKernel(α::T1 = 1.0) where {T1<:Real} = ExponentiatedKernel{floattype(T1)}(α)\n\n@inline exponentiatedkernel(z::T, α::T) where {T<:AbstractFloat} = exp(α*z)\n\n@inline basefunction(::ExponentiatedKernel) = ScalarProduct()\n@inline kappa(κ::ExponentiatedKernel{T}, z::T) where {T} = exponentiatedkernel(z, getvalue(κ.alpha))", "meta": {"hexsha": "e1aac9db822cf8528b754ec00b82660698ea0fc2", "size": 793, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/kernelfunctions/mercer/exponentiated.jl", "max_stars_repo_name": "holtri/MLKernels.jl", "max_stars_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/kernelfunctions/mercer/exponentiated.jl", "max_issues_repo_name": "holtri/MLKernels.jl", "max_issues_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/kernelfunctions/mercer/exponentiated.jl", "max_forks_repo_name": "holtri/MLKernels.jl", "max_forks_repo_head_hexsha": "d7b84b1bd7b9d9ca4cb7c413d0e34b04c09a331c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.0454545455, "max_line_length": 100, "alphanum_fraction": 0.6986128625, "num_tokens": 256, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122756889438, "lm_q2_score": 0.874077222043951, "lm_q1q2_score": 0.8372893008959913}} {"text": "\"\"\"\r\n M = time_to_mean(t, T, Mo=0., epoch=0.)\r\n\r\nComputes the mean anomaly of an orbit with period `T` from a time `t`, the orbit's mean anomaly\r\nat epoch `Mo`, and the epoch.\r\n\"\"\"\r\nfunction time_to_mean(t, T, Mo=0., epoch=0.) \r\n return Mo + 2π*(t - epoch)/T\r\nend\r\n\r\nfunction time_to_mean(t, orb::Orbit)\r\n M = time_to_mean(t, orb.period, orb.Mo, orb.epoch)\r\n if orb.e < 1\r\n return wrap_angle(M)\r\n else\r\n return M\r\n end\r\nend\r\n\r\n\"\"\"\r\n t = mean_to_time(M, T, Mo=0, epoch=0, tmin=nothing)\r\n\r\nComputes the time at which an orbit with period `T`, mean anomaly\r\nat epoch `Mo`, and epoch `epoch` has the specified mean anomaly `M`.\r\n\r\nBecause this time is not uniquely defined for elliptical orbits, `tmin` can be specified\r\nto ensure that the returned time falls between `tmin` and `tmin + T`.\r\n\"\"\"\r\nfunction mean_to_time(M, T, Mo=0, epoch=0, tmin=nothing) \r\n t = epoch + (M-Mo)*T/(2π)\r\n if tmin === nothing\r\n return t\r\n else\r\n return t + T*ceil((tmin-t)/T)\r\n end\r\nend\r\n\r\nmean_to_time(M, orb::Orbit, tmin=nothing) = mean_to_time(M, orb.period, orb.Mo, orb.epoch, orb.e<1 ? tmin : nothing)\r\n\r\n\"\"\"\r\n θ = mean_to_true(M, e)\r\n\r\nComputes the true anomaly of an orbit with eccentricity `e` at the specified mean anomaly `M`.\r\n\"\"\"\r\nfunction mean_to_true(M, e)\r\n if e == 1 # Parabolic case\r\n throw(ArgumentError(\"Parabolic case (e=1) not implemented\"))\r\n elseif e < 1 # Elliptical case\r\n E = kepler(M, e)\r\n return 2*atan(√(1+e)*sin(E/2), √(1-e)*cos(E/2))\r\n else # Hyperbolic case\r\n H = kepler(M, e)\r\n return 2*atan(√(e+1)*sinh(H/2), √(e-1)*cosh(H/2))\r\n end\r\nend\r\n\r\n\"\"\"\r\n θ = time_to_true(t, e, T, Mo=0., epoch=0.)\r\n\r\nComputes the mean anomaly of an orbit with eccentricity `e` and period `T` from a time `t`, the orbit's mean anomaly\r\nat epoch `Mo`, and the epoch.\r\n\"\"\"\r\nfunction time_to_true(t, e, T, Mo=0, epoch=0)\r\n M = time_to_mean(t, T, Mo, epoch)\r\n return mean_to_true(M, e)\r\nend\r\n\r\ntime_to_true(t, orb::Orbit) = time_to_true(t, orb.e, orb.period, orb.Mo, orb.epoch)\r\n\r\n\"\"\"\r\n M = true_to_mean(θ, e)\r\n\r\nComputes the mean anomaly of an orbit with eccentricity `e` at the specified true anomaly `θ`.\r\n\"\"\"\r\nfunction true_to_mean(θ, e)\r\n if e == 1 # Parabolic case\r\n throw(ArgumentError(\"Parabolic case (e=1) not implemented\"))\r\n elseif e < 1 # Elliptical case\r\n E = 2*atan(sin(θ/2)*√(1-e), cos(θ/2)*√(1+e))\r\n return wrap_angle(E - e*sin(E))\r\n else # Hyperbolic case\r\n H = 2*atanh(tan(θ/2)*√((e-1)/(e+1)))\r\n # H = copysign(acosh((e+cos(θ))/(1+e*cos(θ))), θ)\r\n return e*sinh(H) - H\r\n end\r\nend\r\n\r\ntrue_to_mean(θ, orb::Orbit) = true_to_mean(θ, orb.e)\r\n\r\n\"\"\"\r\n t = true_to_time(θ, e, T, Mo=0, epoch=0, tmin=nothing)\r\n\r\nComputes the time at which an orbit with eccentricity `e`, period `T`, mean anomaly\r\nat epoch `Mo`, and epoch `epoch` has the specified true anomaly `θ`.\r\n\r\nBecause this time is not uniquely defined for elliptical orbits, `tmin` can be specified\r\nto ensure that the returned time falls between `tmin` and `tmin + T`.\r\n\"\"\"\r\nfunction true_to_time(θ, e, T, Mo=0, epoch=0, tmin=nothing)\r\n M = true_to_mean(θ, e)\r\n return mean_to_time(M, T, Mo, epoch, e<1 ? tmin : nothing)\r\nend\r\n\r\n# true_to_time(θ, e, a, μ, Mo=0, epoch=0, tmin=nothing) = true_to_time(θ, e, period(a, μ), Mo, epoch, tmin)\r\ntrue_to_time(θ, orb::Orbit, tmin=nothing) = true_to_time(θ, orb.e, orb.period, orb.Mo, orb.epoch, tmin)\r\n", "meta": {"hexsha": "664b60248f45fe8e379dc86bfc188b7a0db5699e", "size": 3519, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/anomaly.jl", "max_stars_repo_name": "tmcgrath325/KeplerTools.jl", "max_stars_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/anomaly.jl", "max_issues_repo_name": "tmcgrath325/KeplerTools.jl", "max_issues_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-08-19T16:28:02.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-19T16:28:17.000Z", "max_forks_repo_path": "src/anomaly.jl", "max_forks_repo_name": "tmcgrath325/KeplerTools.jl", "max_forks_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.1981132075, "max_line_length": 117, "alphanum_fraction": 0.6109690253, "num_tokens": 1151, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9597620585273153, "lm_q2_score": 0.872347368040789, "lm_q1q2_score": 0.8372459057017133}} {"text": "using PyPlot\nusing OffsetArrays\n\nfunction bary_grid(n, a)\n x = OffsetArray{Float64}(undef, 0:n, 0:n)\n y = OffsetArray{Float64}(undef, 0:n, 0:n)\n pts = OffsetArray(range(0, 1, length=n+1), 0:n)\n for i = 0:n\n ξ₁ = pts[i]\n for j = 0:n-i\n ξ₂ = pts[j]\n ξ₃ = 1 - ξ₁ - ξ₂\n x[i,j] = ξ₁ * a[1,1] + ξ₂ * a[1,2] + ξ₃ * a[1,3]\n y[i,j] = ξ₁ * a[2,1] + ξ₂ * a[2,2] + ξ₃ * a[2,3]\n end\n for j = n-i+1:n\n x[i,j] = NaN\n y[i,j] = NaN\n end\n end\n return x, y\nend\n\nfunction bary_func(f, n, a)\n z = OffsetArray{Float64}(undef, 0:n, 0:n)\n pts = OffsetArray(range(0, 1, length=n+1), 0:n)\n ξ = Vector{Float64}(undef, 3)\n for i = 0:n\n ξ[1] = pts[i]\n for j = 0:n-i\n ξ[2] = pts[j]\n ξ[3] = 1 - ξ[1] - ξ[2]\n z[i,j] = f(ξ)\n end\n end\n return z\nend\n\nfunction bary_plot(x, y, z)\n n = size(x, 1) - 1\n xd = [ x[i,n-i] for i = 0:n ]\n yd = [ y[i,n-i] for i = 0:n ]\n zd = [ z[i,n-i] for i = 0:n ]\n idx = [1, 2, 3, 1]\n plot3D(a[1,idx], a[2,idx], zeros(4), \"k\")\n for i = 0:n\n plot3D(x[i,0:n-i], y[i,0:n-i], z[i,0:n-i], \"c\")\n end\n for j = 0:n\n plot3D(x[0:n-j,j], y[0:n-j,j], z[0:n-j,j], \"c\")\n end\n for i = 0:n\n plot3D(xd, yd, zd, \"c\")\n end\n axis(\"off\")\n# xlabel(L\"$x$\")\n# ylabel(L\"$y$\")\nend\n\nfunction midpoints(a)\n prev = [ 3, 1, 2 ]\n succ = [ 2, 3, 1 ]\n m = Matrix{Float64}(undef, 2, 3)\n for i = 1:3\n m[:,i] = ( a[:,prev[i]] + a[:,succ[i]] ) / 2\n end\n return m\nend\n\nψ₁(ξ) = 2 * ξ[1] * ( ξ[1] - 0.5 )\nψ₂(ξ) = 2 * ξ[2] * ( ξ[2] - 0.5 )\nψ₃(ξ) = 2 * ξ[3] * ( ξ[3] - 0.5 )\nψ₄(ξ) = 4 * ξ[2] * ξ[3]\nψ₅(ξ) = 4 * ξ[3] * ξ[1]\nψ₆(ξ) = 4 * ξ[1] * ξ[2]\n\na = Float64[ 0 1 2\n 2 0 3 ]\nm = midpoints(a)\nn = 20\nx, y = bary_grid(n, a)\n\nfigure(1)\nz = bary_func(ψ₁, n, a)\nbary_plot(x, y, z)\nplot3D(a[1,1], a[2,1], \"ro\")\nsavefig(\"psi1.pdf\")\n\nfigure(2)\nz = bary_func(ψ₂, n, a)\nbary_plot(x, y, z)\nplot3D(a[1,2], a[2,2], \"ro\")\nsavefig(\"psi2.pdf\")\n\nfigure(3)\nz = bary_func(ψ₃, n, a)\nbary_plot(x, y, z)\nplot3D(a[1,3], a[2,3], \"ro\")\nsavefig(\"psi3.pdf\")\n\nfigure(4)\nz = bary_func(ψ₄, n, a)\nbary_plot(x, y, z)\nplot3D(m[1,1], m[2,1], \"ro\")\nsavefig(\"psi4.pdf\")\n\nfigure(5)\nz = bary_func(ψ₅, n, a)\nbary_plot(x, y, z)\nplot3D(m[1,2], m[2,2], \"ro\")\nsavefig(\"psi5.pdf\")\n\nfigure(6)\nz = bary_func(ψ₆, n, a)\nbary_plot(x, y, z)\nplot3D(m[1,3], m[2,3], \"ro\")\nsavefig(\"psi6.pdf\")\n", "meta": {"hexsha": "71658b37f8b529e95806166c83c2401f1fce5724", "size": 2490, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/chap6/quad_shape_funcs/quadratic_shape_funcs.jl", "max_stars_repo_name": "billmclean/ComputationalMathsNotes", "max_stars_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-30T21:30:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T21:30:20.000Z", "max_issues_repo_path": "src/chap6/quad_shape_funcs/quadratic_shape_funcs.jl", "max_issues_repo_name": "billmclean/ComputationalMathsNotes", "max_issues_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/chap6/quad_shape_funcs/quadratic_shape_funcs.jl", "max_forks_repo_name": "billmclean/ComputationalMathsNotes", "max_forks_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.1016949153, "max_line_length": 60, "alphanum_fraction": 0.4602409639, "num_tokens": 1231, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9559813538993888, "lm_q2_score": 0.8757869803008765, "lm_q1q2_score": 0.8372360231554893}} {"text": "doc\"\"\"\n second_derivative(f::Function, x::AbstractFloat)\n\n# Description\n\nEvaluate the second derivative of `f` at `x` using finite differences. In\nmathematical notation, we calculate,\n\n```math\n\\frac{f(x + \\epsilon) - 2 f(x) + f(x - \\epsilon)}{\\epsilon^2},\n```\n\nwhere ``\\epsilon`` is chosen to be small enough to approximate the second\nderivative, but not so small as to suffer from extreme numerical inaccuracy.\n\n# Arguments\n\n* `f::Function`: The function to be differentiated.\n* `x::AbstractFloat`: The point at which to evaluate the derivative of `f`. Its\n type must implement `eps`.\n\n# Returns\n\n* `y::Real`: The second derivative of `f` evaluated at `x`.\n\n# Examples\n\n```jl\nimport FiniteDiff: second_derivative\ny = second_derivative(sin, 1.0)\n```\n\"\"\"\n@inline function second_derivative(f::Function, x::AbstractFloat)::Real\n ϵ = step_size(HessianMode(), x)\n return (f(x + ϵ) - 2 * f(x) + f(x - ϵ)) / (ϵ * ϵ)\nend\n\ndoc\"\"\"\n second_derivative!(\n output::AbstractArray,\n f::Function,\n x::AbstractFloat,\n )\n\n# Description\n\nEvaluate the second derivative of `f` at `x` using finite differences. In\nmathematical notation, we calculate,\n\n```math\n\\frac{f(x + \\epsilon) - 2 f(x) + f(x - \\epsilon)}{\\epsilon^2},\n```\n\nwhere ``\\epsilon`` is chosen to be small enough to approximate the second\nderivative, but not so small as to suffer from extreme numerical inaccuracy.\n\nThe first argument, `output`, will be mutated so that the value of the second\nderivative is its first element.\n\n# Arguments\n\n* `output::AbstractArray`: An array whose first element will be mutated.\n* `f::Function`: The function to be differentiated.\n* `x::AbstractFloat`: The point at which to evaluate the derivative of `f`. Its\n type must implement `eps`.\n\n# Returns\n\n* `void::Nothing`: This function is called for its side-effects.\n\n# Examples\n\n```jl\nimport FiniteDiff: second_derivative\ny = Array(Float64, 1)\nsecond_derivative!(y, sin, 1.0)\n```\n\"\"\"\nfunction second_derivative!(\n output::AbstractArray,\n f::Function,\n x::AbstractFloat,\n)::Void\n output[1] = second_derivative(f, x)\n return\nend\n\n\"\"\"\n second_derivative(f::Function; mutates::Bool=false)\n\n# Description\n\nConstruct a new function that will evaluate the second derivative of `f` at\nany point `x`. A keyword argument specifies whether the resulting function\nshould be mutating or non-mutating.\n\n# Arguments\n\n* `f::Function`: The function to be differentiated.\n\n# Keyword Arguments\n\n* `mutates::Bool = false`: Determine whether the resulting function will mutate\n its inputs or will be a pure function.\n\n# Returns\n\n* `void::Nothing`: This function is called for its side-effects.\n\n# Examples\n\n```jl\nimport FiniteDiff: second_derivative\nf′′ = second_derivative(sin)\nf′′(1.0)\n```\n\"\"\"\nfunction second_derivative(f::Function; mutates::Bool=false)::Function\n if mutates\n return (output, x) -> second_derivative!(output, f, x)\n else\n return (x, ) -> second_derivative(f, x)\n end\nend\n", "meta": {"hexsha": "a8b16c95b6a3ed622a28db881e763674bb272c19", "size": 2984, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/second_derivative.jl", "max_stars_repo_name": "johnmyleswhite/FiniteDiff.jl", "max_stars_repo_head_hexsha": "d9c3da50b0c214b7be008b60cb04d6701548ff92", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2017-01-19T14:57:24.000Z", "max_stars_repo_stars_event_max_datetime": "2018-07-27T06:04:07.000Z", "max_issues_repo_path": "src/second_derivative.jl", "max_issues_repo_name": "johnmyleswhite/FiniteDiff.jl", "max_issues_repo_head_hexsha": "d9c3da50b0c214b7be008b60cb04d6701548ff92", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2016-08-17T22:35:39.000Z", "max_issues_repo_issues_event_max_datetime": "2019-06-22T16:09:43.000Z", "max_forks_repo_path": "src/second_derivative.jl", "max_forks_repo_name": "johnmyleswhite/FiniteDiff.jl", "max_forks_repo_head_hexsha": "d9c3da50b0c214b7be008b60cb04d6701548ff92", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-01-16T20:59:17.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-18T22:51:01.000Z", "avg_line_length": 23.872, "max_line_length": 79, "alphanum_fraction": 0.6987265416, "num_tokens": 767, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.953966093674472, "lm_q2_score": 0.8774767986961401, "lm_q1q2_score": 0.8370831139421379}} {"text": "###############################################################################\n#\n# Linear regression of Y~X\n#\n###############################################################################\n\"\"\"\n linear_df_xy(df::DataFrame, intercept::Bool)\n\nLinear fit the X and Y in a given DataFrame, given\n- `df` DataFrame that use `X` and `Y` to store values\n- `intercept` Whether using intercept when making the regression\n\"\"\"\nfunction linear_df_xy(df::DataFrame, intercept::Bool)\n # run the fitting\n if intercept\n lr = lm(@formula(Y ~ X), df);\n else\n lr = lm(@formula(Y ~ 0 + X), df);\n end\n\n return lr\nend\n\n\n\n\n\"\"\"\n line_regress(\n xs::Array,\n ys::Array;\n intercept::Bool = true,\n sorting::Bool = true\n )\n\nMake linear regression and return the fitted results, given\n- `xs` Array of x, can be NaN\n- `ys` Array of y, can be NaN\n- `intercept` Optional: if true use intercept in the fitting\n- `sorting` Optional: if true, sort the values\n\"\"\"\nfunction line_regress(\n xs::Array,\n ys::Array;\n intercept::Bool = true,\n sorting::Bool = true\n)\n # filter out NaN from the lists\n new_x = Float64[];\n new_y = Float64[];\n for (x,y) in zip(xs, ys)\n if ~isnan(x) && ~isnan(y)\n push!(new_x, x);\n push!(new_y, y);\n end\n end\n\n # put new_x and new_y into a DataFrame\n df = DataFrame(X=new_x, Y=new_y);\n sorting ? sort!(df) : nothing;\n\n # run the fitting\n lr = linear_df_xy(df, intercept);\n _r2 = adjr2(lr);\n\n # create items for the struct\n if intercept\n _inter = coef(lr)[1];\n _slope = coef(lr)[2];\n _inter_p = coeftable(lr).cols[4][1];\n _slope_p = coeftable(lr).cols[4][2];\n _inter_ci = confint(lr)[1,:];\n _slope_ci = confint(lr)[2,:];\n else\n _inter = Inf;\n _slope = coef(lr)[1];\n _inter_p = Inf;\n _slope_p = coeftable(lr).cols[4][1];\n _inter_ci = [-Inf, Inf];\n _slope_ci = confint(lr)[1,:];\n end\n\n # calculate the predictions\n if intercept\n mt = [ones(length((df).X)) (df).X];\n else\n mt = (df).X[:,:];\n end\n pred = predict(lr, mt; interval=:confidence);\n df[!,\"predY\" ] = pred.prediction;\n df[!,\"lowerY\"] = pred.lower[:,1];\n df[!,\"upperY\"] = pred.upper[:,1];\n\n return LinearRegressionResult(\n lr,\n _r2,\n _inter,\n _slope,\n _inter_p,\n _slope_p,\n _inter_ci,\n _slope_ci,\n df\n )\nend\n", "meta": {"hexsha": "440a76b132435685da90228841e0bf91292d372b", "size": 2651, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/regression/linear.jl", "max_stars_repo_name": "Yujie-W/PlotPlants.jl", "max_stars_repo_head_hexsha": "8ad793bfccb2f1158d39724d930c69451acb8534", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-05T21:20:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-05T21:20:06.000Z", "max_issues_repo_path": "src/regression/linear.jl", "max_issues_repo_name": "Yujie-W/PlotPlants.jl", "max_issues_repo_head_hexsha": "8ad793bfccb2f1158d39724d930c69451acb8534", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 31, "max_issues_repo_issues_event_min_datetime": "2020-10-01T01:51:29.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-28T15:30:06.000Z", "max_forks_repo_path": "src/regression/linear.jl", "max_forks_repo_name": "Yujie-W/PlotPlants.jl", "max_forks_repo_head_hexsha": "8ad793bfccb2f1158d39724d930c69451acb8534", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.2476190476, "max_line_length": 79, "alphanum_fraction": 0.4903809883, "num_tokens": 690, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9539660976007597, "lm_q2_score": 0.877476785879798, "lm_q1q2_score": 0.8370831051610084}} {"text": "### A Pluto.jl notebook ###\n# v0.16.1\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 987c5c4d-3257-41cf-b8a9-7b3e814ba61b\nbegin\n\tusing Pkg\n\tPkg.add(\"Plots\")\n\tusing Plots\nend\n\n# ╔═╡ 9215d698-38ed-11ec-1e52-45e96b1bc12e\nmd\"\"\"\n# Chapter 1\n## Section 1\n\nTaylor series in the neighborhood of x = a\n\n$$f(x) = \\sum_{n=0}^{\\infty}\\frac{f^{(n)}(a)}{n!}(x-a)^n$$\n\nNear a = 0 and for x = $\\pm$h we can write at the third order\n\n$$f(h) = f_0 + hf' + \\frac{h^2}{2}f''$$\n$$f(-h) = f_0 - hf' + \\frac{h^2}{2}f''$$\n\nIf we sub\n\n$$f(h) - f(-h) = 2hf'$$\n$$\\Rightarrow f' = \\frac{f(h) - f(-h)}{2h}$$\n\nCentered difference\n\"\"\"\n\n# ╔═╡ c7732e76-19b9-4aa6-a61e-5b142dae1d3e\nfunction funcdiff(f, x::T, dx::T)::T where T <: Number\n\t(f(x + dx) - f(x - dx)) / (2 * dx)\nend\n\n# ╔═╡ 5137159c-90a8-49a5-9aa3-5356f3137662\nbegin\n\tx = collect(0:0.01:7)\n\ty = funcdiff.(sin, x, 0.01) \n\tplot(x, y)\nend\n\n# ╔═╡ 72affc38-4df9-48af-98b7-0b1b92d9d980\nfunction arraydiff(x::Vector{T}, dx::T)::Vector{T} where T <: Number\n\t(x[2:end] - x[1:end-1]) / (2 * dx)\nend\n\n# ╔═╡ 42d3c309-bb3a-479b-91fd-3d26adda0555\nbegin\n\tu = collect(0:0.01:7)\n\tv = arraydiff(sin.(x), 0.01) \n\tplot(u[2:end], v)\nend\n\n# ╔═╡ cb996ce0-a6f1-4176-b226-b1e4b322fabb\nmd\"\"\"\nWe can apply this method to other orders of derivations and order of the Taylor series.\n\nIncreasing the order of the series will increase the accuracy but it will also increase the calculation time.\n\nThe value of h also influences the precision\n\nExercice: For the same function and interval, evaluate the mean error with the analytic derivation.\n\"\"\"\n\n# ╔═╡ 7b6412e5-2f4f-409a-b959-6c9c4eb29a9e\nbegin\n\tallh = collect(0.01:0.01:1)\n\talleps = Float64[]\n\tfor h in allh\n\t\tx = 0:h:7\n\t\tcosine = cos.(x)\n\t\tsinus = funcdiff.(sin, x, h)\n\t\teps = sum(sqrt.(cosine.^2 + sinus.^2)) / length(x)\n\t\tappend!(alleps, eps)\n\tend\n\tplot(allh, alleps)\nend\n\n# ╔═╡ Cell order:\n# ╠═987c5c4d-3257-41cf-b8a9-7b3e814ba61b\n# ╟─9215d698-38ed-11ec-1e52-45e96b1bc12e\n# ╠═c7732e76-19b9-4aa6-a61e-5b142dae1d3e\n# ╠═5137159c-90a8-49a5-9aa3-5356f3137662\n# ╠═72affc38-4df9-48af-98b7-0b1b92d9d980\n# ╠═42d3c309-bb3a-479b-91fd-3d26adda0555\n# ╟─cb996ce0-a6f1-4176-b226-b1e4b322fabb\n# ╠═7b6412e5-2f4f-409a-b959-6c9c4eb29a9e\n", "meta": {"hexsha": "25b0c025c2590b340411678f3943a98d25cb319a", "size": 2169, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "differentiation.jl", "max_stars_repo_name": "zazbone/ComputationalPhysics", "max_stars_repo_head_hexsha": "be3c4dd0b75fefb71aab5ccbd91fa1a5e837b598", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "differentiation.jl", "max_issues_repo_name": "zazbone/ComputationalPhysics", "max_issues_repo_head_hexsha": "be3c4dd0b75fefb71aab5ccbd91fa1a5e837b598", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "differentiation.jl", "max_forks_repo_name": "zazbone/ComputationalPhysics", "max_forks_repo_head_hexsha": "be3c4dd0b75fefb71aab5ccbd91fa1a5e837b598", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.0744680851, "max_line_length": 109, "alphanum_fraction": 0.6648224988, "num_tokens": 1041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9539660976007596, "lm_q2_score": 0.8774767826757123, "lm_q1q2_score": 0.837083102104419}} {"text": "using Polynomials\n\n\n\"\"\"\nCalculate Chebyshev polynomials coefficients\n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\nfunction poly_chebyshev{T<:Number}(n, ::Type{T}=Int, var=:x)\n\n if n==0\n return Poly{T}([one(T)], var)\n elseif n==1\n return Poly{T}([zero(T), one(T)], var) \n end\n\n px = Poly{T}([zero(T), one(T)], var)\n T0 = Poly{T}([one(T)], var)\n T1 = px\n \n for i = 2:n\n T2 = 2*px*T1 - T0\n T0 = T1\n T1 = T2\n end\n\n return T1\n\nend\n\n\"\"\"\nCalculate derivative of Chebyshev polynomials coefficients\n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\npoly_dchebyshev{T<:Number}(n, ::Type{T}=Int, var=:x) = polyder(poly_chebyshev(n,T, var))\n\n\n\n\"\"\"\nCalculate the coefficients of Chebyshev polynomials of the second kind\n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\nfunction poly_chebyshev2{T<:Number}(n, ::Type{T}=Int, var=:x)\n\n if n==0\n return Poly{T}([one(T)], var)\n elseif n==1\n return Poly{T}([zero(T), 2*one(T)], var) \n end\n\n px = Poly{T}([zero(T), one(T)], var)\n U0 = Poly{T}([one(T)], var)\n U1 = Poly{T}([zero(T), 2*one(T)], var)\n \n for i = 2:n\n U2 = 2*px*U1 - U0\n U0 = U1\n U1 = U2\n end\n\n return U1\n\nend\n\n\"\"\"\nCalculate the coefficients of the derivative of Chebyshev polynomials of the second kind\n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\npoly_dchebyshev2{T<:Number}(n, ::Type{T}=Int, var=:x) = polyder(poly_chebyshev2(n,T, var))\n\n\n\"\"\"\nCalculate the coefficients of Legendre polynomials \n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\nfunction poly_legendre{T<:Number}(n, ::Type{T}=Float64, var=:x)\n\n\n if n==0\n return Poly{T}([one(T)], var)\n elseif n==1\n return Poly{T}([zero(T), one(T)], var) \n end\n\n px = Poly{T}([zero(T), one(T)], var)\n p0 = Poly{T}([one(T)], var)\n p1 = px\n\n for i = 2:n\n p2 = ( (2i-1)*px*p1 - (i-1)*p0 ) / i\n p0 = p1\n p1 = p2\n end\n\n return p1\nend\n\n\n\"\"\"\nCalculate the coefficients of the derivativ of Legendre polynomials \n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\npoly_dlegendre{T<:Number}(n, ::Type{T}=Int, var=:x) = polyder(poly_legendre(n,T, var))\n\n\n\n\n\n\"\"\"\nCalculate the coefficients of Jacobi polynomials \n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\nfunction poly_jacobi{T<:Number}(n, a, b, ::Type{T}=Float64, var=:x)\n\n ox = one(T)\n zx = zero(T)\n if n==0\n return Poly{T}([one(T)], var)\n elseif n==1\n return Poly{T}([(a-b)/2, (a+b+2)/2], var)\n end\n p0 = Poly{T}([one(T)], var)\n p1 = Poly{T}([(a-b)/2, (a+b+2)/2], var)\n px = Poly{T}([zero(T), one(T)], var)\n \n for i = 1:(n-1)\n\ta1 = 2*(i+1)*(i+a+b+1)*(2*i+a+b);\n\ta2 = (2*i+a+b+1)*(a*a-b*b);\n\ta3 = (2*i+a+b)*(2*i+a+b+1)*(2*i+a+b+2);\n\ta4 = 2*(i+a)*(i+b)*(2*i+a+b+2);\n\tp2 = ox/a1*( (a2 + a3*px)*p1 - a4*p0);\n\n p0 = p1\n p1 = p2\n end\n\n return p1\nend\n\n\n\"\"\"\nCalculate the coefficients of the derivative of Jacobi polynomials \n\nInstead of computing the polynomial at a specific point, \nuse `Polynomials` package to compute the actual polynomial\n\"\"\"\npoly_djacobi{T<:Number}(n, a, b, ::Type{T}=Float64, var=:x) = polyder(poly_jacobi(n,a,b,T,var))\n", "meta": {"hexsha": "c11e2152e6f2784176306c011ac00d7398731a31", "size": 3699, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/polynomials.jl", "max_stars_repo_name": "tkf/Jacobi.jl", "max_stars_repo_head_hexsha": "8a2d769cd78cb0bb9470082d1477fad485866b40", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/polynomials.jl", "max_issues_repo_name": "tkf/Jacobi.jl", "max_issues_repo_head_hexsha": "8a2d769cd78cb0bb9470082d1477fad485866b40", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/polynomials.jl", "max_forks_repo_name": "tkf/Jacobi.jl", "max_forks_repo_head_hexsha": "8a2d769cd78cb0bb9470082d1477fad485866b40", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.9751552795, "max_line_length": 95, "alphanum_fraction": 0.6134090295, "num_tokens": 1185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9539660949832345, "lm_q2_score": 0.8774767794716264, "lm_q1q2_score": 0.8370830967510122}} {"text": "########################################\n###### Vectorized \"dot\" operators ######\n########################################\n\n# For every binary operation like ^, there is a corresponding \"dot\" operation .^ that is \n# automatically defined to perform ^ element-by-element on arrays. For example, [1,2,3] ^ 3 \n# is not defined, since there is no standard mathematical meaning to \"cubing\" a (non-square) array, \n# but [1,2,3] .^ 3 is defined as computing the elementwise (or \"vectorized\") result [1^3, 2^3, 3^3]. \n# Similarly for unary operators like ! or √, there is a corresponding .√ that applies the operator elementwise.\n\n# [1,2,3] .^ 3\nprintln([1,2,3] .^ 3) # [1, 8, 27]\n\n\n", "meta": {"hexsha": "2053e7e5d3cef8267e3e1e93b2849bb68c08bfaf", "size": 676, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "basics/vectorized-dot-op.jl", "max_stars_repo_name": "mauricioriva/learn-julia", "max_stars_repo_head_hexsha": "f4ff81ff412ba4f1243a8cee1123ebc8b3335f01", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "basics/vectorized-dot-op.jl", "max_issues_repo_name": "mauricioriva/learn-julia", "max_issues_repo_head_hexsha": "f4ff81ff412ba4f1243a8cee1123ebc8b3335f01", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "basics/vectorized-dot-op.jl", "max_forks_repo_name": "mauricioriva/learn-julia", "max_forks_repo_head_hexsha": "f4ff81ff412ba4f1243a8cee1123ebc8b3335f01", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.0666666667, "max_line_length": 111, "alphanum_fraction": 0.5931952663, "num_tokens": 191, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9353465116437761, "lm_q2_score": 0.894789464699728, "lm_q1q2_score": 0.8369382044624923}} {"text": "# Rank-based correlations\n#\n# - Spearman's correlation\n# - Kendall's correlation\n#\n\n#######################################\n#\n# Spearman correlation\n#\n#######################################\n\ncorspearman(x::RealVector, y::RealVector) = cor(tiedrank(x), tiedrank(y))\n\ncorspearman(X::RealMatrix, Y::RealMatrix) = cor(mapslices(tiedrank, X, 1), mapslices(tiedrank, Y, 1))\ncorspearman(X::RealMatrix, y::RealVector) = cor(mapslices(tiedrank, X, 1), tiedrank(y))\ncorspearman(x::RealVector, Y::RealMatrix) = cor(tiedrank(x), mapslices(tiedrank, Y, 1))\n\ncorspearman(X::RealMatrix) = (Z = mapslices(tiedrank, X, 1); cor(Z, Z))\n\n\n#######################################\n#\n# Kendall correlation\n#\n#######################################\n\n# Knigh JASA (1966)\n\nfunction corkendall!(x::RealVector, y::RealVector)\n if any(isnan(x)) || any(isnan(y)) return NaN end\n n = length(x)\n if n != length(y) error(\"Vectors must have same length\") end\n \n # Initial sorting\n pm = sortperm(y)\n x[:] = x[pm]\n y[:] = y[pm]\n pm[:] = sortperm(x)\n x[:] = x[pm]\n\n # Counting ties in x and y\n iT = 1\n nT = 0\n iU = 1\n nU = 0\n for i = 2:n\n if x[i] == x[i-1] \n iT += 1\n else\n nT += iT*(iT - 1)\n iT = 1\n end\n if y[i] == y[i-1]\n iU += 1\n else\n nU += iU*(iU - 1)\n iU = 1\n end\n end\n if iT > 1 nT += iT*(iT - 1) end\n nT = div(nT,2)\n if iU > 1 nU += iU*(iU - 1) end\n nU = div(nU,2)\n\n # Sort y after x\n y[:] = y[pm]\n\n # Calculate double ties\n iV = 1\n nV = 0\n jV = 1\n for i = 2:n\n if x[i] == x[i-1] && y[i] == y[i-1] \n iV += 1\n else\n nV += iV*(iV - 1)\n iV = 1\n end\n end\n if iV > 1 nV += iV*(iV - 1) end\n nV = div(nV,2)\n\n nD = div(n*(n - 1),2)\n return (nD - nT - nU + nV - 2swaps!(y)) / (sqrt(nD - nT) * sqrt(nD - nU))\nend\n\n\ncorkendall(x::RealVector, y::RealVector) = corkendall!(float(copy(x)), float(copy(y)))\n\ncorkendall(X::RealMatrix, y::RealVector) = Float64[corkendall!(float(X[:,i]), float(copy(y))) for i in 1:size(X, 2)]\n\ncorkendall(x::RealVector, Y::RealMatrix) = (n = size(Y,2); reshape(Float64[corkendall!(float(copy(x)), float(Y[:,i])) for i in 1:n], 1, n))\n\ncorkendall(X::RealMatrix, Y::RealMatrix) = Float64[corkendall!(float(X[:,i]), float(Y[:,j])) for i in 1:size(X, 2), j in 1:size(Y, 2)]\n\nfunction corkendall(X::RealMatrix)\n n = size(X, 2)\n C = eye(n)\n for j = 2:n\n for i = 1:j-1\n C[i,j] = corkendall!(X[:,i],X[:,j])\n C[j,i] = C[i,j]\n end\n end\n return C\nend\n\n# Auxilliary functions for Kendall's rank correlation\n\nfunction swaps!(x::RealVector)\n n = length(x)\n if n == 1 return 0 end\n n2 = div(n, 2)\n xl = sub(x, 1:n2)\n xr = sub(x, n2+1:n)\n nsl = swaps!(xl)\n nsr = swaps!(xr)\n sort!(xl)\n sort!(xr)\n return nsl + nsr + mswaps(xl,xr)\nend\n\nfunction mswaps(x::RealVector, y::RealVector)\n i = 1\n j = 1\n nSwaps = 0\n n = length(x)\n while i <= n && j <= length(y)\n if y[j] < x[i]\n nSwaps += n - i + 1\n j += 1\n else\n i += 1\n end\n end\n return nSwaps\nend\n\n", "meta": {"hexsha": "19c5346584cabd8ea2c68f0086371cf05b5df8a3", "size": 3238, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/rankcorr.jl", "max_stars_repo_name": "jpata/StatsBase.jl", "max_stars_repo_head_hexsha": "3148a43686962e096f3ccbc063264908e42fb2c5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/rankcorr.jl", "max_issues_repo_name": "jpata/StatsBase.jl", "max_issues_repo_head_hexsha": "3148a43686962e096f3ccbc063264908e42fb2c5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/rankcorr.jl", "max_forks_repo_name": "jpata/StatsBase.jl", "max_forks_repo_head_hexsha": "3148a43686962e096f3ccbc063264908e42fb2c5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.1285714286, "max_line_length": 139, "alphanum_fraction": 0.4873378629, "num_tokens": 1161, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9449947132556618, "lm_q2_score": 0.8856314723088733, "lm_q1q2_score": 0.8369170592247133}} {"text": "### A Pluto.jl notebook ###\n# v0.15.1\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ a6e71018-ab69-42d0-9a15-ffb477b4bd64\nbegin\n\tusing ModelingToolkit\n\tusing DifferentialEquations\n\tusing Plots\nend\n\n# ╔═╡ d4f1e3a9-d1c8-4e3e-8b88-f0a901a894f7\nmd\"\"\"\n# Symbolic ODE Modeling \n\nI recently stumbled upon the [DifferentialEquations.jl](https://github.com/SciML/DifferentialEquations.jl) and [ModelingToolkit.jl](https://github.com/SciML/ModelingToolkit.jl) which are pretty cool packages. \n\nThe idea is that we can represent ordinary differential equation models symbolically using the ModelingToolkit.jl and then use the DifferentialEquations.jl package to find our solutions.\n\nBelow will be a simple example in doing so.\n\"\"\"\n\n# ╔═╡ 656cae30-e450-11eb-266a-f7572588b91b\nmd\"\"\"\n## An example in chemical kinetics\nDetermining the concentration of species involved in reaction kinetics is done by solving a set of ordinary differential equations goverened by [law of mass action](https://en.wikipedia.org/wiki/Law_of_mass_action) which can be expressed as:\n\n```math\n\\begin{align}\n\\frac{dC_i}{dt} &= \\sum_j N_{i,j} K_j \\\\\nK_j &= k_j \\prod_k C_k^{S_j}\n\\end{align}\n```\n\nwhere $N_{i,j}$ is the net stoichiometric value of species $i$ over reactions $j$, $K_j$ is the reaction coefficient, and $S_j$ the stoichiometry of species $k$ in reaction $j$. We can solve the Robertson's 3-species reacting system:\n\n$r_1 = k_1 [A]$\n$r_2 = k_2 [B] [C]$\n$r_3 = k_3 [B]$\n\nThis corresponds to the chemical processes:\n\n$A \\rightarrow B$\n$B + C \\rightarrow A + C$\n$2B \\rightarrow B + C$\n\nOne now needs to determine the concentrations of species A, B, and C as a function of time by solving the ODEs. When a species is consumed/reacted the sign is negative and when it is produced it is positive.\n\n$\\frac{d[A]}{dt} = r_2 - r_1$\n$\\frac{d[B]}{dt} = r_1 - r_2 - r_3$\n$\\frac{d[C]}{dt} = r_3$\n\nThe neat thing is we can solve this in a few lines of code using the [ModelingToolkit.jl](https://github.com/SciML/ModelingToolkit.jl) and [DifferentialEquations.jl](https://github.com/SciML/DifferentialEquations.jl) Julia packages.\n\nThis does the same thing that the [SymPy Code generation tutorial on youtube](https://www.youtube.com/watch?v=5jzIVp6bTy0).\n\"\"\"\n\n# ╔═╡ 53eaf1ae-6d16-4606-b5d2-97fd61ac7da6\nfunction chemicalreaction_ode_solution()\n\t@parameters t k₁ k₂ k₃\n\t@variables A(t) B(t) C(t)\n\tD = Differential(t);\n\t\n\t@named eq = ODESystem([D(A) ~ -A*k₁ + B*C*k₂,\n\t\t\t D(B) ~ A*k₁ - B^2 * k₃ - B*C*k₂,\n\t\t D(C) ~ B^2 * k₃],\n\t t,[A,B,C],[k₁,k₂,k₃])\n\n\t\n\tk0 = [k₁ => 4.0e-2, k₂=>1.0e4, k₃=>3.0e7]\n\tc0 = [A => 1, B=>0, C=>0]\n\tτ = (1.0e-6,1.0e6)\n\tprob = ODEProblem(structural_simplify(eq),\n\t\t c0,τ,k0,jac=true)\n\t\n\t#Use stiff numerical integration solver\n\tsol = solve(prob, TRBDF2())\n\t\n\t#Plotting solution with Plots.jl\n\tp1 = plot(sol,vars=[A,B,C],\n\t\txaxis=:log,\n\t\tlegend=(0.35,0.35),lw=3,size=(1100,400),\n\t\ttitle=\"semi-log scale\")\n\tp2 = plot(sol,vars=[A,B,C],\n\t\txaxis=:log,yaxis=(:log,[1.0e-12,:auto]),\n\t\tlegend=(0.35,0.35),lw=3,size=(1100,400),\n\t\ttitle=\"log-log scale\")\n\ttimep = plot(p1,p2,layout=(1,2),margin=5Plots.mm,\n\t\txlabel=\"time\",ylabel=\"concentration\")\n\t\n\treturn eq,sol,timep\nend;\n\n# ╔═╡ 0267b2d1-f8da-48e8-991b-163497c0cd91\nmd\"\"\"\n!!! note\n\tCreated a function since the scope using Pluto.jl with these packages is a bit strange due to how Pluto.jl evaluates cells.\n\"\"\"\n\n# ╔═╡ 6da13129-b9f4-45c6-89fb-714200bce2d4\neqs,solution,timeplot = chemicalreaction_ode_solution();\n\n# ╔═╡ 65cd1b1f-24df-4a48-bd26-0f394bc06c1f\nmd\"\"\"\nDisplaying the equation generated by the ModelingToolkit:\n\"\"\"\n\n# ╔═╡ 367a7c95-4655-4797-a62e-0f817d527dab\neqs\n\n# ╔═╡ 56ba790d-0c78-46fb-b5b7-4711c77f4b54\nmd\"\"\"\nand now displaying the plots from the solution.\n\"\"\"\n\n# ╔═╡ efe47a33-ea39-4f58-bc26-1517427f43f8\ntimeplot\n\n# ╔═╡ 4f208f0a-7ff4-444d-b271-08f07ec31e72\nmd\"\"\"\nSo thats it! Pretty cool that we can solve this ODE so straightfowardly.\n\n\nThe thing to do is generalize the procedure to create a macro that returns an `ODESystem` for any arbitrary number of reactions and species. This would be similiar to the code below done for `SymPy.jl` but doing it using these packages. \n\n```julia\nusing SymPy\n\nstruct RXN \n\tcoeff::String\n\treactants::Dict\n\tproducts::Dict\nend\n\nfunction getequations(species::Tuple,reactions::Array{RXN})\n\tsymspecies = symbols(join(species,\" \"),real=true,nonnegative=true)\n\tdicspecies = Dict(zip(species,symspecies))\n\tdequations = Dict(s=>Sym(0) for s in species)\n\tcoeffs = Array{Sym}(undef,length(reactions))\n\tfor (i,reaction) in enumerate(reactions)\n\t\tcoeffs[i] = symbols(reaction.coeff)\n\t\tprodstoich = prod([dicspecies[s]^p for (s,p) in reaction.reactants])\n\t\tr = coeffs[i]*prodstoich\n\t\tfor (name,stoich) in reaction.products\n\t\t\tdequations[name] += stoich*r\n\t\tend\n\tend\n\t#Generate array of equations rather than dictionary\n\tequations = [dequations[s] for s in species]\n\treturn (equations,symspecies,coeffs)\nend;\n```\n\"\"\"\n\n# ╔═╡ 4d326031-3ec0-4faa-b6d2-531add859e06\nhtml\"\"\"\n\"\"\"\n\n# ╔═╡ bcbed56c-0c94-11eb-0e8c-2de8d1a20f47\nmd\"\"\"\n\n# Simulation - a small intro\n\n\n## General approach\nWhen a simulation is used to answer a question or to solve a problem, the following steps are involved:\n* Formulate the problem and plan the study\n* Collect the data and formulate the simulation model\n* Check the accuracy of the simulation model (assumptions, limitations etc.)\n* Construct a computer program\n* Test the validity of the simulation model\n* Plan the simulations to be performed\n* Conduct the simulation runs and analyze the results\n* Present the conclusions\n\nDuring this practical session, we will run small simulations going over all these steps.\n\n\n\"\"\"\n\n# ╔═╡ 349b8586-0c96-11eb-2e7b-5b3230d1cf27\nmd\"\"\"\n## Casino game\nA casino offers a coin flipping game with the following rules:\n1. Each play of the game involves repeatedly flipping an unbiased coin \n2. The game stops when the difference between the number of heads tossed and the number of tails is 3\n3. If you decide to play, each flip costs you € 1. You are not allowed to quit during a play of the game.\n4. You receive € 8 at the end of each play of the game.\n\n\nThe problem at hand is to decide whether to play or not and to get an idea of the expected gains and the length of each game. We might also look into the required simulation size and into a parametric study (e.g. influence of the required difference between heads/tails and/or the default gain).\n\n*Tip:* it can be analytically proven that the true mean of the number of flips required for a play of this game is 9. \n\n\n##### Steps in the simulation process\n\n###### Formulate the problem and plan the study:\nthe problem is described in a clear way (cf. suppra). Given these rules, we want to find out\n1. if it is a good idea to play\n2. the value of the expected profit\n\n###### Collect the data and formulate the simulation model:\nLet $X$ be the outcome of one coin flip $\\Rightarrow X\\sim \\mathcal{Be}(1/2)$\n\n###### Check the accuracy of the simulation model:\nOur model is complete, we make no assumptions\n\n###### Construct a computer program\n\"\"\"\n\n# ╔═╡ 3a20552e-0c97-11eb-3f0d-2586b1bb46ed\nbegin\n\t\"\"\"\n\t\tgame(limit::Int=3)\n\t\n\tPlay a game for a given limit. Returns a dict with the scores and the game length\n\t\"\"\"\n\tfunction game(limit::Int=3)\n\t\td = Dict(\"heads\"=>0, \"tails\"=>0, \"length\"=>0)\n\t\twhile abs(d[\"heads\"] - d[\"tails\"]) < limit \n\t\t\td[\"length\"] += 1\n\t\t\tif rand()<=1/2\n\t\t\t\td[\"heads\"]+=1\n\t\t\telse\n\t\t\t\td[\"tails\"]+=1\n\t\t\tend\n\t\tend\n\t\t\n\t\treturn d\n\tend\n\t\n\t\"\"\"\n\t\tsim(N::Int=1000; limit::Int=3)\n\t\n\trun a simulation of the game. \n\t\"\"\"\n\tfunction sim(N::Int=1000; limit::Int=3, gain::Real=8)\n\t\tL = Array{Int64,1}()\n\t\tG = Array{Int64,1}()\n\t\tfor _ in 1:N\n\t\t\tg = game(limit)\n\t\t\tpush!(L,g[\"length\"])\n\t\t\tpush!(G,gain - g[\"length\"])\n\t\tend\n\t\t\n\t\treturn L,G\n\tend\n\n\tnothing\nend\n\n# ╔═╡ b1bae776-0dfe-11eb-291b-53f9a87c4a41\n# other method of creating a game\nbegin\n\tmutable struct Game\n\t\theads::Int\n\t\ttails::Int\n\t\tlength::Int\n\tend\n\t\n\tfunction play!(g::Game, limit::Int=3)\n\t\twhile abs(g.heads - g.tails) < limit \n\t\t\tg.length += 1\n\t\t\tif rand()<=1/2\n\t\t\t\tg.heads += 1\n\t\t\telse\n\t\t\t\tg.tails += 1\n\t\t\tend\n\t\t\t\n\t\tend\n\t\t\n\t\treturn g\n\tend\n\t\n\tg = Game(0,0,0)\n\tplay!(g)\nend\n\n# ╔═╡ 65c6476c-0c98-11eb-2c7d-3bc3ba8d51e2\nmd\"\"\"\n###### Test the validity of the simulation model\nA game should stop when the difference between the amount of heads and tails equals 3\n```julia\nbegin \n\tg = game()\n\t@assert abs(g[\"heads\"] - g[\"tails\"]) == 3\nend\n```\n\"\"\"\n\n# ╔═╡ 5b08b538-0c97-11eb-0319-877b39c4c4ed\nmd\"\"\"\n###### Plan the simulations to be performed\nWe will run a certain number of simulations. At this time we do not have additional information that we can use (optimal number to be determined later).\n\"\"\"\n\n# ╔═╡ 607e4392-0c98-11eb-0a9d-ddfab6b42b26\nmd\"\"\"\n###### Conduct the simulation runs and analyze the results\n\"\"\"\n\n# ╔═╡ 75d35828-0c99-11eb-1888-e9e7be234e89\nbegin\n\t# get data for game length and gains\n\tL,G = sim()\nend;\n\n# ╔═╡ fae1e448-0c98-11eb-3721-dddb169e403f\n# show results\nplot(histogram(L,xlabel=\"Game length\"),histogram(G,xlabel=\"Gains\"),\n label=\"\",grid=false,ylabel=\"Frequency\",size=(600,250) ,bottom_margin=5mm)\n\n# ╔═╡ a39a3a72-0c99-11eb-00df-e92eb7a6ce07\nplot([boxplot(L,ylabel=\"Game length\", ylims=(0,maximum(L)+4)),\n\t boxplot(G,ylabel=\"Gains\")]...,\n label=\"\",grid=false,size=(300,250),xaxis=false)\n\n# ╔═╡ ac87cb8a-0c9a-11eb-21d5-9981e408db29\nmd\"\"\"\n#### Simulation length\nUp to now we have worked with a simulation length 1000. We have no idea about the variation of each simulation result. We investigate this in order to find out how simulation length impacts our estimate.\n\"\"\"\n\n# ╔═╡ 1aa54e8a-0c9b-11eb-3b7c-f3be005991df\nbegin\n\t# evolution of the mean in function of the sample length\n\tx = Array{Int,1}()\n\ty = Array{Float64,1}()\n\tntests = 30\n\tfor N in 10 .^(1:5)\n\t\tfor _ in 1:ntests\n\t\t\t(L,_) = sim(N,limit=3)\n\t\t\tpush!(x,N)\n\t\t\tpush!(y,mean(L))\n\t\tend\n\tend\nend\n\n# ╔═╡ 3236977a-0c9b-11eb-3875-2304eaec0a8e\nbegin\n\tscatter(x,y,xscale=:log10,\n\t\t\txlabel=\"Number of games per simulation\",\n\t\t\tylabel=\"Average game length\",title=\"\",label=\"\")\n\tplot!(x,9*ones(size(x)),color=:red,label=\"\")\n\ttitle!(\"Towards optimal simulation length\\n (30 datapoints per config)\")\nend\n\n# ╔═╡ 7fafde70-0c9c-11eb-0403-9bfb6e42045b\nmd\"\"\"\n#### Parametric study\nsuppose we consider a range of differences [2,4] and a range of gains [4, 12] where we play 1e5 games for each config.\n\"\"\"\n\n# ╔═╡ c8b107ba-0c9d-11eb-11c7-830d112ca82f\nlet\n\tΔ = 1:5\n\tΓ = 0:15\n\tN = Int(1e5)\n\tres = []\n\ti = 1\n\tfor d in Δ\n\t\tfor g in Γ\n\t\t\t_,G = sim(N; limit=d, gain=g)\n\t\t\tpush!(res, mean(G))\n\t\tend\n\tend\n\tblup = reshape(res,:,length(Δ))\n\tp1 = heatmap(Δ, Γ, blup, xlabel=\"diff\", ylabel=\"guaranteed gain\", title=\"mean gain\")\n\tp2 = heatmap(Δ, Γ, blup .>= 0, xlabel=\"diff\", ylabel=\"guaranteed gain\", title=\"positive gain\")\n\tplot(p1,p2, size=(600, 300))\nend\n\n# ╔═╡ 0bfa3ffe-0c9a-11eb-2c99-3d8e57983c4c\nmd\"\"\"\n###### Present the conclusions\nWe find that on average a game takes 9 turns, which will lead you to lose on average € 1. The game could be tempting given that in six out of ten cases the player will actually win (a small amount).\n\"\"\"\n\n# ╔═╡ 8df49b7e-0c9e-11eb-0048-9368c9d7bee9\nmd\"\"\"\n## Project management\nSuppose we want to simulate a industrial process that can be presented as the table shown below:\n\n| Activity | Predecessors | Completion time |\n|----------|--------------|---------------- |\n| A | - | $\\sim \\mathcal{N}(6,1)$ |\n| B | A | $\\sim \\mathcal{U}(6,10)$ |\n| C | A | $\\sim \\mathcal{U}(1.5,2.5)$ |\n| D | B,C | $\\sim \\mathcal{U}(1.5,3)$ |\n| E | D | $\\sim \\mathcal{U}(3,6)$ |\n| F | E | $\\sim \\mathcal{U}(2,5)$ |\n| G | E | $\\sim \\mathcal{U}(3,5)$ |\n| H | F,G | $\\sim \\mathcal{U}(4,7)$ |\n| I | H | $\\sim \\mathcal{U}(5,7)$ |\n| J | H | $\\le 5$ |\n\nWe want to get an idea of:\n1. the mean project completion time\n2. the probability that the project will be finished in less than 36 time units\n3. determine the critical subprocesses with a sensitivity chart (i.e. determine the impact of changing a specific subprocess on the completion time given all the others remain the same)\n\n\"\"\"\n\n# ╔═╡ 7aceebea-0d1d-11eb-05fa-d7374004b881\nmd\"\"\"\n###### Formulate the problem and plan the study\nThe problem and the tasks are described above\n\n###### Collect the data and formulate the simulation Model\nData model (coming from historical data) is known. The duration can be obtained from the a vector of duration times.\n\n###### Construct a computer program\n\"\"\"\n\n# ╔═╡ 8f99cf60-0c9e-11eb-2b6d-b5ca5ed20a65\nbegin\n\t# completion time distributions\n\tD = [Normal(6,1); \n\t\tUniform(6,10); \n\t\tUniform(1.5,2.5); \n\t\tUniform(1.5,4); \t\t \t\t\n\t\tUniform(3,6);\n\t\tUniform(2,5); \n\t\tUniform(3,5); \n\t\tUniform(4,7); \n\t\tUniform(5,7); \n\t\tUniform(0,5)]\n\t\n\t# central positions (for sensitivity analysis)\n\tp = [6; 8; 2; 2.25; 4.5; 3.5; 4; 5.5; 6; 2.5]\n\t\"\"\"\n\t\t duration(t::Array)\n\n\tgenerate a random sample from an array of distributions for the production process\n\t\"\"\"\n\tfunction duration(t::Array)\n\t\tt[1] + max(t[2],t[3]) + t[4] + t[5] + max(t[6], t[7]) + t[8] + max(t[9], t[10])\n\tend\n\t\n\t\"\"\"\n\t\tPM(N::Int)\n\t\n\tRun a Project Management simulation N times that returns a vector of duration times\n\t\"\"\"\n\tfunction PM(D::Array, N::Int=100)\n\t\t# Generate data\n\t\tS = [rand.(D) for _ in 1:N]\n\t\t# get durations\n\t\treturn map(duration, S)\n\tend\n\n\tnothing;\nend\n\n# ╔═╡ 864b050c-0d1e-11eb-03d8-8ffcf23e019e\nmd\"\"\"\n###### Check the accuracy of the simulation model (assumptions, limitations etc.)\nWe use a normal distribution for the first activity, so it is theoretically possible that a negative value for A occurs, which does not make sense. In a similar way, using a uniform distribution sets hard limits for task durations.\n\nYou can make testcases to assert the correctness of the result calculated by the duration function.\n\"\"\"\n\n# ╔═╡ f9562612-0d1e-11eb-1e16-bfa53760c18c\nbegin \n\tv = [0 0 1 0 0 1 0 0 0 0] \n\t@assert duration(v) == 2\nend\n\n# ╔═╡ d18a68b2-0d20-11eb-0f39-150f9bc9e0e9\nmd\"\"\"\n###### Plan the simulations to be performed\nWe will generate a sample of a given length an look at the experimental CDF.\n\"\"\"\n\n# ╔═╡ da7801bc-0ca0-11eb-0480-33f07e16c85b\nlet\n\t# idea mean value\n\tN = 1000; x = 36;\n\tres = PM(D,N)\n\tavg_D = mean(res)\n\t# CDF\n\tp1 = StatsPlots.cdensity(res,label=\"Experimental CDF\",legend=:bottomright)\n\ttitle!(\"P(X <= $(x)): $(sum(res .< x)/N)\")\n\t\n\tplot!([avg_D; avg_D], [0; 1], line=:dash, label=\"mean duration $(round(avg_D,digits=2))\")\n\tylabel!(\"F_{X}(x) \")\n\txlabel!(\"duration\")\n\t# PDF\n\tp2 = StatsPlots.density(res, label=\"Experimental PDF\")\n\txlabel!(\"duration\")\n\tylabel!(\"f_{X}(x)\")\n\t# total plot\n\tplot(p1, p2)\nend\n\n# ╔═╡ 3d5f9648-0d21-11eb-14c4-65704b349ed6\nmd\"\"\"\n###### Conduct the simulation runs and analyze the results\nWe have obtained the results. We can do some additional testing such as validating the normality of our simulation data. \n\n\nThe sensitivity analysis can be done by changing one-factor-at-a-time (OAT), to see what effect this produces on the output. I.e. we move one input variable, keeping others at their baseline (nominal) values and repeat this for each of the other inputs in the same way.\n\"\"\"\n\n# ╔═╡ d8b821a0-0d21-11eb-08bc-f7027b1cbea1\nbegin\n\ts = PM(D,1000)\n\tHypothesisTests.ExactOneSampleKSTest(s,Normal(mean(s),std(s)))\nend\n\n# ╔═╡ d89eb800-0d21-11eb-29d9-c119c9347dbc\nbegin \n\t\"\"\"\n\t\tsensitivity(D, p, i::Int, N::Int=1000)\n\t\n\tGenerate a sample where all values keep their baseline values with the exception of component i\n\t\"\"\"\n\tfunction sensitivity(D, p, i, N::Int=1000)\n\t\t# generate data\n\t\tdata = [[p[1:i-1]; rand(D[i]); p[i+1:end]] for _ in 1:N]\n\t\t# return durations\n\t\tmap(duration,data)\n\tend\n\t\n\tblup = [sensitivity(D,p, i) for i in 1:10]\n\tStatsPlots.boxplot(blup, label=\"\")\n\txlabel!(\"process step\")\n\tylabel!(\"duration boxplot\")\n\tylims!(30,40)\n\txlims!(0,11)\n\txticks!(collect(1:10))\n\ttitle!(\"Sensitivity analysis\")\nend\n\n# ╔═╡ d148f3ea-0d21-11eb-2d67-f1f4c77248d5\nmd\"\"\"\n###### Present the conclusions\nWith respect to the initial goals, we can conclude the following:\n* the mean project completion time is around 37 time units\n* the probability that the project will be finished in less than 36 time units is around 34%\n* processes 3 and 10 do not impact the total project duration. Steps two and four account for the largest variability.\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╟─86a87f27-7b57-4420-8cef-836ac531a254\n# ╠═b77c944b-88e4-421d-a629-0d96bf95ea70\n# ╟─bcbed56c-0c94-11eb-0e8c-2de8d1a20f47\n# ╟─349b8586-0c96-11eb-2e7b-5b3230d1cf27\n# ╠═3a20552e-0c97-11eb-3f0d-2586b1bb46ed\n# ╠═b1bae776-0dfe-11eb-291b-53f9a87c4a41\n# ╟─65c6476c-0c98-11eb-2c7d-3bc3ba8d51e2\n# ╟─5b08b538-0c97-11eb-0319-877b39c4c4ed\n# ╟─607e4392-0c98-11eb-0a9d-ddfab6b42b26\n# ╠═75d35828-0c99-11eb-1888-e9e7be234e89\n# ╠═fae1e448-0c98-11eb-3721-dddb169e403f\n# ╠═a39a3a72-0c99-11eb-00df-e92eb7a6ce07\n# ╟─ac87cb8a-0c9a-11eb-21d5-9981e408db29\n# ╠═1aa54e8a-0c9b-11eb-3b7c-f3be005991df\n# ╠═3236977a-0c9b-11eb-3875-2304eaec0a8e\n# ╟─7fafde70-0c9c-11eb-0403-9bfb6e42045b\n# ╠═c8b107ba-0c9d-11eb-11c7-830d112ca82f\n# ╟─0bfa3ffe-0c9a-11eb-2c99-3d8e57983c4c\n# ╟─8df49b7e-0c9e-11eb-0048-9368c9d7bee9\n# ╟─7aceebea-0d1d-11eb-05fa-d7374004b881\n# ╠═8f99cf60-0c9e-11eb-2b6d-b5ca5ed20a65\n# ╟─864b050c-0d1e-11eb-03d8-8ffcf23e019e\n# ╠═f9562612-0d1e-11eb-1e16-bfa53760c18c\n# ╟─d18a68b2-0d20-11eb-0f39-150f9bc9e0e9\n# ╠═da7801bc-0ca0-11eb-0480-33f07e16c85b\n# ╟─3d5f9648-0d21-11eb-14c4-65704b349ed6\n# ╠═d8b821a0-0d21-11eb-08bc-f7027b1cbea1\n# ╠═d89eb800-0d21-11eb-29d9-c119c9347dbc\n# ╟─d148f3ea-0d21-11eb-2d67-f1f4c77248d5\n", "meta": {"hexsha": "7339026d8ea70fd1ce0c4ebed0156b28c43c57f9", "size": 14047, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Exercises/PS06 - Intro simulation.jl", "max_stars_repo_name": "BenLauwens/ES313.jl", "max_stars_repo_head_hexsha": "5a7553e53c288834f768d26e0d5aa22f9062b6af", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2018-12-17T16:00:26.000Z", "max_stars_repo_stars_event_max_datetime": "2020-01-18T04:09:25.000Z", "max_issues_repo_path": "Exercises/PS06 - Intro simulation.jl", "max_issues_repo_name": "BenLauwens/ES313", "max_issues_repo_head_hexsha": "5a7553e53c288834f768d26e0d5aa22f9062b6af", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Exercises/PS06 - Intro simulation.jl", "max_forks_repo_name": "BenLauwens/ES313", "max_forks_repo_head_hexsha": "5a7553e53c288834f768d26e0d5aa22f9062b6af", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2018-08-27T13:41:05.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T11:00:53.000Z", "avg_line_length": 29.6976744186, "max_line_length": 295, "alphanum_fraction": 0.6825656724, "num_tokens": 5075, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9252299550303295, "lm_q2_score": 0.8856314692902446, "lm_q1q2_score": 0.8194127645048577}} {"text": "using StatisticalRethinking, DataFrames, NamedArrays, OrderedCollections\n\nna = NamedArray(\n [18 134 152; 3 4 7; 21 138 159], \n ( [:vaccinated, :unvaccinated, :total], \n [:varicella, :noninfected, :total]\n ),\n (\"Rows\", \"Cols\")\n)\n\nna |> display\nprintln()\n\n# Risk of varicella among vaccinated children = 18 / 152 = 0.118 = 11.8%\n\nrisk_vac = na[:vaccinated, :varicella] / na[:vaccinated, :total]\n\n# Risk of varicella among unvaccinated children = 3 / 7 = 0.429 = 42.9%\n\nrisk_unvac = na[:unvaccinated, :varicella] / na[:unvaccinated, :total]\n\n# Risk ratio = 0.118 / 0.429 = 0.28\n\nrr = risk_vac / risk_unvac\n\n# EXAMPLE: Calculating Vaccine Effectiveness\n# Calculate the vaccine effectiveness from the varicella data in Table 3.13.\n\n# VE = (42.9 – 11.8) / 42.9 = 31.1 / 42.9 = 72%\n\nve = (risk_unvac - risk_vac) /risk_unvac\n\n# Alternatively, VE = 1 – RR = 1 – 0.28 = 72%\n\n1 - rr |> display\n\n# So, the vaccinated group experienced 72% fewer varicella cases\n# than they would have if they had not been vaccinated.\n\ndf_covid = DataFrame(\n :completed => [true, true, false, false],\n :company => [:pfizer, :moderna, :jj, :astra],\n :unvaccinated => [18261, 15000, missing, missing],\n :vaccinated => [18262, 15000, missing, missing],\n :inf_unv => [162, 185, missing, missing],\n :inf_vac => [8, 11, missing, missing]\n)\n\ndf_covid |> display\n\ndf = df_covid[df_covid.completed, :]\n", "meta": {"hexsha": "96225afb6e0aa751aa7ec850e3a93b00aacc1bad", "size": 1402, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "scripts/varicella.jl", "max_stars_repo_name": "goedman/CovidNYT.jl", "max_stars_repo_head_hexsha": "4fda8a920a9d25241d6a96b0d3ca52b9b2f8add1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "scripts/varicella.jl", "max_issues_repo_name": "goedman/CovidNYT.jl", "max_issues_repo_head_hexsha": "4fda8a920a9d25241d6a96b0d3ca52b9b2f8add1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/varicella.jl", "max_forks_repo_name": "goedman/CovidNYT.jl", "max_forks_repo_head_hexsha": "4fda8a920a9d25241d6a96b0d3ca52b9b2f8add1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.9615384615, "max_line_length": 76, "alphanum_fraction": 0.6569186876, "num_tokens": 482, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9252299509069106, "lm_q2_score": 0.8856314632529872, "lm_q1q2_score": 0.8194127552671768}} {"text": "# -*- coding: utf-8 -*-\n# ---\n# jupyter:\n# jupytext:\n# text_representation:\n# extension: .jl\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.4.2\n# kernelspec:\n# display_name: Julia 1.4.1\n# language: julia\n# name: julia-1.4\n# ---\n\n# # ヒストグラム平坦化\n\n# # 準備\n\n# ## Load Julia Packages\n\nusing Images\nusing TestImages\nusing Plots\n\n# ## read testimage\n\nc = testimage(\"c\")\n\n# ## Visualize gray scale image histogram\n\ndata=reshape(255 .* channelview(c),prod(size(c)))\nhistogram(data, bins=0:255,normalize=:pdf)\n\n# ## Visualize cumulative distribution function (so called CDF)\n\n# +\ni2c = Dict{Int,Int}()\nfor i in 0:255\n i2c[i] = 0 \nend\n\nfor i in Int.(255 .* channelview(c))\n i2c[i] += 1\nend\n# -\n\narr = [i2c[i] for i in 0:255]\ncdf = [sum(arr[begin:i]) for i in 1:length(arr)]\nplot(cdf./length(c), label=\"CDF\")\n\n# # Histogram Equalization\n\n# - $q$ を画像の輝度の分布を表す確率密度関数とする. ヒストグラムを平坦化することは分布が（なるべく)一様分布に近似されれば良い.つまり累積分布関数が $y=ax$ のように原点と積分範囲の右端(画像であれば255=256-1)で $y=1$ となる一次関数になっていれば良い. \n# - 今述べたことを定式化する. $X$ を画像の分布を表す確率変数だとする. $Y$ をヒストグラム平坦化により得られら分布とし滑らかに $Y=f(X)$ と変化したとする. さらに変換が可逆だと仮定する. $p$ を $Y$ の確率密度函数とすれば, 次のことを要請する.\n#\n# $$\n# \\frac{y}{255} = \\int_{0}^{y} p(\\tilde{y})d\\tilde{y} \n# $$\n\n# 一般に $Y=f(X)$ と変換された確率変数の確率密度関数 $p$ は\n#\n# $$\n# p(y) = \\int \\delta(y-f(x))q(x)dx\n# $$\n#\n# である. この一般論を上式の右辺に適用すれば $p(y) = \\frac{q(x)}{f'(x)}\\ \\mathrm{where}\\ y=f(x)$ が従う.\n#\n# $$\n# \\begin{aligned}\n# \\frac{1}{255} &= \\frac{d}{dy}\\left(\\frac{y}{255}\\right) \\\\\n# &= \\frac{d}{dy}\\int_{0}^{y} p(\\tilde{y})d\\tilde{y} \\\\\n# &= p(y) \\\\\n# &= \\frac{q(x)}{f'(x)}\\ \\mathrm{where}\\ y=f(x) \\\\\n# \\end{aligned}\n# $$\n#\n# よって\n#\n# $$\n# f'(x) = 255q(x)\n# $$\n#\n# 要するに欲しい変換は累積分布函数に255(確率変数 $X$ がとる最大値) を乗じたものである.\n#\n# $$\n# f(x) = 255 \\int_{0}^x q(\\tilde{x}) d\\tilde{x}\n# $$\n#\n# これを画像処理に適用することを考えると積分は離散和になる. $q(x)$ は輝度が$x\\in [0, 255]$である確率になる. 平たくいうと $x$ での（正規化された）ヒストグラムの値である．これを踏まえて下記のように実装をしてみよう.\n\nn_pixels = length(c)\nfunction hist_eq(p)\n s = 0\n for i in 0:p\n s+=i2c[i]\n end\n floor(255*s/n_pixels)\nend\n\nnew_c=hist_eq.(255 .* channelview(c))\n\ncolorview(Gray, new_c./255)\n\nhistogram(new_c[:], normalize=:pdf, bins=0:255)\n\n# +\nnew_i2c = Dict{Int,Int}()\nfor i in 0:255\n new_i2c[i] = 0 \nend\n\nfor i in new_c\n new_i2c[i] += 1\nend\n# -\n\n# # 結果の確認\n\narr = [new_i2c[i] for i in 0:255]\ncdf = [sum(arr[begin:i]) for i in 1:length(arr)]\nplot(cdf./length(c))\n# 直線になっている様子がわかるはず\n", "meta": {"hexsha": "20401df0f9b2b814caa3ed5c8e169eddc4a107d6", "size": 2454, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "experiments/notebook/histogram_eq.jl", "max_stars_repo_name": "nash1111/MyWorkflow.jl", "max_stars_repo_head_hexsha": "027f8a176f473bb355d2cd28b881dd55f06f3526", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "experiments/notebook/histogram_eq.jl", "max_issues_repo_name": "nash1111/MyWorkflow.jl", "max_issues_repo_head_hexsha": "027f8a176f473bb355d2cd28b881dd55f06f3526", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-09-06T12:34:13.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-06T12:39:46.000Z", "max_forks_repo_path": "experiments/notebook/histogram_eq.jl", "max_forks_repo_name": "hsugawa8651/MyWorkflow.jl", "max_forks_repo_head_hexsha": "48edcbcc3fd9e425895ef91bea833ef1e5d4c3fe", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.7903225806, "max_line_length": 144, "alphanum_fraction": 0.5969845151, "num_tokens": 1308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297861178929, "lm_q2_score": 0.9099069968764883, "lm_q1q2_score": 0.8193983532843583}} {"text": "# 'cstrt_reinhardt_ngon' provides the vertices coordinates (a,b) of a\r\n# Reinhardt (m*k)-gon\r\nfunction cstrt_reinhardt_ngon(m,k)\r\nif mod(m,2) == 1 && m>=3 && k>=1\r\n a = zeros(m*k-1); b = zeros(m*k-1);\r\n for i = 1:m-1\r\n a[i*k] = sin(2*pi*i/m)/(2*cos(pi/(2*m)));\r\n b[i*k] = (1 - cos(2*pi*i/m))/(2*cos(pi/(2*m)));\r\n end\r\n if k>1\r\n for j = 1:k-1\r\n a[Int(k*(m-1)/2+j)] = cos((1+(m-3)/2+j/k)*(pi/m));\r\n b[Int(k*(m-1)/2+j)] = sin((1+(m-3)/2+j/k)*(pi/m));\r\n for i = 1:Int((m-1)/2)\r\n a[Int((i+(m-1)/2)*k+j)] = a[i*k] + cos((2*i+1+(m-3)/2+j/k)*(pi/m));\r\n b[Int((i+(m-1)/2)*k+j)] = b[i*k] + sin((2*i+1+(m-3)/2+j/k)*(pi/m));\r\n a[Int(m*k-((i+(m-1)/2)*k+j))] = -a[Int((i+(m-1)/2)*k+j)];\r\n b[Int(m*k-((i+(m-1)/2)*k+j))] = b[Int((i+(m-1)/2)*k+j)];\r\n end\r\n end\r\n end\r\n return(a,b)\r\nend\r\nend", "meta": {"hexsha": "3a83568e7d60bbd6be2a0a51318ddbfc3b5555ee", "size": 923, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "cstrt_ngon/cstrt_reinhardt_ngon.jl", "max_stars_repo_name": "cbingane/optigon", "max_stars_repo_head_hexsha": "f585de7a2b01ff1e7a15c566596a7ba8f77f782a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "cstrt_ngon/cstrt_reinhardt_ngon.jl", "max_issues_repo_name": "cbingane/optigon", "max_issues_repo_head_hexsha": "f585de7a2b01ff1e7a15c566596a7ba8f77f782a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "cstrt_ngon/cstrt_reinhardt_ngon.jl", "max_forks_repo_name": "cbingane/optigon", "max_forks_repo_head_hexsha": "f585de7a2b01ff1e7a15c566596a7ba8f77f782a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.4583333333, "max_line_length": 84, "alphanum_fraction": 0.3932827736, "num_tokens": 380, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9732407168145568, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.8192989905707768}} {"text": "#=\nP34 (**) Calculate Euler's totient function phi(m).\nEuler's so-called totient function phi(m) is defined as the number of positive integers r (1 <= r < m) that are coprime to m.\nExample: m = 10: r = 1,3,7,9; thus phi(m) = 4. Note the special case: phi(1) = 1.\n\n* (totient-phi 10)\n4\n\nFind out what the value of phi(m) is if m is a prime number. Euler's totient function plays an important role in one of the most widely used public key cryptography methods (RSA). In this exercise you should use the most primitive method to calculate this function (there are smarter ways that we shall discuss later).\n=#\n\n# From P32 but more terse\nfunction gcd(a, b)\n b == 0 ? a : gcd(b, mod(a, b))\nend\n\n# From P33\nfunction coprime(a, b)\n gcd(a, b) == 1\nend\n\nfunction totient_phi(m)\n # Starting at 2, not 1 (special case)\n coprime_to_m = filter(r -> coprime(r, m), 2:m)\n length(coprime_to_m) + 1\nend\n\n@assert totient_phi(10) == 4\n\nprintln(\"Tests passed: JL-34.jl\")\n", "meta": {"hexsha": "4930b955100a721aa9ce2e256970e57069b62ae4", "size": 967, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "JL-34.jl", "max_stars_repo_name": "microamp/jl-99", "max_stars_repo_head_hexsha": "5d49a7e1617394e6cbc06f1a94fe6230b3025d73", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "JL-34.jl", "max_issues_repo_name": "microamp/jl-99", "max_issues_repo_head_hexsha": "5d49a7e1617394e6cbc06f1a94fe6230b3025d73", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "JL-34.jl", "max_forks_repo_name": "microamp/jl-99", "max_forks_repo_head_hexsha": "5d49a7e1617394e6cbc06f1a94fe6230b3025d73", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.1935483871, "max_line_length": 318, "alphanum_fraction": 0.6866597725, "num_tokens": 302, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.957277806109987, "lm_q2_score": 0.8558511524823263, "lm_q1q2_score": 0.8192873136049853}} {"text": "using DifferentialEquations\nusing Plots\n\n\nmicro_1 = @ode_def Micro1 begin\n dS = r*(1-S/K)*S - β*S*I\n dI = β*S*I-(μ+α)*I\n end β r μ K α\n\nw = 1;\nm = 10;\nβ = 0.0247*m*w^0.44;\nr = 0.6*w^-0.27;\nμ = 0.4*w^-0.26;\nK = 16.2*w^-0.7;\nα = (m-1)*μ;\n\nparms = [β,r,μ,K,α];\ninit = [K,1.];\ntspan = (0.0,10.0);\n\nsir_prob = ODEProblem(micro_1,init,tspan,parms)\n\nsir_sol = solve(sir_prob);\n\n# Plot\nplot(sir_sol,xlabel=\"Time\",ylabel=\"Number\")\n\nm = [5,10,20,40]\nws = 10.^linspace(-3,3,601)\nβs = zeros(601,4)\nfor i = 1:4\n βs[:,i] = 0.0247*m[i]*ws.^0.44\nend\n\n# Plot\nplot(ws,βs,xlabel=\"Weight\",ylabel=\"\\\\beta_min\", xscale=:log10,yscale=:log10, label=[\"m = 5\" \"m = 10\" \"m = 20\" \"m = 40\"],lw=3)\n", "meta": {"hexsha": "96c4eed77f2f33f68e20451f730b17e55ca7148f", "size": 680, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "models/simple_deterministic_models/scaling/scaling.jl", "max_stars_repo_name": "epimodels/epicookbook", "max_stars_repo_head_hexsha": "496088ddc8fc913505d92877e0a687c81976c8f2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "models/simple_deterministic_models/scaling/scaling.jl", "max_issues_repo_name": "epimodels/epicookbook", "max_issues_repo_head_hexsha": "496088ddc8fc913505d92877e0a687c81976c8f2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "models/simple_deterministic_models/scaling/scaling.jl", "max_forks_repo_name": "epimodels/epicookbook", "max_forks_repo_head_hexsha": "496088ddc8fc913505d92877e0a687c81976c8f2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-10-10T12:46:31.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-10T12:46:31.000Z", "avg_line_length": 17.8947368421, "max_line_length": 125, "alphanum_fraction": 0.5720588235, "num_tokens": 325, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9572778012346835, "lm_q2_score": 0.8558511506439707, "lm_q1q2_score": 0.8192873076726341}} {"text": "using Plots\nusing LaTeXStrings\n\n# Probability of v±dv\np(v;m=1.,R=8.3145,T=298.15) = m/(R*T) * v * exp(-m*v^2/(2R*T)) \n\n# Probability of v < x, given m and T\npcum(x;m = 1., T = 298.15, R = 8.3145) = -exp(-m*x^2/(2*R*T)) + 1\n\n# Inverse Cumulative Probability of v < x, given m and T\npinv(x;m=1., T=298.15, R=8.3145) = sqrt(-(2*R*T/m)*log(1-x))\n\n# Start plot\nplot(layout=(1,2))\n\n# Collect x\nx = collect(0:0.1:200)\n\n# Set y with m = 1, T = 298.15K, R = 8.3145 J/Kmol\ny = pcum.(x)\n\n# plot\nplot!(x,y,label=\"\",subplot=1)\nplot!(xlabel=L\"x / m s^{-1}\",ylabel=L\"p~(v
\nEn Julia, las funciones operaran con el cualquier valor que haga sentido.
\nPor ejemplo, `sayhi` funciona con el nombre de este personaje de tele, escrito como entero...\n\"\"\"\n\n# ╔═╡ 10205f9e-6758-44a7-a1ed-f1daae906893\nsayhi(55595472)\n\n# ╔═╡ 5d4f722a-3e68-422c-8dab-4c92be5b9804\nmd\"\"\"\nY `f` va a funcionar en una matriz.\n\"\"\"\n\n# ╔═╡ b9bfd357-22e7-4d75-a251-53e8379f8017\nA = rand(3, 3)\nA\n\n# ╔═╡ 779781b3-5092-4559-9d61-b2e83d2b259a\nf(A)\n\n# ╔═╡ f16551eb-e665-4922-8e98-548a55872ebd\nmd\"\"\"\n`f` funcionará con \"hi\" porque `*` para inputs de cadenas como concatenación.\n\"\"\"\n\n# ╔═╡ b626e299-7afd-4b62-b070-738fff6cb6cb\nf(\"hi\")\n\n# ╔═╡ 2c44a8ae-c8e7-49ad-9de2-0537c56a6f23\nmd\"\"\"\nPor el otro lado, `f` no funcionará sobre un vector. A diferencia de `A^2`, la cual es una operación bien definida, el signifiado de `v^2` para un vector, `v`, es ambigua. \n\"\"\"\n\n# ╔═╡ 747e9a83-7886-4b33-a8d6-fd60293b2a0a\nv = rand(3)\n\n# ╔═╡ fb3682c1-674c-4ea3-a222-103bf960cbe8\nf(v)\n\n# ╔═╡ 81a0e65a-e4fb-427b-b59c-3248eff10b61\nmd\"\"\"\n## Funciones mutantes vs no-mutantes\n\nPor convención, funciones seguidas por un `!` alteran, o bien mutan, sus contenidos y las que carecen de un `!` no lo hacen.\n\nPor ejemplo, `sort` y `sort!`.\n\n\"\"\"\n\n# ╔═╡ 3ce811b3-19aa-4f4f-8cf5-28f0032cc878\nv = [3, 5, 2]\n\n# ╔═╡ 284c7800-9dc2-4f0c-b3c0-da82e020cdfa\nsort(v)\n\n# ╔═╡ eec63ee5-9d20-4892-92df-9d0aeecfb323\nv\n\n# ╔═╡ 99dfcd21-d511-40d5-8119-fdf943bb2708\nmd\"\"\"\n`sort(v)` regresa el arreglo ordenado de `v`, pero `v` no cambia.
\n\nPor otro lado, si corremos `sort!(v)`, el contenido del arreglo es ordenado dentro de `v`.\n\"\"\"\n\n# ╔═╡ 11a3c01b-0ac1-4538-8103-8e7e35eadd3c\nsort!(v)\n\n# ╔═╡ 017e9146-e1ab-46c0-b6e8-e36f14965639\nv\n\n# ╔═╡ ca47c936-e9e4-4039-9ff2-3e6f6f395f89\nmd\"\"\"\n## Broadcasting\n\nSi ponemos `.` entre el nombre de la funcion y su lista de argumento,
\nle estamos diciendo a la función que se \"difunda\"/haga broadcasting sobre los elementos del input.
\n\nPrimero veamos la diferencia entre `f()` y `f.()`.
\n\nPrimero definimos una nueva matriz `A` que hará la diferencia fácil de observar\n\"\"\"\n\n# ╔═╡ 0e5e8129-b03f-4010-8e33-d57d4aa49529\nA = [i + 3*j for j in 0:2, i in 1:3]\n\n# ╔═╡ 96b1c681-280a-43b0-bc00-2abb315549f6\nf(A)\n\n# ╔═╡ ac7f99d1-0eef-43b6-a01a-0da4bf0d5ef5\nmd\"\"\"\nCómo se vio antes, para una matriz, `A`,\n```\nf(A) = A^2 = A * A\n``` \n\n`f.(A)` por el otro lado va a regresar un objeto que contiene el cuadrado de `A[i, j]` en su entrada correspondiente.\n\"\"\"\n\n# ╔═╡ 5d1d011e-2399-4d33-858c-b7b7e89e81b6\nB = f.(A)\n\n# ╔═╡ b5c3116e-f7dd-4ead-a554-c539eb152a32\nA[2, 2]\n\n# ╔═╡ 9144bda4-a4d0-4c30-b551-0324c35761eb\nA[2, 2]^2\n\n# ╔═╡ 918a6647-80e5-4a5a-820c-cdaf6f835fc7\nB[2, 2]\n\n# ╔═╡ 6469d0a9-860b-4bc0-b791-40edc1cea224\nmd\"\"\"\nEsto significa que para `v`, `f.(v)` está definido, pero no para `f(v)` :\n\"\"\"\n\n# ╔═╡ 72658881-9638-4264-8d65-5766423ace70\nv = [1, 2, 3]\n\n# ╔═╡ 5c26b16d-4d73-4b79-b5e4-3b1e20c5cd9d\nf.(v)\n\n# ╔═╡ fb497c48-76e2-4eb1-8e21-785c93f39dc4\nmd\"\"\"\n### Ejercicios\n\n6.1 En vez de broadcastear `f` sobre `v`, pudimos haber hecho `v .^ 2`.\n\nSin declarar una nueva funcion, agrega 1 a cada elemento de una matriz de `3x3` llena de `0`'s.\n\"\"\"\n\n# ╔═╡ f681fc78-679f-45b8-ae95-14b2d9eb6366\n\n\n# ╔═╡ 62c8731f-63a7-4fae-8980-0d0665cb556a\nmd\"\"\"\n\n\"\"\"\n\n# ╔═╡ 7dbb40ed-0b51-4825-bbc8-9e6f4d0c4738\n\n\n# ╔═╡ be113c25-4aec-4ead-a812-50a54aa7428d\nmd\"\"\"\n6.3 Una cifra de César recorre cada letra un número determinado de plazas más adelante en el abecedario. Un corrimiento, o shift, de 1 manda \"A\" a \"B\". Escribe una función llamada `cesar` que toma una cadena como input y un corrimiento y regresa una cadena desencriptada tal que obtengas\n\n```julia\ncesar(\"abc\", 1)\n\"bcd\"\n\ncesar(\"hello\", 4)\n\"lipps\"\n```\n\"\"\"\n\n# ╔═╡ 14e4057a-96f3-4cec-b1f0-e0c01e73dd86\n\n\n# ╔═╡ Cell order:\n# ╟─b7d4165b-d051-441d-b4d1-251eba60a0d2\n# ╟─39434826-4c89-4074-9c9e-770bde3463dd\n# ╠═0adc05a6-8428-4722-94cc-355e65437395\n# ╠═158e67af-59ce-4d4f-8256-f635ffedf5a0\n# ╟─b1bc4d5d-33b5-4698-8fb8-f4126af5463d\n# ╠═eaf96975-d1e6-4369-9fa1-8ab49b69b6d9\n# ╠═7a5019b7-6ae4-4fcd-9859-118a6e000033\n# ╟─e3585fc6-78ed-4ccf-a9f0-3387e41b1193\n# ╠═311a5abb-86ad-411f-9118-8827b65fb833\n# ╠═25594aa9-0d0d-49e6-84cb-33d195e7bd62\n# ╠═99fae709-86fd-4962-a47f-0dadfcebf9fc\n# ╠═c66327e0-3ac1-4078-96df-4c0132eeb5e9\n# ╟─707831b6-e85a-4a11-9648-0af16acf71f7\n# ╠═c71820ac-4ae6-42a1-a1cb-d05ab25afc75\n# ╠═7a5b1e40-def3-4cca-9c81-73addb86aa18\n# ╠═ca7f38ca-1b81-4425-ba04-32c40ba329cd\n# ╠═59382902-5270-44f5-a19e-d8c296e630f3\n# ╟─10880ded-73a9-4165-b993-623d93e08630\n# ╠═10205f9e-6758-44a7-a1ed-f1daae906893\n# ╟─5d4f722a-3e68-422c-8dab-4c92be5b9804\n# ╠═b9bfd357-22e7-4d75-a251-53e8379f8017\n# ╠═779781b3-5092-4559-9d61-b2e83d2b259a\n# ╟─f16551eb-e665-4922-8e98-548a55872ebd\n# ╠═b626e299-7afd-4b62-b070-738fff6cb6cb\n# ╟─2c44a8ae-c8e7-49ad-9de2-0537c56a6f23\n# ╠═747e9a83-7886-4b33-a8d6-fd60293b2a0a\n# ╠═fb3682c1-674c-4ea3-a222-103bf960cbe8\n# ╟─81a0e65a-e4fb-427b-b59c-3248eff10b61\n# ╠═3ce811b3-19aa-4f4f-8cf5-28f0032cc878\n# ╠═284c7800-9dc2-4f0c-b3c0-da82e020cdfa\n# ╠═eec63ee5-9d20-4892-92df-9d0aeecfb323\n# ╟─99dfcd21-d511-40d5-8119-fdf943bb2708\n# ╠═11a3c01b-0ac1-4538-8103-8e7e35eadd3c\n# ╠═017e9146-e1ab-46c0-b6e8-e36f14965639\n# ╟─ca47c936-e9e4-4039-9ff2-3e6f6f395f89\n# ╠═0e5e8129-b03f-4010-8e33-d57d4aa49529\n# ╠═96b1c681-280a-43b0-bc00-2abb315549f6\n# ╟─ac7f99d1-0eef-43b6-a01a-0da4bf0d5ef5\n# ╠═5d1d011e-2399-4d33-858c-b7b7e89e81b6\n# ╠═b5c3116e-f7dd-4ead-a554-c539eb152a32\n# ╠═9144bda4-a4d0-4c30-b551-0324c35761eb\n# ╠═918a6647-80e5-4a5a-820c-cdaf6f835fc7\n# ╟─6469d0a9-860b-4bc0-b791-40edc1cea224\n# ╠═72658881-9638-4264-8d65-5766423ace70\n# ╠═5c26b16d-4d73-4b79-b5e4-3b1e20c5cd9d\n# ╟─fb497c48-76e2-4eb1-8e21-785c93f39dc4\n# ╠═f681fc78-679f-45b8-ae95-14b2d9eb6366\n# ╟─62c8731f-63a7-4fae-8980-0d0665cb556a\n# ╠═7dbb40ed-0b51-4825-bbc8-9e6f4d0c4738\n# ╟─be113c25-4aec-4ead-a812-50a54aa7428d\n# ╠═14e4057a-96f3-4cec-b1f0-e0c01e73dd86\n", "meta": {"hexsha": "785d3c234c1bd80add18fe614d2a2dc9e0d65cff", "size": 7581, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "es-es/intro-to-julia-ES/plutos/plutos2/06.Funciones.jl", "max_stars_repo_name": "miguelraz/JuliaTutorials", "max_stars_repo_head_hexsha": "1d427c70c7254844c9734a20e7402e60a000528f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-11-22T00:48:54.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-10T15:10:03.000Z", "max_issues_repo_path": "es-es/intro-to-julia-ES/plutos/plutos2/06.Funciones.jl", "max_issues_repo_name": "miguelraz/JuliaTutorials", "max_issues_repo_head_hexsha": "1d427c70c7254844c9734a20e7402e60a000528f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "es-es/intro-to-julia-ES/plutos/plutos2/06.Funciones.jl", "max_forks_repo_name": "miguelraz/JuliaTutorials", "max_forks_repo_head_hexsha": "1d427c70c7254844c9734a20e7402e60a000528f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.8737201365, "max_line_length": 287, "alphanum_fraction": 0.7166600712, "num_tokens": 4064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765187126079, "lm_q2_score": 0.8902942203004186, "lm_q1q2_score": 0.8134409238340421}} {"text": "#\n# Non linear model fitting \n#\n# author: Atsushi Sakai\n#\n\nusing PyPlot\nusing ArgMin\n\n\nfunction calc_y(theta, x)\n \ty = theta[1]*exp.(theta[2]*x) .* cos.(theta[3]*x .+ theta[4]);\n\treturn y\nend\n\nfunction main()\n println(PROGRAM_FILE,\" start!!\")\n\ttheta_true = [1, -0.2, 2*pi/5, pi/3] # true parameters\n\n\t# Input data \n\tM = 30\n\txd = [5*rand(M); 5 .+ 15*rand(M)]\n\tyd = calc_y(theta_true, xd)\n\n\tf(theta) = theta[1] * exp.(theta[2]*xd) .* cos.(theta[3] * xd .+ theta[4]) - yd;\n \tDf(theta) = hcat( exp.(theta[2]*xd) .* cos.(theta[3] * xd .+ theta[4]),\n \t\t\t\ttheta[1] * ( xd .* exp.(theta[2]*xd) .* cos.(theta[3] * xd .+ theta[4])),\n \t\t\t\t-theta[1] * ( exp.(theta[2]*xd) .* xd .* sin.(theta[3] * xd .+ theta[4])),\n \t\t\t\t-theta[1] * ( exp.(theta[2]*xd) .* sin.(theta[3] * xd .+ theta[4])) );\n\ttheta0 = [1, 0, 1, 0]; # initial parameters\n\tx = linspace(0, 20, 500);\n\ty1 = calc_y(theta0, x)\n\t\n\ttheta, history = solve_nonlinear_least_square_with_levenberg_marquardt(f, Df, theta0, 1.0)\n\n\tprintln(\"theta_true:\", theta_true)\n\tprintln(\"theta:\",theta)\n \ty = calc_y(theta, x)\n\tplot(xd, yd, \"ob\", label=\"Input data\")\n\tplot(x, y1, \"--r\", label=\"Initial param\")\n\tplot(x, y, \"-r\", label=\"Optimized param\")\n\tlegend()\n\tshow()\n\n println(PROGRAM_FILE,\" Done!!\")\nend\n\nmain()\n\n", "meta": {"hexsha": "d781310d2305cdd9fda365933703bd9e16c02936", "size": 1297, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/nonlinear_model_fitting/nonlinear_model_fitting.jl", "max_stars_repo_name": "AtsushiSakai/ArgMin.jl", "max_stars_repo_head_hexsha": "8584acd49640ac775f5ac5e47860a0d2d4953d54", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "examples/nonlinear_model_fitting/nonlinear_model_fitting.jl", "max_issues_repo_name": "AtsushiSakai/ArgMin.jl", "max_issues_repo_head_hexsha": "8584acd49640ac775f5ac5e47860a0d2d4953d54", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/nonlinear_model_fitting/nonlinear_model_fitting.jl", "max_forks_repo_name": "AtsushiSakai/ArgMin.jl", "max_forks_repo_head_hexsha": "8584acd49640ac775f5ac5e47860a0d2d4953d54", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.94, "max_line_length": 92, "alphanum_fraction": 0.565921357, "num_tokens": 477, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802462567087, "lm_q2_score": 0.8856314632529871, "lm_q1q2_score": 0.8134350044612928}} {"text": "using Distributions, Random, Plots, LaTeXStrings; pyplot()\nRandom.seed!(1)\n\na, c, d = 2, 1, 5\nsig = 0.02\nnext(x,y) = [a*x*(1-x) - x*y, -c*y + d*x*y]\nequibPoint = [(1+c)/d ,(d*(a-1)-a*(1+c))/d]\n\ninitX = [0.8,0.05]\ntEnd,tEndStoch = 100, 10\n\ntraj = [[] for _ in 1:tEnd]\ntrajStoch = [[] for _ in 1:tEndStoch]\ntraj[1], trajStoch[1] = initX, initX\n\nfor t in 2:tEnd\n traj[t] = next(traj[t-1]...)\nend\n\nfor t in 2:tEndStoch\n trajStoch[t] = next(trajStoch[t-1]...) + [rand(Normal(0,sig)),0.0]\nend\n\nscatter([traj[1][1]], [traj[1][2]], \n\tc=:black, ms=10, label=\"Initial state\")\nplot!(first.(traj), last.(traj), \n\tc=:blue, ls=:dash, m=(:dot, 5, Plots.stroke(0)), \n\tlabel=\"Deterministic trajectory\")\nplot!(first.(trajStoch),last.(trajStoch), \n\tc=:green, ls=:dash, m=(:dot, 5, Plots.stroke(0)), \n\tlabel=\"Stochastic trajectory\")\nscatter!([equibPoint[1]], [equibPoint[2]], \n\tc=:red, shape=:cross, ms=10, label=\"Equlibrium point\", \n\txlabel=L\"X_1\", ylabel=L\"X_2\")", "meta": {"hexsha": "38b0a05e7db30ceeeff431208a395e8cedf61727", "size": 951, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "10_chapter/predatorPreyStoch.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "10_chapter/predatorPreyStoch.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "10_chapter/predatorPreyStoch.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 27.9705882353, "max_line_length": 70, "alphanum_fraction": 0.6140904311, "num_tokens": 398, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533107374444, "lm_q2_score": 0.8723473879530492, "lm_q1q2_score": 0.8134232100099826}} {"text": "\nusing ..CommonFunctions\n\n\n@doc raw\"\"\"\n# Hermite's Interpolating Polynomials\n\nHere, we implement a two point Hermite interpolation function which passes\nthrough the function and its first derivative for the interval ``[0,1]``.\nThe polynomial is determined by four constraint equations, matching the\nfunction and its derivative at the points ``0`` and ``1``.\n\nStart by defining the 3rd degree polynomial and its derivative by\n```math\n\\begin{align*}\ng(x) &= a_0 + a_1 x + a_2 x^2 + a_3 x^3 , \\\\\ng'(x) &= a_1 + 2 a_2 x + 3 a_3 x^2 ,\n\\end{align*}\n```\nand apply the constraints\n```math\n\\begin{align*}\ng(0) &= f_0 & & \\Rightarrow & a_0 &= f_0 , \\\\\ng(1) &= f_1 & & \\Rightarrow & a_0 + a_1 + a_2 + a_3 &= f_1 , \\\\\ng'(0) &= f'_0 & & \\Rightarrow & a_1 &= f'_0 , \\\\\ng'(1) &= f'_1 & & \\Rightarrow & a_1 + 2 a_2 + 3 a_3 &= f'_1 . \\\\\n\\end{align*}\n```\nSolving for ``a_0, a_1, a_2, a_3`` leads to\n```math\n\\begin{align*}\na_0 &= f_0 , &\na_1 &= f'_0 , &\na_2 &= - 3 f_0 + 3 f_1 - 2 f'_0 - f'_1 , &\na_3 &= 2 f_0 - 2 f_1 + f'_0 + f'_1 ,\n\\end{align*}\n```\nso that the polynomial ``g(x)`` reads\n```math\ng(x) = f_0 + f'_0 x + (- 3 f_0 + 3 f_1 - 2 f'_0 - f'_1) x^2 + (2 f_0 - 2 f_1 + f'_0 + f'_1) x^3 .\n```\nThe function and derivative values can be factored out, so that ``g(x)`` can be rewritten as\n```math\ng(x) = f_0 (1 - 3 x^2 + 2 x^3) + f_1 (3 x^2 - 2 x^3) + f'_0 (x - 2 x^2 + x^3) + f'_1 (- x^2 + x^3) ,\n```\nor in generic form as\n```math\ng(x) = f_0 a_0(x) + f_1 a_1(x) + f'_0 b_0(x) + f'_1 b_1(x) ,\n```\nwith basis functions\n```math\n\\begin{align*}\na_0 (x) &= 1 - 3 x^2 + 2 x^3 , &\nb_0 (x) &= x - 2 x^2 + x^3 , \\\\\na_1 (x) &= 3 x^2 - 2 x^3 , &\nb_1 (x) &= - x^2 + x^3 .\n\\end{align*}\n```\nThe derivative ``g'(x)`` accordingly reads\n```math\ng'(x) = f_0 a'_0(x) + f_1 a'_1(x) + f'_0 b'_0(x) + f'_1 b'_1(x) ,\n```\nwith\n```math\n\\begin{align*}\na'_0 (x) &= - 6 x + 6 x^2 , &\nb'_0 (x) &= 1 - 4 x + 3 x^2 , \\\\\na'_1 (x) &= 6 x - 6 x^2 , &\nb'_1 (x) &= - 2 x + 3 x^2 .\n\\end{align*}\n```\nThe basis functions ``a_0``and ``a_1`` are associated with the function\nvalues at ``x_0`` and ``x_1``, respectively, while the basis functions\n``b_0`` and ``b_1`` are associated with the derivative values at\n``x_0`` and ``x_1``.\nThe basis functions satisfy the following relations,\n```math\n\\begin{align*}\na_i (x_j) &= \\delta_{ij} , &\nb_i (x_j) &= 0 , &\na'_i (x_j) &= 0 , &\nb'_i (x_j) &= \\delta_{ij} , &\ni,j &= 0, 1 ,\n\\end{align*}\n```\nwhere ``\\delta_{ij}`` denotes the Kronecker-delta, so that\n```math\n\\begin{align*}\ng(0) &= f_0 , &\ng(1) &= f_1 , &\ng'(0) &= f'_0 , &\ng'(1) &= f'_1 .\n\\end{align*}\n```\n\"\"\"\nstruct HermiteInterpolation{T} <: Interpolator{T}\n x₀::T\n x₁::T\n Δx::T\n\n function HermiteInterpolation{T}(x₀, x₁, Δx, d) where {T}\n new(x₀, x₁, Δx)\n end\nend\n\nfunction HermiteInterpolation(x₀::T, x₁::T, Δx::T, d::Int) where {T}\n HermiteInterpolation{T}(x₀, x₁, Δx, d)\nend\n\n\nfunction CommonFunctions.evaluate!(int::HermiteInterpolation{T}, y₀::Vector{T}, y₁::Vector{T}, f₀::Vector{T}, f₁::Vector{T}, x::T, y::Vector{T}) where {T}\n local a₀::T\n local a₁::T\n local b₀::T\n local b₁::T\n\n # Interpolate y values at required locations\n if x == int.x₀\n y .= y₀\n elseif x == int.x₁\n y .= y₁\n else\n a₁ = 3x^2 - 2x^3\n a₀ = 1 - a₁\n b₁ = x^2*(x-1)\n b₀ = x*(1-x)+b₁\n y .= a₀ .* y₀ .+ a₁ .* y₁ .+ b₀ .* int.Δx .* f₀ .+ b₁ .* int.Δx .* f₁\n end\nend\n\nfunction CommonFunctions.evaluate!(int::HermiteInterpolation{T}, y₀::Vector{T}, y₁::Vector{T}, f₀::Vector{T}, f₁::Vector{T}, x::T, y::Vector{T}, f::Vector{T}) where {T}\n local a₀::T\n local a₁::T\n local b₀::T\n local b₁::T\n\n evaluate!(int, y₀, y₁, f₀, f₁, x, y)\n\n # Interpolate f values at required locations\n if x == int.x₀\n f .= f₀\n elseif x == int.x₁\n f .= f₁\n else\n a₁ = (6x - 6x^2) / int.Δx\n a₀ = - a₁\n b₁ = x*(3x-2)\n b₀ = 1-2x+b₁\n f .= b₀ .* f₀ .+ b₁ .* f₁ .+ a₀ .* y₀ .+ a₁ .* y₁\n end\nend\n", "meta": {"hexsha": "a7086ca1186b44a5e01af51ab04319e22d4aff44", "size": 3955, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/interpolation/hermite_interpolation.jl", "max_stars_repo_name": "krystophny/GeometricIntegrators.jl", "max_stars_repo_head_hexsha": "7855e977b014c8ba119f6bb73c6ed9bf96f04b1d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-02-04T11:52:47.000Z", "max_stars_repo_stars_event_max_datetime": "2020-02-04T11:52:47.000Z", "max_issues_repo_path": "src/interpolation/hermite_interpolation.jl", "max_issues_repo_name": "krystophny/GeometricIntegrators.jl", "max_issues_repo_head_hexsha": "7855e977b014c8ba119f6bb73c6ed9bf96f04b1d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/interpolation/hermite_interpolation.jl", "max_forks_repo_name": "krystophny/GeometricIntegrators.jl", "max_forks_repo_head_hexsha": "7855e977b014c8ba119f6bb73c6ed9bf96f04b1d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.0197368421, "max_line_length": 168, "alphanum_fraction": 0.5481668774, "num_tokens": 1687, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.952574129515172, "lm_q2_score": 0.8539127566694177, "lm_q1q2_score": 0.8134152008662714}} {"text": "export Resolvent, qresolvent\n\nabstract type MatrixTransformations end\n\n\"\"\"\n```julia\nResolvent(A::Matrix)\n```\n\n- return the resolvent **function** of `A`, ``R(z)= (A-zI)^{-1}``\n\n# Examples\n\nGet the resolvent **function** of a normalized GOE(500).\n\nAnd evaluate the resolvent at `0`\n```julia\nR = Resolvent(rand(GOE(500),norm = true)) # this a function\n\nR(0)\n\n500×500 Matrix{Float64}:\n 0.460903 -0.384016 0.268931 … -0.59029 -0.178288\n -0.384016 0.489355 -0.366855 0.452094 -0.190179\n 0.268931 -0.366855 -0.697169 -0.221433 0.535448\n 0.388846 0.946668 -0.354459 -0.796717 0.207156\n ⋮ ⋱\n -0.532533 0.950826 -0.366618 0.0926931 -0.288692\n -0.0874979 -0.553262 -0.388048 -0.0156897 0.177695\n -0.59029 0.452094 -0.221433 0.9136 0.343194\n -0.178288 -0.190179 0.535448 0.343194 -0.141701\n```\n\"\"\"\nstruct Resolvent<:MatrixTransformations\n M::Matrix\nend\n\n\nfunction (Resolvent::MatrixTransformations)(z)\n (Resolvent.M-UniformScaling(z))^(-1)\nend\n\nfunction Base.show(io::IO, R::Resolvent) \n print(\"Resolvent{Matrix}\")\nend\n\n\"\"\"\n```julia\nqresolvent(A)\n```\nreturns the quaternion resolvent **function** of A.\n\"\"\"\nfunction qresolvent(A::AbstractMatrix)\n return function f(z::Number,η::Number)\n ([UniformScaling(-η) A-UniformScaling(z) \n A'-UniformScaling(z') UniformScaling(-η)])^(-1)\nend\nend", "meta": {"hexsha": "6bab56c5beb3d74b3d01d979e7aed0ee8cab2150", "size": 1412, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Transformations.jl", "max_stars_repo_name": "weiyang2048/RandomMatrix.jl", "max_stars_repo_head_hexsha": "d112a17d928c84583759177849a76eae1942e947", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-04-27T04:28:10.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-12T17:21:23.000Z", "max_issues_repo_path": "src/Transformations.jl", "max_issues_repo_name": "weiyang607/RandomMatrix.jl", "max_issues_repo_head_hexsha": "d112a17d928c84583759177849a76eae1942e947", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2021-04-13T02:48:58.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-19T22:13:44.000Z", "max_forks_repo_path": "src/Transformations.jl", "max_forks_repo_name": "weiyang607/RandomMatrix.jl", "max_forks_repo_head_hexsha": "d112a17d928c84583759177849a76eae1942e947", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.3448275862, "max_line_length": 65, "alphanum_fraction": 0.6324362606, "num_tokens": 534, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.92522995296862, "lm_q2_score": 0.8791467548438126, "lm_q1q2_score": 0.8134129106366556}} {"text": "################################\n ### Problem Set 0 - ECON 603 ###\n ################################\n\n #Sebastian Melo-Martin\n\n\n# Call some useful packages \nusing Distributions\nusing Random\nusing QuantEcon\nusing Plots\n\n\n############\n# Question 1\n############\n\nfunction tauchen(N::Integer, ρ::T1, σ::T2, μ=zero(promote_type(T1, T2)), n_std::T3=3) where {T1 <: Real, T2 <: Real, T3 <: Real}\n # Discretized space\n a_bar = n_std * sqrt(σ^2 / (1 - ρ^2))\n y = range(-a_bar, stop=a_bar, length=N)\n d = y[2] - y[1]\n\n # Transition probabilities\n Π = zeros(promote_type(T1, T2), N, N)\n for row = 1:N\n # Do end points first\n Π[row, 1] = cdf(Normal(),((y[1] - ρ*y[row] + d/2) / σ))\n Π[row, N] = 1 - cdf(Normal(),((y[N] - ρ*y[row] - d/2) / σ))\n\n # Fill in columns\n for col = 2:N-1\n Π[row, col] = (cdf(Normal(),((y[col] - ρ*y[row] + d/2) / σ)) -\n cdf(Normal(),((y[col] - ρ*y[row] - d/2) / σ)))\n end\n end\n\n return Π, y\n \nend\n\nz = tauchen(3,0.2,0.4)\nprob_tau = z[1]\nybar_tau = @show Vector(z[2])\n\ny_tau = zeros(1000,1)\npos = zeros(1000,1)\ny_tau[1] = ybar_tau[rand(1:3)]\n\nfor i=2:1000\n x=rand()\n for j=1:3\n if y_tau[i-1]==ybar_tau[j]\n pos[i] = j\n end\n end\n prob_row = prob_tau[floor(Int, pos[i]),:]\n thr_1 = prob_row[1]\n thr_2 = thr_1 + prob_row[2]\n if x<=thr_1\n y_tau[i]=ybar_tau[1]\n elseif x>thr_1 && x<=thr_2\n y_tau[i] = ybar_tau[2]\n else\n y_tau[i] = ybar_tau[3]\n end\nend\n\nt = collect(range(1,stop=1000,length=1000))\nplot(t, y_tau)\n\n##############\n# Question 2 #\n##############\n\n#Getting initial vector and probability matrix\nfunction rouwenhorst(N::Integer, ρ::Real, σ::Real, μ::Real)\n\n σ_y = σ / sqrt(1-ρ^2)\n p = (1+ρ)/2\n Θ = [p 1-p; 1-p p]\n ψ = sqrt(N-1) * σ_y\n m = μ / (1 - ρ)\n\n state_values, p = _rouwenhorst(p, p, m, ψ, N)\n\n return p, state_values\nend\n\nfunction _rouwenhorst(p::Real, q::Real, m::Real, Δ::Real, n::Integer)\n if n == 2\n return [m-Δ, m+Δ], [p 1-p; 1-q q]\n else\n _, θ_nm1 = _rouwenhorst(p, q, m, Δ, n-1)\n θN = p *[θ_nm1 zeros(n-1, 1); zeros(1, n)] +\n (1-p)*[zeros(n-1, 1) θ_nm1; zeros(1, n)] +\n q *[zeros(1, n); zeros(n-1, 1) θ_nm1] +\n (1-q)*[zeros(1, n); θ_nm1 zeros(n-1, 1)]\n\n θN[2:end-1, :] ./= 2\n\n return range(m-Δ, stop=m+Δ, length=n), θN\n end\nend\n\nz = rouwenhorst(3,0.2,0.4,0)\nprob_rou = z[1]\nybar_rou = @show Vector(z[2])\n\n#Simulating the time series\ny_rou = zeros(1000,1)\npos = zeros(1000,1)\ny_rou[1] = ybar_rou[rand(1:3)]\n\nfor i=2:1000\n x=rand()\n for j=1:3\n if y_rou[i-1]==ybar_rou[j]\n pos[i] = j\n end\n end\n prob_row = prob_rou[floor(Int, pos[i]),:]\n thr_1 = prob_row[1]\n thr_2 = prob_row[1] + prob_row[2]\n if x<=thr_1\n y_rou[i]=ybar_rou[1]\n elseif x>thr_1 && x<=thr_2\n y_rou[i] = ybar_rou[2]\n else\n y_rou[i] = ybar_rou[3]\n end\nend\n\nt = collect(range(1,stop=1000,length=1000))\nplot(t, y_rou)\n\n##############\n# Question 3 #\n##############\n\nfunction normal_ar(ρ, N)\n d = Normal(0, 0.16)\n y = [0.0 for i = 1:N]\n Random.seed!(641993)\n ϵ = rand(d, N)\n for i in 1:(N-1)\n y[i+1] = ρ*y[i] + ϵ[i] \n end\n return y\nend\n\nρ=0.2\nN=1000\ny_ar = normal_ar(ρ,N)\nt = collect(range(1,stop=1000,length=1000))\n\nplot(t, y_ar)\n\n\n##############\n# Question 4 #\n##############\n\n# Tauchen\nz = tauchen(10,0.2,0.4)\nprob_tau = z[1]\nybar_tau = @show Vector(z[2])\n\ny_tau = zeros(1000,1)\npos = zeros(1000,1)\ny_tau[1] = ybar_tau[rand(1:10)]\n\nfor i=2:1000\n x=rand()\n for j=1:10\n if y_tau[i-1]==ybar_tau[j]\n pos[i] = j\n end\n end\n prob_row = prob_tau[floor(Int, pos[i]),:]\n thr_1 = prob_row[1]\n thr_2 = thr_1 + prob_row[2]\n thr_3 = thr_2 + prob_row[3]\n thr_4 = thr_3 + prob_row[4]\n thr_5 = thr_4 + prob_row[5]\n thr_6 = thr_5 + prob_row[6]\n thr_7 = thr_6 + prob_row[7]\n thr_8 = thr_7 + prob_row[8]\n thr_9 = thr_8 + prob_row[9]\n if x<=thr_1\n y_tau[i]=ybar_tau[1]\n elseif x>thr_1 && x<=thr_2\n y_tau[i] = ybar_tau[2]\n elseif x>thr_2 && x<=thr_3\n y_tau[i] = ybar_tau[3]\n elseif x>thr_3 && x<=thr_4\n y_tau[i] = ybar_tau[4]\n elseif x>thr_4 && x<=thr_5\n y_tau[i] = ybar_tau[5]\n elseif x>thr_5 && x<=thr_6\n y_tau[i] = ybar_tau[6]\n elseif x>thr_6 && x<=thr_7\n y_tau[i] = ybar_tau[7]\n elseif x>thr_7 && x<=thr_8\n y_tau[i] = ybar_tau[8]\n elseif x>thr_8 && x<=thr_9\n y_tau[i] = ybar_tau[9]\n else\n y_tau[i] = ybar_tau[10]\n end\nend\n\nt = collect(range(1,stop=1000,length=1000))\nplot(t, y_tau)\n\n# Rouwenhorst\nz = rouwenhorst(10,0.2,0.4,0)\nprob_rou = z[1]\nybar_rou = @show Vector(z[2])\n\ny_rou = zeros(1000,1)\npos = zeros(1000,1)\ny_rou[1] = ybar_tau[rand(1:10)]\n\nfor i=2:1000\n x=rand()\n for j=1:10\n if y_rou[i-1]==ybar_rou[j]\n pos[i] = j\n end\n end\n prob_row = prob_rou[floor(Int, pos[i]),:]\n thr_1 = prob_row[1]\n thr_2 = thr_1 + prob_row[2]\n thr_3 = thr_2 + prob_row[3]\n thr_4 = thr_3 + prob_row[4]\n thr_5 = thr_4 + prob_row[5]\n thr_6 = thr_5 + prob_row[6]\n thr_7 = thr_6 + prob_row[7]\n thr_8 = thr_7 + prob_row[8]\n thr_9 = thr_8 + prob_row[9]\n if x<=thr_1\n y_rou[i]=ybar_rou[1]\n elseif x>thr_1 && x<=thr_2\n y_rou[i] = ybar_rou[2]\n elseif x>thr_2 && x<=thr_3\n y_rou[i] = ybar_rou[3]\n elseif x>thr_3 && x<=thr_4\n y_rou[i] = ybar_rou[4]\n elseif x>thr_4 && x<=thr_5\n y_rou[i] = ybar_rou[5]\n elseif x>thr_5 && x<=thr_6\n y_rou[i] = ybar_rou[6]\n elseif x>thr_6 && x<=thr_7\n y_rou[i] = ybar_rou[7]\n elseif x>thr_7 && x<=thr_8\n y_rou[i] = ybar_rou[8]\n elseif x>thr_8 && x<=thr_9\n y_rou[i] = ybar_rou[9]\n else\n y_rou[i] = ybar_rou[10]\n end\nend\n\nt = collect(range(1,stop=1000,length=1000))\nplot(t, y_rou)\n\n\n##############\n# Question 5 #\n##############\n\n#= The coding for this question is already included previously in a loop\n =#\n", "meta": {"hexsha": "a09b61a575fa429a6c4978e735507d8913d17ad1", "size": 6245, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "PS_0_first_version.jl", "max_stars_repo_name": "sebaschoo/Second_Term_PS", "max_stars_repo_head_hexsha": "21bc623a99bf60929bf46d07a662cc120e6d6d38", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "PS_0_first_version.jl", "max_issues_repo_name": "sebaschoo/Second_Term_PS", "max_issues_repo_head_hexsha": "21bc623a99bf60929bf46d07a662cc120e6d6d38", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "PS_0_first_version.jl", "max_forks_repo_name": "sebaschoo/Second_Term_PS", "max_forks_repo_head_hexsha": "21bc623a99bf60929bf46d07a662cc120e6d6d38", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.6268115942, "max_line_length": 128, "alphanum_fraction": 0.514011209, "num_tokens": 2476, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109812297141, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.8134103248813424}} {"text": "\"\"\"\n**Adagrad constructor**\n\n```julia\n Adagrad(; η::Real=0.01, ϵ::Real=1e-8)\n```\n\nAlgorithm :\n\n```math\n\\\\begin{align*}\nG_t =& g_t^2\\\\\\\\\n\\\\Delta x_t =& \\\\frac{\\\\eta}{\\\\sqrt{G_t+ϵ}} g_t\n\\\\end{align*}\n```\n\n[Algorithm Reference](http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf)\n\"\"\"\nmutable struct Adagrad <: Optimizer\n opt_type::String\n t::Int64\n ϵ::Float64\n η::Float64\n G_t::AbstractArray\nend\n\n\nfunction Adagrad(; η::Real=0.01, ϵ::Real=1e-8)\n @assert η > 0.0 \"η must be greater than 0\"\n @assert ϵ > 0.0 \"ϵ must be greater than 0\"\n\n Adagrad(\"Adagrad\", 0, ϵ, η, [])\nend\n\nparams(opt::Adagrad) = \"ϵ=$(opt.ϵ), η=$(opt.η)\"\n\nfunction update(opt::Adagrad, g_t::AbstractArray{T}) where {T<:Real}\n # resize squares of gradients\n if opt.t == 0\n opt.G_t = zero(g_t)\n end\n\n # update timestep\n opt.t += 1\n\n # accumulate squares of gradients\n opt.G_t .+= (g_t .^ 2)\n\n δ = opt.η ./ (sqrt.(opt.G_t .+ opt.ϵ)) .* g_t\n\n return δ\nend\n", "meta": {"hexsha": "d72b4e60cfd983a88f807156039e17ea7fc738d3", "size": 983, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/AdaGradOptimizer.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/GradDescent.jl-e1397348-e965-55d8-8fb3-3dd9faf6e4f1", "max_stars_repo_head_hexsha": "a3b2c35983a3cf5c88e1f3bb0df4f43bbacb44c3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 10, "max_stars_repo_stars_event_min_datetime": "2017-08-20T00:43:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-12T03:09:32.000Z", "max_issues_repo_path": "src/AdaGradOptimizer.jl", "max_issues_repo_name": "UnofficialJuliaMirrorSnapshots/GradDescent.jl-e1397348-e965-55d8-8fb3-3dd9faf6e4f1", "max_issues_repo_head_hexsha": "a3b2c35983a3cf5c88e1f3bb0df4f43bbacb44c3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2017-11-19T22:38:05.000Z", "max_issues_repo_issues_event_max_datetime": "2019-10-25T20:33:14.000Z", "max_forks_repo_path": "src/AdaGradOptimizer.jl", "max_forks_repo_name": "UnofficialJuliaMirrorSnapshots/GradDescent.jl-e1397348-e965-55d8-8fb3-3dd9faf6e4f1", "max_forks_repo_head_hexsha": "a3b2c35983a3cf5c88e1f3bb0df4f43bbacb44c3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2017-08-03T16:14:45.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-03T12:48:41.000Z", "avg_line_length": 18.5471698113, "max_line_length": 80, "alphanum_fraction": 0.5890132248, "num_tokens": 382, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109812297141, "lm_q2_score": 0.8558511396138365, "lm_q1q2_score": 0.8134103213869555}} {"text": "#=\nThis is an implementation of the \"Babylonian method\" of approximating square roots.\n(https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)\n=#\nconst steps = 10\nconst example = 20740\n\nfunction sqrt(s)\n x = s/2 # arbitrary initial guess for x ≈ √s\n for i in 1:steps\n x = (x + s/x)/2\n end\n x\nend\n\nprintln(\"√$example ≈ $(sqrt(example)), error = $(abs(sqrt(example)) - (example)^0.5)\")\n", "meta": {"hexsha": "02a85214bd0bf77433ecdddc0fb4324ef0241f5d", "size": 431, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "algorithms/math/babyloniansqrt.jl", "max_stars_repo_name": "landrumb/julia", "max_stars_repo_head_hexsha": "b57d478825c4bea8d0780efab80b7cbe32f3ea45", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 26, "max_stars_repo_stars_event_min_datetime": "2018-10-08T05:29:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-21T09:27:27.000Z", "max_issues_repo_path": "algorithms/math/babyloniansqrt.jl", "max_issues_repo_name": "landrumb/julia", "max_issues_repo_head_hexsha": "b57d478825c4bea8d0780efab80b7cbe32f3ea45", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2018-10-08T05:44:32.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-20T15:33:02.000Z", "max_forks_repo_path": "algorithms/math/babyloniansqrt.jl", "max_forks_repo_name": "landrumb/julia", "max_forks_repo_head_hexsha": "b57d478825c4bea8d0780efab80b7cbe32f3ea45", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-10-26T18:07:08.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-21T09:27:30.000Z", "avg_line_length": 25.3529411765, "max_line_length": 86, "alphanum_fraction": 0.6728538283, "num_tokens": 136, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9615338112885303, "lm_q2_score": 0.8459424411924673, "lm_q1q2_score": 0.8134022596105165}} {"text": "include(\"../utils/format/Heading.jl\")\n\n\n# float32/64\nfloat_64= 1e-2\nfloat_32= 1f-2\nprintln(typeof(float_64), \" \", typeof(float_32))\n#: Float64 Float32\n\n\n# underscore as digit separator\nprintln(100_000, \" \", 0.00_00_5)\n#: 100000 5.0e-5\n\n\n# flotiing point has two zeros - positive and negative\n# both are `=` same but `bitstring` different \nprintln(0.0 == -0.0)\n#: true\nprintln(bitstring(0.0) == bitstring(-0.0))\n#: false\n\n\nHeading.main(\"Special floating point values\")\n\n\nprintln(-5/0)\nprintln(-5/Inf)\n#: -Inf\n#: -0.0\n\n\nprintln(Inf+500)\n#: Inf\n\n\n# as Inf+x (or) Inf-x (or) Inf + Inf all equal to Inf\n# Information is lost. So, Inf - Inf cannot be exactly 0 \nprintln(Inf - Inf)\n#: NaN \n\n\n\n# epsilon: distance between 1.0 and the next larger \n# representable floating-point value i.e smallest distance \n# possible from 1.0\nprintln(eps(Float16))\nprintln(eps(Float32))\nprintln(eps(Float64))\nprintln(eps()) # same as 64\n#: 0.000977\n#: 1.1920929e-7\n#: 2.220446049250313e-16\n#: 2.220446049250313e-16\n\n\nprintln(typemax(BigFloat))\n#: Inf", "meta": {"hexsha": "e44fab0f972316af538cdf295a0bd9a873596e2d", "size": 1026, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "02-Data-Types/02-01-float.jl", "max_stars_repo_name": "rakesh4real/master-julia", "max_stars_repo_head_hexsha": "b7677184296201e29e90bc768385747c82072cf1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "02-Data-Types/02-01-float.jl", "max_issues_repo_name": "rakesh4real/master-julia", "max_issues_repo_head_hexsha": "b7677184296201e29e90bc768385747c82072cf1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "02-Data-Types/02-01-float.jl", "max_forks_repo_name": "rakesh4real/master-julia", "max_forks_repo_head_hexsha": "b7677184296201e29e90bc768385747c82072cf1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.6896551724, "max_line_length": 59, "alphanum_fraction": 0.6910331384, "num_tokens": 346, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505248181418, "lm_q2_score": 0.8991213820004279, "lm_q1q2_score": 0.8133906301019}} {"text": "function bisection(f::Function, xl::Number, xu::Number, ea::Number, maxIter::Int64)\n # Estimate roots of functions using the bisection method\n xr = NaN;\n fxl = f(xl);\n\n for i = 1:maxIter\n xrlast = xr;\n xr = 0.5(xl+xu);\n fxr = f(xr);\n relativeError = abs((xr - xrlast) / xr) * 100;\n\n if fxr * fxl == 0\n break;\n elseif fxr *fxl > 0\n xl = xr;\n fxl = fxr;\n else\n xu = xr;\n end\n\n if relativeError < ea\n break;\n end\n end\n\n return xr;\nend\n", "meta": {"hexsha": "fd351d9d23a8082e39d752a1f657d4e232515ec0", "size": 493, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Julia/Root Estimation/bisection.jl", "max_stars_repo_name": "alexjohnj/numerical-methods", "max_stars_repo_head_hexsha": "152c24a5ab297cf2e7486e96c3f986dbe536537d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Julia/Root Estimation/bisection.jl", "max_issues_repo_name": "alexjohnj/numerical-methods", "max_issues_repo_head_hexsha": "152c24a5ab297cf2e7486e96c3f986dbe536537d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Julia/Root Estimation/bisection.jl", "max_forks_repo_name": "alexjohnj/numerical-methods", "max_forks_repo_head_hexsha": "152c24a5ab297cf2e7486e96c3f986dbe536537d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-16T23:12:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-16T23:12:40.000Z", "avg_line_length": 17.6071428571, "max_line_length": 83, "alphanum_fraction": 0.5476673428, "num_tokens": 177, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9381240142763573, "lm_q2_score": 0.8670357494949105, "lm_q1q2_score": 0.8133870578372755}} {"text": "using SpecialFunctions, Distributions\n\na,b = 0.2, 0.7\nx = 0.75\n\nbetaAB1 = beta(a,b)\nbetaAB2 = (gamma(a)gamma(b))/gamma(a+b)\nbetaAB3 = (factorial(a-1)factorial(b-1))/factorial(a+b-1)\nbetaPDFAB1 = pdf(Beta(a,b),x)\nbetaPDFAB2 = (1/beta(a,b))*x^(a-1) * (1-x)^(b-1)\n\nprintln(\"beta($a,$b) = $betaAB1,\\t$betaAB2,\\t$betaAB3 \")\nprintln(\"betaPDF($a,$b) at $x = $betaPDFAB1,\\t$betaPDFAB2\")", "meta": {"hexsha": "ebcc436c9d839b6330161cd4cb3b859bd2b7c707", "size": 381, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "3_chapter/betaDist.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "3_chapter/betaDist.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "3_chapter/betaDist.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 29.3076923077, "max_line_length": 59, "alphanum_fraction": 0.6404199475, "num_tokens": 168, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9591542887603536, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.8133319103360265}} {"text": "\"\"\"\n frct(signal, α, p)\n\nComputing the α order fractional cosine transform of the input **signal**.\n\n# Example\n\n```julia-repl\njulia> frct([1,2,3], 0.5, 2)\n3-element Vector{ComplexF64}:\n1.707106781186547 + 0.9874368670764581im\n1.5606601717798205 - 1.3964466094067267im\n-0.3535533905932727 - 0.6982233047033652im\n```\n\n### References\n\n```tex\n@article{article,\nauthor = {Pei, Soo-Chang and Yeh, Min-Hung},\nyear = {2001},\nmonth = {07},\npages = {1198 - 1207},\ntitle = {The discrete fractional cosine and sine transforms},\nvolume = {49},\njournal = {Signal Processing, IEEE Transactions on},\ndoi = {10.1109/78.923302}\n}\n```\n\"\"\"\nfunction frct(signal, α, p)\n N = length(signal)\n @views signal = signal[:]\n p = min(max(2, p), N-1)\n E = dFRCT(N,p)\n result = E *(exp.(-im*pi*α*collect(0:N-1)) .*(E' *signal))\n return result\nend\n\nfunction dFRCT(N, p)\n N1 = 2*N-2\n d2 = [1, -2, 1]\n\n d_p = 1\n s = 0\n st = zeros(1, N1)\n\n for k = 1:floor(Int, p/2)\n if typeof(d_p) <: Number\n d_p = @. d2*d_p\n else\n d_p = conv(d2, d_p)\n end\n st[vcat(collect(N1-k+1:N1), collect(1:k+1))] = d_p\n st[1] = 0\n temp = vcat(union(1, collect(1:k-1)), union(1,collect(1:k-1)))\n temp = temp[:] ./collect(1:2*k)\n s = s.+(-1)^(k-1)*prod(temp)*2*st\n end\n\n H = Toeplitz(s[:], s[:]) + diagm(real.(fft(s[:])))\n\n V = hcat(zeros(N-2), zeros(N-2, N-2)+I, zeros(N-2), reverse(zeros(N-2, N-2)+I, dims=1)) ./sqrt(2)\n V = vcat([1 zeros(1, N1-1)], V, [zeros(1, N-1) 1 zeros(1, N-2)])\n\n Ev = V*H*V'\n\n _, ee = eigen(Ev)\n\n E = reverse(ee, dims=2)\n E[end,:] = E[end,:]/sqrt(2)\n return E\nend", "meta": {"hexsha": "58d877bf9f54f33fd4218b71a8ce592bf6b9e117", "size": 1670, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/frct.jl", "max_stars_repo_name": "SciFracX/FractionalTransforms.jl", "max_stars_repo_head_hexsha": "f343419db8e79a3f08ae1d95faae5495f877317b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2021-11-07T22:00:30.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-24T00:49:40.000Z", "max_issues_repo_path": "src/frct.jl", "max_issues_repo_name": "SciFracX/FractionalTransforms.jl", "max_issues_repo_head_hexsha": "f343419db8e79a3f08ae1d95faae5495f877317b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-11-16T15:53:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-10T06:08:16.000Z", "max_forks_repo_path": "src/frct.jl", "max_forks_repo_name": "SciFracX/FractionalTransforms.jl", "max_forks_repo_head_hexsha": "f343419db8e79a3f08ae1d95faae5495f877317b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-08T12:06:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-08T12:06:40.000Z", "avg_line_length": 22.8767123288, "max_line_length": 101, "alphanum_fraction": 0.5556886228, "num_tokens": 653, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.959154281754899, "lm_q2_score": 0.8479677622198946, "lm_q1q2_score": 0.8133319099233319}} {"text": "export rosenbrock, rastrigin\n\n\"\"\"\nThe rastrigin function is notorious for having multiple local minima and only one global optima at (0,0)\n\n\"\"\"\nfunction rastrigin(x::Vector{T}) where {T <: AbstractFloat}\n a = one(eltype(x)) * 10\n return size(x, 1) * a + sum(@. x^2 - a * cos(2 * π * x))\nend\n\nfunction rosenbrock(x::Vector{T}) where {T <: AbstractFloat}\n f = zero(eltype(x))\n a = one(eltype(x))\n b = 100 * a\n for i in 1:(length(x) - 1)\n @inbounds f += b * (x[i + 1] - x[i]^2)^2 + (a - x[i])^2\n end\n return f\nend\n", "meta": {"hexsha": "50fe5f26c2850e4b47778e4e242d34d5d0767006", "size": 541, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/lossfunction.jl", "max_stars_repo_name": "Pramodh-G/VisualOptim.jl", "max_stars_repo_head_hexsha": "e0923e8acc8f91465d61bc75abecc520fc79fd8c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-03-24T09:27:58.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-24T10:54:20.000Z", "max_issues_repo_path": "src/lossfunction.jl", "max_issues_repo_name": "Pramodh-G/VisualOptim.jl", "max_issues_repo_head_hexsha": "e0923e8acc8f91465d61bc75abecc520fc79fd8c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/lossfunction.jl", "max_forks_repo_name": "Pramodh-G/VisualOptim.jl", "max_forks_repo_head_hexsha": "e0923e8acc8f91465d61bc75abecc520fc79fd8c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.7619047619, "max_line_length": 104, "alphanum_fraction": 0.5859519409, "num_tokens": 191, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.959154287592778, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.81333190934596}} {"text": "\nusing JuMP\nusing GLPK\n\n\nG = [\n0 100 30 0 0;\n0 0 20 0 0; \n0 0 0 10 60;\n0 15 0 0 50;\n0 0 0 0 0\n]\n\nn = size(G)[1]\n\nshortest_path = Model(with_optimizer(GLPK.Optimizer))\n\n@variable(shortest_path, x[1:n,1:n], Bin)\n@constraint(shortest_path, [i = 1:n, j = 1:n; G[i,j] == 0], x[i,j] == 0) # Arcs with zero cost are not a part of the path as they do no exist\n@constraint(shortest_path, [i = 1:n; i != 1 && i != 2], sum(x[i,:]) == sum(x[:,i])) # Flow conservation constraint\n@constraint(shortest_path, sum(x[1,:]) - sum(x[:,1]) == 1) # Flow coming out of source = 1\n@constraint(shortest_path, sum(x[2,:]) - sum(x[:,2]) == -1) # Flowing coming out of destination = -1 i.e. Flow entering destination = 1 \n@objective(shortest_path, Min, sum(G .* x))\n\noptimize!(shortest_path)\n@show objective_value(shortest_path);\n@show value.(x);\n\n\nG = [\n6 4 5 0;\n0 3 6 0;\n5 0 4 3;\n7 5 5 5;\n]\n\nn = size(G)[1]\n\nassignment = Model(with_optimizer(GLPK.Optimizer))\n@variable(assignment, y[1:n,1:n], Bin)\n@constraint(assignment, [i = 1:n], sum(y[:,i]) == 1) # One person can only be assigned to one object\n@constraint(assignment, [j = 1:n], sum(y[j,:]) == 1) # One object can only be assigned to one person\n@objective(assignment, Max, sum(G .* y))\n\noptimize!(assignment)\n@show objective_value(assignment);\n@show value.(y);\n\n\nG = [\n0 3 2 2 0 0 0 0 \n0 0 0 0 5 1 0 0 \n0 0 0 0 1 3 1 0 \n0 0 0 0 0 1 0 0 \n0 0 0 0 0 0 0 4 \n0 0 0 0 0 0 0 2 \n0 0 0 0 0 0 0 4 \n0 0 0 0 0 0 0 0 \n]\n\nn = size(G)[1]\n\nmax_flow = Model(with_optimizer(GLPK.Optimizer))\n\n@variable(max_flow, f[1:n,1:n] >= 0)\n@constraint(max_flow, [i = 1:n, j = 1:n], f[i,j] <= G[i,j]) # Capacity constraints\n@constraint(max_flow, [i = 1:n; i != 1 && i != 8], sum(f[i,:]) == sum(f[:,i])) # Flow conservation contraints\n@objective(max_flow, Max, sum(f[1, :]))\n\noptimize!(max_flow)\n@show objective_value(max_flow);\n@show value.(f);\n\n", "meta": {"hexsha": "c3d3c07cc2842eec2ec51c7dbe372e16b42e872c", "size": 1865, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test/modelling/network_flows.jl", "max_stars_repo_name": "carlosal1015/JuMPTutorials.jl", "max_stars_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/modelling/network_flows.jl", "max_issues_repo_name": "carlosal1015/JuMPTutorials.jl", "max_issues_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/modelling/network_flows.jl", "max_forks_repo_name": "carlosal1015/JuMPTutorials.jl", "max_forks_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.2027027027, "max_line_length": 141, "alphanum_fraction": 0.6198391421, "num_tokens": 741, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9591542817548989, "lm_q2_score": 0.8479677602988601, "lm_q1q2_score": 0.8133319080807635}} {"text": "@doc raw\"\"\"\n# Lotka-Volterra Model in 3D\n\nThe Lotka–Volterra model in 3D is an example of a Hamiltonian system with degenerate Poisson structure.\n\nThe equations read\n```math\n\\begin{aligned}\n\\dot{q}_{1} &= q_{1} ( - a_{2} q_{2} + a_{3} q_{3} - b_{2} + b_{3} ) , \\\\\n\\dot{q}_{2} &= q_{2} ( \\hphantom{-} a_{1} q_{1} - a_{3} q_{3} + b_{1} - b_{3} ) , \\\\\n\\dot{q}_{3} &= q_{3} ( - a_{1} q_{1} + a_{2} q_{2} - b_{1} + b_{2} ) , \\\\\n\\end{aligned}\n```\nwhich can be written in Poisson-form as\n```math\n\\dot{q} = P(q) \\nabla H(q) ,\n```\nwith Poisson matrix\n```math\nP(q) = \\begin{pmatrix}\n 0 & - q_{1} q_{2} & \\hphantom{-} q_{1} q_{3} \\\\\n\\hphantom{-} q_{1} q_{2} & 0 & - q_{2} q_{3} \\\\\n- q_{1} q_{3} & \\hphantom{-} q_{2} q_{3} & 0 \\\\\n\\end{pmatrix} ,\n```\nand Hamiltonian\n```math\nH(q) = a_{1} q_{1} + a_{2} q_{2} + a_{3} q_{3} + b_{1} \\ln q_{1} + b_{2} \\ln q_{2} + b_{3} \\ln q_{3} .\n```\n\nReferences:\n* A. M. Perelomov. Selected topics on classical integrable systems,\n Troisième cycle de la physique, expanded version of lectures\n delivered in May 1995.\n\n* Yuri B. Suris. Integrable discretizations for lattice systems: local\n equations of motion and their Hamiltonian properties,\n Rev. Math. Phys. 11, pp. 727–822, 1999.\n\"\"\"\nmodule LotkaVolterra3d\n\n using GeometricBase\n using GeometricEquations\n\n export lotka_volterra_3d_ode\n\n export hamiltonian, casimir\n export compute_energy_error, compute_casimir_error\n\n\n Δt = 0.01\n nt = 1000\n\n const A1=+1.0\n const A2=+1.0\n const A3=+1.0\n\n const B1= 0.0\n const B2=+1.0\n const B3=+1.0\n\n const X0=1.0\n const Y0=1.0\n const Z0=2.0\n\n\n function v₁(t, q)\n q[1] * ( - A2 * q[2] + A3 * q[3] + B2 - B3)\n end\n\n function v₂(t, q)\n q[2] * ( + A1 * q[1] - A3 * q[3] - B1 + B3)\n end\n\n function v₃(t, q)\n q[3] * ( - A1 * q[1] + A2 * q[2] + B1 - B2)\n end\n\n\n const q₀=[X0, Y0, Z0]\n const v₀=[v₁(0, q₀), v₂(0, q₀), v₃(0, q₀)]\n\n\n function hamiltonian(t, q)\n A1*q[1] + A2*q[2] + A3*q[3] - B1*log(q[1]) - B2*log(q[2]) - B3*log(q[3])\n end\n\n hamiltonian_iode(t, q, v) = hamiltonian(t, q)\n\n function casimir(t, q)\n log(q[1]) + log(q[2]) + log(q[3])\n end\n\n\n function lotka_volterra_3d_v(t, q, v)\n v[1] = v₁(t, q)\n v[2] = v₂(t, q)\n v[3] = v₃(t, q)\n nothing\n end\n\n\n function lotka_volterra_3d_ode(q₀=q₀)\n ODE(lotka_volterra_3d_v, q₀; invariants=(h=hamiltonian,))\n end\n\n\n function compute_energy_error(t, q::AbstractDataSeries{<:AbstractVector{T}}) where {T}\n h = DataSeries(T, ntime(q))\n e = DataSeries(T, ntime(q))\n\n for i in axes(q,1)\n h[i] = hamiltonian(t[i], q[i])\n e[i] = (h[i] - h[0]) / h[0]\n end\n\n (h, e)\n end\n\n function compute_casimir_error(t, q::AbstractDataSeries{<:AbstractVector{T}}) where {T}\n c = DataSeries(T, ntime(q))\n e = DataSeries(T, ntime(q))\n\n for i in axes(q,1)\n c[i] = casimir(t[i], q[i])\n e[i] = (c[i] - c[0]) / c[0]\n end\n\n (c, e)\n end\n\nend\n", "meta": {"hexsha": "900a376c6ab349e49086eada05a28a21719d8d40", "size": 3119, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/lotka_volterra_3d.jl", "max_stars_repo_name": "DDMGNI/GeometricProblems.jl", "max_stars_repo_head_hexsha": "367f4cf63ff614c3051c7760b835889fdb7a040f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/lotka_volterra_3d.jl", "max_issues_repo_name": "DDMGNI/GeometricProblems.jl", "max_issues_repo_head_hexsha": "367f4cf63ff614c3051c7760b835889fdb7a040f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2020-02-11T01:23:56.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-24T11:06:23.000Z", "max_forks_repo_path": "src/lotka_volterra_3d.jl", "max_forks_repo_name": "DDMGNI/GeometricProblems.jl", "max_forks_repo_head_hexsha": "367f4cf63ff614c3051c7760b835889fdb7a040f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.4511278195, "max_line_length": 103, "alphanum_fraction": 0.5328630971, "num_tokens": 1250, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9546474207360066, "lm_q2_score": 0.8519528000888386, "lm_q1q2_score": 0.8133145431936284}} {"text": "\"\"\"\r\n basis = basis_rotation(Ω, i, ω)\r\n basis = basis_rotation(xvec, yvec)\r\n basis_rotation() = one(RotMatrix{3, Float64})\r\n\r\nFrom the RAAN `Ω`, inclination `i`, and argument of the periapsis `ω`, computes the rotation matrix that transforms \r\nan orbit's perifocal plane to the inertial frame. For this matrix, the first column is a unit vector that points toward \r\nthe periapsis of the orbit, and the third column is a unit vector that points in the normal direction. \r\n\r\nIf two vectors `xvec` and `yvec` are provided, a rotation matrix is constructed that describes the plane the two vectors\r\ndefine, where `normalize(xvec)` is the first basis vector.\r\n\"\"\"\r\nfunction basis_rotation(Ω, i, ω)\r\n # Ω, ω, i = Float64(Ω), Float64(ω), Float64(i)\r\n # R₁ = [cos(Ω) -sin(Ω) 0.; # rotate around z-axis to match Ω\r\n # sin(Ω) cos(Ω) 0.;\r\n # 0. 0. 1.]\r\n # R₂ = [1. 0. 0.; # rotate around x-axis to match i\r\n # 0. cos(i) -sin(i);\r\n # 0. sin(i) cos(i)]\r\n # R₃ = [cos(ω) -sin(ω) 0.; # rotate around z-axis to match ω\r\n # sin(ω) cos(ω) 0.;\r\n # 0. 0. 1.]\r\n # return R₃*R₂*R₁\r\n return RotZXZ(Ω, i, ω)\r\nend\r\n\r\nfunction basis_rotation(xvec::AbstractVector{<:Real}, yvec::AbstractVector{<:Real})\r\n @assert length(xvec) == length(yvec) == 3\r\n xbasisvec = normalize(xvec)\r\n ybasisvec = normalize(yvec - xbasisvec .* dot(xbasisvec, yvec))\r\n zbasisvec = cross(xbasisvec, ybasisvec)\r\n return RotMatrix(hcat(xbasisvec,ybasisvec,zbasisvec))\r\nend\r\n\r\nbasis_rotation() = one(RotMatrix{3, Float64})\r\n\r\n# methods for switching orbital reference frames\r\n\r\n\"\"\"\r\n pvec = inertial_to_perifocal_bases(vec, basis)\r\n pvec = inertial_to_perifocal_bases(vec, orb)\r\n\r\nTransforms a vector `vec` from the inertial frame to an orbit's perifocal frame,\r\ndefined by `basis`.\r\n\"\"\"\r\ninertial_to_perifocal_bases(vec::AbstractVector{<:Real}, basis::Rotation{3,Float64}) = basis\\vec\r\ninertial_to_perifocal_bases(vec::AbstractVector{<:Real}, orb::Orbit) = inertial_to_perifocal_bases(vec, orb.basis)\r\n\r\n\"\"\"\r\n ivec = perifocal_to_inertial_bases(vec, basis)\r\n ivec = perifocal_to_inertial_bases(vec, orb)\r\n\r\nTransforms a vector `vec` from an orbit's perifocal frame, defined by `basis`, \r\nto the inertial frame.\r\n\"\"\"\r\nperifocal_to_inertial_bases(vec::AbstractVector{<:Real}, basis::Rotation{3,Float64}) = basis*vec\r\nperifocal_to_inertial_bases(vec::AbstractVector{<:Real}, orb::Orbit) = perifocal_to_inertial_bases(vec, orb.basis)\r\n\r\n\"\"\"\r\n rotmat = align_vectors(vec, vecref)\r\n\r\nComputes the rotation matrix to align `vec` with `vecref`.\r\n\"\"\"\r\nfunction align_vectors(vec::AbstractVector{<:Real}, vecref::AbstractVector{<:Real})\r\n nvec, nvecref = normalize(vec), normalize(vecref)\r\n axis = normalize(cross(nvec, nvecref))\r\n angle = wrap_acos(dot(nvec, nvecref))\r\n if angle == 0\r\n return one(RotMatrix{3, Float64})\r\n end\r\n return AngleAxis(angle, axis...)\r\nend\r\n", "meta": {"hexsha": "5230432c9c1e2796e63eb0867e9ee5c6ee864253", "size": 2989, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/basis.jl", "max_stars_repo_name": "tmcgrath325/KeplerTools.jl", "max_stars_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/basis.jl", "max_issues_repo_name": "tmcgrath325/KeplerTools.jl", "max_issues_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-08-19T16:28:02.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-19T16:28:17.000Z", "max_forks_repo_path": "src/basis.jl", "max_forks_repo_name": "tmcgrath325/KeplerTools.jl", "max_forks_repo_head_hexsha": "bb95d6c24ca7e1cea6f7f8057f01a415f5719f6e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.3918918919, "max_line_length": 121, "alphanum_fraction": 0.6634325861, "num_tokens": 923, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545289551958, "lm_q2_score": 0.8577681013541613, "lm_q1q2_score": 0.8132967100922475}} {"text": "module Searching \n\nexport binarysearch, BinarySearchTree, BinaryNode, delete!\nexport size, get, maximum, minimum\n\n# Binary search \nstruct UnsortedException <: Exception end\nBase.showerror(io::IO, e::UnsortedException) = print(io, \"input vecotr v is not sorted\")\n\nmidpoint(lo::Integer, hi::Integer) = lo + div(hi - lo, 2)\n\nfunction binarysearch(key, v::AbstractVector; algo=:recursive)\n issorted(v) || throw(UnsortedException())\n lo, hi = 1, length(v)\n if algo == :recursive\n binarysearch_rec(key, v, lo, hi)\n elseif algo == :iterative\n binarysearch_iter(key, v, lo, hi)\n end\nend\n\nfunction binarysearch_rec(key, v::AbstractVector, lo::Integer, hi::Integer)\n i, j = lo, hi\n if i <= j\n mid = midpoint(i, j)\n if key < v[mid] \n binarysearch_rec(key, v, i, mid-1)\n elseif key > v[mid]\n binarysearch_rec(key, v, mid+1, j)\n else\n return mid\n end\n else\n return -1\n end\nend\n\nfunction binarysearch_iter(key, v::AbstractVector, lo::Integer, hi::Integer)\n i, j = lo, hi\n while i <= j\n mid = midpoint(i, j) \n if key < v[mid]\n j = mid - 1\n elseif key > v[mid]\n i = mid + 1\n else\n return mid\n end\n end\n return -1\nend\n\n# Binary search tree\n\nmutable struct BinaryNode{T} \n val::T\n numnodes::Int\n left::Union{BinaryNode{T}, Nothing}\n right::Union{BinaryNode{T}, Nothing}\n\n # root constructor\n BinaryNode(val::T) where T = new{T}(val, 1, nothing, nothing)\nend\n\nabstract type SearchTree end\n\n\"\"\"\nBinary search tree.\n\"\"\"\nstruct BinarySearchTree{T} <: SearchTree \n root::BinaryNode{T}\nend\n\nfunction BinarySearchTree(v::AbstractVector)\n !isempty(v) || error(\"empty vector\")\n \n bst = BinarySearchTree(BinaryNode(v[1]))\n for val in v[2:end]\n put!(bst, val)\n end\n\n return bst\nend\n\nSearching.size(::Nothing) = 0\nSearching.size(node::BinaryNode{T}) where T = node.numnodes\nSearching.size(tree::BinarySearchTree{T}) where T = Searching.size(tree.root)\n\nfunction updatesize!(node)\n node.numnodes = Searching.size(node.left) + Searching.size(node.right) + 1\n nothing\nend\n\n\"\"\"\n put!(tree, val)\n\nInsert a value into the binary search tree.\n\n put!(node, val)\n\nInsert a value following the node.\n\"\"\"\nfunction put!(node::BinaryNode{T}, val::T) where T\n if val < node.val\n node.left = put!(node.left, val)\n elseif val > node.val \n node.right = put!(node.right, val)\n else\n node.val = val\n end\n updatesize!(node)\n return node\nend\n\nput!(tree::BinarySearchTree{T}, val::T) where T = BinarySearchTree(put!(tree.root, val))\nput!(::Nothing, val) = BinaryNode(val)\n\n\"\"\"\n get(node, val)\n\nSearch for the node by value. Return `nothing` if not found.\n\"\"\"\nfunction get(node::BinaryNode, val)\n if val < node.val\n return get(node.left, val)\n elseif val > node.val\n return get(node.right, val)\n else\n return node\n end\nend\n\nget(tree::BinarySearchTree, val) = get(tree.root, val)\nget(::Nothing, val)= nothing\n\n\"\"\"\n maximum(BinarySearchTree)\n\nReturn node with maximum value.\n\"\"\"\nfunction maximum(node) \n isnothing(node) && return nothing\n\n while !isnothing(node.right)\n node = node.right\n end\n\n return node\nend\n\nmaximum(tree::BinarySearchTree) = maximum(tree.root)\n\nfunction minimum(node)\n isnothing(node) && return nothing\n\n while !isnothing(node.left)\n node = node.left\n end\n\n return node\nend\n\nminimum(tree::BinarySearchTree) = minimum(tree.root)\n\n\"\"\"\n rank(node, val)\n\nFind the rank for value `val`: number of nodes with values less \nthan `val` + 1.\n\"\"\"\nfunction rank(node::BinaryNode, val)\n if val < node.val\n valrank = rank(node.left, val)\n elseif val > node.val\n valrank = rank(node.right, val) + Searching.size(node.left) + 1\n else\n valrank = Searching.size(node.left)\n end\n\n return valrank\nend\n\nrank(::Nothing, val) = 0\nrank(tree::BinarySearchTree, val) = rank(tree.root, val) + 1\n\n\"\"\"\n select(node, rankval)\n\nSelect the node with ranking `rankval`. \nWe assume rankval is in range between 1 to the size of the tree.\n\"\"\"\nfunction select(node::BinaryNode, valrank)\n noderank = Searching.size(node.left) + 1\n if valrank < noderank\n node = select(node.left, valrank)\n elseif valrank > noderank\n node = select(node.right, valrank - noderank)\n else\n # found position with `valrank`\n # return node\n end\n\n return node\nend\n\nselect(::Nothing, valrank) = nothing\nselect(tree::BinarySearchTree, valrank) = select(tree.root, valrank)\n\n\"\"\"\n deletemin!(node)\nFind the minimum node and delete it from the tree.\nWe can find the minimum node by move left recursively, until \nwe find the node with no left child.\n\n\"\"\"\nfunction deletemin!(node)\n if isnothing(node.left)\n node = node.right\n else\n node.left = deletemin!(node.left)\n end\n updatesize!(node)\n return node\nend\n\ndeletemin!(tree::BinarySearchTree) = BinarySearchTree(deletemin!(tree.root))\n\n\"\"\"\n copyval!(tonode, fromnode)\n\nCopy values from `fromnode` to `tonode`.\n\"\"\"\nfunction copyval!(tonode::BinaryNode, fromnode::BinaryNode)\n tonode.val = fromnode.val \n tonode.numnodes = fromnode.numnodes\n nothing\nend\n\n\"\"\"\n delete!(tree, val)\n\nDelete the node from the binary search tree.\nIt returns the successor of the deleted node.\n\n delete!(node, val)\n\nDelete the node with value `val` in the substree \nstarts with `node`.\n\"\"\"\nfunction delete!(node, val)\n if isnothing(node)\n return nothing\n elseif val < node.val\n node.left = delete!(node.left, val)\n elseif val > node.val\n node.right = delete!(node.right, val)\n # found the value and node has only one child node\n elseif isnothing(node.right)\n node = node.left\n elseif isnothing(node.left)\n node = node.right\n # node has both left and right child nodes\n else \n # replace current node with its successor\n # here I cheated using copy value (without links), \n # instead of replacing the object\n # This way we keep the original reference to make the\n # side effect consistent: the returned node and original \"node\"\n # refers to the same object (a substree starting from that node)\n copyval!(node, minimum(node.right)) \n\n # left child remain the same \n # delete the the successor from the right subtree\n # link successor the right child to the pruned subtree\n node.right = deletemin!(node.right)\n end\n\n updatesize!(node)\n return node\nend\n\ndelete!(tree::BinarySearchTree, val) = BinarySearchTree(delete!(tree.root, val))\n\nend # module", "meta": {"hexsha": "0870a3acb17d3eed20773a49ee2edb87688b66ee", "size": 6717, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/searching.jl", "max_stars_repo_name": "Jetafull/AlgorithmsInJulia", "max_stars_repo_head_hexsha": "6e4d8a58e063861967de9260f15d182eb7d35ada", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-03-21T03:27:23.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-21T03:27:23.000Z", "max_issues_repo_path": "src/searching.jl", "max_issues_repo_name": "Jetafull/algorithms-in-julia", "max_issues_repo_head_hexsha": "6e4d8a58e063861967de9260f15d182eb7d35ada", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/searching.jl", "max_forks_repo_name": "Jetafull/algorithms-in-julia", "max_forks_repo_head_hexsha": "6e4d8a58e063861967de9260f15d182eb7d35ada", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.6514084507, "max_line_length": 88, "alphanum_fraction": 0.6471639125, "num_tokens": 1736, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582516374121, "lm_q2_score": 0.8740772351648677, "lm_q1q2_score": 0.8132923760275659}} {"text": "# # Minimization of simple loss functions\n#\n# First we load the required packages.\n\nusing Distributions, LinearAlgebra, Random, Plots\n\nusing EnsembleKalmanProcesses.EnsembleKalmanProcessModule, EnsembleKalmanProcesses.ParameterDistributionStorage\n\n# ## Loss function with single minimum\n#\n# Here, we minimize the loss function\n# ```math\n# G₁(u) = \\|u - u_*\\| ,\n# ```\n# where ``u`` is a 2-vector of parameters and ``u_*`` is given; here ``u_* = (-1, 1)``. \nu★ = [1, -1]\nG₁(u) = [sqrt((u[1] - u★[1])^2 + (u[2] - u★[2])^2)]\nnothing # hide\n\n# We set the seed for pseudo-random number generator for reproducibility.\nrng_seed = 41\nRandom.seed!(rng_seed)\nnothing # hide\n\n# We set a stabilization level, which can aid the algorithm convergence\ndim_output = 1\nstabilization_level = 1e-3\nΓ_stabilization = stabilization_level * Matrix(I, dim_output, dim_output)\n\n# The functional is positive so to minimize it we may set the target to be 0,\nG_target = [0]\nnothing # hide\n\n# ### Prior distributions\n#\n# As we work with a Bayesian method, we define a prior. This will behave like an \"initial guess\"\n# for the likely region of parameter space we expect the solution to live in.\nprior_distributions = [Parameterized(Normal(0, 1)), Parameterized(Normal(0, 1))]\n\nconstraints = [[no_constraint()], [no_constraint()]]\n\nparameter_names = [\"u1\", \"u2\"]\n\nprior = ParameterDistribution(prior_distributions, constraints, parameter_names)\n\n# ### Calibration\n#\n# We choose the number of ensemble members and the number of iterations of the algorithm\nN_ensemble = 20\nN_iterations = 10\nnothing # hide\n\n# The initial ensemble is constructed by sampling the prior\ninitial_ensemble = EnsembleKalmanProcessModule.construct_initial_ensemble(prior, N_ensemble; rng_seed = rng_seed)\n\n# We then initialize the Ensemble Kalman Process algorithm, with the initial ensemble, the\n# target, the stabilization and the process type (for EKI this is `Inversion`, initialized \n# with `Inversion()`). \nensemble_kalman_process =\n EnsembleKalmanProcessModule.EnsembleKalmanProcess(initial_ensemble, G_target, Γ_stabilization, Inversion())\n\n# Then we calibrate by *(i)* obtaining the parameters, *(ii)* calculate the loss function on\n# the parameters (and concatenate), and last *(iii)* generate a new set of parameters using\n# the model outputs:\nfor i in 1:N_iterations\n params_i = get_u_final(ensemble_kalman_process)\n\n g_ens = hcat([G₁(params_i[:, i]) for i in 1:N_ensemble]...)\n\n EnsembleKalmanProcessModule.update_ensemble!(ensemble_kalman_process, g_ens)\nend\n\n# and visualize the results:\nu_init = get_u_prior(ensemble_kalman_process)\n\nanim_unique_minimum = @animate for i in 1:N_iterations\n u_i = get_u(ensemble_kalman_process, i)\n\n plot(\n [u★[1]],\n [u★[2]],\n seriestype = :scatter,\n markershape = :star5,\n markersize = 11,\n markercolor = :red,\n label = \"optimum u⋆\",\n )\n\n plot!(\n u_i[1, :],\n u_i[2, :],\n seriestype = :scatter,\n xlims = extrema(u_init[1, :]),\n ylims = extrema(u_init[2, :]),\n xlabel = \"u₁\",\n ylabel = \"u₂\",\n markersize = 5,\n markeralpha = 0.6,\n markercolor = :blue,\n label = \"particles\",\n title = \"EKI iteration = \" * string(i),\n )\nend\nnothing # hide\n\n# The results show that the minimizer of ``G_1`` is ``u=u_*``. \n\ngif(anim_unique_minimum, \"unique_minimum.gif\", fps = 1) # hide\n\n# ## Loss function with two minima\n\n# Now let's do an example in which the loss function has two minima. We minimize the loss\n# function\n# ```math\n# G₂(u) = \\|u - v_{*}\\| \\|u - w_{*}\\| ,\n# ```\n# where again ``u`` is a 2-vector, and ``v_{*}`` and ``w_{*}`` are given 2-vectors. Here, we take ``v_{*} = (1, -1)`` and ``w_{*} = (-1, -1)``.\n\nv★ = [1, -1]\nw★ = [-1, -1]\nG₂(u) = [sqrt(((u[1] - v★[1])^2 + (u[2] - v★[2])^2) * ((u[1] - w★[1])^2 + (u[2] - w★[2])^2))]\nnothing # hide\n#\n# The procedure is same as the single-minimum example above.\n\n# We set the seed for pseudo-random number generator for reproducibility,\nrng_seed = 10\nRandom.seed!(rng_seed)\nnothing # hide\n\n# A positive function can be minimized with a target of 0,\nG_target = [0]\n\n# We choose the stabilization as in the single-minimum example\n\n# ### Prior distributions\n#\n# We define the prior. We can place prior information on e.g., ``u₁``, demonstrating a belief\n# that ``u₁`` is more likely to be negative. This can be implemented by setting a bias in the\n# mean of its prior distribution to e.g., ``-0.5``:\n\nprior_distributions = [Parameterized(Normal(-0.5, sqrt(2))), Parameterized(Normal(0, sqrt(2)))]\n\nconstraints = [[no_constraint()], [no_constraint()]]\n\nparameter_names = [\"u1\", \"u2\"]\n\nprior = ParameterDistribution(prior_distributions, constraints, parameter_names)\n\n# ### Calibration\n#\n# We choose the number of ensemble members, the number of EKI iterations, construct our initial ensemble and the EKI with the `Inversion()` constructor (exactly as in the single-minimum example):\nN_ensemble = 20\nN_iterations = 20\n\ninitial_ensemble = EnsembleKalmanProcessModule.construct_initial_ensemble(prior, N_ensemble; rng_seed = rng_seed)\n\nensemble_kalman_process =\n EnsembleKalmanProcessModule.EnsembleKalmanProcess(initial_ensemble, G_target, Γ_stabilization, Inversion())\n\n# We calibrate by *(i)* obtaining the parameters, *(ii)* calculating the\n# loss function on the parameters (and concatenate), and last *(iii)* generate a new set of\n# parameters using the model outputs:\nfor i in 1:N_iterations\n params_i = get_u_final(ensemble_kalman_process)\n\n g_ens = hcat([G₂(params_i[:, i]) for i in 1:N_ensemble]...)\n\n EnsembleKalmanProcessModule.update_ensemble!(ensemble_kalman_process, g_ens)\nend\n\n# and visualize the results:\nu_init = get_u_prior(ensemble_kalman_process)\n\nanim_two_minima = @animate for i in 1:N_iterations\n u_i = get_u(ensemble_kalman_process, i)\n\n plot(\n [v★[1]],\n [v★[2]],\n seriestype = :scatter,\n markershape = :star5,\n markersize = 11,\n markercolor = :red,\n label = \"optimum v⋆\",\n )\n\n plot!(\n [w★[1]],\n [w★[2]],\n seriestype = :scatter,\n markershape = :star5,\n markersize = 11,\n markercolor = :green,\n label = \"optimum w⋆\",\n )\n\n plot!(\n u_i[1, :],\n u_i[2, :],\n seriestype = :scatter,\n xlims = extrema(u_init[1, :]),\n ylims = extrema(u_init[2, :]),\n xlabel = \"u₁\",\n ylabel = \"u₂\",\n markersize = 5,\n markeralpha = 0.6,\n markercolor = :blue,\n label = \"particles\",\n title = \"EKI iteration = \" * string(i),\n )\nend\nnothing # hide\n\n# Our bias in the prior shifts the initial ensemble into the negative ``u_1`` direction, and\n# thus increases the likelihood (over different instances of the random number generator) of\n# finding the minimizer ``u=w_*``.\n\ngif(anim_two_minima, \"two_minima.gif\", fps = 1) # hide\n", "meta": {"hexsha": "6bff8f5c6b53a7c849c92f265e7dd8a8dece5410", "size": 6919, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/LossMinimization/loss_minimization.jl", "max_stars_repo_name": "lm2612/EnsembleKalmanProcesses.jl", "max_stars_repo_head_hexsha": "1fee0bc06b9578d4b97851831a280267e0bce7b4", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 25, "max_stars_repo_stars_event_min_datetime": "2021-03-02T08:35:32.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-26T18:50:04.000Z", "max_issues_repo_path": "examples/LossMinimization/loss_minimization.jl", "max_issues_repo_name": "lm2612/EnsembleKalmanProcesses.jl", "max_issues_repo_head_hexsha": "1fee0bc06b9578d4b97851831a280267e0bce7b4", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 129, "max_issues_repo_issues_event_min_datetime": "2021-02-24T17:44:57.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-30T17:38:35.000Z", "max_forks_repo_path": "examples/LossMinimization/loss_minimization.jl", "max_forks_repo_name": "lm2612/EnsembleKalmanProcesses.jl", "max_forks_repo_head_hexsha": "1fee0bc06b9578d4b97851831a280267e0bce7b4", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2021-05-21T20:46:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-03T20:53:46.000Z", "avg_line_length": 31.45, "max_line_length": 195, "alphanum_fraction": 0.6722069663, "num_tokens": 1949, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582516374121, "lm_q2_score": 0.8740772318846386, "lm_q1q2_score": 0.8132923729754497}} {"text": "### A Pluto.jl notebook ###\n# v0.14.5\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 19cc5ec6-f7cb-4736-98c3-498d91eb2414\nusing BarycentricInterpolation\n\n# ╔═╡ 53285b2c-3704-4adc-9bdd-9dea619b5140\nmd\" ## BarycentricInterpolation.jl\"\n\n# ╔═╡ 7ba7dd56-4ee9-4c5c-97a8-5f99a9ba507e\np = Chebyshev2{20}() # create a Chebyshev type 2 polynomial of order 20\n\n# ╔═╡ ddacc062-c4cb-40e1-8645-a1f8619eabea\nx = nodes(p) # get the nodes\n\n# ╔═╡ 7fdcd309-5e3e-4c65-9d8c-96d19fdc923c\ny = sinpi.(x) # generate y values at the nodes\n\n# ╔═╡ 9c9fdfad-44eb-4f5b-a5e0-82eaade02ed1\nx_new = rand()*2 -1 # a random number in [-1, +1]\n\n# ╔═╡ f3f7c1ea-f4aa-4dc2-a630-0fb37e4bdadb\nprintln(interpolate(p, y, x_new) ≈ sinpi(x_new)) # hopefully true!\n\n# ╔═╡ d0ca3867-a266-4299-ac49-9745154b3ad4\nD = differentiation_matrix(p) # get the differentiation matrix\n\n# ╔═╡ 880d1ae8-5a5a-463a-8e35-a73bdc40208a\nprintln(interpolate(p, D*y, x_new) ≈ pi*cospi(x_new)) # hopefully true!\n\n# ╔═╡ c2fcf437-bd86-4b24-8c77-29702cf5373d\nmd\" ## End of BarycentricInterpolation.jl\"\n\n# ╔═╡ Cell order:\n# ╟─53285b2c-3704-4adc-9bdd-9dea619b5140\n# ╠═19cc5ec6-f7cb-4736-98c3-498d91eb2414\n# ╠═7ba7dd56-4ee9-4c5c-97a8-5f99a9ba507e\n# ╠═ddacc062-c4cb-40e1-8645-a1f8619eabea\n# ╠═7fdcd309-5e3e-4c65-9d8c-96d19fdc923c\n# ╠═9c9fdfad-44eb-4f5b-a5e0-82eaade02ed1\n# ╠═f3f7c1ea-f4aa-4dc2-a630-0fb37e4bdadb\n# ╠═d0ca3867-a266-4299-ac49-9745154b3ad4\n# ╠═880d1ae8-5a5a-463a-8e35-a73bdc40208a\n# ╟─c2fcf437-bd86-4b24-8c77-29702cf5373d\n", "meta": {"hexsha": "9b46ed631bf8a48df1147a1d0b3da602999e6aff", "size": 1520, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "notebooks/05/j-05-BarycentricInterpolation.jl", "max_stars_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_stars_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-23T18:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-25T03:32:45.000Z", "max_issues_repo_path": "notebooks/05/j-05-BarycentricInterpolation.jl", "max_issues_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_issues_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-23T21:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-27T23:14:46.000Z", "max_forks_repo_path": "notebooks/05/j-05-BarycentricInterpolation.jl", "max_forks_repo_name": "PtFEM/NumericalMethodsforEngineers.jl", "max_forks_repo_head_hexsha": "e4a997a14adbb86b7efe1586962df39eb9285ebb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2018-10-27T14:13:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-20T18:54:06.000Z", "avg_line_length": 31.6666666667, "max_line_length": 81, "alphanum_fraction": 0.7072368421, "num_tokens": 787, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9399133515091156, "lm_q2_score": 0.865224091265267, "lm_q1q2_score": 0.813235675427566}} {"text": "## Demo: Sonar Equations\n# Demonstrate calculation of the sonar equations, with focus on signal excess and probability of detection.\nusing Plots; gr()\nusing SpecialFunctions\nusing ColorSchemes\n\ninclude(\"../SonarEquations.jl\")\n\n## Single Propagation\nB = 1. # bandwidth\nt = 1. # integration time\nSL = 52. # source level\nNL = 5. # noise level\np_fal = 1e-4 # probability of false alarm\n\nc = 1500. # sound speed\nf = 100. # frequency\nλ = c/f # wavelength\nk = 2π/λ # wavenumber\n\nr₀ = 0. # source range\nz₀ = 100. # source depth\n\nR(r, z) = sqrt.((r - r₀)^2 + (z - z₀)^2) # distance from source\np(r, z) = R(r, z)\\exp(im*k*R(r, z)) # complex pressure\nTL(r, z) = -10log10(abs(p(r, z))) # transmission loss\nDI = 20 # directivity index\n# DT(r, z) = 5log10(d(r, z)*B/t)\nDT(r, z) = SonarEquations.detection_threshold(d(r, z), B, t)\n# SE(r, z) = SL - TL(r, z) - NL + DI - DT(r, z)\nSE(r, z) = SonarEquations.signal_excess_passive(SL, TL(r, z), NL, DI, DT(r, z))\n\n# d(r, z) = B*t*((SL - TL(r, z))/(B*NL))^2\nd(r, z) = SonarEquations.detection_index_gaussian(SL, TL(r, z), NL, B, t)\n# p_dtc(r, z) = erfc(erfcinv(2p_fal) - sqrt(d(r, z)/2))/2\n# POD(r, z) = 100p_dtc(r, z)\nPOD(r, z) = 100SonarEquations.probability_of_detection_gaussian(d(r, z), p_fal)\n\nr = range(1., 1e3, length = 100)\nz = range(1., 400., length = 50)\n\npTL = heatmap(r, z, TL,\n\tseriescolor = cgrad(:jet, rev = true),\n\ttitle = \"TL (dB)\",\n\tyaxis = (\"Depth (m)\", :flip))\n\npd = heatmap(r, z, d,\n\tseriescolor = :jet,\n\ttitle = \"d\",\n\tyaxis = :flip)\n\npSE = heatmap(r, z, SE,\n\tseriescolor = :jet,\n\ttitle = \"SE (dB)\",\n\txaxis = \"Range (m)\",\n\tyaxis = (\"Depth (m)\", :flip))\n\npPOD = heatmap(r, z, POD,\n\tseriescolor = :jet,\n\ttitle = \"POD (%)\",\n\txaxis = \"Range (m)\",\n\tyaxis = :flip)\n\nl = @layout [a b; c d]\n\npt = plot(pTL, pd, pSE, pPOD,\n\tlayout = l)\ndisplay(pt)\nsavefig(pt, \"img/SonarEqs_SimplePropagation.png\")\n\n## Simple Lloyd's Mirror\ninclude(\"../LloydsMirror.jl\")\n\nSL = 50.\nNL = 5.\nB = 1.\nt = 1.\nDI = 20.\np_fal = 1e-2\n\nc = 1500.0\nf = 1e2\nλ = c/f\nr_src = 0.0\nz_src = 2λ\nr = range(0, 3e2, length = 201)\nz = range(0, 10λ, length = 101)\nzTemp = 5λ\n\nTL(r, z) = LloydsMirror.lloydsmirror_singlereflection.(c, f, r_src, r, z_src, z)\nd(r, z) = SonarEquations.detection_index_gaussian(SL, TL(r, z), NL, B, t)\nDT(r, z) = SonarEquations.detection_threshold(d(r, z), B, t)\nSE(r, z) = SonarEquations.signal_excess_passive(SL, TL(r, z), NL, DI, DT(r, z))\nPOD(r, z) = 100SonarEquations.probability_of_detection_gaussian(d(r, z), p_fal)\n\npTL = heatmap(r, z, TL,\n\tseriescolor = cgrad(:jet, rev = true),\n\ttitle = \"TL (dB)\",\n\tyaxis = (\"Depth (m)\", :flip))\n\npd = heatmap(r, z, d,\n\tseriescolor = :jet,\n\ttitle = \"d\",\n\tyaxis = :flip)\n\npSE = heatmap(r, z, SE,\n\tseriescolor = :jet,\n\ttitle = \"SE (dB)\",\n\txaxis = \"Range (m)\",\n\tyaxis = (\"Depth (m)\", :flip))\n\npPOD = heatmap(r, z, POD,\n\tseriescolor = :jet,\n\ttitle = \"POD (%)\",\n\txaxis = \"Range (m)\",\n\tyaxis = :flip)\n\nl = @layout [a b; c d]\n\npt = plot(pTL, pd, pSE, pPOD,\n\tlayout = l)\ndisplay(pt)\nsavefig(pt, \"img/SonarEqs_LloydsMirror.png\")\n\nheatmap(r, z, DT,\n\tyaxis = :flip)\n\n## Multipath Propagation\n\n", "meta": {"hexsha": "33736d75a42c4d80c7a1782d53401a1c1938e301", "size": 3056, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/demo/Demo_SonarEquations.jl", "max_stars_repo_name": "kapple19/OceanAcousticsModelling", "max_stars_repo_head_hexsha": "9fdc1cefa51894661543428022d4fe45cd952e4f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-09-21T11:08:53.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-21T11:08:53.000Z", "max_issues_repo_path": "src/demo/Demo_SonarEquations.jl", "max_issues_repo_name": "kapple19/OceanAcousticsModelling", "max_issues_repo_head_hexsha": "9fdc1cefa51894661543428022d4fe45cd952e4f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/demo/Demo_SonarEquations.jl", "max_forks_repo_name": "kapple19/OceanAcousticsModelling", "max_forks_repo_head_hexsha": "9fdc1cefa51894661543428022d4fe45cd952e4f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-05T02:54:21.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-05T02:54:21.000Z", "avg_line_length": 23.5076923077, "max_line_length": 107, "alphanum_fraction": 0.6178010471, "num_tokens": 1185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133548753619, "lm_q2_score": 0.8652240877899775, "lm_q1q2_score": 0.8132356750736524}} {"text": "### routines related to geometric transforms of a set of coordinates.\n\n\nfunction shiftthenstretch(x,n::Int,T,N)\n return (x-n*T)/N\nend\n\nfunction shiftthencompress(x,n::Int,T,N)\n return (x-n*T)*N\nend\n\n# evaluates f(x/M)\nfunction evalstretchedsignal(x, f::Function, M::Int)\n return f(x/M)\nend\n\n\"\"\"\n convertcompactdomain(x::T, a::T, b::T, c::T, d::T)::T\n\nconverts compact domain x ∈ [a,b] to compact domain out ∈ [c,d].\n\"\"\"\nfunction convertcompactdomain(x::T, a::T, b::T, c::T, d::T)::T where T <: Real\n\n return (x-a)*(d-c)/(b-a)+c\nend\n\nfunction convertcompactdomain(x::Vector{T}, a::Vector{T}, b::Vector{T}, c::Vector{T}, d::Vector{T})::Vector{T} where T <: Real\n\n return collect( convertcompactdomain(x[i], a[i], b[i], c[i], d[i]) for i = 1:length(x) )\nend\n", "meta": {"hexsha": "63230b5db1a06c6c4cf15778bf53767fb4111b6d", "size": 773, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/geometric_transforms.jl", "max_stars_repo_name": "RoyCCWang/RWJuliaUtilities", "max_stars_repo_head_hexsha": "3796a8dd251470a127d4fdb2f2ff19be4a5320aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/geometric_transforms.jl", "max_issues_repo_name": "RoyCCWang/RWJuliaUtilities", "max_issues_repo_head_hexsha": "3796a8dd251470a127d4fdb2f2ff19be4a5320aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/geometric_transforms.jl", "max_forks_repo_name": "RoyCCWang/RWJuliaUtilities", "max_forks_repo_head_hexsha": "3796a8dd251470a127d4fdb2f2ff19be4a5320aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.935483871, "max_line_length": 126, "alphanum_fraction": 0.6364812419, "num_tokens": 249, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133548753619, "lm_q2_score": 0.8652240686758841, "lm_q1q2_score": 0.8132356571080607}} {"text": "module ProtonFunctions\r\n\r\nusing QuadGK\r\nusing Trapz\r\n\r\nexport adiabatic_energy_loss, interaction_rate, hubbles_law, survival_probability\r\n\r\n\"\"\"\r\n `adiabatic_energy_loss(lorentz_initial, z_initial, z_final)`\r\n\r\nCalculates the Lorentz factor of a particle after it propagates from a redshift z_initial to a redshift z_final.\r\nThe particle loses energy due to the adiabatic expansion of the universe.\r\n\r\nBased on eq. (2-4) from arXiv:1505.01347 https://arxiv.org/abs/1505.01347\r\n\"\"\"\r\nfunction adiabatic_energy_loss(lorentz_initial::Real, z_initial::Real, z_final::Real)\r\n integral, err = quadgk(z -> 1/(1+z), z_initial, z_final) # numerical integration using QuadGK.jl\r\n energy_loss = ℯ^(-integral)\r\n return lorentz_initial/energy_loss\r\nend\r\n\r\n\r\n\"\"\"\r\n `interaction_rate(Γ, ϵ, σ)`\r\nCalculates the interaction rate of protons with photons for a given proton Lorentz factor, redshift, photon energy, and interaction cross section.\r\n\r\nInputs:\r\n- Γ: proton Lorentz factor\r\n- z: redshift\r\n- ϵ: photon energy (in the proton rest frame) (eV)\r\n- σ: pγ interaction cross-section (micrometer^2)\r\n\r\nSource: Eq. (10.9) from Cosmic Rays and Particle Physics, 2nd Edition, by Thomas K. Gaisser, Ralph Engel, and Elisa Resconi.\r\n\"\"\"\r\nfunction interaction_rate(Γ::Real, z::Real, ϵ::Vector{T1}, σ::Vector{T2}) where {T1<:Number, T2<:Number}\r\n c = (2.99792e8)*1e9 # speed of light, nanometer/s\r\n kT = 0.234822*1e-9 # boltzmann constant * z=0 CMB temperature, MeV\r\n ħc = .197327*1e-3 # reduced planck constant * speed of light, MeV * nanometer\r\n M = @. ϵ*σ*(-log(1-ℯ^(-ϵ/(2*Γ*kT))))\r\n # trapezoidal integration of M over the array of ϵ\r\n I = trapz(ϵ, M) \r\n @assert I >= 0.0 # An interaction rate will never be negative\r\n rate_at_zero_redshift = (c*kT)/((2*π^2)*(ħc^3)*(Γ^2)) * I\r\n return rate_at_zero_redshift*(1+z)^3\r\nend\r\n\r\n\r\n\"\"\"\r\n 'hubbles_law(z)'\r\n\r\nCalculates dt/dz at the input redshift z. Source: eq. (3) from arXiv:1505.01347 https://arxiv.org/abs/1505.01347\r\n\"\"\"\r\nfunction hubbles_law(z::Real)\r\n H0 = 2.198e-18 # Hubble constant in s^-1\r\n Ωm = 0.3 # matter density parameter\r\n ΩΛ = 0.7 # dark energy density parameter\r\n return -1/(H0*(1+z)*sqrt((1+z^3)*Ωm + ΩΛ))\r\nend\r\n\r\n\r\n\"\"\"\r\n 'survival_probability(p_prev, rate_prev, z_prev, rate, z)'\r\n\r\nThe probability that the proton has not undergone a reaction at each redshift step, approximated\r\nusing the trapezoidal rule. Source: Eq. (7) arXiv:1505.01347 https://arxiv.org/abs/1505.01347\r\n\r\nInputs:\r\n- p_prev = the survival probability at the previous redshift step\r\n- rate_prev = the interaction rate at the previous redshift step\r\n- z_prev = the redshift at the previous redshift step\r\n- rate = the interaction rate at the current redshift step\r\n- z = the redshift at the current redshift step\r\n\"\"\"\r\nfunction survival_probability(p_prev::Real, rate_prev::Real, z_prev::Real, rate::Real, z::Real)\r\n @assert p_prev >= 0 # probabilites cannot be negative\r\n logp = log(p_prev) - (rate_prev*hubbles_law(z_prev) + rate*hubbles_law(z))*(0.5)*(z - z_prev)\r\n return ℯ^logp\r\nend\r\n\r\nend", "meta": {"hexsha": "dc24ed90d8ebf6744268e30f4aedca93174e3471", "size": 3077, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/proton.jl", "max_stars_repo_name": "azeolla/GZK_sim.jl", "max_stars_repo_head_hexsha": "715ab6c4076446a3563ce06c009c33c5067850e8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/proton.jl", "max_issues_repo_name": "azeolla/GZK_sim.jl", "max_issues_repo_head_hexsha": "715ab6c4076446a3563ce06c009c33c5067850e8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/proton.jl", "max_forks_repo_name": "azeolla/GZK_sim.jl", "max_forks_repo_head_hexsha": "715ab6c4076446a3563ce06c009c33c5067850e8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.4625, "max_line_length": 147, "alphanum_fraction": 0.7010074748, "num_tokens": 964, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9683812336438724, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.8131826194311109}} {"text": "\"\"\"Julia program to implement Merge sort\nMerge Sort is a Divide and Conquer algorithm. It divides the input array into two halves, calls itself for the two halves, and then merges the two sorted halves.\nFirst divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. \nFinally all the elements are sorted and merged.\n\"\"\"\nfunction merge_sort(arr)\n if length(arr) <= 1\n return arr\n end\n middle = trunc(Int, length(arr) / 2) # Finding the mid of the array\n L = arr[1:middle]\n R = arr[middle+1:end]\n \n merge_sort(L) # Sorting the first half\n merge_sort(R) # Sorting the second half\n \n # Copy data to temp arrays L[] and R[]\n i = j = k = 1\n while i <= length(L) && j <= length(R)\n if L[i] < R[j]\n arr[k] = L[i]\n i+=1\n else\n arr[k] = R[j]\n j+=1\n end\n k+=1\n end\n # Checking if any element was left\n while i <= length(L)\n arr[k] = L[i]\n i+=1\n k+=1\n end\n while j <= length(R)\n arr[k] = R[j]\n j+=1\n k+=1\n end\n arr\nend\n\nprint(\"How many numbers do you want to sort? \")\nn = readline()\nn = parse(Int64, n)\nif (n <= 0)\n println(\"No numbers to sort!!!\")\n exit()\nend\narr = Int[]\nprint(\"Enter the numbers: \")\narr = [parse(Int, num) for num in split(readline())]\nmerge_sort(arr)\nprint(\"The numbers in sorted order is \")\nprint(arr)\n\n\n\"\"\"\nTime Complexity - O(n*Log n), where 'n' is the size of the array\nSpace Complexity - O(n)\n\nSAMPLE INPUT AND OUTPUT\nHow many numbers do you want to sort? 4\nEnter the numbers: 8 2 7 6 \nThe numbers in sorted order is [2, 6, 7, 8]\n\"\"\"\n\n\n", "meta": {"hexsha": "84d956ba5da8e57e630324665e729669c1e4c9e6", "size": 1742, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Julia/sort/merge_sort.jl", "max_stars_repo_name": "Khushboo85277/NeoAlgo", "max_stars_repo_head_hexsha": "784d7b06c385336425ed951918d1ab37b854d29f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 897, "max_stars_repo_stars_event_min_datetime": "2020-06-25T00:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-24T00:49:31.000Z", "max_issues_repo_path": "Julia/sort/merge_sort.jl", "max_issues_repo_name": "adarshnjena/NeoAlgo", "max_issues_repo_head_hexsha": "77a92858d2bf970054ef31c2f55a6d79917a786a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5707, "max_issues_repo_issues_event_min_datetime": "2020-06-24T17:53:28.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-22T05:03:15.000Z", "max_forks_repo_path": "Julia/sort/merge_sort.jl", "max_forks_repo_name": "adarshnjena/NeoAlgo", "max_forks_repo_head_hexsha": "77a92858d2bf970054ef31c2f55a6d79917a786a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1817, "max_forks_repo_forks_event_min_datetime": "2020-06-25T03:51:05.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-29T05:14:07.000Z", "avg_line_length": 25.2463768116, "max_line_length": 161, "alphanum_fraction": 0.588978186, "num_tokens": 485, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122163480667, "lm_q2_score": 0.8962513793687401, "lm_q1q2_score": 0.8131798254200635}} {"text": "### A Pluto.jl notebook ###\n# v0.14.0\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ b2d786ec-7f73-11ea-1a0c-f38d7b6bbc1e\nmd\"\"\"\n# The Basel problem\n\n_Leonard Euler_ proved in 1741 that the series\n\n```math\n\\frac{1}{1} + \\frac{1}{4} + \\frac{1}{9} + \\cdots\n```\n\nconverges to\n\n```math\n\\frac{\\pi^2}{6}.\n```\n\"\"\"\n\n# ╔═╡ b2d79330-7f73-11ea-0d1c-a9aad1efaae1\nn = 1:100000\n\n# ╔═╡ b2d79376-7f73-11ea-2dce-cb9c449eece6\nseq = n .^ -2\n\n# ╔═╡ b2d792c2-7f73-11ea-0c65-a5042701e9f3\nsqrt(sum(seq) * 6.0)\n\n# ╔═╡ Cell order:\n# ╟─b2d786ec-7f73-11ea-1a0c-f38d7b6bbc1e\n# ╠═b2d792c2-7f73-11ea-0c65-a5042701e9f3\n# ╠═b2d79330-7f73-11ea-0d1c-a9aad1efaae1\n# ╠═b2d79376-7f73-11ea-2dce-cb9c449eece6\n", "meta": {"hexsha": "c3d46b7f2b3c25a0f401671999a35877db5d91cf", "size": 668, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "sample/Basic.jl", "max_stars_repo_name": "Mechachleopteryx/Pluto.jl", "max_stars_repo_head_hexsha": "78fbd9f00a3323208a6df9a4ff0ecbdcf2d9845a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3974, "max_stars_repo_stars_event_min_datetime": "2020-03-06T10:50:12.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T16:39:46.000Z", "max_issues_repo_path": "sample/Basic.jl", "max_issues_repo_name": "Mechachleopteryx/Pluto.jl", "max_issues_repo_head_hexsha": "78fbd9f00a3323208a6df9a4ff0ecbdcf2d9845a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1585, "max_issues_repo_issues_event_min_datetime": "2020-03-01T23:46:18.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T16:04:08.000Z", "max_forks_repo_path": "sample/Basic.jl", "max_forks_repo_name": "Mechachleopteryx/Pluto.jl", "max_forks_repo_head_hexsha": "78fbd9f00a3323208a6df9a4ff0ecbdcf2d9845a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 280, "max_forks_repo_forks_event_min_datetime": "2020-03-28T10:47:48.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-29T19:50:31.000Z", "avg_line_length": 17.5789473684, "max_line_length": 48, "alphanum_fraction": 0.6751497006, "num_tokens": 379, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9284088045171237, "lm_q2_score": 0.8757869948899665, "lm_q1q2_score": 0.8130883569374381}} {"text": "using DataFrames, Plots, Calculus\npyplot()\n\nProjDir = dirname(@__FILE__)\ncd(ProjDir) do\n \n function euler_ex71(f::Function, y0::Float64, x::FloatRange{Float64})\n y = Vector{Real}(length(x))\n y[1] = y0\n h = x.step/x.divisor\n for i in 2:length(x)\n y[i] = y[i-1] + h * f(x[i-1], y[i-1])\n end\n columns=[]\n push!(columns, collect(x))\n push!(columns, y)\n df = DataFrame(columns, [:x, :y])\n end\n\n f(x,y) = 3x^2 + 2x\n fy(x) = x^3+x^2+1\n \n x1 = 1.0:0.25:2.0\n x2 = 1.0:0.1:2.0\n x3 = 0.8:0.1:2.2\n df1 = euler_ex71(f, 3.0, x1)\n df2 = euler_ex71(f, 3.0, x2)\n\n df1 |> display\n println()\n df2 |> display\n println()\n\n global p2\n \n p1 = plot(x3, fy.(x3), ylims=(0.8, 13.0))\n plot!(p1, df1[:x], df1[:y], line=(:dash, 1), marker=(:cross))\n plot!(p1, df2[:x], df2[:y], line=(:dash, 1), marker=(:circle))\n show(p1)\n savefig(\"euler_ex7.1A.png\")\n show(p1)\n gui()\nend", "meta": {"hexsha": "cc0e44ac5d5cade77a1b8e20ab1d18d26e41b84b", "size": 899, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/ch07/IVP/euler_ex7.1a.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/NumericalMethodsforEngineers.jl-00e1d38a-71a9-5665-8612-32ae585a75a3", "max_stars_repo_head_hexsha": "e230c3045d98da0cf789e4a6acdccfbfb21ef49e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-07-23T18:12:52.000Z", "max_stars_repo_stars_event_max_datetime": "2020-11-25T03:32:45.000Z", "max_issues_repo_path": "examples/ch07/IVP/euler_ex7.1a.jl", "max_issues_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_issues_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2018-07-23T21:46:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-27T23:14:46.000Z", "max_forks_repo_path": "examples/ch07/IVP/euler_ex7.1a.jl", "max_forks_repo_name": "OVGULIU/NumericalMethodsforEngineers.jl", "max_forks_repo_head_hexsha": "7ca0b79965a7abd58af29d8dfd1870a954fb3aec", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2018-10-27T14:13:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-20T18:54:06.000Z", "avg_line_length": 20.9069767442, "max_line_length": 71, "alphanum_fraction": 0.552836485, "num_tokens": 396, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9284087926320945, "lm_q2_score": 0.8757869900269367, "lm_q1q2_score": 0.8130883420138045}} {"text": "\"\"\"\nBackground for the example in the README\n\"\"\"\n\nloc_coords=InterpolationFunction(\n :(a0+a1*xi+a2*eta), # expression\n [:xi, :eta], # symbols for coordinates\n [:a0, :a1, :a2], # symbols for interpolation coefficients\n [ # degrees of freedom\n [0, 0] # values at corners (see explanation below)\n ];\n maximum_derivative=1 # we'll only really need first order derivatives\n)\n\nabs_coords=InterpolationFunction(\n :(a0+a1*x+a2*y),\n [:x, :y],\n [:a0, :a1, :a2],\n [\n [0, 0]\n ];\n maximum_derivative=1\n)\n\ndmn=Domain(\n loc_coords, # local coordinates InterpolationFunction struct\n [ # frontier of the domain in each coordinate of loc_coords\n ( # xi\n 0.0, 1.0\n ),\n ( # eta\n 0.0, :(1.0-xi) # linear triangular element\n )\n ],\n [ # points at which the interpolation from local to absolute coordinates\n [0.0, 0.0], # is imposed (triangle corners)\n [1.0, 0.0],\n [0.0, 1.0] # (xi, eta)\n ]\n)\n\nfin=Finel(\n [ # corners\n [1.0, 1.0],\n [2.0, 1.0],\n [2.0, 2.0]\n ],\n dmn # domain of integration\n)\n\nresd=Residual(\n [\n ( # first variable: weight function\n :w, abs_coords, [] # no derivatives calculated\n ),\n ( # second variable: u\n :u, abs_coords, [\n (:du!dx, [1, 0]) # first order derivative in x axis calculated\n ]\n )\n ],\n :(w*(du!dx-v)), # expression for the residual\n 3; # Gauss-Legendre quadrature order\n extra_args=[:v] # v, a constant\n)\n\nresfunc=get_volume_residual_function(resd)\n\nws=[0.0, 1.0, 1.0] # format compatible with abs_coords InterpolationFunction,\nus=[0.5, 1.0, 1.0] # as detailed before\nv=-1.0\n\nR=resfunc([ws, us], fin, v) # and any other extra_args symbols\n\n# Test it!! :)\n@assert isapprox(R, 0.5)\n\nsurface_resd=Residual(\n [\n ( # first variable: weight function\n :w, abs_coords, [] # no derivatives calculated\n ),\n ( # second variable: u\n :u, abs_coords, [\n (:du!dx, [1, 0]) # first order derivative in x axis calculated\n ]\n )\n ],\n :(w.*[du!dx, v]), # expression for the residual\n 3; # Gauss-Legendre quadrature order, now at domain faces\n extra_args=[:v] # v, a constant\n)\n\nsurface_resfunc=get_surface_residual_function(surface_resd)\n\nR=surface_resfunc([ws, us], fin, v)\n\n# test it!! :)\n@assert isapprox(R, 0.25)\n", "meta": {"hexsha": "bdabe3a3517aa66d35f8b6514dda825da5fb6588", "size": 2452, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/READMEexample.jl", "max_stars_repo_name": "pedrosecchi67/SymFinel.jl", "max_stars_repo_head_hexsha": "24cc4110f48af1d33e1ba1837a6436e4d6506c95", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2021-01-11T03:50:16.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-02T15:56:29.000Z", "max_issues_repo_path": "examples/READMEexample.jl", "max_issues_repo_name": "pedrosecchi67/SymFinel.jl", "max_issues_repo_head_hexsha": "24cc4110f48af1d33e1ba1837a6436e4d6506c95", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/READMEexample.jl", "max_forks_repo_name": "pedrosecchi67/SymFinel.jl", "max_forks_repo_head_hexsha": "24cc4110f48af1d33e1ba1837a6436e4d6506c95", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-02-02T15:56:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-02-02T15:56:40.000Z", "avg_line_length": 24.52, "max_line_length": 78, "alphanum_fraction": 0.5672920065, "num_tokens": 735, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.945801274759925, "lm_q2_score": 0.8596637559030338, "lm_q1q2_score": 0.8130710761979943}} {"text": "using Pkg\nPkg.activate(pwd())\n\n# # Wave equation\n# ## Solving the wave equation\n\nstruct Wave\n f\n g\n c\nend\n\n# ### Exercise:\n# \n# Write the function `solve_wave(T, L, wave::Wave; n_t=100, n_x=100)` that solves the wave \n# equation.\n# \n# **Hint**: Follow the procedure from the previous exercise. Discretize time and space, \n# initialize the solution, add the boundary conditions, add the initial conditions and \n# finally, iterate over time.\n# \n# ---\n# ### Solution:\n\nfunction solve_wave(T, L, wave::Wave; n_t=100, n_x=100)\n ts = range(0, T; length=n_t)\n xs = range(0, L; length=n_x)\n Δt = ts[2] - ts[1]\n Δx = xs[2] - xs[1]\n y = zeros(n_t, n_x)\n \n # boundary conditions\n y[:,1] .= wave.f(0)\n y[:,end] .= wave.f(L)\n\n # initial conditions\n y[1,2:end-1] = wave.f.(xs[2:end-1])\n y[2,2:end-1] = y[1,2:end-1] + Δt*wave.g.(xs[2:end-1])\n\n # solution for t = 2Δt, 3Δt, ..., T\n for t in 2:n_t-1, x in 2:n_x-1\n ∂y_xx = (y[t, x+1] - 2*y[t, x] + y[t, x-1])/Δx^2\n y[t+1, x] = c^2 * Δt^2 * ∂y_xx + 2*y[t, x] - y[t-1, x]\n end\n\n return y\nend\n\n# ---\n\nusing Plots\n\nfunction plot_wave(y, file_name; fps = 60, kwargs...)\n anim = @animate for (i, y_row) in enumerate(eachrow(y))\n plot(\n y_row;\n title = \"t = $(i-1)Δt\",\n xlabel = \"x\",\n ylabel = \"y(t, x)\",\n legend = false,\n linewidth = 2,\n kwargs...\n )\n end\n gif(anim, file_name; fps, show_msg = false)\n \n return nothing\nend\n\n# ### Exercise:\n# \n# Solve the wave equation for $L=\\frac32\\pi$, $T=240$, $c=0.02$ and the initial \n# conditions\n# \n# $$\n# \\begin{aligned}\n# f(x) &= 2e^{-(x-\\frac L2)^2} + \\frac{x}{L}, \\\\\n# g(x) &= 0.\n# \\end{aligned}\n# $$\n# \n# Use time discretization with stepsize $\\Delta t=1$ and the space discretization with \n# number of points $n_x=101$ and $n_x=7$ steps. Plot two graphs.\n# \n# ---\n# ### Solution:\n\nf(x,L) = 2*exp(-(x-L/2)^2) + x/L\ng(x) = 0\n\nL = 1.5*pi\nT = 240\nc = 0.02\n\nwave = Wave(x -> f(x,L), g, c)\n\n#+\n\ny1 = solve_wave(T, L, wave; n_t=241, n_x=101)\nplot_wave(y1, joinpath(pwd(), \"lecture_13\", \"wave1.gif\"); ylims=(-2,3), label=\"\")\n\n#+\n\ny2 = solve_wave(T, L, wave; n_t=241, n_x=7)\nplot_wave(y2, joinpath(pwd(), \"lecture_13\", \"wave2.gif\"); ylims=(-2,3), label=\"\")", "meta": {"hexsha": "49075ba3e9ec97ed3a1bd3f162ca4a9ec754c99f", "size": 2299, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "lecture_13/01-ode-solved.jl", "max_stars_repo_name": "JuliaTeachingCTU/Julia-for-Optimization-and-Learning-Scripts", "max_stars_repo_head_hexsha": "8e00299449736e4ccf47c247aa9d80f99a7e5b92", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "lecture_13/01-ode-solved.jl", "max_issues_repo_name": "JuliaTeachingCTU/Julia-for-Optimization-and-Learning-Scripts", "max_issues_repo_head_hexsha": "8e00299449736e4ccf47c247aa9d80f99a7e5b92", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lecture_13/01-ode-solved.jl", "max_forks_repo_name": "JuliaTeachingCTU/Julia-for-Optimization-and-Learning-Scripts", "max_forks_repo_head_hexsha": "8e00299449736e4ccf47c247aa9d80f99a7e5b92", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.8952380952, "max_line_length": 91, "alphanum_fraction": 0.5467594606, "num_tokens": 853, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9458012762876287, "lm_q2_score": 0.8596637451167997, "lm_q1q2_score": 0.8130710673096718}} {"text": "mutable struct FTestResult <: TestResult\n levels::Union{Vector{String}, Nothing}\n n_obs::IntVector\n sds::FloatVector\n statistic::Float64\n dofs::IntVector\n pval::Float64\nend\n\nfunction show(io::IO, ftr::FTestResult)\n println(\"\\n F test to compare two variances\")\n println(\"Data (± SD)\")\n @printf(io, \" %s: ± %.2f\\n\",\n ftr.levels[1], ftr.sds[1])\n @printf(io, \" %s: ± %.2f\\n\",\n ftr.levels[2], ftr.sds[2])\n println(io, \"\\nResult:\\n\", coeftable(ftr))\nend\n\nfunction coeftable(ftr::FTestResult)\n CoefTable(\n [ftr.statistic, ftr.dofs[1], ftr.dofs[2], ftr.pval],\n [\"F\", \"DF₁\", \"DF₂\", \"P-value\"],\n [\"\"],\n 4, 1\n )\nend\n\n\"\"\"\n f_test(x1, x2)\n\nF-test\n\nThe arguments `x1` and `x2` can be numeric `Vector`.\n\"\"\"\nfunction f_test(\n x1::NumVector,\n x2::NumVector;\n levels::Union{Vector{String}, Nothing}=nothing\n)::FTestResult\n # use temporal sample names\n if isnothing(levels)\n @info \"Using the first and second sample names as 'Sample1' and 'Sample2'\"\n levels = [\"Sample1\", \"Sample2\"]\n end\n\n # check types\n if !(x1 isa FloatVector)\n @warn \"$levels[1] is converted to Float64\"\n x1 = convert(Vector{Float64}, x1)\n end\n if !(x2 isa FloatVector)\n @warn \"$levels[2] is converted to Float64\"\n x2 = convert(Vector{Float64}, x2)\n end\n\n # check length\n n1 = length(x1)\n n2 = length(x2)\n n1 > 1 || error(\"size of $levels[1] is not enough: actual size is $n1\")\n n2 > 1 || error(\"size of $levels[2] is not enough: actual size is $n2\")\n\n # perform f-test\n v1 = var(x1)\n v2 = var(x2)\n ν1 = v1 < v2 ? n2-1 : n1-1\n ν2 = v1 < v2 ? n1-1 : n2-1\n f_stat = v1 < v2 ? v2/v1 : v1/v2\n pval = 2.0 * D.ccdf(FDist(ν1, ν2), f_stat)\n res = FTestResult(levels, [n1, n2], [sqrt(v1), sqrt(v2)],\n f_stat, [ν1, ν2], pval)\n return res\nend\n\n\"\"\"\n FTestModel\n\nA `TestModel` type for f-test\n\n# Members\n\n- `X`: model matrix\n- `y`: vector to compare its means\n- `levels`: `String` vector of the unique levels (only 2 elements are allowed)\n\n\"\"\"\nstruct FTestModel{T<:Real} <: TestModel\n x1::AbstractVector{T}\n x2::AbstractVector{T}\n levels::Vector{String}\n function FTestModel(\n X::Matrix{T},\n y::AbstractVector{T},\n levels::Vector{String};\n ) where T\n dummy = X[:,2]\n x1 = y[dummy .== 0]\n x2 = y[dummy .== 1]\n new{T}(x1, x2, levels)\n end\nend\n\nfunction fit(obj::FTestModel)\n f_test(obj.x1, obj.x2, levels=obj.levels)\nend\nfunction fit(\n ::Type{FTestModel},\n X::AbstractMatrix{<:Real},\n y::AbstractVector{<:Real},\n levels::Vector{String}\n)\n ftest_obj = FTestModel(X, y, levels)\n fit(ftest_obj)\nend\n\n\"\"\"\n f_test(X, y)\n\nAn alias for `fit(FTestModel, X, y, levels)`\n\nThe arguments `X` and `y` can be a `Matrix` and a `Vector` or a `Formula` and `DataFrame`.\n\nThe argument `levels` can be a `Vector{String}`.\n\"\"\"\nfunction f_test(\n X::AbstractMatrix{<:Real},\n y::AbstractVector{<:Real}\n)\n @info \"Using the first and second sample names as 'Sample1' and 'Sample2'\"\n fit(FTestModel, X, y, [\"Sample1\", \"Sample2\"])\nend\nfunction f_test(\n formula::FormulaTerm{Term,Term},\n df::DataFrame\n)\n # obtain levels\n f = apply_schema(formula, schema(formula, df))\n levels = DataAPI.levels(df[!,f.rhs.terms[1].sym])\n if eltype(levels) <: Real\n levels = string.(levels)\n end\n\n # model matrix\n model_frame = ModelFrame(formula, df)\n X = ModelMatrix(model_frame).m\n y = response(model_frame)\n\n # perform f-test\n fit(FTestModel, X, y, levels)\nend\n", "meta": {"hexsha": "3be69b5e0d4cfdae322b739591f7641d4e698a2e", "size": 3645, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/ftest.jl", "max_stars_repo_name": "i-kiwamu/StatisticalTests.jl", "max_stars_repo_head_hexsha": "9dfce365ed961b1365f4effc6644081fe3611216", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-10T19:11:18.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-10T19:11:18.000Z", "max_issues_repo_path": "src/ftest.jl", "max_issues_repo_name": "i-kiwamu/StatisticalTests.jl", "max_issues_repo_head_hexsha": "9dfce365ed961b1365f4effc6644081fe3611216", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/ftest.jl", "max_forks_repo_name": "i-kiwamu/StatisticalTests.jl", "max_forks_repo_head_hexsha": "9dfce365ed961b1365f4effc6644081fe3611216", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.1390728477, "max_line_length": 90, "alphanum_fraction": 0.5936899863, "num_tokens": 1186, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541643004809, "lm_q2_score": 0.8633916222765627, "lm_q1q2_score": 0.8130163165388732}} {"text": "\"\"\"\nFile: Kmeans.jl\nAuthors: Reese Pathak, Stephen Boyd\n\nK-means implementation in Julia\n\nParameters:\n list_of_vectors - The vectors to cluster, represented as an array of N n-vectors\n (N column vectors of length n).\n k - The number of clusters (defaults to 2)\n\nReturns:\n representatives - The cluster representatives represented as an array of k\n column vectors of length n.\n labels - An array of N cluster assignments in the same order as data,\n so that the 1st data point's cluster assignment is labels[i].\n The labels themselves index into the representatives list,\n i.e. representatives[labels[1]] returns the first data\n point's representative.\n losses - An array containing the value of J at each iteration.\n\"\"\"\n\nmodule K_means\n\nexport k_means\n\nfunction k_means(list_of_vectors; k=2, epsilon=1e-6)\n losses = Float64[] # array of losses\n # initialize representatives and assignments\n representatives, assignments = initialize_representatives(list_of_vectors, k)\n while has_not_converged(losses, epsilon)\n assignments = partition_data(list_of_vectors, representatives)\n representatives = update_representatives(list_of_vectors, assignments,\n k, representatives)\n push!(losses, loss(list_of_vectors, representatives, assignments))\n end\n return representatives, assignments, losses\nend\n\nfunction has_not_converged(losses, epsilon)\n return !(length(losses) >= 2 && abs(losses[end] - losses[end-1]) <= epsilon)\nend\n# Assigns each data point to the closest representative\nfunction partition_data(data, representatives)\n labels = Int[]\n for i in 1:length(data)\n # calculate the euclidean distance from the ith data point to each representative\n representative_distances = pairwise_distance(data[i], representatives)\n # find the representative closest to the ith data point (argmin)\n distance, representative_index = findmin(representative_distances)\n push!(labels, representative_index)\n end\n return labels\nend\n\n# Calculates each representative as the mean of it's data points\nfunction update_representatives(data, labels, k, old_representatives)\n representatives = []\n for i = 1:k\n # get the data points assigned to the representative\n representative_pts = data[labels .== i]\n # if the representative has assigned data points, set it to be their mean\n if length(representative_pts) > 0\n push!(representatives, mean(representative_pts))\n else # otherwise, leave it alone\n push!(representatives, old_representatives[i])\n end\n end\n return representatives\nend\n\n# Calculates the loss (J) of the given clustering.\nfunction loss(data, representatives, assignments)\n N = length(data)\n J = 0.0\n for i = 1:N\n J += norm(data[i] - representatives[assignments[i]])^2\n end\n return J*(1/N)\nend\n\nfunction initialize_representatives(data, k)\n N = length(data)\n n = length(data[1])\n labels = rand(1:k,N)\n representatives = update_representatives(data, labels, k, [zeros(n) for i=1:k])\n return representatives, labels\nend\n\n# Calculates the Eudclidean distance between vector and each vectors in 'vectors',\n# returning an array.\nfunction pairwise_distance(vector, list_of_vectors)\n return Float64[norm(vector - list_of_vectors[i])\n for i=1:length(list_of_vectors)]\nend\n\nend\n", "meta": {"hexsha": "aec81fdfec5ee13b6b13593843eb7790290b886b", "size": 3461, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Kmeans.jl", "max_stars_repo_name": "reesepathak/mma-functions", "max_stars_repo_head_hexsha": "724e65608faf9da5038606555f4ae964c8ecd997", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Kmeans.jl", "max_issues_repo_name": "reesepathak/mma-functions", "max_issues_repo_head_hexsha": "724e65608faf9da5038606555f4ae964c8ecd997", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Kmeans.jl", "max_forks_repo_name": "reesepathak/mma-functions", "max_forks_repo_head_hexsha": "724e65608faf9da5038606555f4ae964c8ecd997", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-01-08T11:04:54.000Z", "max_forks_repo_forks_event_max_datetime": "2021-01-08T11:04:54.000Z", "avg_line_length": 35.3163265306, "max_line_length": 85, "alphanum_fraction": 0.709043629, "num_tokens": 769, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391579526934, "lm_q2_score": 0.880797085800514, "lm_q1q2_score": 0.8130102004044927}} {"text": "export rand, Ginibre\nimport Base.rand\n\n#Samples a matrix from the Ginibre ensemble\n#This ensemble lives in GL(N, F), the set of all invertible N x N matrices\n#over the field F\n#For beta=1,2,4, F=R, C, H respectively\nstruct Ginibre <: ContinuousMatrixDistribution\n beta::Float64\n N::Integer\nend\n\nfunction rand(W::Ginibre)\n beta, n = W.beta, W.N\n if beta==1\n randn(n,n)\n elseif beta==2\n randn(n,n)+im*randn(n,n)\n elseif beta==4\n Q0=randn(n,n)\n Q1=randn(n,n)\n Q2=randn(n,n)\n Q3=randn(n,n)\n [Q0+im*Q1 Q2+im*Q3;-Q2+im*Q3 Q0-im*Q1]\n else \n error(string(\"beta = \", beta, \" not implemented\"))\n end\nend\n\nfunction jpdf(Z::AbstractMatrix{z}) where {z<:Complex}\n pi^(size(Z,1)^2)*exp(-trace(Z'*Z))\nend\n\n#Dyson Circular orthogonal, unitary and symplectic ensembles\nstruct CircularOrthogonal\n N :: Int64\nend\n\nfunction rand(M::CircularOrthogonal)\n U = rand(Ginibre(2, M.N))\n U * U'\nend\n\nstruct CircularUnitary\n N :: Int64\nend\n\nrand(M::CircularUnitary) = rand(Ginibre(2, M.N))\n\nstruct CircularSymplectic\n N :: Int64\nend\n\nrand(M::CircularSymplectic) = error(\"Not implemented\")\n\n", "meta": {"hexsha": "c921396681639a0e65990d6866ed4326307660d3", "size": 1157, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Ginibre.jl", "max_stars_repo_name": "shashi/RandomMatrices.jl", "max_stars_repo_head_hexsha": "603b9e9acd95133839e804d6ef068cdfee4cbbfb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 56, "max_stars_repo_stars_event_min_datetime": "2017-09-29T08:46:45.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-06T10:05:30.000Z", "max_issues_repo_path": "src/Ginibre.jl", "max_issues_repo_name": "shashi/RandomMatrices.jl", "max_issues_repo_head_hexsha": "603b9e9acd95133839e804d6ef068cdfee4cbbfb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 29, "max_issues_repo_issues_event_min_datetime": "2017-08-18T19:50:53.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-15T08:08:44.000Z", "max_forks_repo_path": "src/Ginibre.jl", "max_forks_repo_name": "shashi/RandomMatrices.jl", "max_forks_repo_head_hexsha": "603b9e9acd95133839e804d6ef068cdfee4cbbfb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 20, "max_forks_repo_forks_event_min_datetime": "2018-01-05T22:05:35.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-02T20:25:41.000Z", "avg_line_length": 20.6607142857, "max_line_length": 74, "alphanum_fraction": 0.6473638721, "num_tokens": 378, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9609517028006208, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.812909831201361}} {"text": "# Random number generators for Julia\r\n# This is a slightly modified and reduced version of a Julia module by John D. Cook.\r\n# http://www.johndcook.com/julia_rng.html\r\n\r\nmodule Random\r\n\r\ninverse_gamma(shape, rate) = 1/gamma(shape, rate)\r\n\r\nfunction gamma(shape, rate)\r\n # Return a random sample from a gamma distribution\r\n # with pdf \\propto x^{shape-1}exp(-rate*x)\r\n\r\n if shape <= 0.0\r\n error(\"Shape parameter must be positive, but shape = $shape\")\r\n end\r\n if rate <= 0.0\r\n error(\"Rate parameter must be positive, but rate = $rate\")\r\n end\r\n \r\n ## Implementation based on \"A Simple Method for Generating Gamma Variables\"\r\n ## by George Marsaglia and Wai Wan Tsang. \r\n ## ACM Transactions on Mathematical Software\r\n ## Vol 26, No 3, September 2000, pages 363-372.\r\n\r\n if shape >= 1.0\r\n d = shape - 1.0/3.0\r\n c = 1.0/sqrt(9.0*d)\r\n while true\r\n x = randn()\r\n v = 1.0 + c*x\r\n while v <= 0.0\r\n x = randn()\r\n v = 1.0 + c*x\r\n end\r\n v = v*v*v\r\n u = rand()\r\n xsq = x*x\r\n if u < 1.0 -.0331*xsq*xsq || log(u) < 0.5*xsq + d*(1.0 - v + log(v))\r\n return d*v/rate\r\n end\r\n end\r\n else\r\n g = gamma(shape+1.0, 1.0)\r\n w = rand()\r\n return g*(w^(1.0/shape))/rate\r\n end\r\nend\r\n\r\n## return a random sample from a chi square distribution\r\n## with the specified degrees of freedom\r\nfunction chi_square(dof)\r\n return gamma(0.5*dof, 0.5)\r\n # This was corrected from gamma(0.5, 2.0*dof) in the original code\r\nend\r\n\r\n## return a random sample from a beta distribution\r\nfunction beta(a, b)\r\n if (a <= 0) || (b <= 0)\r\n error(\"Beta parameters must be positive\")\r\n end\r\n \r\n ## There are more efficient methods for generating beta samples.\r\n ## However such methods are a little more efficient and much more complicated.\r\n ## For an explanation of why the following method works, see\r\n ## http://www.johndcook.com/distribution_chart.html#gamma_beta\r\n\r\n u = gamma(a, 1.0)\r\n v = gamma(b, 1.0)\r\n return u / (u + v)\r\nend\r\n\r\nend\r\n\r\n\r\n\r\n\r\n", "meta": {"hexsha": "e2745c70e7b098078c270f8219d107ebaf28dc84", "size": 2189, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Random.jl", "max_stars_repo_name": "dicai/BayesianMixtures.jl", "max_stars_repo_head_hexsha": "4b71c07b79908891f74aed76f57d4493290c9e63", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Random.jl", "max_issues_repo_name": "dicai/BayesianMixtures.jl", "max_issues_repo_head_hexsha": "4b71c07b79908891f74aed76f57d4493290c9e63", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Random.jl", "max_forks_repo_name": "dicai/BayesianMixtures.jl", "max_forks_repo_head_hexsha": "4b71c07b79908891f74aed76f57d4493290c9e63", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.4285714286, "max_line_length": 85, "alphanum_fraction": 0.5637277296, "num_tokens": 609, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037262250327, "lm_q2_score": 0.8774767842777551, "lm_q1q2_score": 0.8128100149524436}} {"text": "#' ---\n#' title: Facility Location\n#' ---\n\n#' **Originally Contributed by**: Mathieu Tanneau and Alexis Montoison\n\n#+\nusing Random\nusing LinearAlgebra\n\nusing JuMP\nimport GLPK\nusing Plots\n\n#' ## Uncapacitated facility location\n\n#' ### Problem description\n#' \n#' We are given\n#' * $M=\\{1, \\dots, m\\}$ clients\n#' * $N=\\{ 1, \\dots, n\\}$ sites where a facility can be built\n#' \n#' **Decision variables**\n#' Decision variables are split into two categories:\n#' * Binary variable $y_{j}$ indicates whether facility $j$ is built or not\n#' * Binary variable $x_{i, j}$ indicates whether client $i$ is assigned to facility $j$\n#' \n#' **Objective**\n#' The objective is to minimize the total cost of serving all clients.\n#' This costs breaks down into two components:\n#' * Fixed cost of building a facility.\n#' In this example, this cost is $f_{j} = 1, \\ \\forall j$.\n#' * Cost of serving clients from the assigned facility.\n#' In this example, the cost $c_{i, j}$\n#' of serving client $i$ from facility $j$\n#' is the Euclidean distance between the two.\n#' \n#' **Constraints**\n#' * Each customer must be served by exactly one facility\n#' * A facility cannot serve any client unless it is open\n\n#' ### MILP formulation\n#' \n#' The problem can be formulated as the following MILP:\n#' \n#' \\begin{align}\n#' \\min_{x, y} \\ \\ \\ &\n#' \\sum_{i, j} c_{i, j} x_{i, j} + \n#' \\sum_{j} f_{j} y_{j}\\\\\n#' s.t. &\n#' \\sum_{j} x_{i, j} = 1, && \\forall i \\in M\\\\\n#' & x_{i, j} \\leq y_{j}, && \\forall i \\in M, j \\in N\\\\\n#' & x_{i, j}, y_{j} \\in \\{0, 1\\}, && \\forall i \\in M, j \\in N\n#' \\end{align}\n#' \n#' where the first set of constraints ensures\n#' that each client is served exactly once,\n#' and the second set of constraints ensures\n#' that no client is served from an unopened facility.\n\n#' ### Problem data\n\n#+\nRandom.seed!(314)\n\nm = 12 # number of clients\nn = 5 # number of facility locations\n\n# Clients' locations\nXc = rand(m)\nYc = rand(m)\n\n# Facilities' potential locations\nXf = rand(n)\nYf = rand(n)\n\n# Fixed costs\nf = ones(n);\n\n# Distance\nc = zeros(m, n)\nfor i in 1:m\n for j in 1:n\n c[i, j] = norm([Xc[i] - Xf[j], Yc[i] - Yf[j]], 2)\n end\nend\n\n#+\n# Display the data\nscatter(Xc, Yc, label = \"Clients\", markershape=:circle, markercolor=:blue)\nscatter!(Xf, Yf, label=\"Facility\", \n markershape=:square, markercolor=:white, markersize=6,\n markerstrokecolor=:red, markerstrokewidth=2\n)\n\n#' ### JuMP implementation\n\n#+\n# Create a JuMP model\nufl = Model(GLPK.Optimizer)\n\n# Variables\n@variable(ufl, y[1:n], Bin);\n@variable(ufl, x[1:m, 1:n], Bin);\n\n# Each client is served exactly once\n@constraint(ufl, client_service[i in 1:m],\n sum(x[i, j] for j in 1:n) == 1\n);\n\n# A facility must be open to serve a client\n@constraint(ufl, open_facility[i in 1:m, j in 1:n],\n x[i, j] <= y[j]\n)\n\n# Objective\n@objective(ufl, Min, f'y + sum(c .* x));\n\n#+\n# Solve the uncapacitated facility location problem with GLPK\noptimize!(ufl)\nprintln(\"Optimal value: \", objective_value(ufl))\n\n\n#' ### Visualizing the solution\n\n#+\n# The threshold 1e-5 ensure that edges between clients and facilities are drawn when x[i, j] ≈ 1.\nx_ = value.(x) .> 1 - 1e-5\ny_ = value.(y) .> 1 - 1e-5\n\n# Display clients\np = scatter(Xc, Yc, markershape=:circle, markercolor=:blue, label=nothing)\n\n# Show open facility\nmc = [(y_[j] ? :red : :white) for j in 1:n]\nscatter!(Xf, Yf, \n markershape=:square, markercolor=mc, markersize=6,\n markerstrokecolor=:red, markerstrokewidth=2,\n label=nothing\n)\n\n# Show client-facility assignment\nfor i in 1:m\n for j in 1:n\n if x_[i, j] == 1\n plot!([Xc[i], Xf[j]], [Yc[i], Yf[j]], color=:black, label=nothing)\n end\n end\nend\n\ndisplay(p)\n\n\n#' ## Capacitated Facility location\n\n#' ### Problem formulation\n#' \n#' The capacitated variant introduces a capacity constraint on each facility, i.e., clients have a certain level of demand to be served, while each facility only has finite capacity which cannot be exceeded.\n#' \n#' Specifically, let\n#' * $a_{i} \\geq 0$ denote the demand of client $i$\n#' * $q_{j} \\geq 0$ denote the capacity of facility $j$\n#' \n#' The capacity constraints then write\n#' \\begin{align}\n#' \\sum_{i} a_{i} x_{i, j} &\\leq q_{j} y_{j} && \\forall j \\in N\n#' \\end{align}\n#' \n#' Note that, if $y_{j}$ is set to $0$, the capacity constraint above automatically forces $x_{i, j}$ to $0$.\n\n#' Thus, the capacitated facility location can be formulated as follows\n#'\n#' \\begin{align}\n#' \\min_{x, y} \\ \\ \\ &\n#' \\sum_{i, j} c_{i, j} x_{i, j} + \n#' \\sum_{j} f_{j} y_{j}\\\\\n#' s.t. &\n#' \\sum_{j} x_{i, j} = 1, && \\forall i \\in M\\\\\n#' & \\sum_{i} a_{i} x_{i, j} \\leq q_{j} y_{j}, && \\forall j \\in N\\\\\n#' & x_{i, j}, y_{j} \\in \\{0, 1\\}, && \\forall i \\in M, j \\in N\n#' \\end{align}\n#'\n#' For simplicity, we will assume that there is enough capacity to serve the demand,\n#' i.e., there exists at least one feasible solution.\n\n#+\n# Demands\na = rand(1:3, m);\n\n# Capacities\nq = rand(5:10, n);\n\n#+\n# Display the data\nscatter(Xc, Yc, label=nothing,\n markershape=:circle, markercolor=:blue, markersize= 2 .*(2 .+ a)\n)\n\nscatter!(Xf, Yf, label=nothing, \n markershape=:rect, markercolor=:white, markersize= q,\n markerstrokecolor=:red, markerstrokewidth=2\n)\n\n#' ### JuMP implementation\n\n#+\n# Create a JuMP model\ncfl = Model(GLPK.Optimizer)\n\n# Variables\n@variable(cfl, y[1:n], Bin);\n@variable(cfl, x[1:m, 1:n], Bin);\n\n# Each client is served exactly once\n@constraint(cfl, client_service[i in 1:m], sum(x[i, :]) == 1)\n\n# Capacity constraint\n@constraint(cfl, capacity, x'a .<= (q .* y))\n\n# Objective\n@objective(cfl, Min, f'y + sum(c .* x));\n\n#+\n# Solve the problem\noptimize!(cfl)\nprintln(\"Optimal value: \", objective_value(cfl))\n\n#' ### Visualizing the solution\n\n#+\n# The threshold 1e-5 ensure that edges between clients and facilities are drawn when x[i, j] ≈ 1.\nx_ = value.(x) .> 1 - 1e-5\ny_ = value.(y) .> 1 - 1e-5\n\n# Display the solution\np = scatter(Xc, Yc, label=nothing,\n markershape=:circle, markercolor=:blue, markersize= 2 .*(2 .+ a)\n)\n\nmc = [(y_[j] ? :red : :white) for j in 1:n]\nscatter!(Xf, Yf, label=nothing, \n markershape=:rect, markercolor=mc, markersize=q,\n markerstrokecolor=:red, markerstrokewidth=2\n)\n\n# Show client-facility assignment\nfor i in 1:m\n for j in 1:n\n if x_[i, j] == 1\n plot!([Xc[i], Xf[j]], [Yc[i], Yf[j]], color=:black, label=nothing)\n break\n end\n end\nend\n\ndisplay(p)\n\n\n#' ## Further\n#' * [Benders decomposition](https://github.com/JuliaOpt/JuMPTutorials.jl/blob/master/script/optimization_concepts/benders_decomposition.jl)\n#' is a method of choice for solving facility location problems.\n#' * Benchmark instances can be found\n#' [here](https://resources.mpi-inf.mpg.de/departments/d1/projects/benchmarks/UflLib/).\n", "meta": {"hexsha": "e054e097a4785445282ee00720ceb14b37742589", "size": 6709, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "script/modelling/facility_location.jl", "max_stars_repo_name": "ferrolho/JuMPTutorials.jl", "max_stars_repo_head_hexsha": "3e50e6f12d0542fd598d1cdbe0c602a6ae5ea271", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 75, "max_stars_repo_stars_event_min_datetime": "2020-06-15T13:05:14.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-28T12:58:48.000Z", "max_issues_repo_path": "script/modelling/facility_location.jl", "max_issues_repo_name": "ferrolho/JuMPTutorials.jl", "max_issues_repo_head_hexsha": "3e50e6f12d0542fd598d1cdbe0c602a6ae5ea271", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 34, "max_issues_repo_issues_event_min_datetime": "2019-05-27T05:36:48.000Z", "max_issues_repo_issues_event_max_datetime": "2019-08-22T09:52:29.000Z", "max_forks_repo_path": "script/modelling/facility_location.jl", "max_forks_repo_name": "mthelm85/JuMPTutorials.jl", "max_forks_repo_head_hexsha": "b2285a74d05dc0c0e99df2ac999277c34e670afd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 19, "max_forks_repo_forks_event_min_datetime": "2019-10-09T09:32:56.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-02T17:41:21.000Z", "avg_line_length": 25.3169811321, "max_line_length": 207, "alphanum_fraction": 0.6412282009, "num_tokens": 2199, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037221561136, "lm_q2_score": 0.8774767794716264, "lm_q1q2_score": 0.8128100069301267}} {"text": "function incident_field(D, f, n, gridL, cellL)\n ω = 2π * f\n k = n * ω\n\n function efield(x, y)\n r = √(x^2 + y^2 + D^2)\n ℯ ^ (k * r * im) / (4π * r)\n end\n\n grid = range(-gridL/2 + 1/2, gridL/2 - 1/2, length = gridL) .* cellL\n [efield(x, y) for x in grid, y in grid]\nend\n\n\"\"\"\n greens(D, f, ϵ, μ, gridL, cellL)\n\nReturns the Green's function for\nthe near to far transformation.\n\n# Arguments\n- `D`: distance between near and far plane\n- `f`: frequency\n- `ϵ`: relative permittivity\n- `μ`: relative permeability\n- `gridL`: side length of E-field grid (in cells)\n- `cellL`: length of each cell of grid\n\"\"\"\nfunction greens(D, f, ϵ, μ, gridL, cellL)\n ω = 2π * f\n n = √(ϵ*μ)\n k = n * ω\n\n function efield(x, y)\n r = √(x^2 + y^2 + D^2)\n D * (-1 + k * r * im) * ℯ ^ (k * r * im) / (4 * π * r^3)\n end\n\n gridout = range(-gridL, gridL - 1, length = gridL * 2) .* cellL\n g = [efield(x, y) * -μ / ϵ for x in gridout, y in gridout]\n\n g\nend\n", "meta": {"hexsha": "7ce489ff299c2e7b0bc6023573d7d7b4a89be159", "size": 992, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/fields.jl", "max_stars_repo_name": "gaurav-arya/WavePropagation.jl", "max_stars_repo_head_hexsha": "a547e657fcc2ea68d87f35d22f42f3358a352bf2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/fields.jl", "max_issues_repo_name": "gaurav-arya/WavePropagation.jl", "max_issues_repo_head_hexsha": "a547e657fcc2ea68d87f35d22f42f3358a352bf2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/fields.jl", "max_forks_repo_name": "gaurav-arya/WavePropagation.jl", "max_forks_repo_head_hexsha": "a547e657fcc2ea68d87f35d22f42f3358a352bf2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.0697674419, "max_line_length": 72, "alphanum_fraction": 0.5322580645, "num_tokens": 397, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9518632302488963, "lm_q2_score": 0.8539127529517043, "lm_q1q2_score": 0.812808151375337}} {"text": "module GeneralizedCRT\n\nexport crt\nusing Base.Threads\nimport Base.Threads.@spawn\n\nconst THRESHOLD1 = 7 # split longer input arrays to use binary instead of sequential algo\nconst THRESHOLD2 = 15 # split longer input arrays to different cpu threads\n\n\"\"\"\n crt(a, b, p, q)\n\nGiven `0 <= a < p` and `0 <= b < q` with `a == b mod(gcd(p, q)`\nreturn unique solution `x ≡ a mod p` and `x ≡ b mod q` \nwith `0 <= x < lcm(p, q)`.\n\nReturn `x, lcm(p, q)`.\n\"\"\"\nfunction crt(a::A, b::B, p::C, q::D) where {A,B,C,D}\n crt(promote(a, b, p, q)...)\nend\n\nfunction crt(a::T, b::T, p::T, q::T) where T\n c, u = gcdx(p, q)\n q1 = div(q, c)\n u = mod(u, q1)\n iszero(mod(b - a, c)) || throw(ArgumentError(\"remainders must have same rest when diveded by ($a, $b, gcd($p, $q))\"))\n bac = div(b - a, c)\n x = mod(bac * u, q1) * p + a\n lcm = Base.checked_mul(p, q1)\n return x, lcm\nend\n\nfunction crt(ap::AbstractVector{<:Tuple{<:Integer,<:Integer}})\n crt([x[1] for x in ap], [x[2] for x in ap])\nend\ncrt(t::Tuple{S,T}...) where {S<:Integer,T<:Integer} = crt(collect(t))\n\n\"\"\"\n crt(a::Vector, m::Vector; nthreads, splitstreads, splitbinary)\n\nGiven `0 <= a[i] < p[i] for all i` with `a[i] == a[j] mod(gcd(p[i], p[j]) for i != j`,\nreturn unique solution `x ≡ a[i] mod p[i] for all i`\nwith `0 <= x < lcm(p, q)`.\n\nOptional arguments:\nUse `nthreads=nthreads()` threads if `length(a) > splitthreads=($THRESHOLD2)`.\nUse binary algorithm is `length(a) > splitbinary=($THRESHOLD1)`.\n\nReturn `x, lcm(p, q)`.\n\"\"\"\nfunction crt(a::AbstractVector{T}, m::AbstractVector{S}; nt=nthreads()-1, splitthreads=THRESHOLD2, splitbinary=THRESHOLD1) where {S,T}\n # split input array and execute in parallel threads\n n = length(a)\n n == length(m) || throw(ArgumentError(\"vectors of same size required\"))\n if nt <= 1 || n <= splitthreads\n crts(a, m; splitbinary)\n else\n nt = max(nt, nthreads() - 1)\n R = promote_type(S, T)\n ni = max(n ÷ nt, 2)\n nt = (n + ni - 1) ÷ ni\n tasks = Vector{Any}(undef, nt)\n n2 = 0\n nth = nt\n for i = 1:nt\n n1 = n2 + 1\n n2 = n2 + (n - n2) ÷ nth\n nth -= 1\n av = view(a, n1:n2)\n mv = view(m, n1:n2)\n tasks[i] = @spawn crts($av, $mv; splitbinary=$splitbinary)\n end\n a0 = Vector{R}(undef, nt)\n m0 = Vector{R}(undef, nt)\n for i = 1:nt\n res = try\n fetch(tasks[i])\n catch ex\n ex isa TaskFailedException ? throw(ex.task.result) : rethrow()\n end\n a0[i] = first(res)\n m0[i] = last(res)\n end\n x, l = crt(a0, m0)\n x, l\n end\nend\n\n# split input array in halves a recombine output\nfunction crts(a::AbstractVector{T}, m::AbstractVector{S}; splitbinary) where {S,T}\n n = length(a)\n n == length(m) || throw(ArgumentError(\"vectors of same size required\"))\n if n <= max(splitbinary, 3)\n crtm(a, m)\n else\n n2 = (n + 1) ÷ 2\n xI, lcmI = crts(view(a,1:n2), view(m, 1:n2); splitbinary)\n xJ, lcmJ = crts(view(a, n2+1:n), view(m, n2+1:n); splitbinary)\n crt(xI, xJ, lcmI, lcmJ)\n end\nend\n\n# process input array sequentially\nfunction crtm(a::AbstractVector{T}, m::AbstractVector{S}) where {S,T}\n n = length(a)\n # println(\"crtm($n, $(length(m)))\")\n n == length(m) || throw(ArgumentError(\"vectors of same size required\"))\n R = promote_type(S, T)\n n == 0 && return zero(R), one(R)\n xI, lcmI = promote(a[1], m[1])\n xI = _mod(xI, lcmI)\n for i = 2:n\n xI, lcmI = crt(xI, _mod(a[i], m[i]), lcmI, m[i])\n end\n xI, lcmI\nend\n\n_mod(a, b) = 0 <=a < b ? oftype(b, a) : mod(a, b)\n\nend # module\n", "meta": {"hexsha": "8170837d873e2ec61f98283ba35a86ec6390b7e5", "size": 3710, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/GeneralizedCRT.jl", "max_stars_repo_name": "KlausC/GeneralizedCRT.jl", "max_stars_repo_head_hexsha": "cc69f64054bab68839c72ff29756df18b3b36ae5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/GeneralizedCRT.jl", "max_issues_repo_name": "KlausC/GeneralizedCRT.jl", "max_issues_repo_head_hexsha": "cc69f64054bab68839c72ff29756df18b3b36ae5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/GeneralizedCRT.jl", "max_forks_repo_name": "KlausC/GeneralizedCRT.jl", "max_forks_repo_head_hexsha": "cc69f64054bab68839c72ff29756df18b3b36ae5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.4098360656, "max_line_length": 134, "alphanum_fraction": 0.5522911051, "num_tokens": 1270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9518632261523028, "lm_q2_score": 0.8539127492339909, "lm_q1q2_score": 0.8128081443384488}} {"text": "\n## Things outside of the module won't get run when using the package.\n\n#You can create vectors of random normals.\nx = randn(40);\nx'\n# Adding the semicolon to the end hides the output.\n# Vectors are column vectors by default, which take up a good chunk of screen.\n# So often it's nicer to tranpose them.\n\n#You index into arrays with brackets, like in R\nx[3]\nx[end]\nx[2:5]\nx[end-4:end-1]\nx[36:end-1]\n\n#You also index into strings that way.\ngreeting = \"Hello world!\"\ngreeting[3:4]\ngreeting[7:end-1]\n\n#Or arrays of random uniforms.\nu = rand(3,2,2)\nu[:,:,1]\n\n#Standard syntax for randomly sampling is to call rand on the thing you want to sample from.\nrand(1:4, 10)\n\nusing Distributions\ngamma_3_3 = Gamma(3, 3)\nrand(gamma_3_3, 2, 4)\n\n#Julia Arrays are like the arrays/vectors/matrices of R. They will also automatically promote\n[3f0 4.0 2 π]\n\n#The individual elements were\n3f0 |> typeof\n4.0 |> typeof\n2 |> typeof\nπ |> typeof\n#Note that pi is an irrational number! It automatically converts to whatever you want it to be.\n2π\n2π |> typeof\nbig(2)π\nbig(2)π |> typeof\n# |> is the pipe operator in Julia.\n# However, it isn't as powerful or fancy as magrittr's \"%>%\".\n# You can look at Lazy.jl if you want more piping options.\n\n#In \"R\" a scalar is really a length 1 vector.\n#In julia, they are clearly distinct.\nscalar4double = 4.0\nzerodim4double = fill(4.0)\nvector4double = fill(4.0, 1)\nmatrix4double = fill(4.0, 1, 1)\ndim3array4double = fill(4.0, 1, 1, 1)\n\n#They all print differently, but there's also a fundamental difference under the hood.\nisbits(scalar4double)\nisbits(zerodim4double)\nisbits(vector4double)\n\n#Q: Are only numbers bits?\nisbits(gamma_3_3)\n\n#\"isbits\" is low level, referring to how the memory is handled under the hood.\n#\"isbits\" types are raw, they're handled directly, and an array of them will be one solid chunk of memory.\n# This makes them \n#If something is not bits, that normally means it \n#Why would we ever want a zero dim array?\nfunction f!(a::Number)\n a = 2.1\nend\nfunction f!(a::AbstractArray)\n a[:] = 2.1\nend\nf!(scalar4double)\nscalar4double\nf!(zerodim4double)\nzerodim4double\nf!(dim3array4double)\ndim3array4double\n\n# In Julia, variables are not copied.\n# The first f! replaced the input number with another number, but the original scalar4double remained bound to 4\n# The second f! modified the contents of the arrays\n# The arrays stayed the same, but the contents changed.\n# Be weary!\nzerodim2 = zerodim4double\nzerodim2[] = 17.0\nzerodim2\nzerodim4double\n# Julia does not copy unless you ask it to!\n# \"=\" binds a new name to the old variable.\nzerodim3 = copy(zerodim2)\nzerodim3[] = 9.4\nzerodim2\nzerodim3\n\n# It is convention in Julia that whenever a function has side effects -- whenever it modifies inputs\n# to end that function name with a \"bang\"!\n# The mutated arguments are also placed first.\ncopy!(zerodim2, zerodim3)\nzerodim2\nfill!(zerodim2, -1e9)\nzerodim4double\n\n# Also, adding commas makes a column vector.\n[3f0, 4.0, 2, π]\n# For a matrix\n[ 1 2 3 ; 4 5 6]\nreshape(1:6, 2, 3)\n# Note the differences! Order, and\n# In Julia, ranges are lazy -- even when reshaped.\n\n# Also, the colon operator also has low priority.\n# What is this in R?\n# using RCall\n3*1:5+7\n10:3:22\n10:3:22 |> collect\n\n# R is a lazy language, which is great.\n# Julia is eager, but many functions and operations are implemented to be lazy.\n# Array slices can also be lazy\n@views x[4:7]\n# On Julia 0.7, transposing matrices is also lazy. Not yet on 0.6.\n\n\n#If Julia can't promote them all to the same type, it'll give up and create a vector that can hold anything.\n[ 42im 9//2 \"fish\" π ]\n#This isn't a good idea. Don't that.\n\n\n#You can also create undefined arrays. Be cautious, they may contain NaNs, which can infect code if you're not careful!\nArray{Float64}(4, 12)\n\n#Or fill\nfill(1.2, 4, 7)\nfill(gamma_3_3, 2, 4)\n\n#Or if you just want zeros\nzeros(4,7)\nzeros(Int, 4, 7)\n\n\n\n#You can also use comprehensions. Remember x earlier?\nx'\n#Lets use it to create a design matrix!\nX = [xᵢ^j for xᵢ ∈ x, j ∈ 0:4]\n\n#Lets create some fake data! First, lets get a true beta:\nβ_true = [2, 3, -5, -2, 1.2]\n\n#And then add a little noise\ny = X * β_true .+ 0.1randn(40)\n\nβ_hat = X \\ y\n\n# \"\\\" is the left devision operator.\n# It just means use the thing on the left to divide the thing on the right\n2 \\ 4\n# If we tranpose both X and y, we can get dimensions to line up for a right division\ny' / X'\n\n\n# This is of course equivalent to what we're all used to:\ninv(X' * X) * X' * y\n\n# Realstically, we'd always prefer to just use LM and immediately get convenient summary statistics.\nusing DataFrames, GLM\ndf = DataFrame(x = x, x² = X[:,3], x³ = X[:,4], x⁴ = X[:,5], y = y)\nOLS = glm(@formula(y ~ x + x² + x³ + x⁴), df, Normal(), IdentityLink())\n\n\n# There are three ways to define functions.\n# For an anonymous function, use [arguments] -> [function body]\n# For example, if we want a random positive definite matrix\nΣ = randn(20,5) |> x -> x' * x\n\n# If we have more than one argument, we need to wrap them in a parenthesis.\nanon = (a, b) -> sin(a) - exp(b)\nanon(2, 3)\n\n# We can overwrite anon with anything.\nanon = 4\n\n# If we create a proper named function, we cannot replace it with something other than a function.\nf(a) = 2a\nf(4)\nf = 3\nf(a) = 3a\nf(4)\nf(a::Int) = \"Hello world!\"\nf(a, b) = 3a + 2b\nf(4)\nf(4.0)\nf(4, -5)\n\n# Julia uses multiple dispatch.\n# Julia compiles a different specialized version of that function for each combination of types you give it.\n# It then calls the type matching that combination.\n# This also lets you define a different body for the types if you want to.\n# That is, Julia compiles 4 separate functions for \"f\" when we do this:\nf(4, -5)\nf(4.0, -5)\nf(4, -5.0)\nf(4.0, -5.0)\n# And chooses the correct version to use.\n# If we want the compiled function to act differently, we can then give Julia a different function body to use for that type\n# Just like we did for \"Hellow world!\" above\n# For example:\nf(X, β_hat) #Error, dimensions don't line up!\nf(a::Matrix, b::Vector) = b' * a' * a * b\nf(X, β_hat)\nf(4, -5) #Still works!\n# Similar to match.call() in R:\npair = (4,-5)\nf(pair...)\n\n# The long form just makes it easier to write multi-line functions.\n#@inline function f(x, y)\nfunction f(x, y)\n out = zero(promote_type(typeof(x), eltype(y)))\n for yᵢ ∈ y\n out += x * yᵢ\n end\n out\nend\n\n\n@code_llvm f(e, x)\n\n#This is a tuple. A tuple's type is partly defined by its length.\n@code_llvm f(e, (1.2, -9.5, 3.2, 37.2))\n(1.2, -9.5, 3.2, 37.2) |> typeof\n\n@code_llvm f(e, 2.3)\n@code_llvm f(2, 3)\n\nfunction g(f, a, b, c)\n f(a, b) + c\nend\n\n# f is automatically inlined at -O3, not at -O2.\ng(f, 2, 3, 4)\n@code_llvm g(f, 2, 3, 4)\n\n# In Julia, you end a scope or expression via \"end\".\nf(√2) #Does not have the irrational type! It's just taking the square root of 2.\n\n\n# R has \"single dispatch\", which is useful for things like plot, print, and summary methods.\n\n# A simple example why this is useful in Julia is so that we can use \"*\" for matrix multiplication.\n# There are a lot of different methods for \"*\"!\nmethods(*) |> length\n#Run just methods(*) if you want to see the full list!\n\n# This also makes code faster!\n# Compare\nS_big = randn(10_000, 5_000) |> x -> x' * x;\nX_big = randn(5_000, 5_000);\n@time S_big * X_big;\nusing Compat.LinearAlgebra #This line is currently unnecessary\n@time eigfact(S_big);\n# However, we know S is symmetric! What happens if we tell Julia that?\nS_big2 = Symmetric(S);\n@time S_big2 * X_big;\n@time eigfact(S_big2);\n\n#The using Compat.LinearAlgebra line is unnecassary in Julia 0.6\n#However, starting in Julia 0.7 you'll need \"using LinearAlgebra\"\n#Saying \"using Compat.LinearAlgebra\" works on both versions.\n\n# Give Julia more information, and it will automatically pick the best method!\n\n# Multiple dispatch is magical, because\n# a) It makes code easier to read. When you want to multiply, you use \"*\".\n# When you want an eigenfactorization, you use \"eigfact\".\n# No \"eigfact\" here, \"eigfact_symmetric\" there.\n# No map_int here and map_dbl there.\n\n# However, what if we actually want elementise multiplication from \"*\"?\n# In Julia, adding a \".\" automatically broadcosts.\n# A broadcast is like an apply or map statement, except it will line up dimensions.\n\n# Recall\nsize(X)\nsize(y)\ny' * X # Regular matrix-vector product.\nX .* y # Broadcast\n# Both X and y have 40 rows, so it lines these up\n# and then replicates y across each of X's 5 columns.\nX .* β_hat'\n# β_hat' is 1 x 5, so it lines up with X's 5 columns, and is replicated down the 40 rows.\nX .+ 2\n# Here, 2 gets replcicated along both colums and rows.\n\nX .* y' # Does not line up -> error\nX .* β_hat # Does not line up -> error\n\n# This works for any function.\nf.(X)\nf.(X, y)\n\n# If we use multiple ones, they will all fuse.\nf.(X, y) .* β_hat' .+ 2\n\n# Too many dots can be annoying to write. So Julia offers the macro \"@. \"\n@. f(X, y) * β_hat + 2\n\n# Whenever you see something with an \"@\" in Julia, it is a macro.\n# Macros edit your code. Think of them as non-standard evaluation in R.\n# Because they can do extremely weird things, they always have an \"@\" to let you know something funky is going on.\n# A basic example of non-standard evaluation in R is how a lot of functions know the name of the variables you give them.\n# Consider -- straight out of Advanced R:\n\"\nx <- seq(0, 2 * pi, length = 100)\nsinx <- sin(x)\nplot(x, sinx, type = \"l\")\n\"\n# Similarly:\n@show anon;\n# If you ever want to know what it is a macro does to your code, there's a macro for that!\n@macroexpand @show anon\n@macroexpand @. f(X, y) * β_hat + 2\n\n# We can see that \"@.\" does us the favor of adding \".\" to every function call for us.\n\n# We can get creative!\nbunch_o_gammas = @. Gamma(exp(f(X)), exp(-2y))\n\n# What is the meaning of this? Good question!\nmean.(bunch_o_gammas)\n# Maybe we want to do parametric bootstrap?\nsamples = rand.(bunch_o_gammas, 4000)\n\n# Method of Moment's estimator for Gamma\nfunction gammas_MoM(x)\n x̄ = mean(x)\n s² = var(x)\n x̄^2 / s², s² / x̄\nend\nmom_estimators = gammas_MoM.(samples)\nfunction relative_error( estimate, truth )\n @. ( estimate - truth ) / truth\nend\n#Relative errors\n@. relative_error(mom_estimators, params(bunch_o_gammas))\n\n#We can easily condense the steps\nN = 4000\n@. relative_error(gammas_MoM(rand(bunch_o_gammas, N)), params(bunch_o_gammas))\n\n\n\n### Some more array manipulations and fun with broadcasting!\n# Remember Σ from earlier?\nΣ\n# Let's take N samples from a multivariate normal with that as the covariance matrix, and mean:\nμ = 2.* randn(5)\n\nW = chol(Σ) * randn(5, N) .+ μ\n\nmean(W, 2)\ncov(W')\n# We can also recall the definition from multivariate\nS = (W .- mean(W,2)) |> x -> x * x' ./ (size(x,2)-1)\n#and see that it is a pretty good estimator\nΣ\n# The matrix and broadcast syntax makes it easy to weave together array manipulations\n# and apply definitions, like we see in Johnson and Wichern for the sample Covariance matrix.\n\n\n# Remember this?\n[ 42im 9//2 \"fish\" π ]\n# What if we do want to hold a bunch of different types?\n# One straightforward approach is to build our own container!\n\nabstract type four_objects{A,B,C,D} end\nstruct FourThings{A,B,C,D} <: four_objects{A,B,C,D}\n a::A\n b::B\n c::C\n d::D\nend\nfunction FourThings(a::A, b::B, c::C, d::D) where {A,B,C,D}\n FourThings{A,B,C,D}(a,b,c,d)\nend\nft1 = FourThings(42im, 9//2, \"fish\", π)\nisbits(ft1)\nft2 = FourThings(42, 9//2, 4f0, π)\nisbits(ft2)\nmutable struct FourMutableThings{A,B,C,D} <: four_objects{A,B,C,D}\n a::A\n b::B\n c::C\n d::D\nend\nft3 = FourMutableThings(42, 9//2, 4f0, π)\nisbits(ft3)\nstruct MyMatrix{T}\n a::T\n b::T\n c::T\n d::T\n e::T\n f::T\n g::T\n h::T\n i::T\n j::T\n k::T\n l::T\n m::T\n n::T\n o::T\n p::T\nend\n# Then, we get define our operations. For example, multiplication\nfunction Base.:*(x::MyMatrix{T}, y::four) where {T, A,B,C,D,four <: four_objects{A,B,C,D}}\n (\n x.a*y.a + x.e * y.b + x.i * y.c + x.m * y.d,\n x.b*y.a + x.f * y.b + x.j * y.c + x.n * y.d,\n x.c*y.a + x.g * y.b + x.k * y.c + x.o * y.d,\n x.d*y.a + x.h * y.b + x.l * y.c + x.p * y.d,\n )\nend\nv16 = randn(Float32, 16);\nm16 = reshape(v16, 4, 4);\nv4 = randn(Float32, 4);\nmm = MyMatrix(v16...)\nmm * ft2\nmm * ft3\nft4 = FourThings(v4...)\nft5 = FourMutableThings(v4...)\nm16 * v4\nmm * ft4\n\n# What do you expect their performance to be?\n# Built in multiplication, vs our own custom version?\nusing BenchmarkTools\n@benchmark $m16 * $v4\n@benchmark $mm * $ft2\n@benchmark $mm * $ft3\n@benchmark $mm * $ft4\n@benchmark $mm * $ft5\n# Now compare with Julia launched with -O3\n# This is why I like -O3\n\n# More cool features\nusing ForwardDiff\n\nForwardDiff.derivative(f, 2.3)\nForwardDiff.derivative(sin, 2.3)\ncos(2.3)\n", "meta": {"hexsha": "659790d078040f5c7a2426eb8e3bc62db9b39550", "size": 12665, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/DemoScript.jl", "max_stars_repo_name": "JuliaTagBot/IntroJuliaStatsDemo.jl", "max_stars_repo_head_hexsha": "0dfbb027334ae0772610bed971d65ee5ec33b20d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/DemoScript.jl", "max_issues_repo_name": "JuliaTagBot/IntroJuliaStatsDemo.jl", "max_issues_repo_head_hexsha": "0dfbb027334ae0772610bed971d65ee5ec33b20d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/DemoScript.jl", "max_forks_repo_name": "JuliaTagBot/IntroJuliaStatsDemo.jl", "max_forks_repo_head_hexsha": "0dfbb027334ae0772610bed971d65ee5ec33b20d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-02-08T10:55:19.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T10:55:19.000Z", "avg_line_length": 27.5326086957, "max_line_length": 124, "alphanum_fraction": 0.685353336, "num_tokens": 4092, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096227509861, "lm_q2_score": 0.8872045892435128, "lm_q1q2_score": 0.812776661554818}} {"text": "module KernelLibrary\n\nexport linear_kernel, polynomial_kernel, laplacian_kernel, gaussian_kernel, anova_kernel, rbf_kernel, fractrbf_kernel, rationalquad_kernel, multiquad_kernel, invmultiquad_kernel, circular_kernel, spherical_kernel, wave_kernel\nexport power_kernel, log_kernel, spline_kernel, cauchy_kernel, chisquare_kernel, histintersect_kernel, genhistintersect_kernel, gentstudent_kernel, fourier_kernel, sigmoid_kernel, wavelet_kernel, transinvwavelet_kernel\n# Julia implementation of most of the kernel functions from\n# crsouza.blogspot.com/2010/03/kernel-functions-for-machine-learning.html\n\nfunction linear_kernel{T}(x::Array{T}, y::Vector{T}; c::Real=0.)\n\tc=convert(T,c)\n\t(y'*x .+ c)\nend\n\nfunction polynomial_kernel{T}(x::Array{T}, y::Vector{T}, d::Real; α::Real=1., c::Real=1.)\n\td=convert(T,d); α=convert(T,α); c=convert(T,c);\n\t(α.*(y'*x) .+ c).^ d\nend\n\nfunction laplacian_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ)\n\texp(-sum((x.-y).^2.,1).^.5 ./ σ)\nend\n\nfunction gaussian_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ)\n\texp(-sum((x.-y).^2,1) ./ 2*σ.^ 2)\nend\n\nfunction anova_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real, d::Real)\n\td=convert(T,d); σ=convert(T, σ)\n\tsum(exp(-σ*(x.-y).^2).^d,1)\nend\n\nfunction rbf_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ)\n\texp(-sum(abs(x.-y),1) ./ 2*σ.^ 2)\nend\n\nfunction fractrbf_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real, fract::Real)\n\tσ=convert(T, σ); fract=convert(T, fract); \n\texp(-sum(abs(x .- y).^fract,1).^(1/fract) ./ 2*σ.^ 2)\nend\n\nfunction rationalquad_kernel{T}(x::Array{T}, y::Vector{T}, c::Real)\n\tc = convert(T, c)\n\tn = sum((x.-y).^2,1)\n\t1 .- n./(n .+ c)\nend\n\nfunction multiquad_kernel{T}(x::Array{T}, y::Vector{T}, c::Real)\n\tc = convert(T, c)\n\tsqrt(sum((x.-y).^2,1) .+ c^2)\nend\n\nfunction invmultiquad_kernel{T}(x::Array{T}, y::Vector{T}, c::Real)\n\tc = convert(T, c)\n\t1 ./ sqrt(sum((x.-y).^2,1) .+ c^2)\nend\n\nfunction circular_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ)\n\tn = sqrt(sum((x.-y).^2,1))./σ\n\tout = (2/π).*(acos(-n) - n.*sqrt(1 .- n.^2))\n\tout[n.>=σ] = 0\n\tout\nend\n\nfunction spherical_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ)\n\tn = sqrt(sum((x.-y).^2,1))./σ\n\tout = 1 - 1.5.*n + .5.*n.^3\n\tout[n.>=σ] = 0\n\tout\nend\n\nfunction wave_kernel{T}(x::Array{T}, y::Vector{T}, θ::Real)\n\tθ=convert(T, θ)\n\tn = sqrt(sum((x.-y).^2,1))\n\t(θ./n).*sin(n./θ)\nend\n\nfunction power_kernel{T}(x::Array{T}, y::Vector{T}, d::Real)\n\td=convert(T, d)\n\t-sqrt(sum((x.-y).^2,1)).^d\nend\n\nfunction log_kernel{T}(x::Array{T}, y::Vector{T}, d::Real)\n\td=convert(T, d)\n\t-log(sqrt(sum((x.-y).^2.,1)).^d + 1)\nend\n\nfunction spline_kernel{T}(x::Array{T}, y::Vector{T})\n\txy = x.*y\n\tmxy = broadcast(min, x,y)\n z = 1 .+ xy .+ xy.*mxy - ((x.+y)/2).*mxy.^2 + (mxy.^3)/3\n prod(z,1)\nend\n\nfunction cauchy_kernel{T}(x::Array{T}, y::Vector{T}, σ::Real)\n\tσ=convert(T, σ) \n\t1 ./ (1 .+ (sum((x.-y).^2,1) ./ σ.^2))\nend\n\nfunction chisquare_kernel{T}(x::Array{T}, y::Vector{T})\n\t1 .- sum((x.-y).^2 ./ ((x.+y)/2),1)\nend\n\nfunction histintersect_kernel{T}(x::Array{T}, y::Vector{T})\n\tsum(broadcast(min, x,y),1)\nend\n\nfunction genhistintersect_kernel{T}(x::Array{T}, y::Vector{T}, α::Real, β::Real)\n\tα=convert(T, α); β=convert(T, β);\n\tsum(broadcast(min, abs(x).^α,abs(y).^β),1)\nend\n\nfunction gentstudent_kernel{T}(x::Array{T}, y::Vector{T}, d::Real)\n\td=convert(T, d)\n\t1 ./ (1.0 .+ sqrt(sum((x.-y).^2.,1)).^d)\nend\n\nfunction fourier_kernel{T}(x::Array{T}, y::Vector{T}, a::Real)\n\ta=convert(T, a)\n dist = x.-y;\n dist[dist.==0] = eps(eltype(x))\n z = sin((a + 1/2)*dist)./sin(dist/2);\n prod(z,1)\nend\n\nfunction sigmoid_kernel{T}(x::Array{T}, y::Vector{T}; α::Real=1, c::Real=0)\n\tα=convert(T, α); c=convert(T, c);\n\ttanh(α*y'*x + c)\nend\n\nmotherwavelet(x::Array) = cos(1.75.*x).*exp(-(x.^2)./2)\n\nfunction wavelet_kernel{T}(x::Array{T}, y::Vector{T}, a::Real, c::Real)\n\ta=convert(T, a); c=convert(T, c);\n\tprod(motherwavelet((x.-c)./a).*motherwavelet((y.-c)./a),1)\nend\n\nfunction transinvwavelet_kernel{T}(x::Array{T}, y::Vector{T}, a::Real)\n\ta=convert(T, a);\n\tprod(motherwavelet((x.-y)./a),1)\nend\n\n\n\n\n\nend # module\n", "meta": {"hexsha": "4b1c8e2665a405ad31717b5b134066e7b1940015", "size": 4119, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/KernelLibrary.jl", "max_stars_repo_name": "the-moliver/KernelLibrary.jl", "max_stars_repo_head_hexsha": "651d6b44294b8ae7486a83233a770046f0a9525f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2018-10-17T15:59:43.000Z", "max_stars_repo_stars_event_max_datetime": "2019-05-14T19:34:10.000Z", "max_issues_repo_path": "src/KernelLibrary.jl", "max_issues_repo_name": "the-moliver/KernelLibrary.jl", "max_issues_repo_head_hexsha": "651d6b44294b8ae7486a83233a770046f0a9525f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-09-09T15:40:02.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-09T15:40:02.000Z", "max_forks_repo_path": "src/KernelLibrary.jl", "max_forks_repo_name": "the-moliver/KernelLibrary.jl", "max_forks_repo_head_hexsha": "651d6b44294b8ae7486a83233a770046f0a9525f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-05-14T19:34:14.000Z", "max_forks_repo_forks_event_max_datetime": "2019-05-14T19:34:14.000Z", "avg_line_length": 27.2781456954, "max_line_length": 225, "alphanum_fraction": 0.6224811848, "num_tokens": 1541, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9496693702514737, "lm_q2_score": 0.8558511414521923, "lm_q1q2_score": 0.8127756145319084}} {"text": "#### Similarity and dissimilarity measures between images ####\n\n\"\"\"\n`hausdorff_distance(imgA, imgB)` is the modified Hausdorff distance\nbetween binary images (or point sets).\n\nReferences\n----------\nDubuisson, M-P; Jain, A. K., 1994. A Modified Hausdorff Distance for\nObject-Matching.\n\"\"\"\nfunction hausdorff_distance(imgA::AbstractArray, imgB::AbstractArray)\n @assert size(imgA) == size(imgB) \"Images must have the same dimensions\"\n\n # trivial case\n isequal(imgA, imgB) && return 0.\n\n A, B = find(imgA), find(imgB)\n\n # return if there is no object to match\n (isempty(A) || isempty(B)) && return Inf\n\n # grid coordinates (ndims by npoints)\n A = hcat([Float64[ind2sub(size(imgA), a)...] for a in A]...)\n B = hcat([Float64[ind2sub(size(imgB), b)...] for b in B]...)\n\n m = size(A, 2); n = size(B, 2)\n\n D = zeros(m, n)\n for j=1:n, i=1:m\n @inbounds D[i,j] = norm(A[:,i] - B[:,j])\n end\n\n dAB = mean(minimum(D, 2))\n dBA = mean(minimum(D, 1))\n\n max(dAB, dBA)\nend\n", "meta": {"hexsha": "52365b9b162b872e0e4a263bf70382f7b6ece65e", "size": 972, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/distances.jl", "max_stars_repo_name": "JuliaPackageMirrors/Images.jl", "max_stars_repo_head_hexsha": "6a3f1ce49a432567b726696f70e3b650650919ec", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/distances.jl", "max_issues_repo_name": "JuliaPackageMirrors/Images.jl", "max_issues_repo_head_hexsha": "6a3f1ce49a432567b726696f70e3b650650919ec", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/distances.jl", "max_forks_repo_name": "JuliaPackageMirrors/Images.jl", "max_forks_repo_head_hexsha": "6a3f1ce49a432567b726696f70e3b650650919ec", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.9230769231, "max_line_length": 73, "alphanum_fraction": 0.6399176955, "num_tokens": 322, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9539660976007597, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.8127340862478369}} {"text": "using LinearAlgebra, Statistics, Plots\n\nx_vals = [0 0 0; 2 -3 -4];\ny_vals = [0 0 0; 4 3 -3.5];\n\nplot(x_vals, y_vals, arrow = true, color = :blue,\n legend = :none, xlims = (-5, 5), ylims = (-5,5),\n annotations = [(2.2, 4.4, \"[2, 4]\"),\n (-3.3, 3.3, \"[-3, 3]\"),\n (-4.4, -3.85, \"[-4, -3.5]\")],\n xticks = -5:1:5, yticks = -5:1:5,\n framestyle = :origin)\n\n # Illustrate scalar multiplication\n x = [2];\n scalars = [-2 1 2]; \n vals = [0 0 0; x * scalars]\n labels = [(-3.6, -4.2, \"-2x\"), (2.4, 1.8, \"x\"), (4.4, 3.8, \"2x\")]\n \n plot(vals, vals, arrow = true, color = [:red :red :blue],\n legend = :none, xlims = (-5, 5), ylims = (-5, 5),\n annotations = labels, xticks = -5:1:5, yticks = -5:1:5,\n framestyle = :origin)\n\nx = ones(3);\ny = [2, 4, 6];\nx + y;\n4x; # equivalent to 4 * x and 4 .* x\n\ndot(x, y); \nnorm(x); # Norm of x\nsqrt(sum(abs2, x)); # Give the same result\n\n# fixed linear function, to generate a plane\nf(x, y) = 0.2x + 0.1y;\n\n# lines to vectors\nx_vec = [0 0; 3 3]\ny_vec = [0 0; 4 -4]\nz_vec = [0 0; f(3, 4) f(3, -4)]\n\n# draw the plane\nn = 20\ngrid = range(-5, 5, length = n)\nz2 = [ f(grid[row], grid[col]) for row in 1:n, col in 1:n ]\nwireframe(grid, grid, z2, fill = :blues, gridalpha =1 )\nplot!(x_vec, y_vec, z_vec, color = [:blue :green], linewidth = 3, labels = \"\",\n colorbar = false)\n\nA = [1 2\n 3 4]\nA'\nA = ones(3,3);\n2I\nA + I\n\n# Solve Linear Equations\nA = [1.0 2.0; 3.0 4.0];\ny = ones(2,1);\ndet(A);\nA_inv = inv(A)\nx = A_inv * y # solution\nA * x # should equal y (a vector of ones)\nA \\ y # produces the same solution\n\nA = [1.0 2.0; 2.0 1.0];\nevals, evecs = eigen(A);\n\n", "meta": {"hexsha": "2f38dab3d5808e2a3a2f04b23890493461ba7d8c", "size": 1644, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Julia/Linear_Algebra.jl", "max_stars_repo_name": "wongr003/QuantEcon", "max_stars_repo_head_hexsha": "51d809564f8b653ba2e26b0050c0bb74dbb2c221", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Julia/Linear_Algebra.jl", "max_issues_repo_name": "wongr003/QuantEcon", "max_issues_repo_head_hexsha": "51d809564f8b653ba2e26b0050c0bb74dbb2c221", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Julia/Linear_Algebra.jl", "max_forks_repo_name": "wongr003/QuantEcon", "max_forks_repo_head_hexsha": "51d809564f8b653ba2e26b0050c0bb74dbb2c221", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.8260869565, "max_line_length": 78, "alphanum_fraction": 0.5261557178, "num_tokens": 705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9539660976007597, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.8127340862478369}} {"text": "\"\"\"\n uniformprobvec(n)\n\nReturn a uniform probability distribution vector of size `n`.\n\"\"\"\nfunction uniformprobvec(n)\n return ones(n) ./ n\nend\n\n\"\"\"\n uniformtransmat(n)\n\nReturn a stochastic matrix of size `n` with uniform transition probability distributions.\n\"\"\"\nfunction uniformtransmat(n)\n return ones(n, n) ./ n\nend\n\n\"\"\"\n randprobvec(n)\n\nReturn a random probability distribution vector of size `n`.\n\"\"\"\nfunction randprobvec(n)\n p = rand(n)\n return p ./ sum(p)\nend\n\n\"\"\"\n randtransmat(n)\n\nReturn a stochastic matrix of size `n` with random transition probability distributions.\n\"\"\"\nfunction randtransmat(n)\n P = rand(n, n)\n return P ./ sum(P, dims = 2)\nend\n", "meta": {"hexsha": "a16d99b55fd5a3a2a5b1c6a8567b28ade5641dbd", "size": 687, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/utils/randvals.jl", "max_stars_repo_name": "gdalle/PointProcesses.jl", "max_stars_repo_head_hexsha": "32a68c5d263b980af2c38353f29652d97539ec88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2021-08-10T21:17:24.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-30T21:32:43.000Z", "max_issues_repo_path": "src/utils/randvals.jl", "max_issues_repo_name": "gdalle/PointProcesses.jl", "max_issues_repo_head_hexsha": "32a68c5d263b980af2c38353f29652d97539ec88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-08-17T18:16:31.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-18T08:26:54.000Z", "max_forks_repo_path": "src/utils/randvals.jl", "max_forks_repo_name": "gdalle/PointProcesses.jl", "max_forks_repo_head_hexsha": "32a68c5d263b980af2c38353f29652d97539ec88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.0789473684, "max_line_length": 89, "alphanum_fraction": 0.6826783115, "num_tokens": 167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.935346504434783, "lm_q2_score": 0.868826769445233, "lm_q1q2_score": 0.8126540817599638}} {"text": "# using LinearAlgebra\nfunction cart2sph(x::T, y::T, z::T) where {T <: Real}\n hxy = hypot(x, y)\n r = hypot(hxy, z)\n pol = atan(hxy, z)\n az = atan(y, x)\n return [Float64(pol), Float64(az), Float64(r)]\nend\n\nfunction cart2sph(xyz::Point)\n x, y, z = xyz[1], xyz[2], xyz[3]\n return cart2sph(x, y, z)\nend\n\nfunction sph2cart(pol::T, az::T, r::T) where {T <: Real}\n x = r * sin(pol) * cos(az)\n y = r * sin(pol) * sin(az)\n z = r * cos(pol)\n return x, y, z\nend\n\"\"\"\nChange array of vertex coordinates into array of vertex type\n\"\"\"\nfunction array_to_vertex(a::AbstractArray)\n return [Point3f0(vertex) for vertex in a]\nend\n\n\"\"\"\nChange array of face into array of face type\n\"\"\"\nfunction array_to_faces(a::AbstractArray)\n return [TriangleFace{OffsetInteger{-1,UInt32}}(face) for face in a]\nend\n\n\"\"\"\nChange array of array of into matrix\n\"\"\"\nfunction array_of_array_to_matrix(a::AbstractArray)\n return [a[i][j] for i = 1:length(a), j = 1:length(a[1])]\nend\n\"\"\"\nChange matrix to array of array\n\"\"\"\nfunction matrix_to_array_of_array(mat::AbstractMatrix)\n return [mat[i, :] for i = 1:size(mat, 1)]\nend # function]\n\n\"\"\"\nChange matrix to vertices\n\"\"\"\nfunction matrix_to_vertex(mat::AbstractMatrix)\n return [Point3f0(mat[i, :]) for i = 1:size(mat, 1)]\nend # function\n\n\n\"\"\"\nCentering the coordinates of mesh vertices to their mean\n\"\"\"\nfunction centroid(v::AbstractMatrix)\n return sum(v, dims=1) / size(v, 1)\nend\nfunction centroid(verts::Vector{T1}) where {T1}\n return reduce(+, verts) / length(verts)\nend\n\nfunction mesh_centering(verts::Vector{T1}, faces::Vector{T2}) where {T1,T2}\n mean_coord = reduce(+, verts) / length(verts)\n new_verts = similar(verts)\n for i in eachindex(verts)\n new_verts[i] = verts[i] - mean_coord\n end\n return new_verts, faces\nend\nfunction mesh_centering(mesh::Mesh)\n verts = decompose(Point3f0, mesh)\n triangles = decompose(TriangleFace{Int}, mesh)\n verts, triangles = mesh_centering(verts, triangles)\n return Mesh(verts, faces(mesh))\nend\n\n\n\nfunction signed_volume_of_triangle(\n p1::T,\n p2::T,\n p3::T,\n) where {T <: AbstractVector}\n return dot(p1, cross(p2, p3)) / 6.0\nend\n\n\n\"\"\"\nThese functions return the volume of mesh object\n\"\"\"\nfunction mesh_volume(\n verts::Array{Point{3,T1}},\n faces::Array{TriangleFace{T2}},\n) where {T1 <: Real,T2 <: Int}\n total_vol = 0.0\n for face in faces\n vol = signed_volume_of_triangle(\n verts[face[1]],\n verts[face[2]],\n verts[face[3]],\n )\n total_vol += vol\n end\n return abs(total_vol)\nend\n\nfunction mesh_volume(mesh::Mesh)\n verts = decompose(Point3f0, mesh)\n faces = decompose(TriangleFace{Int}, mesh)\n\n return mesh_volume(verts, faces)\nend\n\"\"\"\nThese functions return the copy of mesh object with volume one\n\"\"\"\nfunction volume_normalizing(\n verts::Array{Point{3,T1}},\n faces::Array{TriangleFace{T2}},\n) where {T1 <: Real,T2 <: Int}\n vol = mesh_volume(verts, faces)\n cbrt_vol = cbrt(vol)\n new_verts = similar(verts)\n for i in eachindex(verts)\n new_verts[i] = verts[i] / cbrt_vol\n end\n return new_verts, faces\nend\n\nfunction volume_normalizing2(\n verts::Array{Point{3,T1}},\n faces::Array{TriangleFace{T2}},\n) where {T1 <: Real,T2 <: Int}\n vol = mesh_volume(verts, faces)\n cbrt_vol = cbrt(vol)\n new_verts = similar(verts)\n for i in eachindex(verts)\n new_verts[i] = verts[i] / cbrt_vol\n end\n return new_verts, faces\nend\n\nfunction volume_normalizing(mesh::Mesh)\n verts = decompose(Point3f0, mesh)\n triangles = decompose(TriangleFace{Int}, mesh)\n verts, triangles = volume_normalizing(verts, triangles)\n return Mesh(verts, faces(mesh))\nend\n\"\"\"\nThis function return the mesh object from verts and faces\n\"\"\"\nfunction create_mesh(verts, faces)\n faces = convert(Vector{TriangleFace{OffsetInteger{-1,UInt32}}}, faces)\n return Mesh(verts, faces)\nend\n\"\"\"\nThis function return the angle at vertex v1 from triangle(v1,v2,v3)\n\"\"\"\nfunction calculate_angle(v1, v2, v3)\n v12 = v2 - v1\n v13 = v3 - v1\n v12 = v12 / norm(v12)\n v13 = v13 / norm(v13)\n cos_val = dot(v12, v13)\n angle = acos(cos_val)\n return angle\nend\n# \"\"\"\n# This function return the internal angles of vertices\n# \"\"\"\n# function internal_angles(verts,faces)\n# nrow,ncol = length(faces[1]),length(faces)\n# angles = Matrix{Float64}(undef,nrow,ncol)\n# for j in 1:ncol\n# f = faces[j]\n# for i in 1:nrow\n# v1 = verts[f[i]]\n# idx2 = mod(i+1,nrow) == 0 ? 3 : mod(i+1,nrow)\n# idx3 = mod(i-1,nrow) == 0 ? 3 : mod(i-1,nrow)\n# # @show i\n# # @show idx3\n# # @show f\n# v2 = verts[f[idx2]]\n# v3 = verts[f[idx3]]\n#\n# angles[i,j] = calculate_angle(v1,v2,v3)\n# end\n# end\n# return angles\n# end\n\"\"\"\nThis function return the internal angles of vertices\n\"\"\"\nfunction internal_angles_intrinsic(l)\n s23 = l[:, 1]\n s31 = l[:, 2]\n s12 = l[:, 3]\n\n a23 = acos.((s12.^2 + s31.^2 - s23.^2) ./ (2 * s12 .* s31))\n a31 = acos.((s23.^2 + s12.^2 - s31.^2) ./ (2 * s23 .* s12))\n a12 = acos.((s31.^2 + s23.^2 - s12.^2) ./ (2 * s31 .* s23))\n\n angles = [a23 a31 a12]\n return angles\nend\nfunction internal_angles(verts, faces)\n f1 = [face[1] for face in faces]\n f2 = [face[2] for face in faces]\n f3 = [face[3] for face in faces]\n\n s12 = norm.(verts[f2] - verts[f1])\n s13 = norm.(verts[f3] - verts[f1])\n s23 = norm.(verts[f3] - verts[f2])\n\n l = [s23 s13 s12]\n\n A = internal_angles_intrinsic(l)\nend\n\n\"\"\"\nThis function return the area of triangle made from vectors v1,v2, and v3)\n\"\"\"\nfunction calculate_area(v1, v2, v3)\n v12 = v2 - v1\n v13 = v3 - v1\n area = norm(cross(v12, v13)) / 2.0\n return area\nend\n\"\"\"\nThis function return the surface area)\n\"\"\"\nfunction surface_area(v::AbstractVector, f::AbstractVector)\n nv = length(v)\n nf = length(f)\n\n area = 0\n for i in 1:nf\n l1 = norm(v[f[i][2]] - v[f[i][3]])\n l2 = norm(v[f[i][3]] - v[f[i][1]])\n l3 = norm(v[f[i][1]] - v[f[i][2]])\n s = (l1 + l2 + l3) / 2.0\n square_area = s * (s - l1) * (s - l2) * (s - l3)\n if square_area < 0.0; square_area = 0.0 end\n\n area += sqrt(square_area)\n end\n return area\nend\n\nfunction surface_area_list(v::Matrix{<:AbstractFloat}, f::Matrix{<:Signed})\n f1 = @view f[:, 1]\n f2 = @view f[:, 2]\n f3 = @view f[:, 3]\n\n l1 = sqrt.(sum((view(v, f2, :) - view(v, f3, :)).^2, dims=2))\n l2 = sqrt.(sum((view(v, f3, :) - view(v, f1, :)).^2, dims=2))\n l3 = sqrt.(sum((view(v, f1, :) - view(v, f2, :)).^2, dims=2))\n s = (l1 + l2 + l3) / 2.0\n squared_area = s .* (s - l1) .* (s - l2) .* (s - l3)\n area = sqrt.(squared_area)\n return area\nend\nfunction surface_area(v::Matrix{<:AbstractFloat}, f::Matrix{<:Signed})\n area = surface_area_list(v, f)\n return sum(area)\nend\nfunction surface_area2(v::AbstractVector, f::AbstractVector)\n f1 = [face[1] for face in f]\n f2 = [face[2] for face in f]\n f3 = [face[3] for face in f]\n # F x 3 matrices of triangle edge vectors, named after opposite vertices\n v1 = [v[f3[i]] - v[f2[i]] for i in eachindex(f)]\n v2 = [v[f1[i]] - v[f3[i]] for i in eachindex(f)]\n # v3 = [v[f2[i]] - v[f1[i]] for i in eachindex(f)];\n n = cross.(v1, v2)\n area = norm.(n) / 2.0\n return sum(area)\nend\nfunction surface_area2(v::Matrix{<:AbstractFloat}, f::Matrix{<:Signed})\n f1 = @view f[:, 1]\n f2 = @view f[:, 2]\n f3 = @view f[:, 3]\n # F x 3 matrices of triangle edge vectors, named after opposite vertices\n v1 = view(v, f3, :) - view(v, f2, :)\n v2 = view(v, f1, :) - view(v, f3, :)\n # v3 = v[f2,:] - v[f1,:];\n n = [cross(v1[i, :], v2[i, :]) for i = 1:size(v1, 1)]\n area = norm.(n) / 2.0\n return sum(area)\nend\n\n\nfunction mesh_reorientation(mesh::Mesh, T, N, B)\n verts = decompose(Point3f0, mesh)\n mat_verts = array_of_array_to_matrix(verts)\n new_mat_verts = [T N B] \\ mat_verts'\n new_mat_verts = new_mat_verts'\n new_verts = matrix_to_vertex(new_mat_verts)\n return Mesh(new_verts, faces(mesh))\nend\nfunction mesh_reorientation(mat_verts::AbstractMatrix, T, N, B)\n new_mat_verts = [T N B] \\ mat_verts'\n return new_mat_verts'\nend\n", "meta": {"hexsha": "1e9471b043e7b3952a4a555c675d900e4de6b363", "size": 8302, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/utils_shape.jl", "max_stars_repo_name": "yusri-dh/MovingFrame.jl", "max_stars_repo_head_hexsha": "1be0d7e4ec1f20d898dfbaf9f85e5b9ac86c7b6e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-12-28T07:25:46.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-18T11:59:15.000Z", "max_issues_repo_path": "src/utils_shape.jl", "max_issues_repo_name": "yusri-dh/MovingFrame.jl", "max_issues_repo_head_hexsha": "1be0d7e4ec1f20d898dfbaf9f85e5b9ac86c7b6e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/utils_shape.jl", "max_forks_repo_name": "yusri-dh/MovingFrame.jl", "max_forks_repo_head_hexsha": "1be0d7e4ec1f20d898dfbaf9f85e5b9ac86c7b6e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.9545454545, "max_line_length": 76, "alphanum_fraction": 0.6049144784, "num_tokens": 2714, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9334308147331957, "lm_q2_score": 0.8705972751232809, "lm_q1q2_score": 0.8126423238228242}} {"text": "using Random, Plots, StatsPlots, Distributions, DataFrames, Printf\n# Define calculation of posterior\n\"\"\"\nPosterior Distribution (Gamma dist) for Poisson parameter λ\n\n Inputs\n ------\n data: data (0 or 1)\n α0: shape parameter of prior (Gamma dist)\n β0: inversed scale parameter of prior (Gamma dist)\n prob: credible interval probability (0 < prob < 1)\n\n Outputs\n -------\n results: posterior summary stats table (DataFrames)\n α_star: shape parameter of posterior (Gamma dist)\n β: inversed scale parameter (1 / θ) of posterior (Gamma dist)\n\n\"\"\"\nfunction poisson_stats(data, α0, β0, prob)\n # posterior\n n = size(data)[1]\n sum_data = sum(data)\n α_star = sum_data + α0\n β_star = n + β0\n θ_star = 1.0 / β_star\n d_posterior = Gamma(α_star, θ_star)\n # mean\n mean_λ = mean(d_posterior)\n # median\n median_λ = median(d_posterior)\n # mode\n mode_λ = mode(d_posterior)\n # Analytic Mode for Beta Dist\n # (a - 1.0) / (a + b - 2.0)\n # std\n sd_λ = std(d_posterior)\n # credible interval\n # ci upper boud\n ci_λ_upper = quantile(d_posterior, prob - (1-prob)/2)\n # ci lower boud\n ci_λ_lower = quantile(d_posterior, (1-prob)/2)\n ci_λ = [ci_λ_lower, ci_λ_upper]\n\n posterior_stats = reshape([mean_λ, median_λ, mode_λ, sd_λ, ci_λ], 1, 5)\n stats_string = [\"Mean\", \"Median\", \"Mode\", \"SD\", \"CI\"]\n results = DataFrame(posterior_stats, stats_string)\n return results, α_star, β_star\nend\n\n# Generate moc datas\nλ = 3.0\nn = 50\nRandom.seed!(99)\ndata = rand(Poisson(lam), n)\n\n# Set prior\nα0 = 1.0\nβ0 = 1.0\n\n# Calculate posterior statistics\nprob = 0.95\nresults, α_star, β_star = poisson_stats(data, α0, β0, prob)\nprint(results)\n\n# Visualize Posterior\nθ0, θ_star = 1.0/β0, 1.0/β_star\nplot(Gamma(α0, θ0), xlims=(0, 6), ylims=(0, 1.75), xlabel=\"λ\", ylabel=\"Probability Density\", label=@sprintf(\"prior: Gamma(%.1f, %.1f)\",α0, θ0), linestyle=:dash)\nplot!(Gamma(α_star, θ_star), label=@sprintf(\"prior: Gamma(%.1f, %.2f)\",α_star, θ_star), linestyle=:solid)", "meta": {"hexsha": "068ab8f53e0b6162ebee9305cc216da2ebd57d89", "size": 2023, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "jlbayes_conjugate_poisson.jl", "max_stars_repo_name": "hessihan/julia_bayes_intro", "max_stars_repo_head_hexsha": "e5a6c740dbba4a2794579afde196a4586703fbc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "jlbayes_conjugate_poisson.jl", "max_issues_repo_name": "hessihan/julia_bayes_intro", "max_issues_repo_head_hexsha": "e5a6c740dbba4a2794579afde196a4586703fbc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "jlbayes_conjugate_poisson.jl", "max_forks_repo_name": "hessihan/julia_bayes_intro", "max_forks_repo_head_hexsha": "e5a6c740dbba4a2794579afde196a4586703fbc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.3188405797, "max_line_length": 160, "alphanum_fraction": 0.6544735541, "num_tokens": 678, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.933430803622103, "lm_q2_score": 0.870597271765821, "lm_q1q2_score": 0.8126423110155807}} {"text": "#Maximum Compatibility Pairing \r\nusing JuMP,Gurobi\r\n\r\n#define the model\r\nm =Model(Gurobi.Optimizer)\r\n\r\n#Let n represents the number of cities to be paired\r\nn=4 \r\n\r\n#declaring variables\r\n@variable(m, x[1:n,1:n]>=0)\r\n\r\n#define the compatibility coefficients \r\nc = [\t80 65 83 77;\r\n\t54 87 61 66 ;\r\n\t92 45 53 59 ; \r\n\t70 61 81 76 ]\r\n\r\n#define the objective function\r\n@objective(m, Max, sum(c[i,j]* x[i,j] for i in 1:n for j in 1: n)) #objective is to maximize total compatibility pairing\r\n\r\n#define constraints\r\nfor j = 1: n\r\n @constraint(m ,sum(x[i,j] for i in 1: n) ==1)\r\nend\r\n\r\nfor i = 1: n\r\n @constraint(m ,sum(x[i,j] for j in 1: n) ==1 )\r\nend\r\n\r\nprint(m)\r\n\r\n#Printing the optimal solutions obtained\r\nprintln(\"Optimal Solutions:\")\r\nstatus = JuMP.optimize!(m)\r\nprintln(\" Objective value: \",JuMP.objective_value(m))\r\n\r\nprintln(\"Decision variable values:\")\r\nprintln(\"x = \", JuMP.value.(x))\r\n", "meta": {"hexsha": "4ac7e15886796ac52b599fff82dd88d2812638ea", "size": 892, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Compatibility.jl", "max_stars_repo_name": "Oyekanmi/Optimization-Julia", "max_stars_repo_head_hexsha": "9c89aa41fe14ead83f493c812c9f51d407870d6c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-02-14T14:36:32.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-14T14:36:32.000Z", "max_issues_repo_path": "Compatibility.jl", "max_issues_repo_name": "Oyekanmi/Optimization-Julia", "max_issues_repo_head_hexsha": "9c89aa41fe14ead83f493c812c9f51d407870d6c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Compatibility.jl", "max_forks_repo_name": "Oyekanmi/Optimization-Julia", "max_forks_repo_head_hexsha": "9c89aa41fe14ead83f493c812c9f51d407870d6c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.3, "max_line_length": 121, "alphanum_fraction": 0.6591928251, "num_tokens": 273, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9777138183570425, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.8126200388330933}} {"text": "### A Pluto.jl notebook ###\n# v0.16.1\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 35dd2eca-2446-11ec-1de6-0feeace9d4f7\nmd\"\"\"\n\n#### Flash separation of a binary mixture of component A and component B\n\nA liquid feed stream that contains 60% component A (1) and 40% component B (2) is separated in a Flash Drum operating at P = 1.21 MPa and T = 150$^{\\circ}$C. The saturation pressures of pure component A and B can be modeled using the [Antoine equation](https://en.wikipedia.org/wiki/Antoine_equation): \n\n$$\\begin{eqnarray}\n\\ln\\left(P_{1}^{sat}\\right) &=& 15 -\\frac{3010}{T+250}\\\\\n\\ln\\left(P_{2}^{sat}\\right) &=& 14 -\\frac{2700}{T+205}\n\\end{eqnarray}$$\n\nwhere $T$ denotes the temperature in units of celcius $^{\\circ}$C and $P_{\\star}^{sat}$ denotes the saturation pressure in units of kPa for component $\\star$. \n\n##### Assumptions\n\n1. The liquid and vapor phases of all streams are ideal\n1. The liquid and vapor phases inside the Flash Drum are ideal (Raoult's law)\n1. The liquid and vapor phases inside the Flash Drum reach vapor-liquid equilibrium instantly \n\n##### Solve:\n\n * Calculate the fraction of the input stream that exits the drum as liquid\n * Calculate the fraction of the input stream that exits the drum as vapor\n * Calculate the composition of the liquid ($x_{1}$ and $x_{2}$) and vapor ($y_{1}$ and $y_{2}$) in the exit streams \n\n\"\"\"\n\n# ╔═╡ 92ff1f65-d717-47a8-9cd3-af71c2855210\nPlutoUI.LocalResource(\"./figs/Fig-Flash.png\")\n\n# ╔═╡ 98e43733-8711-4aa4-9081-697dd8865592\nbegin\n\t\n\t# write down some constants -\n\tT = 150 \t\t# units: C\n\tP = 1210 \t\t# units: kPa\n\tz₁ = 0.60 \t\t# units: dimensionless\n\tz₂ = (1-z₁)\t\t# units: dimensionless\n\t\n\t# return -\n\tnothing \nend\n\n# ╔═╡ 792e101b-e736-46d2-bbf3-a1197a0cc654\nmd\"\"\"\n\n##### Governing equations\n\nLet's start by writing down the governing equations. Starting from the total mol balance:\n\n$$\\dot{F} = \\dot{L}+\\dot{V}$$\n\nwe can divide both sides by $\\dot{F}$ to give a relationship which constrains the fraction(s) of liquid and vapor in the outlet streams:\n\n$$\\hat{L}+\\hat{V} = 1$$\n\nFurther, we know that the species mol balance around species $i$ is given by:\n\n$$\\dot{F}z_{i} = \\dot{L}x_{i}+\\dot{V}y_{i}\\qquad{i=1,2}$$\n\nHowever, because we assumed ideal vapor and liquid, we know that the liquid and vapor phase equlibrium compositions are related by the fugacity matching condition: $y_{i}P = x_{i}P_{i}^{sat}$. Lastly, we can sum the matching conditions to arrive at an expression which governs the pressure that is only in $x_{i}$:\n\n$$P = x_{1}P_{1}^{sat} + x_{2}P_{2}^{sat}$$\n\nwhich can be re-written (because we are in a binary mixture) as:\n\n$$P = x_{1}P_{1}^{sat} + \\left(1-x_{1}\\right)P_{2}^{sat}$$\n\n\"\"\"\n\n# ╔═╡ 9e171e7c-953d-4a75-b6f6-cbae930dd1ab\nmd\"\"\"\n\n#### Solve for $x_{1}$ and $x_{2}$:\n\nWe can re-arrange the pressure expression to solve for $x_{1}$:\n\n$$x_{1} = \\frac{P - P_{2}^{sat}}{P_{1}^{sat} - P_{2}^{sat}}$$\n\n\"\"\"\n\n# ╔═╡ 8e13e386-6fbc-4bdf-8eed-de5bd35247da\n# compute P1sat (units: kPa)\nP1_sat = exp(15 - (3010)/(T+250))\n\n# ╔═╡ 45a8e5fc-0929-496b-b01a-0e6f27f8b457\n# compute P2sat (units: kPa)\nP2_sat = exp(14 - (2700)/(T+205))\n\n# ╔═╡ 2bd63a06-0eb6-4e4e-b2b0-78b0ac87e341\n# solve for x₁:\nx₁ = (P - P2_sat)/(P1_sat - P2_sat)\n\n# ╔═╡ 6b3bcfa8-7487-4f84-abbe-185939d91a80\nx₂ = 1 - x₁\n\n# ╔═╡ 50a0b336-e14c-487a-9ba5-25a0024d9766\nmd\"\"\"\n#### Solve for $y_{1}$ and $y_{2}$:\n\nOnce we have the liquid phase composition, we can use the fugacity matching condition to relate the liquid phase and the vapr phase becuase we have asssumed they are in equilibrium i.e., \n\n$$y_{i} = x_{i}\\left(\\frac{P_{i}^{sat}}{P}\\right)\\qquad{i=1,2}$$\n\n\"\"\"\n\n# ╔═╡ 7608d2c9-eb24-4db1-9fb3-452d52770e17\n# solve for y₁:\ny₁ = x₁ * (P1_sat/P)\n\n# ╔═╡ 76f6cf32-8f87-4df0-b1db-a51ee102f709\n# solve for y₂:\ny₂ = x₂* (P2_sat/P)\n\n# ╔═╡ ec5378ba-ca2c-4d31-b176-454b2cd99302\nmd\"\"\"\n\n#### Solve for $\\hat{L}$ and $\\hat{V}$:\n\nNow that we have the composition values in the exit streams, we can use the total mol and species mol balances to estimate the fraction \nof the input feed that exited the unit as liquid $\\hat{L}$ and vapur $\\hat{V}$. If we divide both sides of the species mol balance by $\\dot{F}$ we get the relationship:\n\n$$z_{i} = \\hat{L}x_{i}+\\hat{V}y_{i}\\qquad{i=1,2}$$\n\nwhich we can use (along with the total mol balances) to solve for $\\hat{L}$ and $\\hat{V}$ i.e., \n\n$$\\begin{eqnarray}\n\\hat{L}x_{1}+\\hat{V}y_{1} &=& z_{1} \\\\\n\\hat{L}+\\hat{V} &=& 1\n\\end{eqnarray}$$\n\nThis is a system of 2-equations and 2-unknowns ($\\hat{L}$ and $\\hat{V}$) that can be solved by substitution or by linear algebra. \n\n###### Approach 1: Old school subsitution/elimination approach:\n\nLet's rearrange the total mol expression to get $\\hat{L}$ in terms of $\\hat{V}$:\n\n$$\\hat{L} = 1 - \\hat{V}$$\n\nwhich can be subsituted into the species mol balance (let's use component 1):\n\n$$z_{1} = \\left(1 - \\hat{V}\\right)x_{1}+\\hat{V}y_{1}$$\n\nand solved for $\\hat{V}$:\n\n$$\\hat{V} = \\frac{z_{1} - x_{1}}{y_{1} - x_{1}}$$\n\nOnce we have $\\hat{V}$, we can compute $\\hat{L}$ from the total mol balance.\n\"\"\"\n\n# ╔═╡ bb56b509-5279-40f5-b898-e8c1b29dfb5e\n# compute Vhat -\nVhat = (z₁ - x₁)/(y₁ - x₁)\n\n# ╔═╡ 3f13e0a8-b58b-4ac7-899b-412154acf0cb\n# what is Lhat -\nLhat = 1 - Vhat\n\n# ╔═╡ 745177ba-a88d-496d-b0f6-2e9aca506866\nmd\"\"\"\n###### Approach 2: Fancy linear algebra approach:\n\nWe can rewrite the 2$\\times$2 system of equations into matrix-vector form and then compute the matrix inverse. In particlar, \nthe species and total mass balances for component 1 can be written as:\n\n$$\\begin{pmatrix}\nx_{1} & y_{1} \\\\\n1 & 1 \n\\end{pmatrix}\n\\begin{pmatrix}\n\\hat{L} \\\\\n\\hat{V}\n\\end{pmatrix} = \n\\begin{pmatrix}\nz_{1} \\\\\n1\n\\end{pmatrix}$$\n\nwhich is the form $\\mathbf{A}\\mathbf{x} = \\mathbf{b}$, where the solution takes the form: $\\mathbf{x} = \\mathbf{A}^{-1}\\mathbf{b}$.\n\n\n\"\"\"\n\n# ╔═╡ 4c0a79c3-87b4-4452-ba9e-0de5bd324ac5\n# setup A -\nA = [x₁ y₁ ; 1 1]\n\n# ╔═╡ a835ed0a-a2d0-4a5b-a8a0-dffd7250e34b\n# setup b -\nbV = [z₁ ; 1]\n\n# ╔═╡ 2bf7b85f-3118-4fa9-842d-f2c9f3d16dce\n# compute the unknown (fractions of L and V)\nx = inv(A)*bV\n\n# ╔═╡ b091bbb6-13ce-4b1f-b36b-9964783767d4\nhtml\"\"\"\n\"\"\"\n\n# ╔═╡ 64af17f4-c632-425c-a101-9dfa5e8cbd84\n# julia setup -\nusing PlutoUI\n\n# ╔═╡ 79f99ade-b514-4768-9bc9-bcfa281ce02f\npng_joinpathsplit__FILE__1assetsimagepng = let\n import PlutoUI\n PlutoUI.LocalResource(joinpath(split(@__FILE__, '#')[1] * \".assets\", \"image.png\"))\nend\n\n# ╔═╡ 00000000-0000-0000-0000-000000000001\nPLUTO_PROJECT_TOML_CONTENTS = \"\"\"\n[deps]\nPlutoUI = \"7f904dfe-b85e-4ff6-b463-dae2292396a8\"\n\n[compat]\nPlutoUI = \"~0.7.14\"\n\"\"\"\n\n# ╔═╡ 00000000-0000-0000-0000-000000000002\nPLUTO_MANIFEST_TOML_CONTENTS = \"\"\"\n# This file is machine-generated - editing it directly is not advised\n\n[[Base64]]\nuuid = \"2a0f44e3-6c83-55bd-87e4-b1978d98bd5f\"\n\n[[Dates]]\ndeps = [\"Printf\"]\nuuid = \"ade2ca70-3891-5945-98fb-dc099432e06a\"\n\n[[HypertextLiteral]]\ngit-tree-sha1 = \"72053798e1be56026b81d4e2682dbe58922e5ec9\"\nuuid = \"ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2\"\nversion = \"0.9.0\"\n\n[[IOCapture]]\ndeps = [\"Logging\", \"Random\"]\ngit-tree-sha1 = \"f7be53659ab06ddc986428d3a9dcc95f6fa6705a\"\nuuid = \"b5f81e59-6552-4d32-b1f0-c071b021bf89\"\nversion = \"0.2.2\"\n\n[[InteractiveUtils]]\ndeps = [\"Markdown\"]\nuuid = \"b77e0a4c-d291-57a0-90e8-8db25a27a240\"\n\n[[JSON]]\ndeps = [\"Dates\", \"Mmap\", \"Parsers\", \"Unicode\"]\ngit-tree-sha1 = \"8076680b162ada2a031f707ac7b4953e30667a37\"\nuuid = \"682c06a0-de6a-54ab-a142-c8b1cf79cde6\"\nversion = \"0.21.2\"\n\n[[Logging]]\nuuid = \"56ddb016-857b-54e1-b83d-db4d58db5568\"\n\n[[Markdown]]\ndeps = [\"Base64\"]\nuuid = \"d6f4376e-aef5-505a-96c1-9c027394607a\"\n\n[[Mmap]]\nuuid = \"a63ad114-7e13-5084-954f-fe012c677804\"\n\n[[Parsers]]\ndeps = [\"Dates\"]\ngit-tree-sha1 = \"a8709b968a1ea6abc2dc1967cb1db6ac9a00dfb6\"\nuuid = \"69de0a69-1ddd-5017-9359-2bf0b02dc9f0\"\nversion = \"2.0.5\"\n\n[[PlutoUI]]\ndeps = [\"Base64\", \"Dates\", \"HypertextLiteral\", \"IOCapture\", \"InteractiveUtils\", \"JSON\", \"Logging\", \"Markdown\", \"Random\", \"Reexport\", \"UUIDs\"]\ngit-tree-sha1 = \"d1fb76655a95bf6ea4348d7197b22e889a4375f4\"\nuuid = \"7f904dfe-b85e-4ff6-b463-dae2292396a8\"\nversion = \"0.7.14\"\n\n[[Printf]]\ndeps = [\"Unicode\"]\nuuid = \"de0858da-6303-5e67-8744-51eddeeeb8d7\"\n\n[[Random]]\ndeps = [\"Serialization\"]\nuuid = \"9a3f8284-a2c9-5f02-9a11-845980a1fd5c\"\n\n[[Reexport]]\ngit-tree-sha1 = \"45e428421666073eab6f2da5c9d310d99bb12f9b\"\nuuid = \"189a3867-3050-52da-a836-e630ba90ab69\"\nversion = \"1.2.2\"\n\n[[SHA]]\nuuid = \"ea8e919c-243c-51af-8825-aaa63cd721ce\"\n\n[[Serialization]]\nuuid = \"9e88b42a-f829-5b0c-bbe9-9e923198166b\"\n\n[[UUIDs]]\ndeps = [\"Random\", \"SHA\"]\nuuid = \"cf7118a7-6976-5b1a-9a39-7adc72f591a4\"\n\n[[Unicode]]\nuuid = \"4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5\"\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╠═64af17f4-c632-425c-a101-9dfa5e8cbd84\n# ╟─35dd2eca-2446-11ec-1de6-0feeace9d4f7\n# ╟─92ff1f65-d717-47a8-9cd3-af71c2855210\n# ╠═79f99ade-b514-4768-9bc9-bcfa281ce02f\n# ╠═98e43733-8711-4aa4-9081-697dd8865592\n# ╟─792e101b-e736-46d2-bbf3-a1197a0cc654\n# ╟─9e171e7c-953d-4a75-b6f6-cbae930dd1ab\n# ╠═8e13e386-6fbc-4bdf-8eed-de5bd35247da\n# ╠═45a8e5fc-0929-496b-b01a-0e6f27f8b457\n# ╠═2bd63a06-0eb6-4e4e-b2b0-78b0ac87e341\n# ╠═6b3bcfa8-7487-4f84-abbe-185939d91a80\n# ╟─50a0b336-e14c-487a-9ba5-25a0024d9766\n# ╠═7608d2c9-eb24-4db1-9fb3-452d52770e17\n# ╠═76f6cf32-8f87-4df0-b1db-a51ee102f709\n# ╟─ec5378ba-ca2c-4d31-b176-454b2cd99302\n# ╠═bb56b509-5279-40f5-b898-e8c1b29dfb5e\n# ╠═3f13e0a8-b58b-4ac7-899b-412154acf0cb\n# ╟─745177ba-a88d-496d-b0f6-2e9aca506866\n# ╠═4c0a79c3-87b4-4452-ba9e-0de5bd324ac5\n# ╠═a835ed0a-a2d0-4a5b-a8a0-dffd7250e34b\n# ╠═2bf7b85f-3118-4fa9-842d-f2c9f3d16dce\n# ╟─b091bbb6-13ce-4b1f-b36b-9964783767d4\n# ╟─00000000-0000-0000-0000-000000000001\n# ╟─00000000-0000-0000-0000-000000000002\n", "meta": {"hexsha": "67062886b0e752f1a13cf15a0273639772ab785a", "size": 9750, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "notebooks/Example-FlashSeparation.jl", "max_stars_repo_name": "agansky/ENGRI-1120-Cornell-Varner", "max_stars_repo_head_hexsha": "1a2103ed3a9cd94a677f09083c1bbef27db90a86", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-15T20:05:45.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-15T20:05:45.000Z", "max_issues_repo_path": "notebooks/Example-FlashSeparation.jl", "max_issues_repo_name": "agansky/ENGRI-1120-Cornell-Varner", "max_issues_repo_head_hexsha": "1a2103ed3a9cd94a677f09083c1bbef27db90a86", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/Example-FlashSeparation.jl", "max_forks_repo_name": "agansky/ENGRI-1120-Cornell-Varner", "max_forks_repo_head_hexsha": "1a2103ed3a9cd94a677f09083c1bbef27db90a86", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 11, "max_forks_repo_forks_event_min_datetime": "2021-11-10T20:59:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-19T06:58:07.000Z", "avg_line_length": 28.0172413793, "max_line_length": 314, "alphanum_fraction": 0.6947692308, "num_tokens": 4215, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797124237605, "lm_q2_score": 0.8918110360927155, "lm_q1q2_score": 0.8126001234032965}} {"text": "using Distributions, CSV, DataFrames, LinearAlgebra, LaTeXStrings\n\n# Load observed data\nx_samples_df = CSV.read(\"gmm_observations.csv\")\nx_samples = hcat(x_samples_df[!, :X1], x_samples_df[!, :X2])\n\n# Computes πᵢ for all i with respect to the given parameters in θ\n# Inputs:\n# - θ: A vector containing the current estimates of [α₀, μ₀, Σ₀, μ₁, Σ₁]\n# - x: The (n x 2) matrix of observed X values\n# Ouputs:\n# 𝛑: An n dimensional vector where the ith coordinate is the value of πᵢ\nfunction compute_all_πᵢ(θ, x)\n # Get dataset size\n n = size(x, 1)\n\n # Get current parameters\n α, μ₀, Σ₀, μ₁, Σ₁ = θ\n\n # Set distributions\n 𝒩₁ = MvNormal(μ₁, Σ₁)\n 𝒩₀ = MvNormal(μ₀, Σ₀)\n\n # Compute all πᵢ\n 𝛑 = [α*pdf(𝒩₁, x[i,:])/(α*pdf(𝒩₁, x[i,:]) + (1-α)*pdf(𝒩₀, x[i,:])) for i=1:n]\n\n return 𝛑\nend\n\n# EM algorithm for GMM with observations x and Z ∈ {0,1}\n# Inputs:\n# - θ₀: The initial estimate of the θ parameter\n# - x: The observed X values\n# - iterations: The number of iterations for the EM algorithm\n# Outputs:\n# - θⱼ: The estimate of θ after 1000 iterations\nfunction gmm_EM(θ₀, x, iterations)\n # Get dataset size\n n = size(x, 1)\n\n # Store the value of θ on jth step\n θⱼ = θ₀\n\n for j=1:iterations\n # Get current πᵢ values for all i\n 𝛑 = compute_all_πᵢ(θⱼ, x)\n\n # Store the vector containing 1 - 𝛑\n 𝛑₁ = 1 .- 𝛑\n\n # Get next value of α\n α = sum(𝛑)/n\n\n # Get next value of μ₀\n μ₀ = [dot(𝛑₁, x[:,1]), dot(𝛑₁, x[:,2])]/sum(𝛑₁)\n\n # Get next value of μ₁\n μ₁ = [dot(𝛑, x[:,1]), dot(𝛑, x[:,2])]/sum(𝛑)\n\n # Get next value of Σ₀.\n Σ₀ = sum([(x[i,:] - μ₀)*((x[i,:] - μ₀))'*𝛑₁[i] for i=1:n])/sum(𝛑₁)\n\n # Get next value of Σ₁.\n Σ₁ = sum([(x[i,:] - μ₁)*((x[i,:] - μ₁))'*𝛑[i] for i=1:n])/sum(𝛑)\n\n # Update θ\n θⱼ = [α, μ₀, Σ₀, μ₁, Σ₁]\n\n end\n\n return θⱼ\nend\n\n\n# Initialize EM parameters\nα₀ = 0.3\nμ₀ = [1.0, 2.0]\nμ₁ = [2.0, 3.0]\nΣ₀ = 1.0*Matrix(I, 2, 2)\nΣ₁ = 1.0*Matrix(I, 2, 2)\nθ₀ = [α₀, μ₀, Σ₀, μ₁, Σ₁]\n\nprintln(gmm_EM(θ₀, x_samples, 20))\n\n\n\n## Plots ##\ngr(size=(300,300))\n\nfunction ellipse!(μ,Σ;kw...)\n for i=1:3\n plot!([tuple((i*sqrt(Σ)*[cos(t),sin(t)]+μ)...) for t in range(0,stop=2π,length=100)];\n seriestype=:shape,linealpha=0.5,linecolor=:gray,fillalpha=[0.4,0.2,0.1][i],kw...)\n end\n current()\nend\n\nfunction mixtureplot(X₁,X₂,μ₀,Σ₀,μ₁,Σ₁,Π, plot_title)\n scatter(X₁,X₂;fillalpha=0.5,markerstrokewidth=0.5,\n marker_z=Π,mc=ColorGradient([:lightblue,:orange]),\n colorbar=:false, title=plot_title)\n ellipse!(μ₀,Σ₀,fillcolor=:lightblue)\n ellipse!(μ₁,Σ₁,fillcolor=:orange)\n plot!(;bg=:transparent,xlims=(-4.5,8.5),ylims=(-3.5,10.5),\n leg=false,ticks=:none,ratio=:equal)\nend\n\n\none_step = gmm_EM(θ₀, x_samples, 1)\none_step_Π = compute_all_πᵢ(one_step, x_samples)\nfive_steps = gmm_EM(θ₀, x_samples, 5)\nfive_step_Π = compute_all_πᵢ(ten_steps, x_samples)\ntwenty_steps = gmm_EM(θ₀, x_samples, 20)\ntwenty_step_Π = compute_all_πᵢ(twenty_steps, x_samples)\n\n# Zero step plot\nmixtureplot(x_samples[:,1], x_samples[:,2], θ₀[2], θ₀[3], θ₀[4], θ₀[5], compute_all_πᵢ(θ₀, x_samples), L\"\\theta_0\")\nsavefig(\"../images/gmm_mixtureplot_0.svg\")\n\n# One step plot\nmixtureplot(x_samples[:,1], x_samples[:,2], one_step[2], one_step[3], one_step[4], one_step[5], one_step_Π, L\"\\theta_1\")\nsavefig(\"../images/gmm_mixtureplot_1.svg\")\n\n# Five step plot\nmixtureplot(x_samples[:,1], x_samples[:,2], five_steps[2], five_steps[3], five_steps[4], five_steps[5], five_step_Π, L\"\\theta_5\")\nsavefig(\"../images/gmm_mixtureplot_5.svg\")\n\n# Twenty step plot\nmixtureplot(x_samples[:,1], x_samples[:,2], twenty_steps[2], twenty_steps[3], twenty_steps[4], twenty_steps[5], twenty_step_Π, L\"\\theta_{20}\")\nsavefig(\"../images/gmm_mixtureplot_20.svg\")\n", "meta": {"hexsha": "704061298be541a1c27d4e31b87a10e4090524ba", "size": 3793, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_ex.jl", "max_stars_repo_name": "seanrattana/courses", "max_stars_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2019-08-21T07:33:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-24T15:46:58.000Z", "max_issues_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_ex.jl", "max_issues_repo_name": "seanrattana/courses", "max_issues_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-23T06:04:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-26T12:47:12.000Z", "max_forks_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_ex.jl", "max_forks_repo_name": "seanrattana/courses", "max_forks_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-08-18T21:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T19:14:33.000Z", "avg_line_length": 28.9541984733, "max_line_length": 143, "alphanum_fraction": 0.6274716583, "num_tokens": 1518, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9559813513911654, "lm_q2_score": 0.8499711794579723, "lm_q1q2_score": 0.812556596781775}} {"text": "module NumericalRange\n\nusing LinearAlgebra\nusing Plots\n\n\"\"\"\n nrange(A; nk = 1, thmax = 32, noplot = false)\n\nNumerical range (or field of values) of the matrix `A`.\nEvaluates and plots the numerical range of the\n`nk` largest leading principal submatrices of `A`, using `thmax`\nequally spaced angles in the complex plane.\nThe defaults are `nk = 1` and `thmax = 32`\nThe eigenvalues of `A` are plotted as `x`. The function returns `f` and `e`, \nwhere `f` is the numerical range and `e` is a vector of eigenvalues of `A`.\nSetting `noplot = true` suppresses the plot.\n\nThis function is a direct translation of `fv.m` in Professor Nick Higham's Matrix Computation Toolbox,\nhttps://www.mathworks.com/matlabcentral/fileexchange/2360-the-matrix-computation-toolbox,\nbased on an original routine by A. Ruhe.\n\nReferences:\nR. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge\n University Press, 1991; sec. 1.5.\nA. S. Householder, The Theory of Matrices in Numerical Analysis,\n Blaisdell, New York, 1964; sec. 3.3.\nC. R. Johnson, Numerical determination of the field of values of a\n general complex matrix, SIAM J. Numer. Anal., 15 (1978),\n pp. 595-602.\n\n\n\"\"\"\nfunction nrange(B; nk=1, thmax=32, noplot=false)\n function rq(A, x)\n return x'*A*x/(x'*x)\n end\n thmax -= 1 # the function uses thmax + 1 angles\n\n (n, p) = size(B)\n if n != p\n DimensionMismatch(\"Matrix must be square.\")\n end\n\n z = Matrix{ComplexF64}(undef, 2*thmax + 1, 1)\n F = eigen(B)\n e = F.values\n f = []\n\n # filter out cases where B is Hermitian or skew-Hermitian\n if B == B'\n f = [minimum(e), maximum(e)]\n elseif B == -B'\n e = imag(e)\n f = [minimum(e), maximum(e)]\n e *= im\n f *= im\n else\n for m = 1:nk\n ns = n + 1 - m\n A = B[1:ns, 1:ns]\n for i = 0:thmax\n th = i/thmax*pi\n Ath = exp(im*th)*A\n H = 0.5*(Ath + Ath')\n X = eigen(H, sortby = x -> real(x))\n V = X.vectors\n z[i + 1] = rq(A, V[:, 1])\n z[i + 1 + thmax] = rq(A, V[:, ns])\n end\n f = [f; z]\n end\n # join up the boundary\n f = [f; f[1, :]]\n end\n if thmax == 0\n f = e\n end\n\n if !noplot\n p = plot(real(f), imag(f), framestyle=:origin, aspect_ratio = 1, legend=false)\n scatter!(p, real(e), imag(e), markershape=:x, legend=false)\n display(p)\n end\n\n return f, e\nend\n\nend\n", "meta": {"hexsha": "47e456b3c97c5e846e1395b1eed0d700e931cbd2", "size": 2905, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/NumericalRange.jl", "max_stars_repo_name": "ThomasChaffey/NumericalRange", "max_stars_repo_head_hexsha": "101186af417913e75974b810adeeb59bfba03c00", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-12-15T18:32:29.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-15T18:32:29.000Z", "max_issues_repo_path": "src/NumericalRange.jl", "max_issues_repo_name": "ThomasChaffey/NumericalRange.jl", "max_issues_repo_head_hexsha": "101186af417913e75974b810adeeb59bfba03c00", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-12-15T11:46:13.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-18T13:03:02.000Z", "max_forks_repo_path": "src/NumericalRange.jl", "max_forks_repo_name": "ThomasChaffey/NumericalRange.jl", "max_forks_repo_head_hexsha": "101186af417913e75974b810adeeb59bfba03c00", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.6404494382, "max_line_length": 102, "alphanum_fraction": 0.4960413081, "num_tokens": 777, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418158002492, "lm_q2_score": 0.8791467627598857, "lm_q1q2_score": 0.8124562856918317}} {"text": "using LightGraphs, Distributions, StatsBase, Random, Plots, LaTeXStrings;pyplot()\nRandom.seed!(0)\n\nfunction createNetwork(edges)\n network = Graph(maximum(maximum.(edges)))\n for e in edges\n add_edge!(network, e[1], e[2])\n end\n network\nend\n\nfunction uniformRandomEdge(network)\n outDegrees = length.(network.fadjlist)\n randI = sample(1:length(outDegrees),Weights(outDegrees))\n randJ = rand(network.fadjlist[randI])\n randI, randJ\nend\n\nfunction networkLife(network,source,dest,lambda)\n failureNetwork = copy(network)\n t = 0\n while has_path(failureNetwork, source, dest)\n t += rand(Exponential(1/(failureNetwork.ne*lambda)))\n i, j = uniformRandomEdge(failureNetwork)\n rem_edge!(failureNetwork, i, j)\n end\n t\nend\n\nlambda1, lambda2 = 0.5, 1.0\nroads = [(1,2), (1,3), (2,4), (2,5), (2,3), (3,4), (3,5), (4,5), (4,6), (5,6)]\nsource, dest = 1, 6\nnetwork = createNetwork(roads)\nN = 10^6\n\nfailTimes1 = [ networkLife(network,source,dest,lambda1) for _ in 1:N ]\nfailTimes2 = [ networkLife(network,source,dest,lambda2) for _ in 1:N ]\n\nprintln(\"Edge Failure Rate = $(lambda1): Mean failure time = \",\n\tmean(failTimes1), \" days.\")\nprintln(\"Edge Failure Rate = $(lambda2): Mean failure time = \",\n\tmean(failTimes2), \" days.\")\n\nstephist(failTimes1, bins=200, c=:blue, normed=true, label=L\"\\lambda=0.5\")\nstephist!(failTimes2, bins=200, c=:red, normed=true, label=L\"\\lambda=1.0\", \n xlims=(0,5), ylims=(0,1.1), xlabel=\"Network Life Time\", ylabel = \"Density\")", "meta": {"hexsha": "d5627b2d198906e436095fcdadf7ce93984a2ff3", "size": 1502, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "10_chapter/networkLifeTime.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "10_chapter/networkLifeTime.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "10_chapter/networkLifeTime.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 32.652173913, "max_line_length": 81, "alphanum_fraction": 0.6704394141, "num_tokens": 480, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418178895029, "lm_q2_score": 0.8791467548438124, "lm_q1q2_score": 0.812456280213018}} {"text": "### A Pluto.jl notebook ###\n# v0.12.20\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ b9f84364-6b75-11eb-183f-9358dcc0e4d0\nbegin\n\tusing Test\n\tusing LinearAlgebra\nend\n\n# ╔═╡ 9c44adc6-6b7c-11eb-20b5-11ad83fee84b\nArrayFloatOrIntDim2 = Array{<:Union{Float64, Int64}, 2}\n\n# ╔═╡ 95a22bc2-6b77-11eb-2cd5-c50658a942b9\nmd\"\"\"\n## Gram - Schmidt\n\"\"\"\n\n# ╔═╡ 3d8a1668-6b70-11eb-293c-ab5ce07a3ce5\nbegin\n\tfunction GramSchmidt(A::ArrayFloatOrIntDim2)\n\t\tm, n = size(A)\n\t\tQ = zeros(m, n)\n\t\tR = zeros(n, n)\n\t\tfor j = 1:n\n\t\t\tv = A[:,j]\n\t\t\tfor i = 1:j-1\n\t\t\t\tR[i, j] = Q[:, i]' * v\n\t\t\t\tv = v - R[i, j] * Q[:, i]\n\t\t\tend\n\t\t\tR[j, j] = norm(v)\n\t\t\tQ[:,j] = v / R[j,j]\n\t\tend\n\t\treturn (Q, R)\n\tend\nend\n\n# ╔═╡ 7356cb64-6b71-11eb-01f7-91f369a9fb74\nbegin\n\tA = [\n\t\t1 0\n\t\t1 1\n\t\t1 3\n\t\t1 4\n\t]\n\tQ, R = GramSchmidt(A)\n\t\n\t@test A == round.(Int, Q * R)\nend\n\n# ╔═╡ 78c7863c-6b7c-11eb-0be7-cbfb3c368065\nmd\"\"\"\n## Householder (probably wrong)\n\"\"\"\n\n# ╔═╡ 83f357d4-6b7c-11eb-11b6-5ddf77041fe5\nbegin\n\tfunction householder(A::ArrayFloatOrIntDim2)\n\t\tA = convert(Array{Float64}, A)\n\t\tm, n = size(A)\n\t\tU = zeros(m, n)\n\t\tfor k = 1:n\n\t\t\tw = A[k:m, k]\n\t\t\twₙ = norm(w)\n\t\t\tw[1] = (w[1] - wₙ)\n\t\t\tu = w / wₙ\n\t\t\tU[k:m, k] = u\n\t\t\tA[k:m, k:n] = A[k:m, k:n] - 2 * u * (u' * A[k:m, k:n])\n\t\tend\n\t\tR = triu(A[:, 1:n])\n\t\treturn U, R\n\tend\nend\n\n# ╔═╡ 43133fa8-6b7d-11eb-331e-73d6eabd7d83\nbegin\n\tU, Rₕ = householder(A)\nend\n\n# ╔═╡ Cell order:\n# ╠═b9f84364-6b75-11eb-183f-9358dcc0e4d0\n# ╠═9c44adc6-6b7c-11eb-20b5-11ad83fee84b\n# ╟─95a22bc2-6b77-11eb-2cd5-c50658a942b9\n# ╠═3d8a1668-6b70-11eb-293c-ab5ce07a3ce5\n# ╠═7356cb64-6b71-11eb-01f7-91f369a9fb74\n# ╟─78c7863c-6b7c-11eb-0be7-cbfb3c368065\n# ╠═83f357d4-6b7c-11eb-11b6-5ddf77041fe5\n# ╠═43133fa8-6b7d-11eb-331e-73d6eabd7d83\n", "meta": {"hexsha": "ddb466026217be6d21e2cd2a4f078f0a6a3df9b3", "size": 1698, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "2.FrameworkForAppliedMath/3.LeastSquaresForRectangularMatrix/LectureNotes.jl", "max_stars_repo_name": "nickovchinnikov/Computational-Science-and-Engineering", "max_stars_repo_head_hexsha": "45620e432c97fce68a24e2ade9210d30b341d2e4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2021-01-14T08:00:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-31T14:00:11.000Z", "max_issues_repo_path": "2.FrameworkForAppliedMath/3.LeastSquaresForRectangularMatrix/LectureNotes.jl", "max_issues_repo_name": "nickovchinnikov/Computational-Science-and-Engineering", "max_issues_repo_head_hexsha": "45620e432c97fce68a24e2ade9210d30b341d2e4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2.FrameworkForAppliedMath/3.LeastSquaresForRectangularMatrix/LectureNotes.jl", "max_forks_repo_name": "nickovchinnikov/Computational-Science-and-Engineering", "max_forks_repo_head_hexsha": "45620e432c97fce68a24e2ade9210d30b341d2e4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-25T15:21:40.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-25T15:21:40.000Z", "avg_line_length": 18.6593406593, "max_line_length": 57, "alphanum_fraction": 0.617196702, "num_tokens": 925, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.960361158630024, "lm_q2_score": 0.8459424314825853, "lm_q1q2_score": 0.8124102536329153}} {"text": "\"\"\"\n copy_eigenvalues!(λs, A) → λs\n\nPuts the eigenvalues of a quasi-upper triangular matrix A in the λs vector.\n\"\"\"\nfunction copy_eigenvalues!(λs, A::AbstractMatrix{T}, range = OneTo(size(A, 2)), tol = eps(real(T))) where {T}\n i = first(range)\n\n @inbounds while i < last(range)\n if is_offdiagonal_small(A, i, tol)\n λs[i] = A[i, i]\n i += 1\n else\n # Conjugate pair\n d = A[i,i] * A[i+1,i+1] - A[i,i+1] * A[i+1,i]\n x = (A[i,i] + A[i+1,i+1]) / 2\n y = sqrt(complex(x*x - d))\n λs[i + 0] = x + y\n λs[i + 1] = x - y\n i += 2\n end\n end\n\n @inbounds if i == last(range)\n λs[i] = A[i, i] \n end\n\n return λs\nend\n\n\"\"\"\n eigenvalue(R, i) → λ\n \nGet the `i`th eigenvalue of R. NOTE: assumes `i` points to the start of\na block.\n\"\"\"\nfunction eigenvalue(R, i)\n n = minimum(size(R))\n\n @inbounds begin\n if i == n || iszero(R[i+1,i])\n return complex(R[i,i])\n else\n d = R[i,i] * R[i+1,i+1] - R[i,i+1] * R[i+1,i]\n x = (R[i,i] + R[i+1,i+1]) / 2\n y = sqrt(complex(x*x - d))\n return x + y\n end\n end\nend\n\n\"\"\"\n eigenvalues(A::AbstractMatrix{T}) → Vector{complex(T)}\n\nComputes the eigenvalues of the matrix A. Assumes that A is quasi-upper triangular.\nThe eigenvalues are returned in complex arithmetic, even if their imaginary\npart is 0.\n\"\"\"\neigenvalues(A::AbstractMatrix{T}, tol = eps(real(T))) where {T} =\n copy_eigenvalues!(Vector{complex(T)}(undef, size(A, 2)), A, OneTo(size(A, 2)), tol)\n\n\"\"\"\n partialeigen(P::PartialSchur) → (Vector{<:Union{Real,Complex}}, Matrix{<:Union{Real,Complex}})\n\nTransforms a partial Schur decomposition into an eigendecomposition.\n\n!!! note\n\n For real-symmetric and Hermitian matrices the Schur vectors coincide with \n the eigenvectors, and hence it is not necessary to call this function in \n that case.\n\nThe method still relies on LAPACK to compute the eigenvectors of the (quasi)\nupper triangular matrix `R` from the partial Schur decomposition.\n\n!!! note\n\n This method is currently type unstable for real matrices, since we have not\n yet decided how to deal with complex conjugate pairs of eigenvalues. E.g.\n if almost all eigenvalues are real, but there are just a few conjugate \n pairs, should all eigenvectors be complex-valued?\n\"\"\"\nfunction partialeigen(P::PartialSchur)\n vals, vecs = eigen(P.R)\n return vals, P.Q*vecs\nend\n", "meta": {"hexsha": "a16d6b4b20ec8aba4a9b55e561ecf1cfc6244fa6", "size": 2505, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/eigvals.jl", "max_stars_repo_name": "andreasnoack/ArnoldiMethod.jl", "max_stars_repo_head_hexsha": "d44bac4022e5e39f9fa80f6398091c41180ec5a8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 41, "max_stars_repo_stars_event_min_datetime": "2018-08-28T14:04:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-29T02:18:47.000Z", "max_issues_repo_path": "src/eigvals.jl", "max_issues_repo_name": "andreasnoack/ArnoldiMethod.jl", "max_issues_repo_head_hexsha": "d44bac4022e5e39f9fa80f6398091c41180ec5a8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 37, "max_issues_repo_issues_event_min_datetime": "2018-05-21T12:53:55.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-27T08:58:38.000Z", "max_forks_repo_path": "src/eigvals.jl", "max_forks_repo_name": "andreasnoack/ArnoldiMethod.jl", "max_forks_repo_head_hexsha": "d44bac4022e5e39f9fa80f6398091c41180ec5a8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2018-09-10T19:35:02.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-03T11:48:46.000Z", "avg_line_length": 28.7931034483, "max_line_length": 109, "alphanum_fraction": 0.6003992016, "num_tokens": 744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9294404018582426, "lm_q2_score": 0.874077230244524, "lm_q1q2_score": 0.81240269213361}} {"text": "# Description:\n# Dimensions: d\n#\n# The Michalewicz function has d! local minima, and it is multimodal. The parameter m defines the steepness of they valleys and ridges; a larger m leads to a more difficult search. The recommended value of m is m = 10. The function's two-dimensional form is shown in the plot above.\n#\n# Input Domain:\n# The function is usually evaluated on the hypercube xi ∈ [0, π], for all i = 1, …, d.\n#\n# Global Minima:\n# https://www.sfu.ca/~ssurjano/michal3.png\n\nfunction michal(xx, m = 10)\n\n d = length(xx);\n sum = 0;\n\n for ii = 1:d\n \txi = xx[ii];\n \tnew = sin(xi) * (sin(ii * xi^2 / pi))^(2 * m);\n \tsum = sum + new;\n end\n\n y = -sum;\n\n return [y]\nend\n", "meta": {"hexsha": "3f62f3208c6b22c99b9a8e489bf46976e5b35c62", "size": 709, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "michalewicz.jl", "max_stars_repo_name": "cagataycali/bacterial-foraging-optimization-algorithm", "max_stars_repo_head_hexsha": "b84ac0667707e36dbaa4a50457074245b174b491", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "michalewicz.jl", "max_issues_repo_name": "cagataycali/bacterial-foraging-optimization-algorithm", "max_issues_repo_head_hexsha": "b84ac0667707e36dbaa4a50457074245b174b491", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "michalewicz.jl", "max_forks_repo_name": "cagataycali/bacterial-foraging-optimization-algorithm", "max_forks_repo_head_hexsha": "b84ac0667707e36dbaa4a50457074245b174b491", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.2592592593, "max_line_length": 282, "alphanum_fraction": 0.6262341326, "num_tokens": 217, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9626731105140615, "lm_q2_score": 0.8438951104066293, "lm_q1q2_score": 0.8123951308827572}} {"text": "# Load Julia packages (libraries) needed\n\nusing StatisticalRethinking\n#gr(size=(600,600));\n\n# ### snippet 2.3\n\n# Define a grid\n\nN = 20\np_grid = range( 0 , stop=1 , length=N )\n\n# Define the (uniform) prior\n\nprior = ones( 20 );\n\n# Compute likelihood at each value in grid\n\nlikelihood = [pdf(Binomial(9, p), 6) for p in p_grid]\nlikelihood[1:5]\n\n# Compute product of likelihood and prior\n\nunstd_posterior = likelihood .* prior;\n\n# Standardize the posterior, so it sums to 1\n\nposterior = unstd_posterior ./ sum(unstd_posterior);\n\n# ### snippet 2.4\n\np1 = plot( p_grid , posterior ,\n xlab=\"probability of water\" , ylab=\"posterior probability\",\n lab = \"interpolated\", title=\"20 points\" )\np2 = scatter!( p1, p_grid , posterior, lab=\"computed\" )\n\n# ### snippet 2.5\n\nprior1 = [p < 0.5 ? 0 : 1 for p in p_grid]\nprior2 = [exp( -5*abs( p - 0.5 ) ) for p in p_grid]\n\np3 = plot(p_grid, prior1, \n xlab=\"probability of water\" , ylab=\"posterior probability\",\n lab = \"semi_uniform\", title=\"Other priors\" )\nscatter!(p3, p_grid, prior1, lab = \"semi_uniform grid point\")\nplot!(p3, p_grid, prior2, lab = \"double_exponential\" )\nscatter!(p3, p_grid, prior2, lab = \"double_exponential grid point\" )\n\n# End of `02/clip-03-05.jl`\n", "meta": {"hexsha": "fc092a506d548573d9fa585994e3a2bd897df3d1", "size": 1212, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "scripts/02/clip-03-05.jl", "max_stars_repo_name": "UnofficialJuliaMirror/StatisticalRethinking.jl-2d09df54-9d0f-5258-8220-54c2a3d4fbee", "max_stars_repo_head_hexsha": "08ee7b4244edcb2c94f4410829372e7d5082cb7a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "scripts/02/clip-03-05.jl", "max_issues_repo_name": "UnofficialJuliaMirror/StatisticalRethinking.jl-2d09df54-9d0f-5258-8220-54c2a3d4fbee", "max_issues_repo_head_hexsha": "08ee7b4244edcb2c94f4410829372e7d5082cb7a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/02/clip-03-05.jl", "max_forks_repo_name": "UnofficialJuliaMirror/StatisticalRethinking.jl-2d09df54-9d0f-5258-8220-54c2a3d4fbee", "max_forks_repo_head_hexsha": "08ee7b4244edcb2c94f4410829372e7d5082cb7a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.24, "max_line_length": 69, "alphanum_fraction": 0.6757425743, "num_tokens": 389, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026618464796, "lm_q2_score": 0.8856314692902446, "lm_q1q2_score": 0.8123921041949501}} {"text": "using LinearAlgebra\nusing Plots\nusing Random\nusing BenchmarkTools\nusing JuMP\n\nfunction kernelf(X1, X2, l=1.0, sigma_f=1.0)\n\n # Isotropic squared exponential kernel. Computes\n # a covariance matrix from points in X1 and X2.\n\n #Σ = zeros(size(X1,1), size(X2,1))\n Σ = Array{Float64,2}(undef, size(X1,1), size(X2,1))\n for i in 1:size(Σ,1)\n for j in 1:size(Σ,2)\n Σ[i,j] = sigma_f^2*exp(-(X1[i]-X2[j])^2/(2*l^2))\n end\n end\n\n return Σ\n\nend\n\nfunction plot_gp(μ, Σ, X, X_train=nothing, Y_train=nothing)\n σ = 1.96*sqrt.(diag(Σ))\n p = plot([X X], [μ μ], fillrange=[μ+σ μ-σ], fillalpha=0.3, c=:blue, label=\"\")\n plot!(X, μ, c=:blue, label=\"GP mean\")\n if X_train!=nothing && Y_train!=nothing\n plot!(X_train, Y_train, seriestype=\"scatter\", color=\"red\", label=\"samples\")\n end\n display(p)\nend\n\nfunction kernelTest()\n X1 = collect(range(-5, step=1, 5))\n X2 = collect(range(-5, step=1, 5)) .+ 0.1.*randn(size(X1))\n plot(X1, label=\"original data\")\n plot!(X2, label=\"corrupted data\")\n μ = zeros(size(X1))\n Σ = kernelf(X1, X1)\n plot_gp(μ, Σ, X1)\nend\n\nfunction posteriorPredictor(X_s, X_train, Y_train, l=1.0, sigma_f=1.0, sigma_y=1e-8, K_inv=nothing)\n K_s = kernelf(X_train, X_s, l, sigma_f)\n K_ss = kernelf(X_s, X_s, l, sigma_f) + 1e-8*I\n if K_inv == nothing\n K = kernelf(X_train, X_train, l, sigma_f) + sigma_y^2*I\n K_inv = inv(K)\n end\n\n μ_s = K_s'*K_inv*Y_train\n #mu_s = K_s.T.dot(K_inv).dot(Y_train)\n Σ_s = K_ss - K_s'*K_inv*K_s\n #cov_s = K_ss - K_s.T.dot(K_inv).dot(K_s)\n\n return μ_s, Σ_s, K_inv\nend\n\nfunction testPredictor()\n lb = 0;\n ub = 5;\n n = 5;\n sigma_y = 0.0\n X_train = [-4 -3 -2 -1 1 4]'\n X_train = rand(n) .*(ub-lb) .+lb\n Y_train = sin.(X_train) + sigma_y*randn(size(X_train,1))\n X = collect(range(lb, ub, length=n))\n μ_s, Σ_s = posteriorPredictor(X, X_train, Y_train, 1.0, 1.0, sigma_y)\n plot_gp(μ_s, Σ_s, X, X_train, Y_train)\n plot!(X, sin.(X), lw=:2, color=:black, label=\"true function\")\nend\n\nfunction hyperParameterOptimization(X_train, Y_train, noise, naive=true)\n model = Model()\n @variable(theta[1:2] >= 1e-5) #hyperparam variables\n if naive\n end\nend\n\n\nfunction f(x)\n return 8*sin.(x) + 1/2*x.^2 + 1/8*x.^3 + 1/32*x.^4 - 1/128*x.^5\nend\n\n#kernelTest()\n#testPredictor()\n\nlb = 1;\nub = 7;\nn = 100;\nl = 1.0;\nsigma_f = 1.0\nsigma_y = 0.9\n#X_train = [-4 -3 -2 -1 1 4]'\n#X_train = rand(n).*(ub-lb) .+ lb\nX_train = 1.1*randn(n) .+ (ub - lb)/2 .+ lb\nY_train = f(X_train) + sigma_y*randn(size(X_train,1))\nX = collect(range(lb, ub, length=n))\nif sigma_y != 0.0\n μ_s, Σ_s, K_inv = posteriorPredictor(X, X_train, Y_train, l, sigma_f, sigma_y)\nelse\n μ_s, Σ_s, K_inv = posteriorPredictor(X, X_train, Y_train, l, sigma_f)\nend\nprintln(\"Timing training + inference\")\n@time posteriorPredictor(X, X_train, Y_train, 1.0, 1.0, sigma_y)\nprintln(\"Timing inference\")\n@time posteriorPredictor(X, X_train, Y_train, 1.0, 1.0, sigma_y, K_inv)\nplot_gp(μ_s, Σ_s, X, X_train, Y_train)\nplot!(X, f(X), lw=:2, color=:black, label=\"true function\", xlim=(lb,ub))\n", "meta": {"hexsha": "8dc0256a804f27f8e41aed8e6498ca6bef51d435", "size": 3111, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "my_gp_implementation.jl", "max_stars_repo_name": "gbarsih/Safe-Optimal-Rendezvous", "max_stars_repo_head_hexsha": "6acc9f11c092c81dbe19daeb0ff7363f7ab5bd5f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "my_gp_implementation.jl", "max_issues_repo_name": "gbarsih/Safe-Optimal-Rendezvous", "max_issues_repo_head_hexsha": "6acc9f11c092c81dbe19daeb0ff7363f7ab5bd5f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "my_gp_implementation.jl", "max_forks_repo_name": "gbarsih/Safe-Optimal-Rendezvous", "max_forks_repo_head_hexsha": "6acc9f11c092c81dbe19daeb0ff7363f7ab5bd5f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.027027027, "max_line_length": 99, "alphanum_fraction": 0.6194149791, "num_tokens": 1202, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026573249612, "lm_q2_score": 0.8856314662716159, "lm_q1q2_score": 0.812392097421555}} {"text": "# Integration using a function\r\nfunction trapz(f, a, b, h)\r\n x = collect(Float64, a:h:b);\r\n suma = 0\r\n for j ∈ 2:(length(x) - 1)\r\n val = f(x[j])\r\n suma += val\r\n end\r\n I = (b - a) * (f(x[1]) + 2 * suma + f(x[end])) / (2 * (length(x) - 1));\r\n return I\r\nend\r\n\r\n\r\n# Integration of a table\r\nfunction trapz(x::Array{Float64,1}, y::Array{Float64,1})\r\n suma = 0;\r\n for j ∈ 2:(length(x) - 1)\r\n suma += y[j]\r\n end\r\n I = (x[end] - x[1]) * (y[1] + 2 * suma + y[end]) / (2 * (length(x) - 1));\r\n return I\r\nend\r\n\r\n\r\n\r\n\r\n\r\n\r\n", "meta": {"hexsha": "bcd2985c8505d64def1d5d951b3ba7639ebb2433", "size": 563, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/trapz.jl", "max_stars_repo_name": "leogabac/FiscomTools.jl", "max_stars_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/trapz.jl", "max_issues_repo_name": "leogabac/FiscomTools.jl", "max_issues_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/trapz.jl", "max_forks_repo_name": "leogabac/FiscomTools.jl", "max_forks_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.4137931034, "max_line_length": 78, "alphanum_fraction": 0.4493783304, "num_tokens": 208, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026482819238, "lm_q2_score": 0.8856314738181875, "lm_q1q2_score": 0.8123920963352467}} {"text": "# some simpler (and faster) implementations for root finding\n#\n# Not exported\n#\n# These avoid the setup costs of the `find_zero` method, so should be faster\n# though they will take similar number of function calls.\n#\n# `Roots.bisection(f, a, b)` (Bisection).\n# `Roots.secant_method(f, xs)` (Order1) secant method\n# `Roots.dfree(f, xs)` (Order0) more robust secant method\n#\n\n## Bisection\n##\n## Essentially from Jason Merrill https://gist.github.com/jwmerrill/9012954\n## cf. http://squishythinking.com/2014/02/22/bisecting-floats/\n## This also borrows a trick from https://discourse.julialang.org/t/simple-and-fast-bisection/14886\n## where we keep x1 so that y1 is negative, and x2 so that y2 is positive\n## this allows the use of signbit over y1*y2 < 0 which avoid < and a multiplication\n## this has a small, but noticeable impact on performance.\n\"\"\"\n bisection(f, a, b; [xatol, xrtol])\n\nPerforms bisection method to find a zero of a continuous\nfunction.\n\nIt is assumed that (a,b) is a bracket, that is, the function has\ndifferent signs at a and b. The interval (a,b) is converted to floating point\nand shrunk when a or b is infinite. The function f may be infinite for\nthe typical case. If f is not continuous, the algorithm may find\njumping points over the x axis, not just zeros.\n\n\nIf non-trivial tolerances are specified, the process will terminate\nwhen the bracket (a,b) satisfies `isapprox(a, b, atol=xatol,\nrtol=xrtol)`. For zero tolerances, the default, for Float64, Float32,\nor Float16 values, the process will terminate at a value `x` with\n`f(x)=0` or `f(x)*f(prevfloat(x)) < 0 ` or `f(x) * f(nextfloat(x)) <\n0`. For other number types, the A42 method is used.\n\n\"\"\"\nfunction bisection(f, a::Number, b::Number; xatol=nothing, xrtol=nothing)\n\n x1, x2 = adjust_bracket(float.((a,b)))\n T = eltype(x1)\n\n\n atol = xatol == nothing ? zero(T) : abs(xatol)\n rtol = xrtol == nothing ? zero(one(T)) : abs(xrtol)\n CT = iszero(atol) && iszero(rtol) ? Val(:exact) : Val(:inexact)\n\n x1, x2 = float(x1), float(x2)\n y1, y2 = f(x1), f(x2)\n\n _unitless(y1 * y2) >= 0 && error(\"the interval provided does not bracket a root\")\n\n if isneg(y2)\n x1, x2, y1, y2 = x2, x1, y2, y1\n end\n\n xm = Roots._middle(x1, x2) # for possibly mixed sign x1, x2\n ym = f(xm)\n\n while true\n\n if has_converged(CT, x1, x2, xm, ym, atol, rtol)\n return xm\n end\n\n if isneg(ym)\n x1, y1 = xm, ym\n else\n x2, y2 = xm, ym\n end\n\n xm = Roots.__middle(x1,x2)\n ym = f(xm)\n\n\n end\n\nend\n\n# -0.0 not returned by __middle, so isneg true on [-Inf, 0.0)\n@inline isneg(x::T) where {T <: AbstractFloat} = signbit(x)\n@inline isneg(x) = _unitless(x) < 0\n\n@inline function has_converged(::Val{:exact}, x1, x2, m, ym, atol, rtol)\n iszero(ym) && return true\n isnan(ym) && return true\n x1 != m && m != x2 && return false\n return true\nend\n\n@inline function has_converged(::Val{:inexact}, x1, x2, m, ym, atol, rtol)\n iszero(ym) && return true\n isnan(ym) && return true\n val = abs(x1 - x2) <= atol + max(abs(x1), abs(x2)) * rtol\n return val\nend\n\n\n\"\"\"\n secant_method(f, xs; [atol=0.0, rtol=8eps(), maxevals=1000])\n\nPerform secant method to solve f(x) = 0.\n\nThe secant method is an iterative method with update step\ngiven by b - fb/m where m is the slope of the secant line between\n(a,fa) and (b,fb).\n\nThe inital values can be specified as a pair of 2, as in `(a,b)` or\n`[a,b]`, or as a single value, in which case a value of `b` is chosen.\n\nThe algorithm returns m when `abs(fm) <= max(atol, abs(m) * rtol)`.\nIf this doesn't occur before `maxevals` steps or the algorithm\nencounters an issue, a value of `NaN` is returned. If too many steps\nare taken, the current value is checked to see if there is a sign\nchange for neighboring floating point values.\n\nThe `Order1` method for `find_zero` also implements the secant\nmethod. This one will be faster, as there are fewer setup costs.\n\nExamples:\n\n```julia\nRoots.secant_method(sin, (3,4))\nRoots.secant_method(x -> x^5 -x - 1, 1.1)\n```\n\nNote:\n\nThis function will specialize on the function `f`, so that the inital\ncall can take more time than a call to the `Order1()` method, though\nsubsequent calls will be much faster. Using `FunctionWrappers.jl` can\nensure that the initial call is also equally as fast as subsequent\nones.\n\n\"\"\"\nfunction secant_method(f, xs; atol=zero(float(real(first(xs)))), rtol=8eps(one(float(real(first(xs))))), maxevals=100)\n\n if length(xs) == 1 # secant needs x0, x1; only x0 given\n a = float(xs[1])\n\n h = eps(one(real(a)))^(1/3)\n da = h*oneunit(a) + abs(a)*h^2 # adjust for if eps(a) > h\n b = a + da\n\n else\n a, b = promote(float(xs[1]), float(xs[2]))\n end\n secant(f, a, b, atol, rtol, maxevals)\nend\n\n\nfunction secant(f, a::T, b::T, atol=zero(T), rtol=8eps(T), maxevals=100) where {T}\n nan = (0a)/(0a)\n cnt = 0\n\n fa, fb = f(a), f(b)\n fb == fa && return nan\n\n uatol = atol / oneunit(atol) * oneunit(real(a))\n adjustunit = oneunit(real(fb))/oneunit(real(b))\n\n while cnt < maxevals\n m = b - (b-a)*fb/(fb - fa)\n fm = f(m)\n\n iszero(fm) && return m\n isnan(fm) && return nan\n abs(fm) <= adjustunit * max(uatol, abs(m) * rtol) && return m\n if fm == fb\n sign(fm) * sign(f(nextfloat(m))) <= 0 && return m\n sign(fm) * sign(f(prevfloat(m))) <= 0 && return m\n return nan\n end\n\n a,b,fa,fb = b,m,fb,fm\n\n cnt += 1\n end\n\n return nan\nend\n\n\"\"\"\n muller(f, xᵢ; xatol=nothing, xrtol=nothing, maxevals=100)\n muller(f, xᵢ₋₂, xᵢ₋₁, xᵢ; xatol=nothing, xrtol=nothing, maxevals=100)\n\n> *Muller’s method* generalizes the secant method, but uses quadratic\n> interpolation among three points instead of linear interpolation between two.\n> Solving for the zeros of the quadratic allows the method to find complex\n> pairs of roots.\n> Given three previous guesses for the root `xᵢ₋₂`, `xᵢ₋₁`, `xᵢ`, and the values\n> of the polynomial `f` at those points, the next approximation `xᵢ₊₁` is produced.\n\nExcerpt and the algorithm taken from\n\n> W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery\n> *Numerical Recipes in C*, Cambridge University Press (2002), p. 371\n\nConvergence here is decided by `xᵢ₊₁ ≈ xᵢ` using the tolerances specified,\nwhich both default to `eps(one(typeof(abs(xᵢ))))^4/5` in the appropriate units.\nEach iteration performs three evaluations of `f`.\nThe first method picks two remaining points at random in relative proximity of `xᵢ`.\n\nNote that the method may return complex result even for real intial values\nas this depends on the function.\n\nExamples:\n```\nmuller(x->x^3-1, 0.5, 0.5im, -0.5) # → -0.500 + 0.866…im\nmuller(x->x^2+2, 0.0, 0.5, 1.0) # → ≈ 0.00 - 1.41…im\nmuller(x->(x-5)*x*(x+5), rand(3)...) # → ≈ 0.00\nmuller(x->x^3-1, 1.5, 1.0, 2.0) # → 2.0, Not converged\n```\n\n\"\"\"\nmuller(f, x₀::T; kwargs...) where T = muller(f, (rand(T, 2).*x₀)..., x₀; kwargs...)\n\nmuller(f, xᵢ₋₂, xᵢ₋₁, xᵢ; kwargs...) =\n muller(f, promote(xᵢ₋₂, xᵢ₋₁, xᵢ)...; kwargs...)\n\nfunction muller(f, oldest::T, older::T, old::T;\n xatol=nothing, xrtol=nothing, maxevals=300) where {T}\n @assert old ≠ older ≠ oldest ≠ old # we want q to be non-degenerate\n xᵢ₋₂, xᵢ₋₁, xᵢ = oldest, older, old\n fxᵢ₋₂, fxᵢ₋₁ = f(xᵢ₋₂), f(xᵢ₋₁) \n\n RT = typeof(abs(oldest))\n atol = xatol !== nothing ? xatol : oneunit(RT) * (eps(one(RT)))^(4/5)\n rtol = xrtol !== nothing ? xrtol : eps(one(RT))^(4/5)\n\n for i in 1:maxevals÷3\n # one evaluation per iteration\n fxᵢ = f(xᵢ)\n x = muller_step(xᵢ₋₂, xᵢ₋₁, xᵢ, fxᵢ₋₂, fxᵢ₋₁, fxᵢ)\n\n if isnan(x)\n @warn \"The algorithm might not have converged, stopping at i=$i:\" abs(xᵢ - xᵢ₋₁)\n return xᵢ\n end\n\n # @debug \"Iteration $i:\" xᵢ₋₂ xᵢ₋₁ xᵢ x abs(x-xᵢ)\n xᵢ₋₂, xᵢ₋₁, xᵢ = xᵢ₋₁, xᵢ, x\n fxᵢ₋₂, fxᵢ₋₁ = fxᵢ₋₁, fxᵢ\n #stopping criterion\n isapprox(xᵢ, xᵢ₋₁, atol=atol, rtol=rtol) && return xᵢ\n end\n @warn \"The algorithm might not have converged, maxevals limit hit:\" abs(xᵢ₋₁- xᵢ)\n return xᵢ\nend\n\nfunction muller_step(a,b,c,fa,fb,fc)\n a,b,c = promote(a,b,c)\n q = qq(a, b, c)\n q² = q^2\n q1 = q+one(q)\n\n A = q*fc - q*q1*fb + q²*fa\n B = (q1+q)*fc - q1^2*fb + q²*fa\n C = q1*fc\n\n den = let\n Δ = B^2 - 4A*C\n typeof(Δ) <: Real && Δ < 0 && throw(\n DomainError(Δ, \"Discriminant is negative and the function most likely has complex roots. You might want to call muller with complex input.\"))\n Δ = √Δ\n d⁺ = B + Δ\n d⁻ = B - Δ\n abs(d⁺) > abs(d⁻) ? d⁺ : d⁻\n end\n return c - (c - b)*2C/den\nend\n\n@inline qq(a, b, c) = (c - b)/(b - a)\n\n\"\"\"\n newton((f, f'), x0; xatol=nothing, xrtol=nothing, maxevals=100)\n newton(fΔf, x0; xatol=nothing, xrtol=nothing, maxevals=100)\n\nNewton's method.\n\nFunction may be passed in as a tuple (f, f') *or* as function which returns (f,f/f').\n\nExamples:\n```\nnewton((sin, cos), 3.0)\nnewton(x -> (sin(x), sin(x)/cos(x)), 3.0, xatol=1e-10, xrtol=1e-10)\n```\n\nNote: unlike the call `newton(f, fp, x0)`--which dispatches to a method of `find_zero`, these\ntwo interfaces will specialize on the function that is passed in. This means, these functions\nwill be faster for subsequent calls, but may be slower for an initial call.\n\nConvergence here is decided by x_n ≈ x_{n-1} using the tolerances specified, which both default to\n`eps(T)^4/5` in the appropriate units.\n\n\"\"\"\nstruct TupleWrapper{F, Fp}\nf::F\nfp::Fp\nend\n(F::TupleWrapper)(x) = begin\n u, v = F.f(x), F.fp(x)\n return (u, u/v)\nend\n\nnewton(f::Tuple, x0; kwargs...) = newton(TupleWrapper(f[1],f[2]), x0; kwargs...)\nfunction newton(f, x0; xatol=nothing, xrtol=nothing, maxevals = 100)\n\n x = float(x0)\n T = typeof(x)\n atol = xatol != nothing ? xatol : oneunit(T) * (eps(one(T)))^(4/5)\n rtol = xrtol != nothing ? xrtol : eps(one(T))^(4/5)\n\n\n xo = Inf\n for i in 1:maxevals\n\n fx, Δx = f(x)\n iszero(fx) && return x\n\n x -= Δx\n\n if isapprox(x, xo, atol=atol, rtol=rtol)\n return x\n end\n\n xo = x\n end\n\n error(\"No convergence\")\nend\n\n\n\n\n## This is basically Order0(), but with different, default, tolerances employed\n## It takes more function calls, but works harder to find exact zeros\n## where exact means either iszero(fx), adjacent floats have sign change, or\n## abs(fxn) <= 8 eps(xn)\n\"\"\"\n dfree(f, xs)\n\nA more robust secant method implementation\n\nSolve for `f(x) = 0` using an alogorithm from *Personal Calculator Has Key\nto Solve Any Equation f(x) = 0*, the SOLVE button from the\n[HP-34C](http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1979-12.pdf).\n\nThis is also implemented as the `Order0` method for `find_zero`.\n\nThe inital values can be specified as a pair of two values, as in\n`(a,b)` or `[a,b]`, or as a single value, in which case a value of `b`\nis computed, possibly from `fb`. The basic idea is to follow the\nsecant method to convergence unless:\n\n* a bracket is found, in which case bisection is used;\n\n* the secant method is not converging, in which case a few steps of a\n quadratic method are used to see if that improves matters.\n\nConvergence occurs when `f(m) == 0`, there is a sign change between\n`m` and an adjacent floating point value, or `f(m) <= 2^3*eps(m)`.\n\nA value of `NaN` is returned if the algorithm takes too many steps\nbefore identifying a zero.\n\n# Examples\n\n```julia\nRoots.dfree(x -> x^5 - x - 1, 1.0)\n```\n\n\"\"\"\nfunction dfree(f, xs)\n\n if length(xs) == 1\n a = float(xs[1])\n fa = f(a)\n\n h = eps(one(a))^(1/3)\n da = h*oneunit(a) + abs(a)*h^2 # adjust for if eps(a) > h\n b = float(a + da)\n fb = f(b)\n else\n a, b = promote(float(xs[1]), float(xs[2]))\n fa, fb = f(a), f(b)\n end\n\n\n nan = (0*a)/(0*a) # try to preserve type\n cnt, MAXCNT = 0, 5 * ceil(Int, -log(eps(one(a)))) # must be higher for BigFloat\n MAXQUAD = 3\n\n if abs(fa) > abs(fb)\n a,fa,b,fb=b,fb,a,fa\n end\n\n # we keep a, b, fa, fb, gamma, fgamma\n quad_ctr = 0\n while !iszero(fb)\n cnt += 1\n\n if sign(fa) * sign(fb) < 0\n return bisection(f, a, b)\n end\n\n # take a secant step\n gamma = float(b - (b-a) * fb / (fb - fa))\n # modify if gamma is too small or too big\n if iszero(abs(gamma-b))\n gamma = b + 1/1000 * abs(b-a) # too small\n elseif abs(gamma-b) >= 100 * abs(b-a)\n gamma = b + sign(gamma-b) * 100 * abs(b-a) ## too big\n end\n fgamma = f(gamma)\n\n # change sign\n if sign(fgamma) * sign(fb) < 0\n return bisection(f, gamma, b)\n end\n\n # decreasing\n if abs(fgamma) < abs(fb)\n a,fa, b,fb = b, fb, gamma, fgamma\n quad_ctr = 0\n cnt < MAXCNT && continue\n end\n\n gamma = float(quad_vertex(a,fa,b,fb,gamma,fgamma))\n fgamma = f(gamma)\n # decreasing now?\n if abs(fgamma) < abs(fb)\n a,fa, b,fb = b, fb, gamma, fgamma\n quad_ctr = 0\n cnt < MAXCNT && continue\n end\n\n\n quad_ctr += 1\n if (quad_ctr > MAXQUAD) || (cnt > MAXCNT) || iszero(gamma - b) || isnan(gamma)\n bprev, bnext = prevfloat(b), nextfloat(b)\n fbprev, fbnext = f(bprev), f(bnext)\n sign(fb) * sign(fbprev) < 0 && return b\n sign(fb) * sign(fbnext) < 0 && return b\n for (u,fu) in ((b,fb), (bprev, fbprev), (bnext, fbnext))\n abs(fu)/oneunit(fu) <= 2^3*eps(u/oneunit(u)) && return u\n end\n return nan # Failed.\n end\n\n if abs(fgamma) < abs(fb)\n b,fb, a,fa = gamma, fgamma, b, fb\n else\n a, fa = gamma, fgamma\n end\n\n end\n b\n\nend\n", "meta": {"hexsha": "18e7803ed599a285e93bd7c29b9996f0421fb8fc", "size": 13846, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/simple.jl", "max_stars_repo_name": "alecloudenback/Roots.jl", "max_stars_repo_head_hexsha": "299e5ead40ad16d112d461ab2a736ccc44eeafd0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/simple.jl", "max_issues_repo_name": "alecloudenback/Roots.jl", "max_issues_repo_head_hexsha": "299e5ead40ad16d112d461ab2a736ccc44eeafd0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/simple.jl", "max_forks_repo_name": "alecloudenback/Roots.jl", "max_forks_repo_head_hexsha": "299e5ead40ad16d112d461ab2a736ccc44eeafd0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.776344086, "max_line_length": 153, "alphanum_fraction": 0.6050122779, "num_tokens": 4776, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361652391386, "lm_q2_score": 0.8902942363098472, "lm_q1q2_score": 0.8123366589130644}} {"text": "\nexport _build, _get_kernel, funcs, r\nusing Statistics\nusing LinearAlgebra\nusing StaticArrays\n#########################################################\n# Radial Basis functions #\n#########################################################\n\nphi_MQ(r, e) = sqrt(1 + (e .* r) .^ 2)\nphi_InvMQ(r, e) = phi_MQ(r, e)^-1\nphi_Q(r, e) = phi_MQ(r, e)^-2\nphi_Gauss(r, e) = exp(-(e .* r)^2)\nphi_Multiquadric(r, e) = -sqrt(1 + r .^ 2)\nphi_InvMultiquadric(r, e) = 1 / (-sqrt(1 + r .^ 2))\nphi_thin_plate_spline(r, e) = r .^ 2 .* log(r + 1e-5)\nphi_cubic(r, e) = (r + 1) .^ -3\nphi_quintic(r, e) = (r + 1) .^ -5\nfuncs = Dict([(\"MQ\", phi_MQ),\n (\"InvMQ\", phi_InvMQ),\n (\"Quadratic\", phi_Q),\n (\"Gaussian\", phi_Gauss),\n (\"minusMQ\", phi_Multiquadric),\n (\"minusInvMQ\", phi_InvMultiquadric),\n (\"Cubic\", phi_cubic),\n (\"Quintic\", phi_quintic),\n (\"ThinPlate\", phi_thin_plate_spline)\n])\n\n\"\"\"\nThe function to retrive the basis for interpolation\n -> Input:\n :basis: String with the name of the function\n being selected. The available options are:\n [\"MQ\" , \"InvMQ\" , \"Quadratic , \"Gaussian\"]\n ->Output:\n :basis function: two float parameters=> e (the epsilopn of the fuction)\n r (Distance form center)\n\n\"\"\"\nfunction _get_kernel(basis::String = \"MQ\"; func_dict = funcs)\n if basis ∉ keys(func_dict)\n throw(DomainError(basis, \"Basis function not implemented\\n If you want your own function set it using φ parameter\"))\n end\n func_dict[basis]\nend\n\n# Distance calculation function \nr(x, xi) = norm(x .- xi)\n\n\"\"\"\nbuild the onterpolation matrix\n\n-> Input:\n :basis: ::Matrix with data to be interpolated\n->Output:\n :Weights vector: Vector containing the weights of the interpolation\n\"\"\"\nfunction _build(X, y; φ = _get_kernel[\"QM\"], ϵ = 1.0)\n A = ones((size(X)[1], size(X)[1]))\n for i = 1:size(X)[1], j = 1:size(X)[1]\n A[i, j] = φ(r(X[i, :], X[j, :]), ϵ)\n end\n sigm_y = std(y)\n if std(y) != 0\n newy = (y .- mean(y)) / std(y)\n w = (pinv(A) * newy)\n else\n w = (pinv(A) * y)\n end\n w\nend", "meta": {"hexsha": "73e6f73a329c84b1e6c31eeaa9d4c6ae73955a71", "size": 2171, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/utils/RBFutils.jl", "max_stars_repo_name": "munozariasjm/RbfInterpolationTools.jl", "max_stars_repo_head_hexsha": "f9889bf24e774761a973db25c9a9540970e5cfbc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-03T19:26:24.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-03T19:26:24.000Z", "max_issues_repo_path": "src/utils/RBFutils.jl", "max_issues_repo_name": "munozariasjm/RbfInterpolationTools.jl", "max_issues_repo_head_hexsha": "f9889bf24e774761a973db25c9a9540970e5cfbc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2022-01-03T01:08:49.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-16T22:40:06.000Z", "max_forks_repo_path": "src/utils/RBFutils.jl", "max_forks_repo_name": "munozariasjm/RbfInterpolationTools.jl", "max_forks_repo_head_hexsha": "f9889bf24e774761a973db25c9a9540970e5cfbc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.1527777778, "max_line_length": 124, "alphanum_fraction": 0.5393827729, "num_tokens": 676, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122720843811, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.8122787165614551}} {"text": "# Methods to provide grids and distance-matrices for\n# different topologies of Kohonen maps (SOMs).\n\n\n\"\"\"\n gridRectangular(xdim, ydim)\n\nCreate coordinates of all neurons on a rectangular SOM.\n\nThe return-value is an array of size (Number-of-neurons, 2) with\nx- and y- coordinates of the neurons in the first and second\ncolumn respectively.\nThe distance between neighbours is 1.0.\nThe point of origin is bottom-left.\nThe first neuron sits at (0,0).\n\"\"\"\nfunction gridRectangular(xdim, ydim)\n\n grid = zeros(Float64, (xdim*ydim, 2))\n for ix in 1:xdim\n for iy in 1:ydim\n\n grid[ix+(iy-1)*xdim, 1] = ix-1\n grid[ix+(iy-1)*xdim, 2] = iy-1\n end\n end\n return grid\nend\n\n\n\n\"\"\"\n gridHexagonal(xdim, ydim)\n\nCreate coordinates of all neurons on a hexagonal SOM.\n\nThe return-value is an array of size (Number-of-neurons, 2) with\nx- and y- coordinates of the neurons in the first and second\ncolumn respectively.\nThe distance between neighbours is 1.0.\nThe point of origin is bottom-left.\nThe first neuron sits at (0,0).\n\"\"\"\nfunction gridHexagonal(xdim, ydim)\n\n grid = zeros(Float64, (xdim*ydim, 2))\n rowHeight = √(1-0.25)\n\n for ix in 1:xdim\n for iy in 1:ydim\n\n if iy % 2 != 0 # 1st, 3rd, ... line\n grid[ix+(iy-1)*xdim, 1] = ix - 1\n else # 2nd, 4th, ... line\n grid[ix+(iy-1)*xdim, 1] = ix - 0.5\n end\n\n grid[ix+(iy-1)*xdim, 2] = (iy-1) * rowHeight\n end\n end\n return grid\nend\n\n\n\"\"\"\n gridSpherical(numNeurons)\n\nCreate coordinates of dim neurons on a spherical SOM.\n\nThe neurons are almost equally distributed on a sphere by\napplying Fibonacci's spirals.\nThe radius of the sphere is adapted to get a distance of neighbours\nof about 1.0.\nThe return-value is an array of size (Number-of-neurons, 3) with\nx-, y- and z- coordinates of the neurons in the first, second and third\ncolumn respectively.\nThe point of origin is bottom-left.\nThe center of the sphere sits at (0,0).\n\"\"\"\nfunction gridSpherical(numNeurons)\n\n r = √(numNeurons/(4 * π))\n grid = zeros(Float64, (numNeurons, 3))\n\n # iterate Fibonacci spiral:\n for i in 1:(numNeurons)\n py = (i-1) * 2/numNeurons - 1 + (1/numNeurons)\n pr = √(1-py^2)\n ϕ = (i-1) * π * (3-√5)\n px = cos(ϕ) * pr\n pz = sin(ϕ) * pr\n\n #println(\"raw: $px $py $pz\")\n\n # shift point from origin to atom location:\n px = r * px\n py = r * py\n pz = r * pz\n\n #println(\"scaled: $px $py $pz\")\n grid[i,:] = [px, py, pz]\n end\n return grid\nend\n\n# verify sphere:\n# x = gridSpherical(50)\n# R\"\"\"\n# library(rgl)\n# plot3d($x)\n# #readline(prompt=\"Press [enter] to continue\")\n# \"\"\"\n\n\n\"\"\"\n distMatrix(grid::Array, toroidal::Bool)\n\nReturn the distance matrix for a non-toroidal or toroidal SOM.\n\n# Arguments\n- `grid`: coordinates of all neurons as generated by\n one of the `grid-`functions with x-coordinates in 1st column\n and y-coordinates in 2nd column.\n- `toroidal`: true for a toroidal SOM.\n\"\"\"\nfunction distMatrix(grid::Array, toroidal::Bool)\n\n X = 1\n Y = 2\n xdim = maximum(grid[:,X]) - minimum(grid[:,X]) + 1.0\n ydim = maximum(grid[:,Y]) - minimum(grid[:,Y]) + 1.0\n\n numNeurons = nrow(grid)\n\n dm = zeros(Float64, (numNeurons,numNeurons))\n for i in 1:numNeurons\n for j in 1:numNeurons\n Δx = abs(grid[i,X] - grid[j,X])\n Δy = abs(grid[i,Y] - grid[j,Y])\n\n if toroidal\n Δx = min(Δx, xdim-Δx)\n Δy = min(Δy, ydim-Δy)\n end\n\n dm[i,j] = √(Δx^2 + Δy^2)\n end\n end\n # show(STDOUT, \"text/plain\", grid)\n # show(STDOUT, \"text/plain\", dm)\n return dm\nend\n\n\n\n\"\"\"\n distMatrixSphere(grid::Array)\n\nReturn the distance matrix for a spherical SOM.\n\n# Arguments\n- `grid`: coordinates of all neurons as gereated by\n one of the `grid-`functions with x-coordinates in 1st,\n y-coordinates in 2nd and z-coordinates in 3rd column.\n Point of origin must be in the center of the spere.\n\"\"\"\nfunction distMatrixSphere(grid::Array)\n\n r = (maximum(grid) - minimum(grid)) / 2\n\n numNeurons = nrow(grid)\n\n dm = zeros(Float64, (numNeurons,numNeurons))\n for i in 1:numNeurons\n for j in 1:numNeurons\n len_i = norm(grid[i,:])\n len_j = norm(grid[j,:])\n bog = grid[i,:] ⋅ grid[j,:] / (len_i * len_j)\n if bog > 1\n bog = 1.0\n end\n α = acos(bog)\n dm[i,j] = r * α\n end\n end\n return dm\nend\n", "meta": {"hexsha": "7ac99912a8847d4f0362f96a44a9cf0355fb8d5b", "size": 4619, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/grids.jl", "max_stars_repo_name": "UnofficialJuliaMirror/SOM.jl-172a5912-0be3-5bce-b68a-ea4f60b69b05", "max_stars_repo_head_hexsha": "51b9073c8e9f896f415b938ea15a65ea3b32558e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2018-10-31T10:17:26.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-09T17:55:05.000Z", "max_issues_repo_path": "src/grids.jl", "max_issues_repo_name": "andreasdominik/SOM.jl", "max_issues_repo_head_hexsha": "a0a85496bb7c15921bfeed636e9ff37ee8a138f0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 18, "max_issues_repo_issues_event_min_datetime": "2018-04-25T06:25:09.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-08T17:14:57.000Z", "max_forks_repo_path": "src/grids.jl", "max_forks_repo_name": "andreasdominik/SOM.jl", "max_forks_repo_head_hexsha": "a0a85496bb7c15921bfeed636e9ff37ee8a138f0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 11, "max_forks_repo_forks_event_min_datetime": "2018-05-07T16:14:30.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-08T10:45:10.000Z", "avg_line_length": 24.5691489362, "max_line_length": 71, "alphanum_fraction": 0.5886555532, "num_tokens": 1409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9407897558991953, "lm_q2_score": 0.8633916222765627, "lm_q1q2_score": 0.8122699935669777}} {"text": "# Locally Linear Embedding (LLE)\n# ------------------------\n# Nonlinear dimensionality reduction by locally linear embedding,\n# Roweis, S. & Saul, L., Science 290:2323 (2000)\n\n#### LLE type\nimmutable LLE{T <: Real} <: SpectralResult\n k::Int\n λ::AbstractVector{T}\n proj::Projection{T}\n\n LLE{T}(k::Int, λ::AbstractVector{T}, proj::Projection{T}) = new(k, λ, proj)\nend\n\n## properties\noutdim(M::LLE) = size(M.proj, 1)\nprojection(M::LLE) = M.proj\n\neigvals(M::LLE) = M.λ\nneighbors(M::LLE) = M.k\n\n## show & dump\nfunction show(io::IO, M::LLE)\n print(io, \"LLE(indim = $(indim(M)), outdim = $(outdim(M)), neighbors = $(neighbors(M)))\")\nend\n\nfunction dump(io::IO, M::LLE)\n show(io, M)\n println(io, \"eigenvalues: \")\n Base.showarray(io, M.λ', header=false, repr=false)\n println(io)\n println(io, \"projection:\")\n Base.showarray(io, M.proj, header=false, repr=false)\nend\n\n## interface functions\nfunction transform{T<:Real}(::Type{LLE}, X::DenseMatrix{T}; d::Int=2, k::Int=12)\n n = size(X, 2)\n\n # Construct NN graph\n D, E = find_nn(X, k)\n\n # Select largest connected component\n cc = components(E)\n c = cc[indmax(map(size, cc))]\n if length(cc) == 1\n c = cc[1]\n else\n c = cc[indmax(map(size, cc))]\n # renumber edges\n R = Dict(c, 1:length(c))\n Ec = zeros(Int,k,length(c))\n for i = 1 : length(c)\n Ec[:,i] = map(i->get(R,i,0), E[:,c[i]])\n end\n E = Ec\n X = X[:,c]\n end\n\n if k > d\n warn(\"K>D: regularization will be used\")\n tol = 1e-5\n else\n tol = 0\n end\n\n # Reconstruct weights\n W = zeros(k, n)\n for i = 1 : n\n Z = X[:, E[:,i]] .- X[:,i]\n C = Z'*Z\n C = C + eye(k, k) * tol * trace(C)\n wi = vec(C\\ones(k, 1))\n W[:, i] = wi./sum(wi)\n end\n\n # Compute embedding\n M = speye(n,n)\n for i = 1 : n\n w = W[:, i]\n jj = E[:, i]\n M[i,jj] = M[i,jj] - w'\n M[jj,i] = M[jj,i] - w\n M[jj,jj] = M[jj,jj] + w*w'\n end\n\n λ, V = decompose(M, d)\n return LLE{T}(k, λ, scale!(V', sqrt(n)))\nend\n", "meta": {"hexsha": "75ebc020b240c9f45973ca14604901ba95fffe77", "size": 2108, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/lle.jl", "max_stars_repo_name": "JuliaPackageMirrors/ManifoldLearning.jl", "max_stars_repo_head_hexsha": "4f994ed07f1f7cd62c5b36efae9db473a5031ac3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/lle.jl", "max_issues_repo_name": "JuliaPackageMirrors/ManifoldLearning.jl", "max_issues_repo_head_hexsha": "4f994ed07f1f7cd62c5b36efae9db473a5031ac3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/lle.jl", "max_forks_repo_name": "JuliaPackageMirrors/ManifoldLearning.jl", "max_forks_repo_head_hexsha": "4f994ed07f1f7cd62c5b36efae9db473a5031ac3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.4222222222, "max_line_length": 93, "alphanum_fraction": 0.5128083491, "num_tokens": 709, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9407897509188344, "lm_q2_score": 0.863391611731321, "lm_q1q2_score": 0.8122699793461204}} {"text": "#=\n A/B test\n\n From http://rpubs.com/rasmusab/exercise_2_bayesian_ab_testing\n \"\"\"\n Exercise 2: Bayesian A/B testing for Swedish Fish Incorporated with Stan\n Rasmus Bååth\n\n Swedish Fish Incorporated is the largest Swedish company delivering fish by mail order,\n but you probably already knew that. The marketing department have done a pilot study and\n tried two different marketing methods:\n\n A: Sending a mail with a colorful brochure that invites people to sign up for a one year\n salmon subscription.\n\n B: Sending a colorful brochure that invites people to sign up for a one year\n salmon subscription nd that includes a free salmon.\n\n The marketing department sent out 16 mails of type A and 16 mails of type B.\n Six Danes that received a mail of type A signed up for one year of salmon, and\n ten Danes that received a mail of type B signed up!\n\n The marketing department now wants to know, which method should we use, A or B?\n\n At the bottom of this document you’ll find a solution. But try yourself first!\n Question I: Build a Bayesian model in Stan that answers the question: What is the\n probability that method B is better than method A?\n \"\"\"\n\n See ~/stan/ab_testing.R\n ~/blog/ab_testing.blog\n ~/psi/ab_testing.psi\n ~/webppl/ab_testing.wppl\n=#\n\nusing Turing, StatsPlots, DataFrames\n# using ReverseDiff, Zygote, Tracker\n# Turing.setadbackend(:reversediff)\n# Turing.setadbackend(:zygote)\n# Turing.setadbackend(:tracker)\n\ninclude(\"jl_utils.jl\")\n\n#=\nummary Statistics\n parameters mean std naive_se mcse ess rhat\n Symbol Float64 Float64 Float64 Float64 Float64 Float64\n\n rateA 0.3893 0.1126 0.0003 0.0009 15774.8462 1.0004\n rateB 0.6121 0.1108 0.0003 0.0008 16943.7123 1.0001\n\nQuantiles\n parameters 2.5% 25.0% 50.0% 75.0% 97.5%\n Symbol Float64 Float64 Float64 Float64 Float64\n\n rateA 0.1839 0.3088 0.3857 0.4659 0.6183\n rateB 0.3847 0.5397 0.6153 0.6918 0.8148\n\nmean(rateB > rateA): 0.91461875\nmean(rateB - rateA): 0.22276908208667892\n=#\n@model function ab_testing(nA=16,nB=16,obsSA=6, obsSB=10)\n # nA = 16 # Number of sent mail\n # nB = 16\n\n # obsSA = 6; # Number of signments (observed)\n # obsSB = 10\n\n rateA ~ Beta(1,1) # priors\n rateB ~ Beta(1,1)\n\n obsSA ~ Binomial(nA,rateA) # likelihood\n obsSB ~ Binomial(nB,rateB)\n\n # This variant don't work!\n # sA ~ Binomial(nA,rateA) # likelihood\n # sB ~ Binomial(nB,rateB)\n # sA == obsSA || begin Turing.@addlogprob! -Inf; return; end\n # sB == obsSB || begin Turing.@addlogprob! -Inf; return; end\n\nend\n\nmodel = ab_testing(16,16,6,10)\n# model = ab_testing(160,160,60,100)\nnum_chains = 4\n# chains = sample(model, Prior(), MCMCThreads(), 1000, num_chains)\nchains = sample(model, MH(), MCMCThreads(), 40_000, num_chains)\n# chains = sample(model, MH(), 40_000)\n\n# chains = sample(model, PG(20), MCMCThreads(), 10_000, num_chains)\n# chains = sample(model, IS(), MCMCThreads(), 10_000, num_chains)\n\n# chains = sample(model, Gibbs(MH(:sA,:sB),NUTS(1000,0.65,:rateA,:rateB)), MCMCThreads(), 40_000, num_chains)\n\ndisplay(chains)\n# display(plot(chains))\n\ndf = DataFrame(chains);\nprintln(\"mean(rateB > rateA): \", mean(df[!,\"rateB\"] .> df[!,\"rateA\"]))\nprintln(\"mean(rateB - rateA): \", mean(df[!,\"rateB\"] .- df[!,\"rateA\"]))\n", "meta": {"hexsha": "a065fb28dfc2c5d6e05dc3a943f926f5a5caafdb", "size": 3417, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "julia/turing/ab_testing.jl", "max_stars_repo_name": "Wikunia/hakank", "max_stars_repo_head_hexsha": "030bc928d2efe8dcbc5118bda3f8ae9575d0fd13", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "julia/turing/ab_testing.jl", "max_issues_repo_name": "Wikunia/hakank", "max_issues_repo_head_hexsha": "030bc928d2efe8dcbc5118bda3f8ae9575d0fd13", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "julia/turing/ab_testing.jl", "max_forks_repo_name": "Wikunia/hakank", "max_forks_repo_head_hexsha": "030bc928d2efe8dcbc5118bda3f8ae9575d0fd13", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.8316831683, "max_line_length": 109, "alphanum_fraction": 0.6701785192, "num_tokens": 1102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297887874625, "lm_q2_score": 0.90192067059785, "lm_q1q2_score": 0.8122064309965283}} {"text": "# Benchmarking the NLsolve.fixedpoint() function in the use case of: https://lectures.quantecon.org/jl/mccall_model_with_separation.html\n\n# Dependencies. \nusing Distributions # So that some method overwritten error from NLsolve doesn't intermingle with the benchmarks. \n\n# Wages (global since used in constructor)\nconst n = 60 # n possible outcomes for wage\nconst default_w_vec = linspace(10, 20, n) # wages between 10 and 20\nconst a, b = 600, 400 # shape parameters\nconst dist = BetaBinomial(n-1, a, b)\nconst default_p_vec = pdf.(dist, support(dist))\n\n# Model Struct (global since types can't be defined in local scope)\nmutable struct McCallModel{TF <: AbstractFloat,\n TAV <: AbstractVector{TF},\n TAV2 <: AbstractVector{TF}}\n α::TF # Job separation rate\n β::TF # Discount rate\n γ::TF # Job offer rate\n c::TF # Unemployment compensation\n σ::TF # Utility parameter\n w_vec::TAV # Possible wage values\n p_vec::TAV2 # Probabilities over w_vec\n\n McCallModel(α::TF=0.2,\n β::TF=0.98,\n γ::TF=0.7,\n c::TF=6.0,\n σ::TF=2.0,\n w_vec::TAV=default_w_vec,\n p_vec::TAV2=default_p_vec) where {TF, TAV, TAV2} =\n new{TF, TAV, TAV2}(α, β, γ, c, σ, w_vec, p_vec)\nend\n\n\n# Implementation from McCall with separation lecture on QuantEcon \nfunction default_quantecon()\n # Utility function. \n function u(c::Real, σ::Real)\n if c > 0\n return (c^(1 - σ) - 1) / (1 - σ)\n else\n return -10e6\n end\n end\n # Instantiate model object. \n mcm = McCallModel()\n # Define the function to be used in the iteration loop. \n \"\"\"\n A function to update the Bellman equations. Note that V_new is modified in\n place (i.e, modified by this function). The new value of U is returned.\n \"\"\"\n function update_bellman!(mcm::McCallModel,\n V::AbstractVector,\n V_new::AbstractVector,\n U::Real)\n # Simplify notation\n α, β, σ, c, γ = mcm.α, mcm.β, mcm.σ, mcm.c, mcm.γ\n\n for (w_idx, w) in enumerate(mcm.w_vec)\n # w_idx indexes the vector of possible wages\n V_new[w_idx] = u(w, σ) + β * ((1 - α) * V[w_idx] + α * U)\n end\n\n U_new = u(c, σ) + β * (1 - γ) * U +\n β * γ * dot(max.(U, V), mcm.p_vec)\n\n return U_new\n end\n # Define a function to carry out the iteration loop. \n function solve_mccall_model(mcm::McCallModel;\n tol::AbstractFloat=1e-5,\n max_iter::Integer=2000)\n\n V = ones(length(mcm.w_vec)) # Initial guess of V\n V_new = similar(V) # To store updates to V\n U = 1.0 # Initial guess of U\n i = 0\n error = tol + 1\n\n while error > tol && i < max_iter\n U_new = update_bellman!(mcm, V, V_new, U)\n error_1 = maximum(abs, V_new - V)\n error_2 = abs(U_new - U)\n error = max(error_1, error_2)\n V[:] = V_new\n U = U_new\n i += 1\n end\n\n return V, U\n end\n # Get results. \n return solve_mccall_model(mcm)\nend \n\n# Implementation of the above, but with one vector that holds the state of the calculation [U V]\nfunction stacked_quantecon()\n # Utility function. \n function u(c::Real, σ::Real)\n if c > 0\n return (c^(1 - σ) - 1) / (1 - σ)\n else\n return -10e6\n end\n end\n # Instantiate model object. \n mcm = McCallModel()\n # Function for each loop of the iteration. \n \"\"\"\n An (in-place) stacked implementation of the Bellman operator for the McCall model with separation. By convention, U is at the top of the vector. \n bellman_operator!(newVec::AbstractVector, oldVec::AbstractVector, model::McCallModel = mcm)\n \"\"\"\n function bellman_operator!(newVec, oldVec, model = mcm) \n # Unpack parameters. \n α, β, σ, c, γ = mcm.α, mcm.β, mcm.σ, mcm.c, mcm.γ\n # Add new V(w) values to newVec.\n for (w_idx, w) in enumerate(mcm.w_vec)\n # w_idx indexes the vector of possible wages\n newVec[w_idx + 1] = u(w, σ) + β * ((1 - α) * oldVec[1 + w_idx] + α * oldVec[1])\n end\n # Add new U value to newVec. \n newVec[1] = u(c, σ) + β * (1 - γ) * oldVec[1] + β * γ * dot(max.(oldVec[1], oldVec[2:end]), mcm.p_vec)\n end\n # Function to carry out the iteration. \n \"\"\"\n Modified version of solve_mccall_model(), which uses our bellman_operator! function. \n solve_mccall_model_inplace(mcm::McCallModel; tol::AbstractFloat=1e-5, max_iter::Integer=2000)\n \"\"\" \n function solve_mccall_model(mcm; tol=1e-5, max_iter=2000)\n oldVec = ones(length(mcm.w_vec) + 1) # Initial guess for states. Same as those in the QE implementation. \n newVec = similar(oldVec) # To store updates. \n iter = 0 # Initialize the iteration counter. \n error = tol + 1 # Initialize the error. \n\n while error > tol && iter < max_iter\n bellman_operator!(newVec, oldVec, mcm)\n error = maximum(abs.(newVec - oldVec))\n iter += 1\n oldVec .= newVec \n end \n\n return newVec\n end\n # Get results. \n return solve_mccall_model(mcm)\nend\n\n# Implementation of stacked_quantecon(), but using the NLsolve fixed point method. m = 0. \nfunction fixedpoint_iteration()\n # Utility function. \n function u(c::Real, σ::Real)\n if c > 0\n return (c^(1 - σ) - 1) / (1 - σ)\n else\n return -10e6\n end\n end\n # Instantiate model object. \n mcm = McCallModel()\n # Function for each loop of the iteration. \n \"\"\"\n An (in-place) stacked implementation of the Bellman operator for the McCall model with separation. By convention, U is at the top of the state vector. \n bellman_operator!(newVec::AbstractVector, oldVec::AbstractVector, model::McCallModel = mcm)\n \"\"\"\n function bellman_operator!(newVec, oldVec, model = mcm) \n # Unpack parameters. \n α, β, σ, c, γ = mcm.α, mcm.β, mcm.σ, mcm.c, mcm.γ\n # Add new V(w) values to newVec.\n for (w_idx, w) in enumerate(mcm.w_vec)\n # w_idx indexes the vector of possible wages\n newVec[w_idx + 1] = u(w, σ) + β * ((1 - α) * oldVec[1 + w_idx] + α * oldVec[1])\n end\n # Add new U value to newVec. \n newVec[1] = u(c, σ) + β * (1 - γ) * oldVec[1] + β * γ * dot(max.(oldVec[1], oldVec[2:end]), mcm.p_vec)\n end\n # Solve \n init = ones(length(mcm.w_vec)+1)\n return fixedpoint(bellman_operator!, init; ftol = 1e-5, iterations = 2000)\nend \n\n# Using fixedpoint() for variable m. \nfunction fixedpoint_anderson_m(m)\n # Utility function. \n function u(c::Real, σ::Real)\n if c > 0\n return (c^(1 - σ) - 1) / (1 - σ)\n else\n return -10e6\n end\n end\n # Instantiate model object. \n mcm = McCallModel()\n # Function for each loop of the iteration. \n \"\"\"\n An (in-place) stacked implementation of the Bellman operator for the McCall model with separation. By convention, U is at the top of the state vector. \n bellman_operator!(newVec::AbstractVector, oldVec::AbstractVector, model::McCallModel = mcm)\n \"\"\"\n function bellman_operator!(newVec, oldVec, model = mcm)\n # Unpack parameters. \n α, β, σ, c, γ = mcm.α, mcm.β, mcm.σ, mcm.c, mcm.γ\n # Add new V(w) values to newVec.\n for (w_idx, w) in enumerate(mcm.w_vec)\n # w_idx indexes the vector of possible wages\n newVec[w_idx + 1] = u(w, σ) + β * ((1 - α) * oldVec[1 + w_idx] + α * oldVec[1])\n end\n # Add new U value to newVec. \n newVec[1] = u(c, σ) + β * (1 - γ) * oldVec[1] + β * γ * dot(max.(oldVec[1], oldVec[2:end]), mcm.p_vec)\n end\n # Solve \n init = ones(length(mcm.w_vec)+1)\n return fixedpoint(bellman_operator!, init; m = m, ftol = 1e-5, iterations = 2000)\nend \n\nfunction fixedpoint_newton()\n # Utility function. \n function u(c::Real, σ::Real)\n if c > 0\n return (c^(1 - σ) - 1) / (1 - σ)\n else\n return -10e6\n end\n end\n # Instantiate model object. \n mcm = McCallModel()\n # Function for each loop of the iteration. \n \"\"\"\n An (in-place) stacked implementation of the Bellman operator for the McCall model with separation. By convention, U is at the top of the state vector. \n bellman_operator!(newVec::AbstractVector, oldVec::AbstractVector, model::McCallModel = mcm)\n \"\"\"\n function bellman_operator!(newVec, oldVec, model = mcm)\n # Unpack parameters. \n α, β, σ, c, γ = mcm.α, mcm.β, mcm.σ, mcm.c, mcm.γ\n # Add new V(w) values to newVec.\n for (w_idx, w) in enumerate(mcm.w_vec)\n # w_idx indexes the vector of possible wages\n newVec[w_idx + 1] = u(w, σ) + β * ((1 - α) * oldVec[1 + w_idx] + α * oldVec[1])\n end\n # Add new U value to newVec. \n newVec[1] = u(c, σ) + β * (1 - γ) * oldVec[1] + β * γ * dot(max.(oldVec[1], oldVec[2:end]), mcm.p_vec)\n end\n # Solve \n init = ones(length(mcm.w_vec)+1)\n return fixedpoint(bellman_operator!, init; method = :newton, ftol = 1e-5, iterations = 2000)\nend ", "meta": {"hexsha": "6b6e047e17af5211ba03273c35c06b95dbe67e8c", "size": 9502, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "julia_sandbox/QuantEcon-tests/mccall_utilities.jl", "max_stars_repo_name": "ajozefiak/computing_and_datascience", "max_stars_repo_head_hexsha": "ecc5a369adc5784e85614f3f74ae25ca1aede1f4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2018-07-27T00:56:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-15T00:20:26.000Z", "max_issues_repo_path": "julia_sandbox/QuantEcon-tests/mccall_utilities.jl", "max_issues_repo_name": "ajozefiak/computing_and_datascience", "max_issues_repo_head_hexsha": "ecc5a369adc5784e85614f3f74ae25ca1aede1f4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 96, "max_issues_repo_issues_event_min_datetime": "2017-12-22T18:26:34.000Z", "max_issues_repo_issues_event_max_datetime": "2019-05-13T22:53:47.000Z", "max_forks_repo_path": "julia_sandbox/QuantEcon-tests/mccall_utilities.jl", "max_forks_repo_name": "ajozefiak/computing_and_datascience", "max_forks_repo_head_hexsha": "ecc5a369adc5784e85614f3f74ae25ca1aede1f4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 14, "max_forks_repo_forks_event_min_datetime": "2018-01-05T19:05:21.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-19T17:01:09.000Z", "avg_line_length": 38.7836734694, "max_line_length": 155, "alphanum_fraction": 0.5697747843, "num_tokens": 2811, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206791658465, "lm_q2_score": 0.9005297794439687, "lm_q1q2_score": 0.8122064302851743}} {"text": "A = [1 2 3; -4 -5 -6]\n\nA[2,3]\n\nlength(A)\n\nsize(A,1)\n\nsize(A,2)\n\nsumA = 0\nfor i = 1:size(A,1)\n for j = 1:size(A,2)\n sumA += A[i,j]\n end\nend\n\nprintln(\"Sum of the elements in A = \", sumA)\n\nsumA = 0\nfor i = 1:size(A,1), j = 1:size(A,2)\n sumA += A[i,j]\nend\n\nprintln(\"Sum of the elements in A = \", sumA)\n\nsumAsquared = 0\nfor a in A\n sumAsquared += a^2\nend\n\nprint(\"Sum of the squares of the elements in A = \", sumAsquared)\n\nC = ones(5,3) # 5-by-4 matrix with all elements == 1\n\nones(5) # 1-d array (can be used as a column vector)\n\nones(5,1) # 2-d array, column vector\n\nones(1,5) # 2-d array, row vector\n\nD = [A; C] # Vertical concatenation - arrays must have same # columns\nE = [zeros(5) C] # Horizontal concatenation - arrays must have same # rows\nF = [A ones(2,2); zeros(1,5)] # General vertical-horizontal concatenation - all dimensions must match\n\nG = [1:5 ones(Int64,5) 101:105]\n\nB = A.^2 .- 3A # Apply function elementwise\n\nB = @. A^2 - 3A # Easier syntax - entire expression evaluated elementwise\n\nA[1,:] # Row 1\n\nA[:,1] # Column 1\n\nA[:,2:3] # Columns 2-3\n\nA[:,[1,3]] .= 0 # Set columns 1 and 3 to zero\nA\n\nA[:,2:3] = 2A[:,1:2] # Set columns 2 and 3 to twice of columns 1 and 2\nA\n\n\n", "meta": {"hexsha": "8865a94821067e3866474b06074ecc8269508bc2", "size": 1327, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "textbook/_build/jupyter_execute/content/Arrays_and_Dictionaries/Multi_Dimensional_Arrays.jl", "max_stars_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_stars_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "textbook/_build/jupyter_execute/content/Arrays_and_Dictionaries/Multi_Dimensional_Arrays.jl", "max_issues_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_issues_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "textbook/_build/jupyter_execute/content/Arrays_and_Dictionaries/Multi_Dimensional_Arrays.jl", "max_forks_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_forks_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.4153846154, "max_line_length": 102, "alphanum_fraction": 0.5546345139, "num_tokens": 472, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793453, "lm_q2_score": 0.9032941988938413, "lm_q1q2_score": 0.8121711315145564}} {"text": "using InstantiateFromURL\n# optionally add arguments to force installation: instantiate = true, precompile = true\ngithub_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.8.0\")\n\nusing LinearAlgebra, Statistics\nusing Plots, Distributions, Random, Statistics\ngr(fmt = :png, size = (900, 500))\n\nfunction ksl(distribution, n = 100)\n title = nameof(typeof(distribution))\n observations = rand(distribution, n)\n sample_means = cumsum(observations) ./ (1:n)\n μ = mean(distribution)\n plot(repeat((1:n)', 2),\n [zeros(1, n); observations'], label = \"\", color = :grey, alpha = 0.5)\n plot!(1:n, observations, color = :grey, markershape = :circle,\n alpha = 0.5, label = \"\", linewidth = 0)\n hline!([μ], color = :black, linewidth = 1.5, linestyle = :dash, grid = false, label = [\"Mean\"])\n plot!(1:n, sample_means, linewidth = 3, alpha = 0.6, color = :green,\n label = \"Sample mean\")\n return plot!(title = title)\nend\n\n# Example distributions\ndistributions = [TDist(10), Beta(2,2), Gamma(5,2), Poisson(4), LogNormal(0.5), Exponential(1)]\n\nRandom.seed!(0); #reproducible results\nksl(Normal())\nplot(ksl.(sample(distributions, 3, replace = false))..., layout = (3, 1), legend = false)\n\n# Cauchy distribution may not converge as E|X| < ∞ not satisfied (heavy tailed)\n\n\nfunction plotmeans(distribution::Distribution{Univariate,Continuous};n=1000)\n sample_mean = cumsum(rand(distribution,n)) ./ (1:n)\n plot(1:n, sample_mean, label = \"sample mean\")\n hline!([0], color = :black, linestyle = :dash)\nend\n\nRandom.seed!(0);\nplotmeans(Cauchy(),n=10000)\n\n\nusing StatsPlots\n\nfunction simulation1(distribution::Distribution{Univariate,Continuous}; n=250, k = 10_000)\n σ = std(distribution)\n y = rand(distribution,n,k)\n y .-= mean(distribution)\n y = mean(y,dims=1)\n y = √n *vec(y)\n density(y,label = \"Empirical Distribution\")\n return plot!(Normal(0,σ), linestyle = :dash, label = \"Normal(0.0, $(σ^2))\")\nend\n\nRandom.seed!(0);\nsimulation1(Exponential(0.5), n=100)\n\nfunction simulation2(distribution = Beta(2, 2), n = 5, k = 10_000)\n y = rand(distribution, k, n)\n for col in 1:n\n y[:,col] += rand([-0.5, 0.6, -1.1], k)\n end\n y = (y .- mean(distribution)) ./ std(distribution)\n y = cumsum(y, dims = 2) ./ sqrt.(1:5)' # return grid of data\nend\n\nys = simulation2()\nplots = [] # would preallocate in optimized code\nfor i in 1:size(ys, 2)\n p = density(ys[:, i], linealpha = i, title = \"n = $i\")\n push!(plots, p)\nend\n\nplot(plots..., legend = false)\n\n# Exercise 1\n\nfunction exercise1(a,b; g = sin, g′ = cos, n=250, k=10_000)\n distribution = Uniform(a,b)\n μ, σ = mean(distribution), std(distribution)\n gμ = sin.(mean(distribution))\n X = rand(distribution,n,k)\n X = mean(X, dims = 1)\n X = vec(X)\n y = √n .* vec(g.(X) .- g.(μ))\n density(y,label = \"Empirical Distribution\")\n return plot!(Normal(0,g′(μ).*σ), linestyle = :dash, label = \"Asymptotic\")\nend\n\nexercise1(0.0,π/2,k=15000)\n\n\n# Exercise 2\nfunction exercise2(;n = 250, k = 50_000, dw = Uniform(-1, 1), du = Uniform(-2, 2))\n vw = var(dw)\n vu = var(du)\n Σ = [vw vw; vw vu+vw]\n Q = inv(sqrt(Σ))\n\n function generate_data(dw, du, n)\n dw = rand(dw, n)\n X = [dw dw + rand(du, n)]\n return sqrt(n) * mean(X, dims = 1)\n end\n\n X = mapreduce(x -> generate_data(dw, du, n), vcat, 1:k)\n X = sqrt(n) .* mean(X, dims=1)\n X = Q*X'\n X = sum(abs2, X, dims = 1)\n X = vec(X)\n density(X, label = \"\", xlim = (0, 10))\n return plot!(Chisq(2), color = :black, linestyle = :dash,\n label = \"Chi-squared with 2 degrees of freedom\", grid = false)\nend\nexercise2()\n", "meta": {"hexsha": "7fd1069b04df5a6e91a19d8a84559c4063ee755b", "size": 3663, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "ToolsandTechniques_LLNCLT.jl", "max_stars_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_stars_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ToolsandTechniques_LLNCLT.jl", "max_issues_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_issues_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ToolsandTechniques_LLNCLT.jl", "max_forks_repo_name": "shanemcmiken/quantecon-notebooks-julia", "max_forks_repo_head_hexsha": "c9968403bc866fe1f520762619055bd21e09ad10", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.3076923077, "max_line_length": 99, "alphanum_fraction": 0.6161616162, "num_tokens": 1184, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765187126079, "lm_q2_score": 0.8887587890727754, "lm_q1q2_score": 0.8120380363752464}} {"text": "\r\n# Identity matrix\r\neye(m::Int64) = Diagonal(ones(m))\r\n\r\nunpack(s, f::Symbol) = reduce(hcat, getproperty.(s, f))\r\n\r\nfunction unpack(s, f::Vector{Symbol})\r\n A = []\r\n for i in f\r\n push!(A, reduce(hcat, getproperty.(s, i)))\r\n end\r\n return A\r\nend\r\n\r\nunpack_matrix(s, f) = getproperty.(s, f)\r\n\r\nfunction unpack_matrix(s, f::Vector{Symbol})\r\n A = []\r\n for i in f\r\n push!(A, getproperty.(s, i))\r\n end\r\n return A\r\nend\r\n\r\n_return_type(P::Matrix, C) = Matrix(C)\r\n_return_type(P, C) = Diagonal(C)\r\n\r\n_square_root_filter(filter::CholeskyFactorization, P) = _return_type(P, cholesky(P).L)\r\n\r\nfunction _square_root_filter(filter::ModifiedCholeskyFactorization, P)\r\n\tF = _modified_cholesky(P)\r\n\t_return_type(P, F.U * sqrt.(F.D))\r\nend\r\n\r\nfunction _square_root_filter(filter::SVDFactorization, P)\r\n\t_C = svd(Matrix(P))\r\n\treturn _return_type(P, _C.U * sqrt(Diagonal(_C.S)))\r\nend\r\n\r\n\"\"\"\r\n\r\n\tF = _modified_cholesky(P)\r\n\r\nComputes modified Cholesky factors F.U and F.D of a symmetric positive definite matrix P,\r\nsuch that F.U is unit upper triangular and F.D is diagonal. Thus, and P = F.U*F.D*F.U'.\r\n\r\nReference: Grewal - Kalman filtering theory and practice\r\n\r\n\"\"\"\r\nfunction _modified_cholesky(P::AbstractMatrix)\r\n\tisposdef(P) || throw(\"input matrix must be positive-definite. Modified Cholesky factorization failed.\")\r\n\tn = size(P, 1)\r\n\t_P = (P .+ P') ./ 2.0 # take symmetric part\r\n\tU, D = similar(_P), similar(_P)\r\n\tfor j in n:-1:1\r\n \t\tfor i in j:-1:1\r\n\t\t\ts = P[i, j]\r\n\t\t\tfor k in j+1:n\r\n\t\t\t\ts -= U[i, k]*D[k, k]*U[j, k]\r\n\t\t\tend\r\n\t\t\tif i==j\r\n\t\t\t\tD[j, j] = s\r\n\t\t\t\tU[j, j] = 1.0\r\n\t\t\telse\r\n\t\t\t\tU[i, j] = s / D[j, j]\r\n\t\t\tend\r\n\t\tend\r\n\tend\r\n\treturn (U=U, D=D)\r\nend\r\n", "meta": {"hexsha": "cc7025df42aa0b77c3f12e007cf54ce2a6c0aa93", "size": 1685, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Utils.jl", "max_stars_repo_name": "lnacquaroli/KalmanFilter.jl", "max_stars_repo_head_hexsha": "9746b6b1079c499323f777ecf7077a995ec49f78", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Utils.jl", "max_issues_repo_name": "lnacquaroli/KalmanFilter.jl", "max_issues_repo_head_hexsha": "9746b6b1079c499323f777ecf7077a995ec49f78", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Utils.jl", "max_forks_repo_name": "lnacquaroli/KalmanFilter.jl", "max_forks_repo_head_hexsha": "9746b6b1079c499323f777ecf7077a995ec49f78", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.7323943662, "max_line_length": 105, "alphanum_fraction": 0.6195845697, "num_tokens": 539, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218434359676, "lm_q2_score": 0.8807970811069351, "lm_q1q2_score": 0.8120260687071251}} {"text": "function load_distributions_samples()\nreturn quote\n\nusing .Distributions: Distribution, Uniform, Normal\nimport .Distributions\n\n# ======================================================\n# Sampling from a uniform distribution on balls/spheres\n# ======================================================\n\n\"\"\"\n _sample_unit_nsphere_muller!(D::Vector{Vector{N}}, n::Int, p::Int;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N}\n\nDraw samples from a uniform distribution on an ``n``-dimensional unit\nsphere using Muller's method.\n\n### Input\n\n- `D` -- output, vector of points\n- `n` -- dimension of the sphere\n- `p` -- number of random samples\n- `rng` -- (optional, default: `GLOBAL_RNG`) random number generator\n- `seed` -- (optional, default: `nothing`) seed for reseeding\n\n### Output\n\nA vector of `nsamples` vectors.\n\n### Algorithm\n\nThis function implements Muller's method of normalised Gaussians [1] to uniformly\nsample over the ``n``-dimensional sphere ``S^n`` (which is the bounding surface\nof the ``n``-dimensional unit ball).\n\nGiven ``n`` canonical Gaussian random variables ``Z₁, Z₂, …, Z_n``, the\ndistribution of the vectors\n\n```math\n\\\\dfrac{1}{α}\\\\left(z₁, z₂, …, z_n\\\\right)^T,\n```\nwhere ``α := \\\\sqrt{z₁² + z₂² + … + z_n²}``, is uniform over ``S^n``.\n\n[1] Muller, Mervin E. *A note on a method for generating points uniformly on\n n-dimensional spheres.* Communications of the ACM 2.4 (1959): 19-20.\n\"\"\"\nfunction _sample_unit_nsphere_muller!(D::Vector{Vector{N}}, n::Int, p::Int;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N}\n rng = reseed(rng, seed)\n Zdims = [Normal() for _ in 1:n] # normal distributions, one for each dimension\n v = Vector{N}(undef, n) # sample direction\n @inbounds for j in 1:p\n α = zero(N)\n for i in 1:n\n v[i] = rand(rng, Zdims[i])\n α += v[i]^2\n end\n D[j] = v ./ sqrt(α)\n end\n return D\nend\n\n\"\"\"\n _sample_unit_nball_muller!(D::Vector{Vector{N}}, n::Int, p::Int;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N}\n\nDraw samples from a uniform distribution on an ``n``-dimensional unit ball\nusing Muller's method.\n\n### Input\n\n- `D` -- output, vector of points\n- `n` -- dimension of the ball\n- `p` -- number of random samples\n- `rng` -- (optional, default: `GLOBAL_RNG`) random number generator\n- `seed` -- (optional, default: `nothing`) seed for reseeding\n\n### Output\n\nA vector of `nsamples` vectors.\n\n### Algorithm\n\nThis function implements Muller's method of normalised Gaussians [1] to\nsample from the interior of the ball.\n\nGiven ``n`` Gaussian random variables ``Z₁, Z₂, …, Z_n`` and a uniformly\ndistributed random variable ``r`` with support in ``[0, 1]``, the distribution\nof the vectors\n\n```math\n\\\\dfrac{r^{1/n}}{α} \\\\left(z₁, z₂, …, z_n\\\\right)^T,\n```\nwhere ``α := \\\\sqrt{z₁² + z₂² + … + z_n²}``, is uniform over the\n``n``-dimensional unit ball.\n\n[1] Muller, Mervin E. *A note on a method for generating points uniformly on\n n-dimensional spheres.* Communications of the ACM 2.4 (1959): 19-20.\n\"\"\"\nfunction _sample_unit_nball_muller!(D::Vector{Vector{N}}, n::Int, p::Int;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N}\n\n rng = reseed(rng, seed)\n Zdims = [Normal() for _ in 1:n] # normal distributions, one for each dimension\n Zrad = Uniform() # distribution to pick random radius\n one_over_n = one(N)/n\n v = Vector{N}(undef, n) # sample direction\n @inbounds for j in 1:p\n α = zero(N)\n for i in 1:n\n v[i] = rand(rng, Zdims[i])\n α += v[i]^2\n end\n r = rand(rng, Zrad)\n β = r^one_over_n / sqrt(α)\n D[j] = v .* β\n end\n return D\nend\n\n# ========================\n# Sampling from a LazySet\n# ========================\n\n\"\"\"\n Sampler\n\nAbstract type for defining new sample methods.\n\n### Notes\n\nAll subtypes should implement a `_sample!` method.\n\"\"\"\nabstract type Sampler end\n\n\"\"\"\n sample(X::LazySet{N}, num_samples::Int;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N}\n\nSampling of an arbitrary `LazySet` `X`.\n\n### Input\n\n- `X` -- lazyset\n- `num_samples` -- number of random samples\n- `sampler` -- Sampler used (default: `RejectionSampler`)\n- `rng` -- (optional, default: `GLOBAL_RNG`) random number generator\n- `seed` -- (optional, default: `nothing`) seed for reseeding\n\n### Output\n\nA vector of `num_samples` vectors.\nIf `num_samples` is not passed one sample as a single vector is returned.\n\n### Algorithm\n\nSee the documentation of `RejectionSampler`.\n\"\"\"\nfunction sample(X::LazySet{N}, num_samples::Int;\n sampler=RejectionSampler,\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N<:Real}\n require(:Distributions; fun_name=\"sample\")\n @assert isbounded(X) \"this function requires that the set `X` is bounded\"*\n \", but it is not\"\n\n D = Vector{Vector{N}}(undef, num_samples) # preallocate output\n _sample!(D, sampler(X); rng=rng, seed=seed)\n return D\nend\n\n# without argument, returns a single element (instead of a singleton)\nfunction sample(X::LazySet{N}; kwargs...) where {N<:Real}\n return sample(X, 1; kwargs...)[1]\nend\n\n# =====================\n# Rejection Sampling\n# =====================\n\n\"\"\"\n RejectionSampler{S<:LazySet, D<:Distribution} <: Sampler\n\nType used for rejection sampling of an arbitrary `LazySet` `X`.\n\n### Fields\n\n- `X` -- lazyset\n- `box_approx` -- Distribution from which the sample is drawn\n\n### Algorithm\n\nDraw a sample ``x`` from a uniform distribution of a box-overapproximation of the\noriginal set ``X`` in all ``n`` dimensions. The function rejects a drawn sample ``x``\nand redraws as long as the sample is not contained in the original set ``X``,\ni.e., ``x ∉ X``.\n\"\"\"\nstruct RejectionSampler{S<:LazySet, D<:Distribution} <: Sampler\n X::S\n box_approx::Vector{D}\nend\n\nfunction RejectionSampler(X, distribution=Uniform)\n B = box_approximation(X)\n canonical_support = hcat(low(B), high(B))\n dims = size(canonical_support, 1)\n box_approx = [distribution(canonical_support[i,:]...) for i = 1:dims]\n return RejectionSampler(X, box_approx)\nend\n\n\"\"\"\n _sample!(D::Vector{Vector{N}},\n sampler::RejectionSampler;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N<:Real}\n\nSample points using rejection sampling.\n\n### Input\n\n- `D` -- output, vector of points\n- `sampler` -- Sampler from which the points are sampled\n- `rng` -- (optional, default: `GLOBAL_RNG`) random number generator\n- `seed` -- (optional, default: `nothing`) seed for reseeding\n\n### Output\n\nA vector of `num_samples` vectors.\n\"\"\"\nfunction _sample!(D::Vector{Vector{N}},\n sampler::RejectionSampler;\n rng::AbstractRNG=GLOBAL_RNG,\n seed::Union{Int, Nothing}=nothing) where {N<:Real}\n rng = reseed(rng, seed)\n @inbounds for i in 1:length(D)\n w = rand.(Ref(rng), sampler.box_approx)\n while w ∉ sampler.X\n w = rand.(Ref(rng), sampler.box_approx)\n end\n D[i] = w\n end\n return D\nend\n\nend # quote\nend # function load_distributions_samples()\n", "meta": {"hexsha": "b4a0c8e1071d1b4f5cc8192a660dc761d117de02", "size": 7627, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Utils/samples.jl", "max_stars_repo_name": "UnofficialJuliaMirror/LazySets.jl-b4f0291d-fe17-52bc-9479-3d1a343d9043", "max_stars_repo_head_hexsha": "375c103128089b12d3c08f7011d24fdbffc65c47", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/Utils/samples.jl", "max_issues_repo_name": "UnofficialJuliaMirror/LazySets.jl-b4f0291d-fe17-52bc-9479-3d1a343d9043", "max_issues_repo_head_hexsha": "375c103128089b12d3c08f7011d24fdbffc65c47", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Utils/samples.jl", "max_forks_repo_name": "UnofficialJuliaMirror/LazySets.jl-b4f0291d-fe17-52bc-9479-3d1a343d9043", "max_forks_repo_head_hexsha": "375c103128089b12d3c08f7011d24fdbffc65c47", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.2658730159, "max_line_length": 85, "alphanum_fraction": 0.6037760587, "num_tokens": 2048, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9425067244294588, "lm_q2_score": 0.8615382094310355, "lm_q1q2_score": 0.8120055557416664}} {"text": "using Random, PyPlot, StatsBase, Combinatorics\nRandom.seed!(1)\nfunction bruteSetsProbabilityAllMiss(n)\n omega = collect(permutations(1:n))\n matchEvents = []\n\n for i in 1:n\n event = []\n for p in omega\n if p[i] == i\n push!(event,p)\n end\n end\n push!(matchEvents,event)\n end\n\n noMatch = setdiff(omega,union(matchEvents...))\n return length(noMatch)/length(omega)\nend\n\nfunction formulaCalcAllMiss(n)\n return sum([(-1)^k/factorial(k) for k in 0:n])\nend\n\nfunction mcAllMiss(n,N)\n function envelopeStuffer()\n envelopes = Random.shuffle!(collect(1:n))\n return sum([envelopes[i] == i for i in 1:n]) == 0\n end\n\n data = [envelopeStuffer() for _ in 1:N]\n return sum(data)/N\nend\n\nN = 10^6\n\nprintln(\"n\\tBrute Force\\tFormula\\t\\tMonte Carlo\\t\\tAnalytic\",)\nfor n in 1:8\n bruteForce = bruteSetsProbabilityAllMiss(n)\n fromFormula = formulaCalcAllMiss(n)\n fromMC = mcAllMiss(n,N)\n println(n,\"\\t\",round(bruteForce,digits=4),\"\\t\\t\",round(fromFormula,digits=4),\n\t\"\\t\\t\",round(fromMC,digits=4),\"\\t\\t\",round(1/MathConstants.e,digits=4))\nend", "meta": {"hexsha": "54974d9c972955c0a6a7d4b99001535683a12576", "size": 1137, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/stats/secretaryEnvelopes.jl", "max_stars_repo_name": "mikiec84/SemanticModels.jl", "max_stars_repo_head_hexsha": "f81baf0789cc547375f300429d0fd49c866d5339", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-01-21T15:52:41.000Z", "max_stars_repo_stars_event_max_datetime": "2020-01-21T15:52:41.000Z", "max_issues_repo_path": "examples/stats/secretaryEnvelopes.jl", "max_issues_repo_name": "mikiec84/SemanticModels.jl", "max_issues_repo_head_hexsha": "f81baf0789cc547375f300429d0fd49c866d5339", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/stats/secretaryEnvelopes.jl", "max_forks_repo_name": "mikiec84/SemanticModels.jl", "max_forks_repo_head_hexsha": "f81baf0789cc547375f300429d0fd49c866d5339", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.8409090909, "max_line_length": 81, "alphanum_fraction": 0.6341248901, "num_tokens": 341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.942506716354847, "lm_q2_score": 0.8615382040983515, "lm_q1q2_score": 0.8120055437589891}} {"text": "function four1(data::Array{T,1}, isign::Int64 = 1) where {T<:Number}\n\tnn = length(data)\n\tnnlog = log2(nn)\n\tif ceil(nnlog) != nnlog\n\t\terror(\"Length of data vector must be a power of 2.\")\n\tend\n\tj::Int64 = 1\n\tres::Array{Complex{Float64},1} = copy(data)\n\n\tfor i = 1:nn\n\t\tif j > i\n\t\t\tres[j], res[i] = data[i], data[j]\n\t\tend\n\t\tm = nn/2\n\t\twhile m >= 2 && j > m\n\t\t\tj -= m\n\t\t\tm = m/2\n\t\tend\n\t\tj += m\n\tend\n\n\tmmax = 1;\n\twhile nn > mmax\n\t\tistep = 2*mmax\n\t\ttheta = isign*(pi/mmax)\n\t\twp = -2*sin(theta/2)^2 + sin(theta)*im\n\t\tw = 1 + 0im\n\n\t\tfor m = 1:mmax\n\t\t\tfor i = m:istep:nn\n\t\t\t\tj = i + mmax\n\t\t\t\ttemp = w*res[j]\n\t\t\t\tres[j] = res[i] - temp\n\t\t\t\tres[i] += temp\n\t\t\tend\n\t\t\tw += w*wp\n\t\tend\n\t\tmmax = istep\n\tend\n\n\tres\nend\n\ninput_size = 2048\nFs = 1000\nT = 1/Fs # Sampling period \n\nt = T*(0:(input_size-1))\nX = 0.7*sin.(2*pi*50*t) + sin.(2*pi*120*t)\n\n@time Y = four1(X, 1)\n\n;\n", "meta": {"hexsha": "8904ba3274bd337847198c0456f6167788b44fe6", "size": 869, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "temp/fft/julia_fft.jl", "max_stars_repo_name": "Ashymad/praca.inz", "max_stars_repo_head_hexsha": "dbb1fba10e421c3610bb66a06b7601d2ca4366c6", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "temp/fft/julia_fft.jl", "max_issues_repo_name": "Ashymad/praca.inz", "max_issues_repo_head_hexsha": "dbb1fba10e421c3610bb66a06b7601d2ca4366c6", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "temp/fft/julia_fft.jl", "max_forks_repo_name": "Ashymad/praca.inz", "max_forks_repo_head_hexsha": "dbb1fba10e421c3610bb66a06b7601d2ca4366c6", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 16.0925925926, "max_line_length": 68, "alphanum_fraction": 0.5327963176, "num_tokens": 370, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9553191335436404, "lm_q2_score": 0.849971175657575, "lm_q1q2_score": 0.8119937270662639}} {"text": "my_number = 1.2 # Assignment\n\npi # Pre-defined variable\n\nMy_number # Error - was defined without capital M\n\nδ = 0.001 # Unicode characters allowed (type \\delta and press tab)\n\nmy_number_2 = (my_number + 1) * δ # Use pre-defined variables in expressions\n\ncounter = 1 # counter = 1\n\ncounter += 1 # counter = counter + 1\n\ncounter ^= 4 # counter = counter ^ 4\n\nangle = 60 # degrees\n\nangle *= π / 180 # convert to radians\n\nx = 3\nprintln(2x^2 - 3x + 1)\nprintln(1.5x^2 - .5x + 1)\nprintln(2(x-1)^2 - 3(x-1) + 1)\n\nprintln(360 / 2*π) # Left first - equals 180*π\nprintln(360 / 2π) # Right first - equals 180/π\n\nprintln(2^2x) # 2^(2x), not (4)x\nprintln(2x^2) # 2*(x^2), not (2x)^2\n\n#=\n Experiments with spheres\n=#\n\nradius1 = 100 # Outer sphere radius\nradius2 = 50 # Inner sphere radius \n\n# Compute the volume of each sphere\nvol1 = 4π * radius1^3 / 3\nvol2 = 4π * radius2^3 / 3\n\n# Compute the volume of a hollow sphere\nvol_hollow = vol1 - vol2\n", "meta": {"hexsha": "d302df7ad46c8bf214ae2a7dd9cc3aa00bde2d09", "size": 1093, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "textbook/_build/jupyter_execute/content/Introduction/Variables_And_Assignments.jl", "max_stars_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_stars_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "textbook/_build/jupyter_execute/content/Introduction/Variables_And_Assignments.jl", "max_issues_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_issues_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "textbook/_build/jupyter_execute/content/Introduction/Variables_And_Assignments.jl", "max_forks_repo_name": "NoseKnowsAll/NoseKnowsAll.github.io", "max_forks_repo_head_hexsha": "b2cff3e33cc2087770fb4aecb38b7925ad8d6e5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.2888888889, "max_line_length": 83, "alphanum_fraction": 0.56084172, "num_tokens": 354, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9553191309994468, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.8119937230884764}} {"text": "### A Pluto.jl notebook ###\n# v0.14.4\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 892fb737-6fc0-4438-b586-039a440b01f9\nusing LinearAlgebra, SparseArrays\n\n# ╔═╡ 069cf190-77cc-4757-9db3-1945ddd07438\nusing DifferentialEquations, Plots\n\n# ╔═╡ 7bc3e874-b7be-11eb-0792-cf4e21e8c8b2\nmd\"# Heat equation with Proportional control\n\nConsider a one-dimensional rod $\\Omega = (0,L)$ and time $t \\in (0,T).$ \n\"\n\n# ╔═╡ 1158de2e-b7a6-43f3-a4e6-2ef9598a4ca5\nL = 0.1 # Length of rod\n\n# ╔═╡ 30cebec3-9c70-4c6a-94a8-fdcbe6cda783\nTf = 2000.0 # Final simulation time\n\n# ╔═╡ 93fb0f90-c454-410f-aa5e-cd8e6c055dc0\nmd\"The rod has the physical properties\n- thermal conductivity $\\lambda$,\n- specific heat capacity $c$,\n- mass density $\\rho$.\n\nThe diffusivity constant is given by $\\alpha = \\frac{\\lambda}{c \\rho}$. The heat equation is noted as\n\n$\\dot{\\vartheta}(t,x) = \\alpha \\frac{\\partial^2}{\\partial x^2} \\vartheta(t,x)$\n\nfor $(t,x) \\in (0,T) \\times \\Omega$ with initial condition\n\n$\\vartheta(0,x) = \\vartheta_{0}(x)$ \n\nfor $x \\in \\overline{\\Omega}$.\" \n\n# ╔═╡ b294bbfc-45e7-4a15-a839-5ceb56746be7\nbegin\n\tλ = 45.0; # Thermal conductivity\n\tρ = 7800.0; # Mass density\n\tcap = 480.0; # Specific heat capacitivity\nend\n\n# ╔═╡ 543b3f13-4cef-46f1-9141-172b81e7d48b\nα = λ/(cap * ρ) # Diffusivity\n\n# ╔═╡ 1deb117d-21f7-4329-a7db-2be26e43129a\nmd\"## Boundary conditions \n\nOnly on the right side of the rod, heat transfer\n\n$\\phi_{t}(t,x=L) = - h (\\vartheta(t,x=L) - \\vartheta_{amb})$\n\nand heat radiation\n\n$\\phi_{r}(t,x=L) = - \\epsilon \\varrho (\\vartheta(t,x=L)^4 - \\vartheta_{amb}^4)$\n\nis assumed to operate. Parameter $\\vartheta_{amb}$ describes the ambient temperature, $h>0$ is called the heat transfer coefficient, $\\epsilon \\in (0,1)$ is called emissivity and $\\varrho\\approx 5.67 \\cdot 10^{-8}$ is known as Stefan-Boltzmann constant. For simplicity, $k=\\epsilon \\cdot \\varrho$. \n\nThe sum of $\\phi_{t}$ and $\\phi_{r}$ \n\n$\\phi_{out}(t,L) = \\phi_{t}(t,L) + \\phi_{r}(t,L) = - h (\\vartheta(t,L) - \\vartheta_{amb}) - k (\\vartheta(t,L)^4 - \\vartheta_{amb}^4)$\n\nis the flux from the rod to the environment.\"\n\n# ╔═╡ 78292b42-a203-496c-a143-def3ff128f69\nbegin\n\th = 10.0; # Heat transfer coefficient\n\tϵ = 0.6; # Emissivity\n\tsb = 5.67*10^(-8) # Stefan-Boltzmann constant\n\tk = ϵ * sb; \t # Radiation coefficient\nend\n\n# ╔═╡ 0ecea33b-bf24-48eb-b4b4-bbf288e62e6b\nθamb = 298.0 # Ambient temperature in Kelvin\n\n# ╔═╡ 3a059f15-d850-468a-bb1d-f42ef5b24609\nϕout(θ) = -h * (θ - θamb) - k*(θ^4 - θamb^4) \n\n# ╔═╡ 7202aa44-b393-4e0b-9c34-2a9b9376194c\nmd\"### Induced heat\n\nOn the left side of the rod, a heat source is assumed as \n\n$\\phi_{in}(t,x=0) = b ~ u(t)$\n\nwith constant $b>0$, input signal \n\n$u(t) = K_{p} ~ e(t)$\n\nand error\n\n$e(t) = y_{ref}(t) - y(t).$\n\nThe output is considered as the temperature on the right side, e.g.\n\n$y(t) = \\vartheta(t, x=L)$\n\nand the reference is as an arbitrary fixed value \n\n$y_{ref}(t) = y_{ref} > 0.$\n\nThus, the natural (Robin) boundary conditions are noted as \n\n$\\lambda ~ \\left.\\frac{\\partial}{\\partial x} \\vartheta(\\cdot, x) \\right\\rvert_{x = 0} \\cdot \\vec{n} ~=~ \\phi_{in}(t,0) = b ~ u(t)$\n\nand\n\n$\\lambda ~ \\left. \\frac{\\partial}{\\partial x} \\vartheta(\\cdot, x) \\right\\rvert_{x = L} \\cdot \\vec{n} ~=~ \\phi_{out}(t,x) = - h (\\vartheta(t,L) - \\vartheta_{amb}) - k (\\vartheta(t,L)^4 - \\vartheta_{amb}^4)$\n\nwith outer [normal vector](https://en.wikipedia.org/wiki/Normal_(geometry)) $\\vec{n}$ on the left or right boundary. Here, the normal vector $\\vec{n} = -1$ on the left boundary at $x=0$, and $\\vec{n} = 1$ on the right boundary at $x=L$.\"\n\n# ╔═╡ 2775433b-d8b1-48be-9b60-08641db15803\nb = 1;\n\n# ╔═╡ 4c59909b-ffb6-4dfc-94ab-142d41deeeab\nKp = 10^3 # Proportional gain\n\n# ╔═╡ 4f0c3671-d3bf-4e36-a1f3-8e8d0ac0e454\nyref = 400.0 # Reference temperature\n\n# ╔═╡ 23343fa0-6033-498d-b089-d43e4382e332\nu_in(err) = Kp * err # input signals\n\n# ╔═╡ daa17037-fd28-460c-97f5-826742b9a419\nmd\"### Spatial approximation\n\nThe one-dimensional rod is discretized as a one-dimensional grid with N points and the finite discretization is noted by\n\n$\\Delta x = \\frac{L}{N-1}.$\n\nThe resulting grid points have the position $x^{0} = 0$, $x^{n} = n~\\Delta x$ and $x^{N-1} = L$. The second order derivative is approximated using the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) as \n\n$\\left. \\frac{\\partial^2 f(x)}{\\partial x^2} \\right\\rvert_{x = \\tilde{x}} \\approx \\frac{1}{\\Delta x^2} \\left[ f(\\tilde{x} - \\Delta x) - 2 f(\\tilde{x}) + f(\\tilde{x} + \\Delta x) \\right].$\n\nThis [Finite Difference](https://en.wikipedia.org/wiki/Finite_difference_method) scheme is written in matrix-vector form as \n\n$\\frac{\\partial^2 \\vartheta(t,x)}{\\partial x^2} \\approx \\frac{1}{\\Delta x^2}\n\\begin{pmatrix}\n-2 & 1 & 0 & \\cdots & & 0 \\\\\n1 & -2 & 1 & 0 & \\cdots & \\vdots \\\\\n0 & 1 & -2 & 1 & \\ddots & \\\\\n & & \\ddots & \\ddots & \\ddots & & \\\\\n& & & 1 & -2 & 1 \\\\\n& & & & 1 & -2 \\\\\n\\end{pmatrix}\n~\n\\begin{pmatrix}\n\\vartheta(t, x^{0}) \\\\\n\\vartheta(t, x^{1}) \\\\\n\\vdots \\\\\n\\vartheta(t, x^{N-1}) \\\\\n\\end{pmatrix}.$\n\"\n\n# ╔═╡ 0309bf5e-4033-4832-86ed-146aea0863c0\nN = 101 # Number of grid elements\n\n# ╔═╡ 5c9ec071-6140-4af6-a418-df79c10daf5e\nΔx = L/(N-1) # Finite discretization \n\n# ╔═╡ 905337c5-9eee-4b67-b215-bfd7ca905e4c\n# Diffusion matrix\nM = spdiagm(-1 => ones(N-1), 0 => -2*ones(N), 1 => ones(N-1));\n\n# ╔═╡ f0e9f197-303b-48a6-a540-9b11513050d4\nMatrix(M)[1:5,1:5]\n\n# ╔═╡ 0e155080-2cfe-45c0-8166-ba91f95d6f8b\nmd\"### Approximated boundary conditions\n\nThe spatial approximation of the heat equation \n\n$\\frac{\\partial^2 \\vartheta(t,x)}{\\partial x^2} \\approx \\frac{1}{\\Delta x^2} \\left( \\vartheta(\\cdot, x^{n-1}) - 2 ~ \\vartheta(\\cdot, x^{n}) + \\vartheta(\\cdot, x^{n+1}) \\right)$\n\ncan not be evaluated directly at the grid points $x^{0}$ and $x^{N-1}$ because they depend on values from the not-existing grid points $x^{-1}$ and $x^{N}$. Therefore, the Neumann boundary condition is used to find $\\vartheta(\\cdot, x^{-1})$ and $\\vartheta(\\cdot, x^{N})$.\n\nThe Robin boundary condition states that the heat flux at both sides is described by\n\n$\\lambda \\left. \\frac{\\partial \\vartheta(\\cdot,x)}{\\partial x} \\right\\rvert_{x = 0} \\cdot \\vec{n} = -1 \\cdot \\lambda \\frac{\\vartheta(\\cdot, x^{1}) - \\vartheta(\\cdot, x^{-1})}{2 \\Delta x} = \\phi_{in}(t,x^{0})$\n\non the left boundary and \n\n$\\lambda \\left. \\frac{\\partial \\vartheta(\\cdot,x)}{\\partial x} \\right\\rvert_{x = L} \\cdot \\vec{n} = \\lambda \\frac{\\vartheta(\\cdot, x^{N}) - \\vartheta(\\cdot, x^{N-2})}{2 \\Delta x} = \\phi_{out}(t,x^{N-1})$\n\non the right boundary. Reformulating both equations lead to \n\n$\\vartheta(\\cdot, x^{-1}) = \\vartheta(\\cdot, x^{1}) + \\frac{2 \\Delta x}{\\lambda} ~ \\phi_{in}(t,x^{0})$\n\nand\n\n$\\vartheta(\\cdot, x^{N}) = \\vartheta(\\cdot, x^{N-2}) + \\frac{2 \\Delta x}{\\lambda} ~ \\phi_{out}(t,x^{N-1}).$\n\nOn the left boundary one yields \n\n$\\left. \\frac{\\partial^2 \\vartheta(t,x)}{\\partial x^2} \\right\\rvert_{x = 0} \\approx \\frac{1}{\\Delta x^2} \\left( - 2 ~ \\vartheta(\\cdot, x^{0}) + 2 ~ \\vartheta(\\cdot, x^{1}) \\right) + \\frac{2}{\\lambda ~ \\Delta x} \\phi_{in}(t,x^{0})$\n\nand analog on the right boundary\n\n$\\left. \\frac{\\partial^2 \\vartheta(t,x)}{\\partial x^2} \\right\\rvert_{x = 0} \\approx \\frac{1}{\\Delta x^2} \\left(2 ~ \\vartheta(\\cdot, x^{N-2}) - 2 ~ \\vartheta(\\cdot, x^{N-1}) \\right) + \\frac{2}{\\lambda ~ \\Delta x} \\phi_{out}(t,x^{N-1}).$\n\nThe induced and emitted flux are noted as \n\n$\\Phi_{in}(t) = \n\\begin{pmatrix}\nb \\\\\n0 \\\\\n\\vdots \\\\\n0 \\\\\n\\end{pmatrix}\n~ u(t)\n\n\\quad \\text{and} \\quad\n\n\\Phi_{out}(t) = \n\\begin{pmatrix}\n0 \\\\\n\\vdots \\\\\n0 \\\\\n1 \\\\\n\\end{pmatrix}\n~\\phi_{out}(t, x^{N-1})$\n\nand the diffusion matrix is described by\n\n$M = \n\\begin{pmatrix}\n-2 & 2 & 0 & \\cdots & & 0 \\\\\n1 & -2 & 1 & 0 & \\cdots & \\vdots \\\\\n0 & 1 & -2 & 1 & \\ddots & \\\\\n & & \\ddots & \\ddots & \\ddots & & \\\\\n& & & 1 & -2 & 1 \\\\\n& & & & 2 & -2 \\\\\n\\end{pmatrix}.$\n\"\n\n\n# ╔═╡ c61962e8-86bd-45ed-a62d-ffc812d8e35e\n# First row\nM[1,2] = 2;\n\n# ╔═╡ 33baab03-fbc1-4944-bc16-e4022b051c69\n# Last row\nM[end,end-1] = 2;\n\n# ╔═╡ facdbabf-6e2b-418a-a07e-36188244e585\nmd\"### Heat equation in State-space representation\n\nAfter the spatial approximation the heat equation has a form of \n\n$\\dot{\\theta}(t) = \\frac{\\alpha}{\\Delta x^2} ~ M ~ \\theta(t) + \\frac{2 \\alpha}{\\Delta x} ~ \\Phi_{in}(t) + \\frac{2 \\alpha}{\\Delta x} ~ \\Phi_{out}(t) \\tag{1}$\n\nwith M as diffusion matrix, temperature vector $\\quad \\theta(t) = \\left( \\vartheta(t, x^{0}), \\cdots, \\vartheta(t, x^{N-1}) \\right)^{\\top}$. \n\nThe original ODE $(1)$ is recasted in [state-space representation](https://en.wikipedia.org/wiki/State-space_representation) as\n\n$\\dot{\\theta}(t) = A ~ \\theta(t) + B ~ u(t) + E ~ w(t)$\n$y(t) = C \\theta(t)$\n\nwith \n- system matrix $A = \\frac{\\alpha}{\\Delta x^2} ~ M$, \n- input vector \n$B = \\frac{2 ~ \\alpha}{\\lambda ~ \\Delta x} \n\\begin{pmatrix}\nb \\\\\n0 \\\\\n\\vdots \\\\\n0\n\\end{pmatrix},$\n\n- disturbance vector \n$E = \\frac{2 ~ \\alpha}{\\lambda ~ \\Delta x} \n\\begin{pmatrix}\n0 \\\\\n\\vdots \\\\\n0 \\\\\n1\n\\end{pmatrix},$\n\n- emitted flux $w(t) = \\phi_{out}(t,x^{N-1})$ and\n- output vector $C = (0, \\cdots, 0, 1)$\n\nto describe the input-state-output behaviour including the integral controller. The emitted heat is considered as a disturbance. The PI controller is introduced as \n\n$u(t) = K_{p} ~ e(t)$ \n\nwith error \n$e(t) = y_{ref}(t) - y(t) = y_{ref}(t) - C ~ \\theta(t)$\n\n\nThe thermal dynamics with PI control is described by\n\n$\\dot{\\theta}(t) = A ~ \\theta(t) + B ~ K_{p} ~ e(t) + E ~ w(t).$\n\nThis is an ordinary differential equation (ODE) that can be solved with common solvers like forward Euler method or Runge-Kutta scheme.\n\n\"\n\n# ╔═╡ 5ef85c8f-cb69-42df-a098-ab98903a5f32\nA = α/(Δx^2) * M\n\n# ╔═╡ 25bb45ab-55af-4b8b-a11b-05be68631b78\nbegin\n\tB = spzeros(N); # Input vector\n\tB[1] = 2 * (α/(λ*Δx)) * b\n\tE = spzeros(N); # Disturbance vector\n\tE[end] = 2 * (α/(λ*Δx)) * b\n\tC = spzeros(1,N); # Output vector\n\tC[end] = 1;\nend\n\n# ╔═╡ 81a2d204-7386-4ed7-af89-7c076940817b\nw(θ) = ϕout(θ)\n\n# ╔═╡ 113d8c87-f4e6-4f1b-b188-879e7ad1e146\nfunction heat_eq!(dx,x,p,t)\n global u_hist\n \n err = yref - x[end]\n u = u_in(err)\n \n\n dx .= A * x + B * u + E * ϕout( x[end] ) # Integration of temperatures\n \n end\n\n# ╔═╡ d9138271-79fc-4b7c-8501-e8d8f3304cd5\nmd\"### Initial Conditions\n\nFor simplicity, the initial data of the heat equation is assumed as $\\vartheta_{0}(x)= 1000$ Kelvin for all $x \\in \\left[0,L\\right]$. This function is approximated to gain the initial conditions of the ODE \n\n$\\theta(0) = \\left( 300, \\cdots , 300 \\right)^{\\top}.$\n\"\n\n# ╔═╡ db6248c7-24b8-4f44-82d3-4892203297b5\nθ₀ = 300.0 * ones(N)\n\n# ╔═╡ 837632eb-b4db-47ec-8e4a-32a0fe81060b\nmd\"## Simulation\"\n\n# ╔═╡ 3f382fe5-2aa6-4d82-b939-977985ae8bbb\nΔt = 10^(-2) # Sampling time\n\n# ╔═╡ 02ba374a-15b4-4481-baf6-c1a3ce45ed5e\ntspan = (0.0, Tf)\n\n# ╔═╡ 0db82493-4dda-4992-aa5b-63734b556a90\nprob = ODEProblem( heat_eq!, θ₀, tspan ) # ODE Problem\n\n# ╔═╡ f7dcb8a6-0eb4-4762-942e-53362d4fe1ba\nsol = solve(prob,Euler(),dt=Δt,progress=true, saveat=1.0, save_start=true); # Solving the ODE\n\n# ╔═╡ 9c38b2b3-1622-41f3-a2c2-ff79b552e454\n# 1-dimensional grid\nxspan = 0 : Δx : L\n\n# ╔═╡ ba66c778-6c1c-4e32-934d-e8aca0eb1c66\nheatmap(sol.t, xspan, sol[1:end,1:end], xaxis=\"Time [s]\", yaxis=\"Position [m]\", title=\"Evolution of temperature\")\n\n# ╔═╡ 3dcfed19-8235-4309-b1c7-8d4e30ba6701\nplot(sol.t,[sol[1,:], sol[end-1,:]], label=[\"Left\" \"Right\"], title=\"Temperature at the left/right end\", legend=:bottomright)\n\n# ╔═╡ Cell order:\n# ╟─7bc3e874-b7be-11eb-0792-cf4e21e8c8b2\n# ╠═1158de2e-b7a6-43f3-a4e6-2ef9598a4ca5\n# ╠═30cebec3-9c70-4c6a-94a8-fdcbe6cda783\n# ╟─93fb0f90-c454-410f-aa5e-cd8e6c055dc0\n# ╠═b294bbfc-45e7-4a15-a839-5ceb56746be7\n# ╠═543b3f13-4cef-46f1-9141-172b81e7d48b\n# ╟─1deb117d-21f7-4329-a7db-2be26e43129a\n# ╠═78292b42-a203-496c-a143-def3ff128f69\n# ╠═0ecea33b-bf24-48eb-b4b4-bbf288e62e6b\n# ╠═3a059f15-d850-468a-bb1d-f42ef5b24609\n# ╟─7202aa44-b393-4e0b-9c34-2a9b9376194c\n# ╠═2775433b-d8b1-48be-9b60-08641db15803\n# ╠═4c59909b-ffb6-4dfc-94ab-142d41deeeab\n# ╠═4f0c3671-d3bf-4e36-a1f3-8e8d0ac0e454\n# ╠═23343fa0-6033-498d-b089-d43e4382e332\n# ╟─daa17037-fd28-460c-97f5-826742b9a419\n# ╠═892fb737-6fc0-4438-b586-039a440b01f9\n# ╠═0309bf5e-4033-4832-86ed-146aea0863c0\n# ╠═5c9ec071-6140-4af6-a418-df79c10daf5e\n# ╠═905337c5-9eee-4b67-b215-bfd7ca905e4c\n# ╠═f0e9f197-303b-48a6-a540-9b11513050d4\n# ╟─0e155080-2cfe-45c0-8166-ba91f95d6f8b\n# ╠═c61962e8-86bd-45ed-a62d-ffc812d8e35e\n# ╠═33baab03-fbc1-4944-bc16-e4022b051c69\n# ╟─facdbabf-6e2b-418a-a07e-36188244e585\n# ╠═5ef85c8f-cb69-42df-a098-ab98903a5f32\n# ╠═25bb45ab-55af-4b8b-a11b-05be68631b78\n# ╠═81a2d204-7386-4ed7-af89-7c076940817b\n# ╠═113d8c87-f4e6-4f1b-b188-879e7ad1e146\n# ╟─d9138271-79fc-4b7c-8501-e8d8f3304cd5\n# ╠═db6248c7-24b8-4f44-82d3-4892203297b5\n# ╠═837632eb-b4db-47ec-8e4a-32a0fe81060b\n# ╠═069cf190-77cc-4757-9db3-1945ddd07438\n# ╠═3f382fe5-2aa6-4d82-b939-977985ae8bbb\n# ╠═02ba374a-15b4-4481-baf6-c1a3ce45ed5e\n# ╠═0db82493-4dda-4992-aa5b-63734b556a90\n# ╠═f7dcb8a6-0eb4-4762-942e-53362d4fe1ba\n# ╠═9c38b2b3-1622-41f3-a2c2-ff79b552e454\n# ╠═ba66c778-6c1c-4e32-934d-e8aca0eb1c66\n# ╠═3dcfed19-8235-4309-b1c7-8d4e30ba6701\n", "meta": {"hexsha": "bd0590b17496d44f5e28d1ee1d84dfa6f3d551e1", "size": 12973, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/control/prop_control.jl", "max_stars_repo_name": "stephans3/IntroHeatEquation.jl", "max_stars_repo_head_hexsha": "37e9c9ededeb7cdfc2ffb0c11ac1661d157a0288", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/control/prop_control.jl", "max_issues_repo_name": "stephans3/IntroHeatEquation.jl", "max_issues_repo_head_hexsha": "37e9c9ededeb7cdfc2ffb0c11ac1661d157a0288", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/control/prop_control.jl", "max_forks_repo_name": "stephans3/IntroHeatEquation.jl", "max_forks_repo_head_hexsha": "37e9c9ededeb7cdfc2ffb0c11ac1661d157a0288", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.7965686275, "max_line_length": 298, "alphanum_fraction": 0.6571340476, "num_tokens": 5763, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802395624257, "lm_q2_score": 0.8840392771633079, "lm_q1q2_score": 0.8119726070715487}} {"text": "using Random, Distributions, KernelDensity, Plots; pyplot()\nRandom.seed!(0)\n\nmu1, sigma1 = 10, 5\nmu2, sigma2 = 40, 12\ndist1, dist2 = Normal(mu1,sigma1), Normal(mu2,sigma2)\np = 0.3\nmixRv() = (rand() <= p) ? rand(dist1) : rand(dist2)\nmixPDF(x) = p*pdf(dist1,x) + (1-p)*pdf(dist2,x)\n\nn = 2000\ndata = [mixRv() for _ in 1:n]\n\nkdeDist = kde(data)\n\nxGrid = -20:0.1:80\npdfKDE = pdf(kdeDist,xGrid)\n\nplot(xGrid, pdfKDE, c=:blue, label=\"KDE PDF\")\nstephist!(data, bins=50, c=:black, normed=:true, label=\"Histogram\")\np1 = plot!(xGrid, mixPDF.(xGrid), c=:red, label=\"Underlying PDF\",\n xlims=(-20,80), ylims=(0,0.035), legend=:topleft,\n xlabel=\"X\", ylabel = \"Density\")\n\nhVals = [0.5,2,10]\nkdeS = [kde(data,bandwidth=h) for h in hVals]\nplot(xGrid, pdf(kdeS[1],xGrid), c = :green, label= \"h=$(hVals[1])\")\nplot!(xGrid, pdf(kdeS[2],xGrid), c = :blue, label= \"h=$(hVals[2])\")\np2 = plot!(xGrid, pdf(kdeS[3],xGrid), c = :purple, label= \"h=$(hVals[3])\",\n xlims=(-20,80), ylims=(0,0.035), legend=:topleft, \n xlabel=\"X\", ylabel = \"Density\")\nplot(p1,p2,size = (800,400))", "meta": {"hexsha": "e074b6e2b5f906ca411deaedd0086f34cf6e0c53", "size": 1056, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "4_chapter/KDE.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "4_chapter/KDE.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "4_chapter/KDE.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 33.0, "max_line_length": 74, "alphanum_fraction": 0.625, "num_tokens": 431, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9465966717067253, "lm_q2_score": 0.8577681068080749, "lm_q1q2_score": 0.8119604350007026}} {"text": "function sigmoid(z)\n 1 ./ (1 .+ e .^ (-z))\nend\n\nfunction sigmoid_grad(z)\n sigmoid(z) .* (1 - sigmoid(z))\nend\n\nfunction init_theta(num_input, num_output, nums_hidden)\n num_units = [num_input, nums_hidden..., num_output]\n layers = length(num_units)\n Theta = [rand(num_units[i], num_units[i-1]+1) - 0.5 for i = 2:layers]\nend\n\nfunction cost(Y, Y_pred)\n len = size(Y, 1)\n J = (-1.0 / len ) * sum((Y .* log(Y_pred)) + ((1-Y) .* log(1-Y_pred)))\nend\n\nfunction forward_prop(X, Theta)\n A = {}\n H = {}\n push!(H, X)\n push!(A, zeros(1, 1))\n\n for t=1:length(Theta)\n # add bias\n H[t] = [ones(size(H[t], 1), 1) H[t]]\n # pre-activation\n push!(A, H[t] * Theta[t]')\n # activation\n push!(H, sigmoid(A[t+1]))\n end\n A, H, H[end]\nend\n\nfunction back_prop(X, Y, Theta)\n len = size(X, 1)\n T = length(Theta)\n\n Theta_grad = {}\n for i=1:T\n push!(Theta_grad, zeros(size(Theta[i])))\n end\n\n A, H, Y_pred = forward_prop(X, Theta)\n\n delta_N = {}\n \n for t=1:T\n push!(delta_N, {})\n end\n\n delta = Y_pred - Y\n\n delta_N[T] = delta\n\n for t=0:T-2\n delta = (delta*Theta[T-t][:, 2:end]) .* sigmoid_grad(A[T-t])\n delta_N[T-t-1] = delta\n end\n\n for t=1:T\n Theta_grad[t] = Theta_grad[t] + delta_N[t]' * H[t]\n end\n \n Y_pred, Theta_grad\nend\n\nfunction fit(X, Y, Theta, learning_rate, epochs)\n J_list = zeros(epochs)\n for i=1:epochs\n Y_pred, Theta_grad = back_prop(X, Y, Theta)\n J_list[i] = cost(Y_train, Y_pred)\n for t=1:length(Theta)\n Theta[t] = Theta[t] - learning_rate*Theta_grad[t]\n end\n end\n Theta, J_list\nend\n\nfunction predict(X, Theta)\n _, _, y = forward_prop(X, Theta)\n y \nend\n", "meta": {"hexsha": "a6827e46bee07d0ce6931a8dc41876e2ac37e8ca", "size": 1622, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Neu.jl", "max_stars_repo_name": "rinuboney/Neu.jl", "max_stars_repo_head_hexsha": "5382ac83b650cf418a69536455dfb16af92aabfa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2015-05-29T13:21:59.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-27T10:07:35.000Z", "max_issues_repo_path": "src/Neu.jl", "max_issues_repo_name": "rinuboney/Neu.jl", "max_issues_repo_head_hexsha": "5382ac83b650cf418a69536455dfb16af92aabfa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/Neu.jl", "max_forks_repo_name": "rinuboney/Neu.jl", "max_forks_repo_head_hexsha": "5382ac83b650cf418a69536455dfb16af92aabfa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-09-04T01:30:15.000Z", "max_forks_repo_forks_event_max_datetime": "2019-09-04T01:30:15.000Z", "avg_line_length": 18.8604651163, "max_line_length": 71, "alphanum_fraction": 0.5918618989, "num_tokens": 571, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9597620550745211, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.8119034446506279}} {"text": "\r\ninclude(\"../../util/src/util_module.jl\")\r\n\r\n\r\nmodule dev\r\n\r\n\r\n\r\nmodule matrix_vector_prod\r\nusing LinearAlgebra\r\n\r\nfunction matrix_vector_prod_default(A::Matrix{T}, x::Vector{T}) where {T<:Float64}\r\n sz_a = size(A)\r\n sz_b = size(x)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 1\r\n error(\"B is not a vector\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n c = A * x\r\n return c\r\nend\r\n\r\n\r\nfunction matrix_vector_prod_inner(A, x)\r\n # A in R^m-by-p\r\n # x in R^p-by-1\r\n # c = A * x in R^m-by-1\r\n sz_a = size(A)\r\n sz_b = size(x)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 1\r\n error(\"B is not a vector\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n\r\n c = zeros(eltype(A), sz_a[1])\r\n @inbounds @simd for ii = 1:sz_a[1]\r\n # c[ii] = A[ii, :]' * x\r\n c[ii] = dot( @view(A[ii, :]), x)\r\n end\r\n return c\r\nend\r\n\r\nfunction matrix_vector_prod_inner_loop(A, x)\r\n # A in R^m-by-p\r\n # x in R^p-by-1\r\n # c = A * x in R^m-by-1\r\n sz_a = size(A)\r\n sz_b = size(x)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 1\r\n error(\"B is not a vector\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n\r\n c = zeros(eltype(A), sz_a[1])\r\n @inbounds @simd for ii = 1:sz_a[1]\r\n inner_sum = zero(eltype(x))\r\n @inbounds @simd for jj = 1:sz_a[2]\r\n inner_sum += A[ii, jj] * x[jj] \r\n end\r\n c[ii] = inner_sum\r\n end\r\n return c\r\nend\r\n\r\nfunction matrix_vector_prod_outer(A, x)\r\n # A in R^m-by-p\r\n # x in R^p-by-1\r\n # c = A * x in R^m-by-1\r\n sz_a = size(A)\r\n sz_b = size(x)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 1\r\n error(\"B is not a vector\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n\r\n c = zeros(eltype(A), sz_a[1])\r\n @inbounds @simd for jj = 1:sz_a[2]\r\n c .+= @view( A[:, jj] ) .* x[jj]\r\n end\r\n return c\r\nend\r\n\r\nfunction matrix_vector_prod_outer_loop(A, x)\r\n # A in R^m-by-p\r\n # x in R^p-by-1\r\n # c = A * x in R^m-by-1\r\n sz_a = size(A)\r\n sz_b = size(x)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 1\r\n error(\"B is not a vector\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n\r\n c = zeros(eltype(A), sz_a[1])\r\n @inbounds @simd for jj = 1:sz_a[2]\r\n @inbounds @simd for ii = 1:sz_a[1]\r\n c[ii] += A[ii, jj] * x[jj]\r\n end\r\n end\r\n return c\r\nend\r\n\r\n\r\n\r\nfunction matrix_mult_inner(A, B)\r\n # A in R^m-by-p\r\n # B in R^p-by-n\r\n # C = A * B in R^m-by-n\r\n # [ma, p] = size(A)\r\n # [, p] = size(A)\r\n sz_a = size(A)\r\n sz_b = size(B)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 2\r\n error(\"B is not a matrix\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n \r\n C = zeros(eltype(A), sz_a[1], sz_b[2])\r\n for ii = 1:sz_a[1]\r\n for jj = 1:sz_b[2]\r\n C[ii, jj] = dot( A[ii, :], B[:, jj] )\r\n # C[ii, jj] = A[ii, :]' * B[:, jj]\r\n end\r\n end\r\n return C\r\nend\r\n\r\nfunction matrix_mult_outer(A, B)\r\n # A in R^m-by-p\r\n # B in R^p-by-n\r\n # C = A * B in R^m-by-n\r\n # [ma, p] = size(A)\r\n # [, p] = size(A)\r\n sz_a = size(A)\r\n sz_b = size(B)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 2\r\n error(\"B is not a matrix\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n \r\n C = zeros(eltype(A), sz_a[1], sz_b[2])\r\n for kk = 1:sz_a[2]\r\n C += A[:, kk] * B[kk, :]'\r\n end\r\n return C\r\nend\r\n\r\nfunction matrix_mult_default(A, B)\r\n # A in R^m-by-p\r\n # B in R^p-by-n\r\n # C = A * B in R^m-by-n\r\n # [ma, p] = size(A)\r\n # [, p] = size(A)\r\n sz_a = size(A)\r\n sz_b = size(B)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 2\r\n error(\"B is not a matrix\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n C = A * B\r\n return C\r\nend\r\n\r\nfunction matrix_mult_inner_loop(A, B)\r\n # A in R^m-by-p\r\n # B in R^p-by-n\r\n # C = A * B in R^m-by-n\r\n # [ma, p] = size(A)\r\n # [, p] = size(A)\r\n sz_a = size(A)\r\n sz_b = size(B)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 2\r\n error(\"B is not a matrix\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n C = zeros(eltype(A), sz_a[1], sz_b[2])\r\n\r\n for ii = 1:sz_a[1]\r\n for jj = 1:sz_b[2]\r\n inner_sum = zero(eltype(A))\r\n for kk = 1:sz_a[2]\r\n inner_sum += A[ii, kk] * B[kk, jj]\r\n end\r\n C[ii, jj] = inner_sum\r\n end\r\n end\r\n return C\r\nend\r\n\r\nfunction matrix_mult_outer_loop(A, B)\r\n #\r\n # Algo probably wrong\r\n #\r\n # A in R^m-by-p\r\n # B in R^p-by-n\r\n # C = A * B in R^m-by-n\r\n # [ma, p] = size(A)\r\n # [, p] = size(A)\r\n sz_a = size(A)\r\n sz_b = size(B)\r\n \r\n if length(sz_a) != 2\r\n error(\"A is not a matrix\")\r\n end\r\n if length(sz_b) != 2\r\n error(\"B is not a matrix\")\r\n end\r\n if sz_a[2] != sz_b[1]\r\n error(\"A and B are not compatible (size mismatch) \")\r\n end\r\n C = zeros(eltype(A), sz_a[1], sz_b[2])\r\n\r\n #=\r\n for kk = 1:sz_a[2]\r\n for ii = 1:sz_a[1]\r\n C[ii, kk] += A[ii, kk] * B[kk, ii]\r\n end\r\n end\r\n =#\r\n for ii = 1:sz_b[2]\r\n for kk = 1:sz_a[2]\r\n C[:, ii] += A[:, kk] * B[kk, ii]\r\n end\r\n end\r\n\r\n return C\r\nend\r\n\r\n\r\nend # dev\r\n\r\n\r\nmodule devtest\r\nusing ..dev\r\nusing BenchmarkTools\r\n\r\nfunction benchmark_matrix_vector_prod_default(A, x)\r\n b = @benchmark dev.matrix_vector_prod_default($A, $x)\r\n\r\n println(\"=== matrix_vector_prod_default ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\n\r\nfunction benchmark_matrix_vector_prod_inner(A, x)\r\n b = @benchmark dev.matrix_vector_prod_inner($A, $x)\r\n\r\n println(\"=== matrix_vector_prod_inner ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_vector_prod_outer(A, x)\r\n b = @benchmark dev.matrix_vector_prod_outer($A, $x)\r\n\r\n println(\"=== matrix_vector_prod_outer ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_vector_prod_inner_loop(A, x)\r\n b = @benchmark dev.matrix_vector_prod_inner_loop($A, $x)\r\n\r\n println(\"=== matrix_vector_prod_inner_loop ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_vector_prod_outer_loop(A, x)\r\n b = @benchmark dev.matrix_vector_prod_outer_loop($A, $x)\r\n\r\n println(\"=== matrix_vector_prod_outer_loop ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\n\r\nfunction benchmark_all()\r\n\r\n # size definitions\r\n # m = 1000\r\n # p = 5000\r\n\r\n # m = 10\r\n # p = 500000\r\n\r\n m = 100000\r\n p = 50\r\n\r\n # allocate the matrices\r\n A = rand(m, p)\r\n x = rand(p)\r\n\r\n benchmark_matrix_vector_prod_default(A, x)\r\n benchmark_matrix_vector_prod_inner(A, x)\r\n benchmark_matrix_vector_prod_outer(A, x)\r\n benchmark_matrix_vector_prod_inner_loop(A, x)\r\n benchmark_matrix_vector_prod_outer_loop(A, x)\r\n\r\nend\r\n\r\n\r\n\r\nend\r\n\r\nmodule devtest_matrix\r\n using ..dev\r\n using BenchmarkTools\r\n\r\n\r\n\r\nfunction benchmark_matrix_mult_inner(A, B)\r\n b = @benchmark dev.matrix_mult_inner($A, $B)\r\n\r\n println(\"=== matrix_mult_inner ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\n end\r\n\r\nfunction benchmark_matrix_mult_outer(A, B)\r\n b = @benchmark dev.matrix_mult_outer($A, $B)\r\n\r\n println(\"=== matrix_mult_outer ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_mult_default(A, B)\r\n b = @benchmark dev.matrix_mult_default($A, $B)\r\n\r\n println(\"=== matrix_mult_default ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_mult_inner_loop(A, B)\r\n b = @benchmark dev.matrix_mult_inner_loop($A, $B)\r\n\r\n println(\"=== matrix_mult_inner_loop ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\nfunction benchmark_matrix_mult_outer_loop(A, B)\r\n b = @benchmark dev.matrix_mult_outer_loop($A, $B)\r\n\r\n println(\"=== matrix_mult_outer_loop ===\")\r\n show(\r\n stdout,\r\n MIME(\"text/plain\"),\r\n b )\r\n println(\"\\n\\n\")\r\nend\r\n\r\n\r\n\r\n\r\n\r\nfunction benchmark_all()\r\n # size definitions\r\n m = 100\r\n p = 500\r\n n = 100\r\n\r\n # allocate the matrices\r\n A = rand(m, p)\r\n B = rand(p, n)\r\n\r\n\r\n benchmark_matrix_mult_inner(A, B)\r\n benchmark_matrix_mult_outer(A, B)\r\n benchmark_matrix_mult_default(A, B)\r\n benchmark_matrix_mult_inner_loop(A, B)\r\n # benchmark_matrix_mult_outer_loop(A, B)\r\nend\r\n\r\n\r\n \r\n \r\n # multiply\r\n #= \r\n Ci = dev.matrix_mult_inner(A, B)\r\n Co = dev.matrix_mult_outer(A, B)\r\n Cd = A * B;\r\n\r\n println( all( abs.( Ci .- Co ) .<= 1e-10 ) ) =#\r\n\r\nend\r\n\r\n#= \r\nmodule plottest\r\n \r\nusing PyPlot\r\nPyPlot.pygui(true)\r\n\r\nx = LinRange(-2, 2, 30) # [1, 2, 3]\r\ny = LinRange(-4, 1, 30) # [10, 20, 30]\r\nz = [i.^2 + j.^2 for i in x, j in y]\r\nz2 = [50, 51, 52]\r\n\r\nPyPlot.figure()\r\n# PyPlot.plot3D(x, y, z2)\r\nPyPlot.plot_surface(x, y, z, alpha = 0.5)\r\n\r\n\r\nend =#\r\n\r\n\r\n", "meta": {"hexsha": "98fa81549ec9bc275009f4cd949d836feeb2b8d8", "size": 10426, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "__lib__/math/linalg/test/matrix_vector_product_tryouts.jl", "max_stars_repo_name": "HomoModelicus/julia", "max_stars_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "__lib__/math/linalg/test/matrix_vector_product_tryouts.jl", "max_issues_repo_name": "HomoModelicus/julia", "max_issues_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "__lib__/math/linalg/test/matrix_vector_product_tryouts.jl", "max_forks_repo_name": "HomoModelicus/julia", "max_forks_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.2342158859, "max_line_length": 83, "alphanum_fraction": 0.5038365624, "num_tokens": 3178, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9343951552333004, "lm_q2_score": 0.8688267830311354, "lm_q1q2_score": 0.8118275368012268}} {"text": "using Distributions, Plots; pyplot()\n\ntriangDist = TriangularDist(0,2,1)\nxGrid = 0:0.1:2\nN = 10^6\ninverseSampledData = quantile.(triangDist,rand(N))\n\nhistogram( inverseSampledData, bins=30, normed=true,\n\tylims=(0,1.1), label=\"Inverse transform data\")\nplot!( xGrid, pdf.(triangDist,xGrid), c=:red, lw=4, \n\txlabel=\"x\", label=\"PDF\", ylabel = \"Density\", legend=:topright)", "meta": {"hexsha": "0aa22e56473e9271f326ad0f4f633bc401184962", "size": 367, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "3_chapter/triangularDistInv.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "3_chapter/triangularDistInv.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "3_chapter/triangularDistInv.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 33.3636363636, "max_line_length": 63, "alphanum_fraction": 0.7138964578, "num_tokens": 130, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533107374443, "lm_q2_score": 0.870597270087091, "lm_q1q2_score": 0.811791306811689}} {"text": "## compose_matrix.jl : methods to compose example matrices\n\n\"\"\"\nanisotropic2d(ϵ, n, m)\n\n2d anisotropic Laplacian on [0,1]^2 using n x m points and anisotropy number ϵ\n\"\"\"\nanisotropic2d(ϵ, n, m) = stencil2mat(SArray{Tuple{3,3}}([0 -m^2 0 -ϵ*n^2 2(ϵ*n^2+m^2) -ϵ*n^2 0 -m^2 0]), n, m)\n\n\"\"\"\nanisotropic2d(ϵ, β, n, m)\n\n2d rotated anisotropic Laplacian on [0,1]^2 using n x m points with anisotropy number ϵ and angle β\n\"\"\"\nanisotropic2d(ϵ, β, n, m) = stencil2mat(SArray{Tuple{3,3}}([0 0 0 -n^2*(ϵ*cos(β)^2+sin(β)^2) 2n^2*(ϵ*cos(β)^2+sin(β)^2) -n^2*(ϵ*cos(β)^2+sin(β)^2) 0 0 0]), n, m) + stencil2mat(SArray{Tuple{3,3}}([n*m/2*((ϵ-1)*cos(β)*sin(β)) 0 -n*m/2*((ϵ-1)*cos(β)*sin(β)) 0 0 0 -n*m/2*((ϵ-1)*cos(β)*sin(β)) 0 n*m/2*((ϵ-1)*cos(β)*sin(β))]), n, m) + stencil2mat(SArray{Tuple{3,3}}([0 -m^2*(ϵ*sin(β)^2+cos(β)^2) 0 0 2m^2*(ϵ*sin(β)^2+cos(β)^2) 0 0 -m^2*(ϵ*sin(β)^2+cos(β)^2) 0]), n, m)\n\n\n\"\"\"\nanisotropic3d(ϵ₁, ϵ₂, n, m, l)\n\n3d anisotropic Laplacian on [0,1]^3 using n x m x l points and anisotropy numbers ϵ₁ and ϵ₂\n\"\"\"\nanisotropic3d(ϵ₁, ϵ₂, n, m, l) = stencil2mat(SArray{Tuple{3,3,3}}([0 0 0 0 -l^2 0 0 0 0 0 -ϵ₂*m^2 0 -ϵ₁*n^2 2(ϵ₁*n^2+ϵ₂*m^2+l^2) -ϵ₁*n^2 0 -ϵ₂*m^2 0 0 0 0 0 -l^2 0 0 0 0]), n, m, l)\n", "meta": {"hexsha": "808df6d4dd6c4fdaef682cb0f20c3fddab5f92d5", "size": 1199, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/compose_matrix.jl", "max_stars_repo_name": "PieterjanRobbe/NotSoSimpleMultigrid.jl", "max_stars_repo_head_hexsha": "877de2f5be2c96bc9c7796bf321434858ba9659d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/compose_matrix.jl", "max_issues_repo_name": "PieterjanRobbe/NotSoSimpleMultigrid.jl", "max_issues_repo_head_hexsha": "877de2f5be2c96bc9c7796bf321434858ba9659d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/compose_matrix.jl", "max_forks_repo_name": "PieterjanRobbe/NotSoSimpleMultigrid.jl", "max_forks_repo_head_hexsha": "877de2f5be2c96bc9c7796bf321434858ba9659d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.9583333333, "max_line_length": 464, "alphanum_fraction": 0.5954962469, "num_tokens": 667, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9643214460461697, "lm_q2_score": 0.8418256472515683, "lm_q1q2_score": 0.8117905254763851}} {"text": "\"\"\"\n _orientation(x::Tuple{<:Number, <:Number}, y::Tuple{<:Number, <:Number}, z::Tuple{<:Number, <:Number})\n\nUtility function for `convex_hull`. Given an ordered triplet, this function returns if three\npoints are collinear, oriented in clock-wise or anticlock-wise direction. \n\"\"\"\nfunction _orientation(x::Tuple{<:Number, <:Number}, y::Tuple{<:Number, <:Number}, z::Tuple{<:Number, <:Number})\n \n a = ((y[2] - x[2]) * (z[1] - y[1]) - (y[1] - x[1]) * (z[2] - y[2]))\n\n if isapprox(a, 0, atol=1e-6)\n return 0 # collinear\n elseif a > 0 \n return 1 # clock-wise\n else \n return 2 # counterclock-wise\n end\n\nend\n\n\"\"\"\n _lt_filter(a::Tuple{<:Number, <:Number}, b::Tuple{<:Number, <:Number})\n\nUtility function for sorting step in `convex_hull`. Given two points, `a` and `b`, this function \nreturns true if `a` has larger polar angle (counterclock-wise direction) than `b` w.r.t. first point `chull_p1`. \n\"\"\"\nfunction _lt_filter(a::Tuple{<:Number, <:Number}, b::Tuple{<:Number, <:Number})\n o = QCO._orientation(chull_p1, a, b)\n\n if o == 0\n if LA.norm(collect(chull_p1 .- b))^2 >= LA.norm(collect(chull_p1 .- a))^2\n return true\n else\n return false\n end\n elseif o == 2\n return true\n else\n return false\n end\n \nend\n\n\"\"\"\n convex_hull(points::Vector{T}) where T<:Tuple{Number, Number}\n\nGraham's scan algorithm to compute the convex hull of a finite set of `n` points in a plane \nwith time complexity `O(n*log(n))`. Given a vector of tuples of co-ordinates, this function returns a \nvector of tuples of co-ordinates which form the convex hull of the given set of points. \n\nSources: https://doi.org/10.1016/0020-0190(72)90045-2\n https://en.wikipedia.org/wiki/Graham_scan \n\"\"\"\nfunction convex_hull(points::Vector{T}) where T<:Tuple{Number, Number}\n \n num_points = size(points)[1]\n\n if num_points == 0\n Memento.error(_LOGGER, \"Atleast one point is necessary for evaluating the convex hull\")\n elseif num_points <= 2 \n return points\n end\n\n min_y = points[1][2]\n min = 1\n \n # Bottom-most or else the left-most point when tie\n for i in 2:num_points\n y = points[i][2]\n if (y < min_y) || (y == min_y && points[i][1] < points[min][1])\n min_y = y\n min = i\n end\n end\n \n # Placing the bottom/left-most point at first position\n points[1], points[min] = points[min], points[1]\n\n global chull_p1 = points[1]\n\n points = Base.sort(points, lt = QCO._lt_filter)\n\n # If two or more points are collinear with chull_p1, remove all except the farthest from chull_p1\n arr_size = 2\n for i in 2:num_points\n while (i < num_points) && (QCO._orientation(chull_p1, points[i], points[i+1]) == 0)\n i += 1\n end\n points[arr_size] = points[i]\n arr_size += 1\n end\n\n if arr_size <= 3\n return []\n end\n\n chull = []\n append!(chull, [points[1]])\n append!(chull, [points[2]])\n append!(chull, [points[3]])\n \n for i in 4:(arr_size-1)\n while (size(chull)[1] > 1) && (QCO._orientation(chull[end - 1], chull[end], points[i]) != 2)\n pop!(chull)\n end\n append!(chull, [points[i]])\n end\n \n return chull\nend", "meta": {"hexsha": "fa4242a6aaa84276362e9219a9716e22c1ca220a", "size": 3293, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/chull.jl", "max_stars_repo_name": "lockwo/QuantumCircuitOpt.jl", "max_stars_repo_head_hexsha": "952fcd8ca37ee9d1670699390629f8993097883e", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 23, "max_stars_repo_stars_event_min_datetime": "2021-06-30T06:23:46.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-06T18:08:09.000Z", "max_issues_repo_path": "src/chull.jl", "max_issues_repo_name": "lockwo/QuantumCircuitOpt.jl", "max_issues_repo_head_hexsha": "952fcd8ca37ee9d1670699390629f8993097883e", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 47, "max_issues_repo_issues_event_min_datetime": "2021-06-30T05:55:10.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-25T00:15:34.000Z", "max_forks_repo_path": "src/chull.jl", "max_forks_repo_name": "lockwo/QuantumCircuitOpt.jl", "max_forks_repo_head_hexsha": "952fcd8ca37ee9d1670699390629f8993097883e", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2021-07-12T20:17:13.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-06T18:08:40.000Z", "avg_line_length": 29.9363636364, "max_line_length": 113, "alphanum_fraction": 0.6000607349, "num_tokens": 992, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.914900950352329, "lm_q2_score": 0.8872045892435128, "lm_q1q2_score": 0.8117043218558375}} {"text": "## Exercise 4-7\n## Make a more general version of circle called arc that takes an additional parameter angle, which determines what fraction of a circle to draw. angle is in units of degrees, so when angle = 360, arc should draw a complete circle.\nusing ThinkJulia\n\nprintln(\"Ans: \")\n\nfunction arc(turtle::Turtle, radius::Int, angle)\n len = 2 * π * radius * angle / 360 # finding the length of the arc 2πr * a/360\n n = Int(trunc(len) / 3) + 1 # number of times the turtle will rotate and draw \n\n len /= n\n angle /= n \n\n for i in 1:n\n forward(turtle, len)\n turn(turtle, -angle)\n end\nend\n\n@svg begin \n arc(Turtle(), 200, 270)\nend\n\nprintln(\"End.\")\n", "meta": {"hexsha": "527d43a706446b68f6b46552574ca3275eaaed2c", "size": 680, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Chapter4/ex7.jl", "max_stars_repo_name": "yashppawar/ThinkJuliaExercises.jl", "max_stars_repo_head_hexsha": "72145b969dc51ebcac413a10004175ebc63cd5c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-02-13T14:11:30.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-13T14:11:30.000Z", "max_issues_repo_path": "Chapter4/ex7.jl", "max_issues_repo_name": "yashppawar/ThinkJuliaExercises.jl", "max_issues_repo_head_hexsha": "72145b969dc51ebcac413a10004175ebc63cd5c2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter4/ex7.jl", "max_forks_repo_name": "yashppawar/ThinkJuliaExercises.jl", "max_forks_repo_head_hexsha": "72145b969dc51ebcac413a10004175ebc63cd5c2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.2, "max_line_length": 231, "alphanum_fraction": 0.6588235294, "num_tokens": 195, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9740426450627306, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.8116936970454903}} {"text": "using Symbolics\nusing SatisfiabilityInterface\n\n\nvars = @variables x, y, z, w\n\nconstraints = [\n x ∈ 1:2\n y ∈ 2:5\n z ∈ 3:7\n z == x * y\n]\n\nprob = DiscreteCSP(constraints)\n\nsolve(prob)\n\nall_solutions(prob)\n\n\n###\n\n\nvars = @variables x, y, z, w\n\nconstraints = [\n x ∈ 1:2\n y ∈ 2:5\n z ∈ 3:7\n w ∈ 4:6\n z == x + y\n w == z + x\n]\n\nprob = DiscreteCSP(constraints)\nsolve(prob)\nall_solutions(prob)\n\n\n###\n\nconstraints = [\n x ∈ 1:2\n y ∈ 2:5\n z ∈ 3:7\n z == 2x\n]\n\nall_solutions(DiscreteCSP(constraints))\n\n###\n\nvars = @variables x, y, z \n\n# constraints = [\n# x ∈ 1:4\n# y ∈ 2:5\n# z == x + y\n# z ≤ 4\n# ]\n\n\n\n### Subset sum Problem\n\n# https://en.wikipedia.org/wiki/Subset_sum_problem\n\n\n# Given a set of integers like {−7, −3, −2, 5, 8}, \n# do a subset sum to 0?\n\ns = [-7, -3, -2, 5, 8]\nn = length(s)\nx = [Num(Variable(:x, i)) for i in 1:n]\n\nconstraints = [\n [x[i] ∈ (0:1) for i in 1:n]\n sum(x[i] * s[i] for i in 1:n) == 0\n sum(x) ≠ 0 # at least one is not 0\n ]\n\nprob = DiscreteCSP(constraints)\n\nall_solutions(prob)\n\n\n#####\n\n@variables x, y, z\n\nconstraints = [\n x ∈ 1:2\n y ∈ 2:5\n z ∈ 3:7\n z == 2x\n]\n\n\n\nprob = DiscreteCSP(constraints)\nall_solutions(prob)\n\n## Example from https://github.com/newptcai/BeeEncoder.jl\n\nfunction make_vector(name::Symbol, range)\n return Num.(Variable.(name, range))\nend\n\n@variables x, y, z, w\nb = make_vector(:b, 1:4)\n# @variables b[1:4]\n\n\n¬(x::Num) = 1 - x # assumes a Boolean variable\n\n\nconstraints = [\n x ∈ 0:5\n y ∈ -4:9\n z ∈ -5:10\n w ∈ 0:10\n\n [bb ∈ 0:1 for bb ∈ b]\n \n x + y == z\n \n b[1] == ¬b[2]\n b[2] == 1\n\n ¬b[1] + b[2] + ¬b[3] + b[4] == w\n]\n\nprob = DiscreteCSP(constraints)\nsolve(prob)\n\n# @time length(all_solutions(prob)) # 1184 solutions in 38 seconds\n\n\n\n### \n\nconstraints = [\n x ∈ -5:5\n y ∈ -4:9\n x - y^2 == 3\n # sum(b) == w\n\n]\n\n\nprob = DiscreteCSP(constraints) \nall_solutions(prob)\n\n### \n\n@variables x, y\n\nconstraints = [\n x ∈ -5:5\n y ∈ -4:9\n 10x + y == 9\n # sum(b) == w\n]\n\nprob = DiscreteCSP(constraints) \nall_solutions(prob)\n\n\n", "meta": {"hexsha": "022f5254f31ff8b8c27d032a31604903f6c1905e", "size": 2090, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/examples.jl", "max_stars_repo_name": "Wimmerer/SatisfiabilityInterface.jl", "max_stars_repo_head_hexsha": "b2148fadf45dca6304e9f4b571d1e3ea54c85ba4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2021-01-15T22:50:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-31T22:24:42.000Z", "max_issues_repo_path": "examples/examples.jl", "max_issues_repo_name": "Wimmerer/SatisfiabilityInterface.jl", "max_issues_repo_head_hexsha": "b2148fadf45dca6304e9f4b571d1e3ea54c85ba4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-06-26T21:09:18.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-08T12:12:43.000Z", "max_forks_repo_path": "examples/examples.jl", "max_forks_repo_name": "Wimmerer/SatisfiabilityInterface.jl", "max_forks_repo_head_hexsha": "b2148fadf45dca6304e9f4b571d1e3ea54c85ba4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-06-25T20:22:17.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T09:47:46.000Z", "avg_line_length": 12.3668639053, "max_line_length": 68, "alphanum_fraction": 0.5320574163, "num_tokens": 812, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9381240211961401, "lm_q2_score": 0.8652240947405564, "lm_q1q2_score": 0.8116875069938009}} {"text": "#=\nIt is possible to show that the square root of two can be expressed as an infinite continued fraction.\n\n√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...\n\nBy expanding this for the first four iterations, we get:\n\n1 + 1/2 = 3/2 = 1.5\n1 + 1/(2 + 1/2) = 7/5 = 1.4\n1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...\n1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...\n\nThe next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.\n\nIn the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?\n=#\nfunction calc()\n p = BigInt(1)\n q = BigInt(1)\n count = 1\n total = 0\n while count <= 1000\n np = p + 2q\n nq = p + q\n p,q = np,nq\n if length(string(p)) > length(string(q))\n total = total + 1\n end\n count = count + 1\n end\n total\nend\n@time println(calc())\n", "meta": {"hexsha": "1e4d6f69098d05e7aaafab4074336718338da1d3", "size": 971, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Solutions/p57.jl", "max_stars_repo_name": "daniel-beard/JuliaProjectEuler", "max_stars_repo_head_hexsha": "5c990f7dde3b1b09f23a11ab2c1f42b3e5337854", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2015-02-01T15:56:04.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-08T03:01:04.000Z", "max_issues_repo_path": "Solutions/p57.jl", "max_issues_repo_name": "daniel-beard/JuliaProjectEuler", "max_issues_repo_head_hexsha": "5c990f7dde3b1b09f23a11ab2c1f42b3e5337854", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Solutions/p57.jl", "max_forks_repo_name": "daniel-beard/JuliaProjectEuler", "max_forks_repo_head_hexsha": "5c990f7dde3b1b09f23a11ab2c1f42b3e5337854", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2016-08-22T18:22:41.000Z", "max_forks_repo_forks_event_max_datetime": "2016-08-22T18:22:41.000Z", "avg_line_length": 28.5588235294, "max_line_length": 208, "alphanum_fraction": 0.6189495366, "num_tokens": 349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.93812402119614, "lm_q2_score": 0.8652240738888188, "lm_q1q2_score": 0.8116874874322849}} {"text": "# nuclear Norm (times a constant)\n\nexport NuclearNorm\n\n\"\"\"\n**Nuclear norm**\n\n NuclearNorm(λ=1.0)\n\nReturns the function\n```math\nf(X) = \\\\|X\\\\|_* = λ ∑_i σ_i(X),\n```\nwhere `λ` is a positive parameter and ``σ_i(X)`` is ``i``-th singular value of matrix ``X``.\n\"\"\"\nstruct NuclearNorm{R <: Real} <: ProximableFunction\n lambda::R\n function NuclearNorm{R}(lambda::R) where {R <: Real}\n if lambda < 0\n error(\"parameter λ must be nonnegative\")\n else\n new(lambda)\n end\n end\nend\n\nis_convex(f::NuclearNorm) = true\n\nNuclearNorm(lambda::R=1.0) where {R <: Real} = NuclearNorm{R}(lambda)\n\nfunction (f::NuclearNorm{R})(X::AbstractMatrix{T}) where {R <: Real, T <: Union{R, Complex{R}}}\n F = svd(X);\n return f.lambda * sum(F.S);\nend\n\nfunction prox!(Y::AbstractMatrix{T}, f::NuclearNorm{R}, X::AbstractMatrix{T}, gamma::R=one(R)) where {R <: Real, T <: Union{R, Complex{R}}}\n F = svd(X)\n S_thresh = max.(zero(R), F.S .- f.lambda*gamma)\n rankY = findfirst(S_thresh .== zero(R))\n if rankY === nothing\n rankY = minimum(size(X))\n end\n Vt_thresh = view(F.Vt, 1:rankY, :)\n U_thresh = view(F.U, :, 1:rankY)\n # TODO: the order of the following matrix products should depend on the shape of x\n M = S_thresh[1:rankY] .* Vt_thresh\n mul!(Y, U_thresh, M)\n return f.lambda * sum(S_thresh);\nend\n\nfun_name(f::NuclearNorm) = \"nuclear norm\"\nfun_dom(f::NuclearNorm) = \"AbstractArray{Real,2}, AbstractArray{Complex,2}\"\nfun_expr(f::NuclearNorm) = \"X ↦ λ∑σ_i(X)\"\nfun_params(f::NuclearNorm) = \"λ = $(f.lambda)\"\n\nfunction prox_naive(f::NuclearNorm, X::AbstractMatrix{T}, gamma=1.0) where T\n F = svd(X)\n S = max.(0, F.S .- f.lambda*gamma)\n Y = F.U * (Diagonal(S) * F.Vt)\n return Y, f.lambda * sum(S)\nend\n", "meta": {"hexsha": "f82d928d1074e64514ed18ffca155f7cc6bef13e", "size": 1706, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/functions/nuclearNorm.jl", "max_stars_repo_name": "lostella/ProximalOperators.jl", "max_stars_repo_head_hexsha": "9935cd0c24b325667e688b5c35703dde71bfd3a7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-12-15T13:57:52.000Z", "max_stars_repo_stars_event_max_datetime": "2018-12-15T13:57:52.000Z", "max_issues_repo_path": "src/functions/nuclearNorm.jl", "max_issues_repo_name": "lostella/ProximalOperators.jl", "max_issues_repo_head_hexsha": "9935cd0c24b325667e688b5c35703dde71bfd3a7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/functions/nuclearNorm.jl", "max_forks_repo_name": "lostella/ProximalOperators.jl", "max_forks_repo_head_hexsha": "9935cd0c24b325667e688b5c35703dde71bfd3a7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.5161290323, "max_line_length": 139, "alphanum_fraction": 0.6395076202, "num_tokens": 585, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9381240194661945, "lm_q2_score": 0.865224070413529, "lm_q1q2_score": 0.8116874826752415}} {"text": "using Plots\nusing Printf\n\n\"\"\"\n untildivergence(f, x, maxiter=50)\n\nIterate `f` on `x` until its module is greater than 2 or `maxiter` iterations have been done.\n\nReturns the number of iterations.\n\"\"\"\nfunction untildivergence(f, x, maxiter = 50)\n i = 0\n while abs(x) < 2 && (i += 1) <= maxiter\n x = f(x)\n end\n return i\nend\n\n\"\"\"\n julia_set_wh(f; kwargs)\n\nApproximate the Julia set of `f` in a grid using the escape time algorithm.\n\nThe grid is defined by its bounds `xmin`, `xmax`, `ymin` and `ymax` as well as its width `w` and height `h` in pixels. The optional parameter `maxiter` is passed on to `untildivergence`.\n\"\"\"\nfunction julia_set_wh(f; xmin::Real=-2, xmax::Real=2, ymin::Real=-2, ymax::Real=2, w::Int=1000, h::Int=1000, maxiter::Int=50)\n img = Array{Int,2}(undef, h, w)\n x = LinRange(xmin, xmax, w)\n y = LinRange(ymin, ymax, h)\n # creating labels for use with `heatmap`\n xticks = LinRange(1,w,11)\n yticks = LinRange(1,h,11)\n xlabels = map(z->@sprintf(\"%.2f\",z), LinRange(xmin,xmax,11))\n ylabels = map(z->@sprintf(\"%.2f\",z), LinRange(ymin,ymax,11))\n\n for i in 1:w, j in 1:h\n img[j,i] = untildivergence(f, x[i]+im*y[j], maxiter)\n end\n return img, xticks, xlabels, yticks, ylabels\nend\n\n\"\"\"\n julia_set(f, cellsize; kwargs)\n\nApproximate the Julia of `f` set in a grid of cells of length `cellsize`.\n\nThe grid is defined by the same parameters as in `julia_set_wh`, except its width and height in pixels are computed with `cellsize`.\n\"\"\"\nfunction julia_set(f, cellsize::Real; xmin=-2, xmax=2, ymin=-2, ymax=2, maxiter=50)\n w = Integer(round(abs(xmax-xmin)/cellsize))\n h = Integer(round(abs(ymax-ymin)/cellsize))\n return julia_set_wh(f, xmin=xmin, xmax=xmax, ymin=ymin, ymax=ymax, w=w, h=h, maxiter=maxiter)\nend\n\n\"\"\"\n mandelbrot_set_wh(f; kwargs)\n\nApproximate the Mandelbrot set of `f` in a grid using the escape time algorithm.\n\nThe grid is defined by its bounds `xmin`, `xmax`, `ymin` and `ymax` as well as its width `w` and height `h` in pixels. The optional parameter `maxiter` is passed on to `untildivergence`.\n\"\"\"\nfunction mandelbrot_set_wh(f; xmin::Real=-2, xmax::Real=2, ymin::Real=-2, ymax::Real=2, w::Integer=1000, h::Integer=1000, maxiter::Integer=50)\n img = Array{Int,2}(undef, h, w)\n x = LinRange(xmin, xmax, w)\n y = LinRange(ymin, ymax, h)\n # creating labels for use with `heatmap`\n xticks = LinRange(1,w,11)\n yticks = LinRange(1,h,11)\n xlabels = map(z->@sprintf(\"%.2f\",z), LinRange(xmin,xmax,11))\n ylabels = map(z->@sprintf(\"%.2f\",z), LinRange(ymin,ymax,11))\n\n for i in 1:w, j in 1:h\n img[j,i] = untildivergence(f(x[i]+im*y[j]), 0, maxiter)\n end\n return img, xticks, xlabels, yticks, ylabels\nend\n\n\"\"\"\n mandelbrot_set(f, cellsize; kwargs)\n\nApproximate the Mandelbrot of `f` set in a grid of cells of length `cellsize`.\n\nThe grid is defined by the same parameters as in `julia_set_wh`, except its width and height in pixels are computed with `cellsize`.\n\"\"\"\nfunction mandelbrot_set(f, cellsize; xmin=-2, xmax=2, ymin=-2, ymax=2, maxiter=50)\n w = Integer(round(abs(xmax-xmin)/cellsize))\n h = Integer(round(abs(ymax-ymin)/cellsize))\n return mandelbrot_set_wh(f, xmin=xmin, xmax=xmax, ymin=ymin, ymax=ymax, w=w, h=h, maxiter=maxiter)\nend\n\n# the traditional julia/mandelbrot function\nf(c) = z -> z^2 + c\n\nimg, xticks, xlabels, yticks, ylabels = mandelbrot_set(f, 0.00005, xmin=-2.01, xmax = -1.8, ymin = -0.1, ymax = 0.1)\n\nimg, xticks, xlabels, yticks, ylabels = julia_set(f(im), 0.0005, xmin=-2, xmax=2, ymin=-2,ymax=2)\n\n\nheatmap(img,\n xticks = (xticks, xlabels),\n yticks = (yticks, ylabels),\n aspect_ratio=:equal\n )\n\n# alternative, for better looking images\nusing Images, Colors\ng = Gray.(img/51)\n\ng = Gray.(.!(img .>= 50))\n\nmset = img .>= 50\n\n#save(\"julia_minus2.png\", g)\n\ninclude(\"FractalDimensions.jl\")\n\nB = boundary(mset)\nGray.(B)\npoints = array_to_points(B)\n\nd, N, ε = box_counting_dimension(points, 10)\n\nC, r = correlation_dimension(points)\n\nloglog_regression(C,r)\n", "meta": {"hexsha": "08464181aeae0d994865f4c94a7367f8ebae772a", "size": 4045, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "mandelbrot.jl", "max_stars_repo_name": "csimal/chaos-and-fractals", "max_stars_repo_head_hexsha": "4b189b0fde62d480f1f2e750b976de54bd9d2297", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "mandelbrot.jl", "max_issues_repo_name": "csimal/chaos-and-fractals", "max_issues_repo_head_hexsha": "4b189b0fde62d480f1f2e750b976de54bd9d2297", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "mandelbrot.jl", "max_forks_repo_name": "csimal/chaos-and-fractals", "max_forks_repo_head_hexsha": "4b189b0fde62d480f1f2e750b976de54bd9d2297", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.1031746032, "max_line_length": 186, "alphanum_fraction": 0.6637824475, "num_tokens": 1333, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9381240090865197, "lm_q2_score": 0.8652240756264639, "lm_q1q2_score": 0.8116874785848764}} {"text": "### Gali's Book Chapter 3 model\n### We use this as a consistency check of gensys implementation\n### By Daniel Coutinho and Gilberto Boaretto\n\n#### Parameters from Gali's book example, chapter 3, page 52, first edition\n\nbet = 0.99\nsig = 1\nphi = 1\nalfa = 1/3\nepsilon = 6\ntheta = 2/3\nphi_pi = 1.5\nphi_y = 0.5/4\nrho_v = 0.5\n\ntrue_pars = (alfa = alfa, bet = bet, epsilon = epsilon, theta = theta, sig = sig, s2 = 1, phi = phi, phi_pi = phi_pi, phi_y = phi_y, rho_v = rho_v)\n\nTHETA = (1-alfa)/(1-alfa+alfa*epsilon)\nlamb = (1-theta)*(1-bet*theta)/theta*THETA\nkappa = lamb*(sig+(phi+alfa)/(1-alfa))\n\n## Gali's give analytical expressions for the effects of a shock in the monetary policy over inflation and output gap,chapter 3, page 51, first edition\n\nLAMBDA_v = ((1-bet*rho_v)*(sig*(1-rho_v)+phi_y)+kappa*(phi_pi-rho_v))^(-1)\n\ny_irf(v) = -(1-bet*rho_v)*LAMBDA_v*v\npi_irf(v) = -kappa*LAMBDA_v*v\nr_irf(v) = sig*(1-rho_v)*(1-bet*rho_v)*LAMBDA_v*v\ni_irf(v) = (sig*(1-rho_v)*(1-bet*rho_v) - rho_v*kappa)*LAMBDA_v*v\nv_irf(v,uu) = rho_v*v + uu\n\n## Test drive: lets see if this gives the same IRFs that we get in Gali's book\n## This is just for debugging, so comment it out when using it to check gensys consistency\n\n#using Plots\n\nirfs_true = zeros(15,5)\n\nirfs_true[1,1] = 0\n\nuu = 0.25\n\nfor t in 2:15\n irfs_true[t,4] = v_irf(irfs_true[t-1,4],uu)\n irfs_true[t,1] = pi_irf(irfs_true[t,4])\n irfs_true[t,2] = y_irf(irfs_true[t,4])\n irfs_true[t,3] = i_irf(irfs_true[t,4])\n irfs_true[t,5] = r_irf(irfs_true[t,4])\n global uu = 0\nend\n\n#plot(irfs[2:13,1], label = \"v\")\n#plot(irfs[2:13,2]*4, label = \"pi\")\n#plot(irfs[2:13,3], label = \"y\")\n#plot(4*irfs[2:13,4], label = \"i\")\n#plot(4*irfs[2:13,5], label = \"r\")\n", "meta": {"hexsha": "fbf283a5c69ba38d20e448536dc30c8bb0816133", "size": 1703, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "galis_example.jl", "max_stars_repo_name": "danmrc/DSGE-HMC", "max_stars_repo_head_hexsha": "04601e6d7c5a878ea0812d48e974b3f9c91c90ce", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "galis_example.jl", "max_issues_repo_name": "danmrc/DSGE-HMC", "max_issues_repo_head_hexsha": "04601e6d7c5a878ea0812d48e974b3f9c91c90ce", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "galis_example.jl", "max_forks_repo_name": "danmrc/DSGE-HMC", "max_forks_repo_head_hexsha": "04601e6d7c5a878ea0812d48e974b3f9c91c90ce", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-11-20T09:15:46.000Z", "max_forks_repo_forks_event_max_datetime": "2019-11-20T09:15:46.000Z", "avg_line_length": 29.3620689655, "max_line_length": 151, "alphanum_fraction": 0.6617733412, "num_tokens": 677, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582612793112, "lm_q2_score": 0.8723473796562744, "lm_q1q2_score": 0.8116828261065403}} {"text": "# AAA algorithm from the paper \"The AAA Alorithm for Rational Approximation\"\n# by Y. Nakatsukasa, O. Sete, and L.N. Trefethen, SIAM Journal on Scientific Computing\n# 2018 \n\nstruct AAAapprox{T <: AbstractArray} <: BRInterp\n x::T\n f::T\n w::T\n errvec::T\nend\n\n(a::AAAapprox)(zz) = reval(zz, a.x, a.f, a.w)\n\n# Handle function inputs as well\nfunction aaa(Z::AbstractArray{T,1}, F::S; tol=1e-13, mmax=100, verbose=false, clean=true) where {T, S<:Function}\n aaa(Z, F.(Z), tol=tol, mmax=mmax, verbose=verbose, clean=clean)\nend\n\n\"\"\"aaa rational approximation of data F on set Z\n r = aaa(Z, F; tol, mmax, verbose, clean)\n\n Input: Z = vector of sample points\n F = vector of data values at the points in Z\n tol = relative tolerance tol, set to 1e-13 if omitted\n mmax: max type is (mmax-1, mmax-1), set to 100 if omitted\n verbose: print info while calculating default = false\n clean: detect and remove Froissart doublets default = true\n\n Output: r = an AAA approximant as a callable struct with fields\n z, f, w = vectors of support pts, function values, weights\n errvec = vector of errors at each step\n\n NOTE: changes from matlab version:\n switched order of Z and F in function signature\n added verbose and clean boolean flags\n pol, res, zer = vectors of poles, residues, zeros are now only calculated on demand\n by calling prz(z::AAAapprox)\n\"\"\"\nfunction aaa(Z::AbstractArray{T,1}, F::AbstractArray{S,1}; tol=1e-13, mmax=100,\n verbose=false, clean=true) where {S, T}\n # filter out any NaN's or Inf's in the input\n keep = isfinite.(F)\n F = F[keep]\n Z = Z[keep]\n \n M = length(Z) # number of sample points\n mmax = min(M, mmax) # max number of support points\n \n reltol = tol*norm(F, Inf)\n SF = spdiagm(M, M, 0 => F) # left scaling matrix\n \n F, Z = promote(F, Z)\n P = promote_type(S, T)\n \n J = [1:M;]\n z = P[] # support points\n f = P[] # function values at support points\n C = P[]\n w = P[]\n \n errvec = P[]\n R = F .- mean(F)\n @inbounds for m = 1:mmax\n j = argmax(abs.(F .- R)) # select next support point\n push!(z, Z[j])\n push!(f, F[j])\n deleteat!(J, findfirst(isequal(j), J)) # update index vector\n\n # next column of Cauchy matrix\n C = isempty(C) ? reshape((1 ./ (Z .- Z[j])), (M,1)) : [C (1 ./ (Z .- Z[j]))]\n\n Sf = diagm(f) # right scaling matrix\n A = SF * C - C * Sf # Loewner matrix\n G = svd(A[J, :])\n \n # A[J, :] might not have full rank, so svd can return fewer than m columns\n #w = G.V[:, m] \n w = G.V[:, end] # weight vector = min sing vector\n \n N = C * (w .* f) # numerator \n D = C * w\n R .= F\n R[J] .= N[J] ./ D[J] # rational approximation\n \n err = norm(F - R, Inf)\n verbose && println(\"Iteration: \", m, \" err: \", err)\n errvec = [errvec; err] # max error at sample points\n err <= reltol && break # stop if converged\n end\n r = AAAapprox(z, f, w, errvec)\n\n # remove Frois. doublets if desired. We do this in place\n if clean\n pol, res, zer = prz(r) # poles, residues, and zeros\n ii = findall(abs.(res) .< 1e-13) # find negligible residues\n length(ii) != 0 && cleanup!(r, pol, res, zer, Z, F)\n end\n return r\nend\n\n\nfunction prz(r::AAAapprox)\n z, f, w = r.x, r.f, r.w \n m = length(w)\n B = diagm(ones(m+1))\n B[1, 1] = 0.0\n E = [0.0 transpose(w); ones(m) diagm(z)]\n pol, _ = eigen(E, B)\n pol = pol[isfinite.(pol)] \n dz = 1e-5 * exp.(2im*pi*[1:4;]/4)\n \n # residues\n res = r(pol .+ transpose(dz)) * dz ./ 4 \n \n E = [0 transpose(w .* f); ones(m) diagm(z)]\n zer, _ = eigen(E, B)\n zer = zer[isfinite.(zer)]\n pol, res, zer\nend\n\n\nfunction reval(zz, z, f, w)\n # evaluate r at zz\n zv = size(zz) == () ? [zz] : vec(zz) \n CC = 1.0 ./ (zv .- transpose(z)) # Cauchy matrix\n r = (CC * (w .* f)) ./ (CC * w) # AAA approx as vector\n r[isinf.(zv)] .= sum(f .* w) ./ sum(w)\n \n ii = findall(isnan.(r)) # find values NaN = Inf/Inf if any\n @inbounds for j in ii\n if !isnan(zv[j]) && ((v = findfirst(isequal(zv[j]), z)) !== nothing)\n r[j] = f[v] # force interpolation there\n end\n end\n r = size(zz) == () ? r[1] : reshape(r, size(zz)) # the AAA approximation\nend\n\n\n# Only calculate the updated z, f, and w\nfunction cleanup!(r, pol, res, zer, Z, F)\n z, f, w = r.x, r.f, r.w\n m = length(z)\n M = length(Z)\n ii = findall(abs.(res) .< 1e-13) # find negligible residues\n ni = length(ii)\n ni == 0 && return\n println(\"$ni Froissart doublets. Number of residues = \", length(res))\n\n @inbounds for j = 1:ni\n azp = abs.(z .- pol[ii[j]] )\n jj = findall(isequal(minimum(azp)), azp)\n deleteat!(z, jj) # remove nearest support points\n deleteat!(f, jj)\n end \n\n @inbounds for j = 1:length(z)\n jj = findall(isequal(z[j]), Z)\n deleteat!(F, jj)\n deleteat!(Z, jj)\n end\n m = m - length(ii)\n SF = spdiagm(M-m, M-m, 0 => F)\n Sf = diagm(f)\n C = 1 ./ (Z .- transpose(z))\n A = SF*C - C*Sf\n G = svd(A)\n w[:] .= G.V[:, m]\n println(\"cleanup: \", size(z), \" \", size(f), \" \", size(w))\n return nothing\nend\n", "meta": {"hexsha": "e2a19e30650aefafbce67e0e9eef186f92abef62", "size": 5678, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/aaa.jl", "max_stars_repo_name": "macd/BaryRational.jl", "max_stars_repo_head_hexsha": "6e5bb75e6fa3146bea855d2412fb45fecf9a8103", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2021-01-28T20:12:23.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-05T03:34:03.000Z", "max_issues_repo_path": "src/aaa.jl", "max_issues_repo_name": "macd/BaryRational.jl", "max_issues_repo_head_hexsha": "6e5bb75e6fa3146bea855d2412fb45fecf9a8103", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-01-22T09:05:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-01-22T09:05:02.000Z", "max_forks_repo_path": "src/aaa.jl", "max_forks_repo_name": "macd/BaryRational.jl", "max_forks_repo_head_hexsha": "6e5bb75e6fa3146bea855d2412fb45fecf9a8103", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.2046783626, "max_line_length": 113, "alphanum_fraction": 0.5149700599, "num_tokens": 1765, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582612793112, "lm_q2_score": 0.8723473796562744, "lm_q1q2_score": 0.8116828261065403}} {"text": "#############################\n## Numerical experiments associated with section 5.4.1\n#############################\n## Nonlinear equations are treated by first defining an objective function, then using standard iterative methods\n## such as Newton's method via NLsolve.jl. Using more sophisticated methods, linesearches etc. can sometimes result in significant speed boosts.\n##\n\nusing NLsolve, ApproxFun, MultivariateOrthogonalPolynomials, BandedMatrices, BlockBandedMatrices, Plots\nusing SparseVolterraExamples\n\n#####################################################\n## Problem in Equation (25), u_1(x) = exp(x)\n#####################################################\n####\n## Define g(x), the nonlinearity and the kernel\ng(x) = exp(x)+1/3*x*(1-exp(3*x))\n gFun = Fun(g,Jacobi(0,1,0..1))\n f(u) = u^3\n K(x,y) = x\n n = 15 # Desired polynomial order of approximation for accuracy\n####\n## Define the objective function and compute the solution using a standard Newton method\nobjective(x) = triNonLinearVolterraObjective(x,f,K,gFun,n)\nsol = nlsolve(objective,zeros(n),method=:newton,ftol=1e-8)\n####\n## Plot function aganinst analytic solution\nplot(Fun(Jacobi(0,1,0..1),sol.zero),grid=false,legend=:topleft,xlabel=\"x\",ylabel=\"u(x)\",label=\"sparse method\")\nplot!(exp,0,1,grid=false,legend=:topleft,xlabel=\"x\",ylabel=\"u(x)\",label=\"analytic exp(x)\")\n####\n## Or plot error. A simple loop over polynomial degree can then produce Figure 7(a).\nplot(Fun(Jacobi(0,1,0..1),sol.zero)-Fun(x->exp(x),Jacobi(0,1,0..1)),grid=false,legend=:bottomleft,xlabel=\"x\",ylabel=\"error\",label=false)\n\n#####################################################\n## Problem in Equation (26), u_2(x) = sin(x)\n#####################################################\n####\n## Define g(x), the nonlinearity and the kernel\ng(x) = sin(x)+1/4*sin(x)^2-1/4*x^2\n gFun = Fun(g,Jacobi(0,1,0..1))\n f(u) = u^2\n K(x,y) = x-y\n n = 14 # Desired polynomial order of approximation for accuracy\n####\n## Define the objective function and compute the solution using a standard Newton method\nobjective(x) = triNonLinearVolterraObjective(x,f,K,gFun,n)\nsol = nlsolve(objective,zeros(n),method=:newton,ftol=1e-15)\n####\n## Plot function aganinst analytic solution\nplot(Fun(Jacobi(0,1,0..1),sol.zero),grid=false,legend=:topleft,xlabel=\"x\",ylabel=\"u(x)\",label=\"sparse method\")\nplot!(sin,0,1,grid=false,legend=:topleft,xlabel=\"x\",ylabel=\"u(x)\",label=\"analytic sin(x)\")\n####\n## Or plot error. A simple loop over polynomial degree can then produce Figure 7(b).\nplot(Fun(Jacobi(0,1,0..1),sol.zero)-Fun(x->sin(x),Jacobi(0,1,0..1)),grid=false,legend=:bottomleft,xlabel=\"x\",ylabel=\"error\",label=false)\n", "meta": {"hexsha": "8ed89d5863a8f4e660f0bb43f2daacac6b24ddec", "size": 2643, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/Section 5-4-1 - Nonlinear Volterra integral equations.jl", "max_stars_repo_name": "TSGut/SparseVolterraExamples.jl", "max_stars_repo_head_hexsha": "277733c70dfe78b65d1d9ad289de9dbeacda27b7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-18T16:58:42.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-18T16:58:42.000Z", "max_issues_repo_path": "examples/Section 5-4-1 - Nonlinear Volterra integral equations.jl", "max_issues_repo_name": "TSGut/SparseVolterraExamples.jl", "max_issues_repo_head_hexsha": "277733c70dfe78b65d1d9ad289de9dbeacda27b7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-02-09T15:39:49.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-09T15:59:37.000Z", "max_forks_repo_path": "examples/Section 5-4-1 - Nonlinear Volterra integral equations.jl", "max_forks_repo_name": "TSGut/SparseVolterraExamples.jl", "max_forks_repo_head_hexsha": "277733c70dfe78b65d1d9ad289de9dbeacda27b7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.9444444444, "max_line_length": 144, "alphanum_fraction": 0.6382898222, "num_tokens": 737, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109798251322, "lm_q2_score": 0.8539127548105611, "lm_q1q2_score": 0.8115680579846832}} {"text": "using LinearAlgebra\n\n\"\"\"\n is_schur(A::AbstractMatrix)\n\nCheck if a square matrix is Schur, i.e. the eigenvalues lie within the unit circle.\n\n### Input\n- `A` -- square matrix\n\n### Ouput\nBoolean value indicating if the matrix `A` is Schur.\n\"\"\"\nis_schur(A::AbstractMatrix) = maximum(abs.(eigvals(A))) < 1.0\n", "meta": {"hexsha": "f60bcbf1c71afa53d5d0c94586fedf8e765f35b8", "size": 306, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/helper/control_systems.jl", "max_stars_repo_name": "mforets/InvariantSets", "max_stars_repo_head_hexsha": "fc6e61e404c80e959932336199fa3508fe2c2c35", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-08-10T20:47:55.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-31T06:13:51.000Z", "max_issues_repo_path": "src/helper/control_systems.jl", "max_issues_repo_name": "mforets/InvariantSets", "max_issues_repo_head_hexsha": "fc6e61e404c80e959932336199fa3508fe2c2c35", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-05-19T13:03:05.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-22T07:46:51.000Z", "max_forks_repo_path": "src/helper/control_systems.jl", "max_forks_repo_name": "ueliwechsler/InvariantSets.jl", "max_forks_repo_head_hexsha": "680cb87bd61cb26f3c1cfe97eaf27009d38ac016", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-05-18T12:26:10.000Z", "max_forks_repo_forks_event_max_datetime": "2020-05-18T12:26:10.000Z", "avg_line_length": 20.4, "max_line_length": 83, "alphanum_fraction": 0.6960784314, "num_tokens": 91, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9504109770159682, "lm_q2_score": 0.8539127566694177, "lm_q1q2_score": 0.81156805735258}} {"text": "export solarposition, zenith, zenith_azimuth, sines_and_cosines\n\n# Calculate the position of the sun as (zenith,azimuth) in degrees for a location\n# on earth given as (latitude,longitude) in degrees. The DateTime should be given\n# as Universal Time, e.g. UTC. \nfunction solarposition_wiki(datetime::DateTime, latitude, longitude)\n\t# Position of the Sun as seen from Earth.\n\t# Ported from Matlab, original helpstring below.\n\n\t# This is the most basic algorithm. It is documented in Seinfeld &\n\t# Pandis, Duffie & Beckman and Wikipedia.\n\t#\n\t# [ANGLES,PROJECTION] = SOLARPOSITION(DATE,TIME,LATITUDE,LONGITUDE,TIME_ZONE)\n\t# returns ZENITH & AZIMUTH for all DATE & TIME pairs at LATITUDE, LONGITUDE.\n\t# ANGLES = [ZENITH,AZIMUTH] and PROJECTION = [PHI_X, PHI_Y]\n\t# PHI_X is projection on x-z plane & PHI_Y is projection on y-z plane.\n\t# DATETIME can be string, vector [YEAR, MONTH, DAY, HOURS, MINUTES, SECONDS],\n\t# cellstring or matrix N x [YEAR, MONTH, DAY, HOURS, MINUTES, SECONDS] for N\n\t# times.\n\t# LATITUDE [degrees] and LONGITUDE [degrees] are the coordinates of the site.\n\t# TIME_ZONE [hours] of the site.\n\t# ROTATION [degrees] clockwise rotation of system relative to north.\n\t# DST [logical] flag for daylight savings time, typ. from March to November\n\t# in the northern hemisphere.\n\t#\n\t# References:\n\t# http://en.wikipedia.org/wiki/Solar_azimuth_angle\n\t# http://en.wikipedia.org/wiki/Solar_elevation_angle\n\t#\n\t# Mark A. Mikofski\n\t# Copyright (c) 2013\n\n\n\n\t# Equation of Time\n\t# \"Alternative calculation\" on wikipedia:\n\t# https://en.wikipedia.org/wiki/Equation_of_time#Calculating_the_equation_of_time\n\t# Supposedly accurate to 6 seconds (0.1 minutes)\n\n\tyear = Dates.year(datetime)\n\t# date in days starting at zero on 1 January (the subtraction produces a difference in milliseconds)\n\td_ms = datetime - DateTime(year,1,1)\n\td = d_ms.value/1000/3600/24\n\n\t# Earth's mean angular orbital velocity [degrees/day]\n\tw = 360/365.24\n\t# angle the earth moves on its orbit at its average speed from the December solstice\n\t# 10 is the approximate number of days from the December solstice to January 1\n\ta = w * (d+10)\n\t# The angle the Earth moves from the solstice to date D, including a first-order correction\n\t# for the Earth's orbital eccentricity, 0.0167. The number 2 is the number of days from 1 January\n\t# to the date of the Earth's perihelion.\n\tb = a + (360/pi*0.0167)*sind(w*(d-2))\n\t# C is the difference between the angles moved at mean speed, and at the corrected speed projected\n\t# onto the equatorial plane, and divided by 180 to get the difference in \"half turns\". The value 23.44°\n\t# is the obliquity (tilt) of the Earth's axis. The subtraction gives the conventional sign to the equation of time. \n\tc = (a - atand(tand(b)/cosd(23.44))) / 180\n\t# For any given value of x, arctan x (sometimes written as tan−1 x) has multiple values, differing from each other\n\t# by integer numbers of half turns. This may cause C to be wrong by an integer number of half turns. The excess\n\t# half turns are removed in the next step of the calculation to give the equation of time:\n\tET = 720*(c - round(c))\n\n\t# approximate solar time [hours]\n\tsolarTime = 24*(d - floor(d)) + longitude*24/360 + ET/60\n\tt_h = 15*(solarTime - 12) # [degrees] hour angle\n\n\t# declination [degrees]\n\t# accurate to 0.2 degrees, see https://en.wikipedia.org/wiki/Position_of_the_Sun#Calculations\n\tdelta = -asind(sind(23.44)*cosd(b))\n\tzenith = acosd(sind(latitude)*sind(delta) + cosd(latitude)*cosd(delta)*cosd(t_h)) # [degrees] zenith\n\n\t# azimuth [0, 180], absolute value measured from due south, so east = west = 90,\n\t# south = 0, north = 180\n\tcos_phi = (cosd(zenith)*sind(latitude) - sind(delta)) ./ (sind(zenith)*cosd(latitude)) # cosine(azimuth)\n\tphi_south = acosd(clamp(cos_phi,-1,1))\n\t#phi_south = acosd(cos_phi);\n\n\t# azimuth [0, 360], measured clockwise from due north,\n\t# so east = 90, south = 180, and west = 270 degrees\n\tazimuth = 180 + sign(t_h)*phi_south # Shift domain to 0-360 deg\n\n\t#angles = [theta, phi]; # [degrees] zenith, azimuth\n\treturn zenith, azimuth, solarTime, delta\nend\n\nfunction solarposition_wiki_faster(datetime::DateTime, latitude, longitude)\n\tyear = Dates.year(datetime)\n\td_ms = datetime - DateTime(year,1,1)\n\td = d_ms.value/1000/3600/24\n\tw = 360/365.24\n\ta = w * (d+10)\n\tb = a + (360/pi*0.0167)*sind(w*(d-2))\n\tc = (a - atand(tand(b)/cosd(23.44))) / 180\n\tET = 720*(c - round(c))\n\tsolarTime = 24*(d - floor(d)) + longitude*24/360 + ET/60\n\tt_h = 15*(solarTime - 12) # [degrees] hour angle\n\tdelta = -asind(sind(23.44)*cosd(b))\n\tslat, clat = sind(latitude), cosd(latitude)\n\tsdel, cdel = sind(delta), cosd(delta)\n\tczen = slat*sdel + clat*cdel*cosd(t_h)\n\tzenith = acosd(czen) # [degrees] zenith\n\n\t# azimuth [0, 180], absolute value measured from due south, so east = west = 90,\n\t# south = 0, north = 180\n\tcos_phi = (czen*slat - sdel) / (sind(zenith)*clat) # cosine(azimuth)\n\tphi_south = acosd(clamp(cos_phi,-1,1))\n\t#phi_south = acosd(cos_phi);\n\n\t# azimuth [0, 360], measured clockwise from due north,\n\t# so east = 90, south = 180, and west = 270 degrees\n\tazimuth = 180 + sign(t_h)*phi_south # Shift domain to 0-360 deg\n\n\t#angles = [theta, phi]; # [degrees] zenith, azimuth\n\treturn zenith, azimuth, solarTime, delta\nend\n\n\n\n# Splits the DateTime into (year, month, day) (as integers) and the hour (as a floating point)\nfunction splitdatetime(datetime::DateTime)\n\t# date = floor(datetime, Dates.Day)\n\ty, m, d = Dates.yearmonthday(datetime)\n\treturn y, m, d, (datetime - DateTime(y,m,d)).value/1000/3600\nend\n\n# time correction term, difference between Terrestrial Time and Universal Time, see Grena (2012) section 2.1.\n# Conversion to TT is necessary since UT follows the rotation of the earth, which is not uniform.\n# t [days] days from January 1, 2060 (in UT)\n# returns Δτ [seconds]\ndifferenceTTUT(t) = 96.4 + 0.00158*t\n\n# Calculate time variable needed for Grena (2012) algorithms, see section 3.1.\n# The time t is the input DateTime expressed as days from January 1, 2060 (in UT).\n# returns t [days]\ncalctime(datetime::DateTime) = (datetime-DateTime(2060,1,1)).value/1000/3600/24\n\n# Same result as above, but this is the function in Grena's paper. The Julia version is faster.\nfunction calctime_grena(datetime::DateTime)\n\ty, m, d, h = splitdatetime(datetime)\n\tif m <= 2\n\t\tm += 12\n\t\ty -= 1\n\tend\n\treturn trunc(365.25*(y-2000)) + trunc(30.6001*(m+1)) - trunc(0.01*y) + d + 0.0416667*h - 21958\nend\n\n# Full version of algorithm 1 of Grena (2012), see section 3.2.\n# Calculate the position of the sun as (zenith,azimuth) in radians for a location\n# on earth given as (latitude,longitude) in degrees. The DateTime should be given\n# as Universal Time, e.g. UTC. \nfunction solarposition_grena1(datetime::DateTime, latitude, longitude)\n\tω = 0.017202786 # [1/day]\n\tt = calctime(datetime)\n\tt_e = t + 1.1574e-5 * differenceTTUT(t)\n\ts1, c1 = sin(ω*t_e), cos(ω*t_e)\n\ts2, c2 = 2*s1*c1, (c1+s1)*(c1-s1)\n\tα = mod(-1.38880 + 1.72027920e-2*t_e + 3.199e-2*s1 - 2.65e-3*c1 + 4.050e-2*s2 + 1.525e-2*c2, 2π)\n\tδ = 6.57e-3 + 7.347e-2*s1 - 3.9919e-1*c1 + 7.3e-4*s2 - 6.60e-3*c2\n\tH = mod(1.75283 + 6.3003881*t + deg2rad(longitude) - α + π, 2π) - π\n\tsϕ, cϕ = sind(latitude), cosd(latitude)\n\tsδ, cδ = sin(δ), cos(δ)\n\tsH, cH = sin(H), cos(H)\n\tse0 = sϕ*sδ + cϕ*cδ*cH\n\tep = asin(se0) - 4.26e-5*sqrt(1-se0^2) #cos(e0)\n\tazimuth = atan(sH, cH*sϕ - sδ*cϕ/cδ)\t#azimuth = 0 towards south and positive direction towards west\n\tzenith = π/2 - ep\n\treturn zenith, azimuth\t\t\t\t# radians\n\t# return zenith, azimuth, δ, H\t\t# radians\n\t# return rad2deg(zenith), rad2deg(mod(azimuth+π,2π)), rad2deg(H)/15+12, rad2deg(δ) # degrees, same azimuth origin as solarposition0\nend\n\n# First half of algorithm 1 of Grena (2012), see section 3.2.\n# Calculate the position of the sun as (declination, hour angle) in radians for a location\n# on earth given by its longitude in degrees. The DateTime should be given\n# as Universal Time, e.g. UTC. \nfunction solarposition(datetime::DateTime, longitude)\n\tω = 0.017202786 # [1/day]\n\tt = calctime(datetime)\n\tt_e = t + 1.1574e-5 * differenceTTUT(t)\n\ts1, c1 = sin(ω*t_e), cos(ω*t_e)\n\ts2, c2 = 2*s1*c1, (c1+s1)*(c1-s1)\n\tα = mod(-1.38880 + 1.72027920e-2*t_e + 3.199e-2*s1 - 2.65e-3*c1 + 4.050e-2*s2 + 1.525e-2*c2, 2π)\n\tδ = 6.57e-3 + 7.347e-2*s1 - 3.9919e-1*c1 + 7.3e-4*s2 - 6.60e-3*c2\n\tH = mod(1.75283 + 6.3003881*t + deg2rad(longitude) - α + π, 2π) - π\n\treturn δ, H\nend\n\n# Full version of algorithm 1 of Grena (2012), see section 3.2.\n# Calculate the position of the sun as (zenith, azimuth) in degrees for a location\n# on earth given as (latitude,longitude) in degrees. The DateTime should be given\n# as Universal Time, e.g. UTC. \nfunction solarposition(datetime::DateTime, latitude, longitude)\n\tδ, H = solarposition(datetime, longitude)\n\tzenith, azimuth = zenith_azimuth(latitude, sines_and_cosines(δ, H)...)\n\treturn rad2deg(zenith), rad2deg(azimuth)\nend\n\nsines_and_cosines(δ, H) = sin(δ), cos(δ), sin(H), cos(H)\n\nfunction zenith_azimuth(latitude, sδ, cδ, sH, cH)\n\tsϕ, cϕ = sind(latitude), cosd(latitude)\n\tse0 = sϕ*sδ + cϕ*cδ*cH\n\tazimuth = atan(sH, cH*sϕ - sδ*cϕ/cδ)\t# azimuth = 0 towards south and positive direction towards west\n\tzenith = π/2 - asin(se0) + 4.26e-5*sqrt(1-se0^2) # sqrt(1-se0^2) = cos(e0), but slightly faster\n\treturn zenith, azimuth\t\t# radians\nend\n\nfunction zenith(latitude, sδ, cδ, sH, cH)\n\tsϕ, cϕ = sind(latitude), cosd(latitude)\n\tse0 = sϕ*sδ + cϕ*cδ*cH\n\treturn π/2 - asin(se0) + 4.26e-5*sqrt(1-se0^2) # sqrt(1-se0^2) = cos(e0), but slightly faster\nend\n\nconst solarconstant = 1361 # W/m2\n\n# Annual variation of solar insolation outside the atmosphere (W/m2):\n# https://www.itacanet.org/the-sun-as-a-source-of-energy/part-2-solar-energy-reaching-the-earths-surface/#2.1.-The-Solar-Constant\nsolarinsolation(dt::DateTime) = solarconstant * (1 + 0.034*cos(2*π*(dt - DateTime(year(dt))).value/1000/3600/24/365.25))\n\n# dt = DateTime(2019,4,15,13,46)\n# δ, H = solarposition(dt, 12)\t# return rad2deg(zenith), rad2deg(mod(azimuth+π,2π)), rad2deg(H)/15+12, rad2deg(δ)\n\n# pos = sines_and_cosines(δ, H)\n# zen = zenith(58, pos...)\n\n", "meta": {"hexsha": "a7782c47fcb9be3b114cf0f0e62b0ce6e2c8d5a6", "size": 10016, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/solarposition.jl", "max_stars_repo_name": "henrikhodel/GlobalEnergyGIS", "max_stars_repo_head_hexsha": "2dad081ff3565a551d03f2ef31c67413f545a5e4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/solarposition.jl", "max_issues_repo_name": "henrikhodel/GlobalEnergyGIS", "max_issues_repo_head_hexsha": "2dad081ff3565a551d03f2ef31c67413f545a5e4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/solarposition.jl", "max_forks_repo_name": "henrikhodel/GlobalEnergyGIS", "max_forks_repo_head_hexsha": "2dad081ff3565a551d03f2ef31c67413f545a5e4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.3185840708, "max_line_length": 132, "alphanum_fraction": 0.7023761981, "num_tokens": 3428, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109784205502, "lm_q2_score": 0.8539127548105611, "lm_q1q2_score": 0.8115680567852928}} {"text": "using StatsBase, Distributions, Plots; pyplot()\n\nfunction prn(lambda)\n k, p = 0, 1\n while p > MathConstants.e^(-lambda)\n k += 1\n p *= rand()\n end\n return k-1\nend\n\nxGrid, lambda, N = 0:16, 5.5, 10^6\n\npDist = Poisson(lambda)\nbPmf = pdf.(pDist,xGrid)\ndata = counts([prn(lambda) for _ in 1:N],xGrid)/N\n\nplot( xGrid, data, \n\tline=:stem, marker=:circle, \n\tc=:blue, ms=10, msw=0, lw=4, label=\"MC estimate\")\nplot!( xGrid, bPmf, line=:stem, \n\tmarker=:xcross, c=:red, ms=6, msw=0, lw=2, label=\"PMF\",\n\tylims=(0,0.2), xlabel=\"x\", ylabel=\"Probability of x events\")", "meta": {"hexsha": "f087d55e0d4fa942045c0ceabc9b3bf9121815f9", "size": 577, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "3_chapter/poisson.jl", "max_stars_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_stars_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 988, "max_stars_repo_stars_event_min_datetime": "2018-06-21T00:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T01:37:47.000Z", "max_issues_repo_path": "3_chapter/poisson.jl", "max_issues_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_issues_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 41, "max_issues_repo_issues_event_min_datetime": "2019-02-20T05:06:27.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-23T16:53:08.000Z", "max_forks_repo_path": "3_chapter/poisson.jl", "max_forks_repo_name": "Yoshinobu-Ishizaki/StatsWithJuliaBook", "max_forks_repo_head_hexsha": "4c704e96d87b91e680122a6b6fa2d2083c70ea88", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 264, "max_forks_repo_forks_event_min_datetime": "2018-07-31T03:11:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T16:12:13.000Z", "avg_line_length": 25.0869565217, "max_line_length": 61, "alphanum_fraction": 0.6135181976, "num_tokens": 223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9525741254760638, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.8115481917011729}} {"text": "function identical_matrix(n::Integer)\n ret = zeros(n, n)\n for itr1 in 1 : n\n ret[itr1, itr1] = 1\n end\n return ret\nend\n\nfunction exchange_row!(mat::Matrix, a::Integer, b::Integer)\n # if $a$ or $b$ are out of bound, do nothing\n if a > size(mat, 1) || b > size(mat, 1)\n return mat\n end\n # exchange row $a$ and row $b$\n for itr1 in 1 : size(mat, 2)\n mat[a, itr1], mat[b, itr1] = mat[b, itr1], mat[a, itr1]\n end\n return mat\nend\n\nfunction similarity_transformation!(\n mat::Matrix, a::Integer, b::Integer)\n # if $a$ or $b$ are out of bound, do nothing\n if a > size(mat, 1) || b > size(mat, 1)\n return mat\n end\n # taking advantage of the matrix, the transformation\n # is just exchanging part of the row $a$ and row $b$\n for itr1 in 1 : (min(a, b) - 1)\n mat[a, itr1], mat[b, itr1] = mat[b, itr1], mat[a, itr1]\n end\n return mat\nend\n\nfunction plu(mat::Matrix)\n # make a copy of the original matrix, which will finally become $U$\n U = float(copy(mat))\n # # U = copy(mat)\n # it's assumed that the matrix has $m$ rows and $n$ columns\n m, n = size(U)\n # initialize two identical matrices as $P$ and $L$\n P = identical_matrix(m)\n L = identical_matrix(m)\n # $i$ and $j$ are used to represent iterators of row and column,\n # respectively\n i = j = 1\n while i <= m && j <= n\n # find the entry with largest absolute value\n imax, vmax = 0.0, 0.0\n for itr1 in i : m\n if vmax < abs(U[itr1, j])\n imax, vmax = itr1, abs(U[itr1, j])\n end\n end\n (vmax > 0) ? (k = imax) : (j += 1; continue)\n # left multiply $U$ and $P$ with a permutation matrix\n if k != i\n exchange_row!(U, i, k)\n exchange_row!(P, i, k)\n end\n # left and right multiply $L$ with the same matrix\n similarity_transformation!(L, i, k)\n # `forward elimination'\n for itr1 in (i + 1) : m\n scaler = U[itr1, j] / U[i, j]\n for itr2 in j : n\n U[itr1, itr2] -= U[i, itr2] * scaler\n end\n # right multiply $L$ with an elimination matrix\n L[itr1, i] = scaler\n end\n i += 1\n j += 1\n end\n return P, L, U\nend", "meta": {"hexsha": "74958734cd446d48782150a0c451ea67e45700c1", "size": 2303, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "linear-algebra/plu.jl", "max_stars_repo_name": "Shimushushushu/Etude-Rhapsody", "max_stars_repo_head_hexsha": "cdca63ca92a44f0f31e716cf4acd3e0de17c9181", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "linear-algebra/plu.jl", "max_issues_repo_name": "Shimushushushu/Etude-Rhapsody", "max_issues_repo_head_hexsha": "cdca63ca92a44f0f31e716cf4acd3e0de17c9181", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "linear-algebra/plu.jl", "max_forks_repo_name": "Shimushushushu/Etude-Rhapsody", "max_forks_repo_head_hexsha": "cdca63ca92a44f0f31e716cf4acd3e0de17c9181", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.3026315789, "max_line_length": 71, "alphanum_fraction": 0.5327833261, "num_tokens": 727, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133481428691, "lm_q2_score": 0.8633916117313211, "lm_q1q2_score": 0.811513300540854}} {"text": "using ThreeBodyProblem\nusing DifferentialEquations\nusing Plots\nusing LinearAlgebra\n# using ForwardDiff\n\n# The first step is to define the system. ThreeBodyProblem.jl has built-in functions for common systems like Sun/Jupiter.\nsys = sun_jupiter()\n\n## There's even a recipe to plot the system for us!\nplot(sys)\n\n## We can see the Lagrange Points, but where are the Sun and Jupiter? Turns out that the distance between them is so much larger than the radius of either one that it makes it hard to see them on a truly scaled picture. Let's scale the Sun and Jupiter to make them visible.\nplot(sys, scaled = true)\n\n## That's better! Now let's compute the L1 and L2 Lagrange points of our system\n\nL1, L2 = computeLpts(sys)\n# Declare state vectors for L1 and L2 (with zero velocity)\nrv1 = [L1; zeros(3)] # state at Lagrange point L1 (zero velocity added on)\nrv2 = [L2; zeros(3)] # state at Lagrange point L2\n\n## The Lagrange points are equilibrium points in our dynamics. This means that an object placed there perfectly will stay there forever. But, Lagrange points L1, L2, and L3 are unstable, meaning if you perturb the object slightly, it may fall away. This is how we will generate the invariant manifolds.\n## We first need to linearize the dynamics about these Lagrange points to determine which directions are stable and unstable\n\nΦ₀ = I(6) # The initial State Transition Matrix (STM)\nw₀ = vcat(rv1,reshape(Φ₀,36,1)) # Augmented state with state and STM\n# Set up the ODE problem\ntspan = (0.,1.)\nprob = ODEProblem(CR3BPstm!,w₀,tspan,sys)\nsol = solve(prob, reltol=1e-6)\nΦₜ = Matrix(reshape(sol.u[end][7:42],6,6))\n\n# last time step should match very closely with rv1\nrvₜ = sol.u[end][1:6]\nerr = norm(rv1 - rvₜ)\n\n# Compute the eigenvectors of the Monodromy Matrix\nΛ,V = eigen(Φₜ,sortby=isreal) # Λ is vector of eigenvalues, V is matrix of eigenvectors\nYw = real(V[:,findall(isreal, Λ)]) # Eigenvectors corresponding to real eigenvalues\nΛ = Λ[findall(isreal, Λ)] # Purely real eigenvalues (have 0.0 imaginary component)\nYws = Yw[:,findmin(real(Λ))[2]] # Eigenvector associated with stable eigenvalue λ < 1\nYwu = Yw[:,findmax(real(Λ))[2]] # Eigenvector associated with unstable eigenvalue λ > 1\n\n## We perturb our states in the stable and unstable directions\nα = 1e-6\n\n# L1 points\nrv1up = rv1 + α*Ywu # unstable manifold + side\nrv1un = rv1 - α*Ywu # unstable manifold - side\nrv1sp = rv1 + α*Yws # stable manifold + side\nrv1sn = rv1 - α*Yws # stable manifold - side\n\n\n# Integrate trajectories for 2 Jupiter periods\ntf = 2*2π\ntspan_forward = (0.,tf) # make sure to add decimal points so tspan contains floating point values, not integers\ntspan_back = (0.,-tf)\nmyreltol = 1e-12\n\n# Set up the ODE Problems\nprob1up = ODEProblem(CR3BPdynamics!,rv1up,tspan_forward,sys) # Unstable positive\nprob1un = ODEProblem(CR3BPdynamics!,rv1un,tspan_forward,sys) # Unstable negative\nprob1sp = ODEProblem(CR3BPdynamics!,rv1sp,tspan_back,sys) # Stable positive\nprob1sn = ODEProblem(CR3BPdynamics!,rv1sn,tspan_back,sys) # Stable negative\n\n# Solutions to the ODEs\nsol1up = solve(prob1up, reltol=myreltol)\nsol1un = solve(prob1un, reltol=myreltol)\nsol1sp = solve(prob1sp, reltol=myreltol)\nsol1sn = solve(prob1sn, reltol=myreltol)\n\nplot(sys,scaled=true)\nplot!(sol1up,vars=(1,2),label=\"Wu+\",linecolor=:red)\nplot!(sol1un,vars=(1,2),label=\"Wu-\",linecolor=:magenta)\nplot!(sol1sp,vars=(1,2),label=\"Ws+\",linecolor=:blue)\nplot!(sol1sn,vars=(1,2),label=\"Ws-\",linecolor=:cyan)\nplot!(aspect_ratio=1,ylims=[-1,1],xlims=[-0.6,1.3],legend=:topleft,flip=false)\n\n## We can see two strands moving in towards the sun, but the rest is a jumbled mess. Let's zoom in to see what's going on. Also, I need to make sure things are scaled correctly, or Jupiter will look way too big.\nplot(sys)\nplot!(sol1up,vars=(1,2),label=\"Wu+\",linecolor=:red)\nplot!(sol1un,vars=(1,2),label=\"Wu-\",linecolor=:magenta)\nplot!(sol1sp,vars=(1,2),label=\"Ws+\",linecolor=:blue)\nplot!(sol1sn,vars=(1,2),label=\"Ws-\",linecolor=:cyan)\nplot!(aspect_ratio=1,ylims=[-0.04,0.04],xlims=[0.93,1.06],legend=:outerright,flip=false)\n\n# L2 points\nrv2up = rv2 + α*Ywu # unstable manifold + side\nrv2un = rv2 - α*Ywu # unstable manifold - side\nrv2sp = rv2 + α*Yws # stable manifold + side\nrv2sn = rv2 - α*Yws # stable manifold - side\n\n## Now we look at the L2 point manifolds\nprob2up = ODEProblem(CR3BPdynamics!,rv2up,tspan_forward,sys) # Unstable positive\nprob2un = ODEProblem(CR3BPdynamics!,rv2un,tspan_forward,sys) # Unstable negative\nprob2sp = ODEProblem(CR3BPdynamics!,rv2sp,tspan_back,sys) # Stable positive\nprob2sn = ODEProblem(CR3BPdynamics!,rv2sn,tspan_back,sys) # Stable negative\n\nsol2up = solve(prob2up, reltol=1e-6)\nsol2un = solve(prob2un, reltol=1e-6)\nsol2sp = solve(prob2sp, reltol=1e-6)\nsol2sn = solve(prob2sn, reltol=1e-6)\n\nplot(sys, scaled=true)\nplot!(sol2up,vars=(1,2),label=\"Wu+\",linecolor=:red)\nplot!(sol2un,vars=(1,2),label=\"Wu-\",linecolor=:magenta)\nplot!(sol2sp,vars=(1,2),label=\"Ws+\",linecolor=:blue)\nplot!(sol2sn,vars=(1,2),label=\"Ws-\",linecolor=:cyan)\nplot!(aspect_ratio=1,ylims=[-1.0,1.0],xlims=[-0.3,1.5],legend=:topright,flip=false)\n## This time there are two strands moving out away from the Sun, along with another jumble around Jupiter. Let's zoom in again to take a look.\nplot(sys)\nplot!(sol2up,vars=(1,2),label=\"Wu+\",linecolor=:red)\nplot!(sol2un,vars=(1,2),label=\"Wu-\",linecolor=:magenta)\nplot!(sol2sp,vars=(1,2),label=\"Ws+\",linecolor=:blue)\nplot!(sol2sn,vars=(1,2),label=\"Ws-\",linecolor=:cyan)\nplot!(aspect_ratio=1,ylims=[-0.04,0.04],xlims=[0.93,1.07],legend=:outerright,flip=false)\n", "meta": {"hexsha": "d4cf80dfb6795d587b1e517743c99c286a4e4e9f", "size": 5521, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "example/5_LagrangePointManifolds.jl", "max_stars_repo_name": "jared711/ThreeBodyProblem_old.jl", "max_stars_repo_head_hexsha": "492c1fc34aa5d0a7b003ce224429a634230438d5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-07-28T01:21:10.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-28T22:34:55.000Z", "max_issues_repo_path": "example/5_LagrangePointManifolds.jl", "max_issues_repo_name": "jared711/ThreeBodyProblem_old.jl", "max_issues_repo_head_hexsha": "492c1fc34aa5d0a7b003ce224429a634230438d5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "example/5_LagrangePointManifolds.jl", "max_forks_repo_name": "jared711/ThreeBodyProblem_old.jl", "max_forks_repo_head_hexsha": "492c1fc34aa5d0a7b003ce224429a634230438d5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 46.7881355932, "max_line_length": 302, "alphanum_fraction": 0.738815432, "num_tokens": 1838, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133430934989, "lm_q2_score": 0.8633916152464017, "lm_q1q2_score": 0.8115132994851414}} {"text": "################################################################################\n#\n# The LLL-Reduction Algorithm\n#\n# Implementration of algorithm described in :\n# Lenstra, A.K., Lenstra, H.W. & Lovász, L. Math. Ann. (1982)\n# 261: 515. https://doi.org/10.1007/BF01457454\n#\n################################################################################\n\n\"\"\"\n lll(b₀::AbstractMatrix{Float64})::AbstractMatrix{Float64}\n\nPerforms Lenstra-Lenstra-Lovasv reduction on the n vectors in b₀.\n\nParameters\n----------\n- b₀ : vector of n vectors of n Float64\n\nReturns\n-------\n- The LLL reduction of the vectors in `b₀`\n\"\"\"\nfunction lll(b₀::Vector{Vector{Float64}})::Vector{Vector{Float64}}\n b = deepcopy(b₀) # preserve argument by working on a copy\n n = length(b)\n\n # initialize mu and B with zeros\n μ = zeros(n,n)\n B = zeros(n)\n\n # function (*) of page 521\n function ☼(l)\n abs(μ[k,l]) < 0.5 && return\n r = round(μ[k,l])\n b[k] .-= r * b[l]\n μ[k,1:l-1] .-= r * μ[l,1:l-1]\n μ[k,l] -= r\n end\n\n b₂ = similar(b)\n for t in 1:n\n b₂[t] = copy(b[t])\n for j in 1:t-1\n μ[t,j] = dot(b[t], b₂[j]) / B[j]\n b₂[t] .-= μ[t,j] * b₂[j]\n end\n B[t] = dot(b₂[t], b₂[t])\n end\n\n k = 2\n while k <= n\n ☼(k-1)\n if B[k] >= (0.75 - μ[k,k-1]^2) * B[k-1]\n foreach(☼, k-2:-1:1)\n k += 1\n else\n μₒ = μ[k,k-1]\n Bₒ = B[k] + μₒ^2 * B[k-1]\n μ[k,k-1] = μₒ * B[k-1] / Bₒ\n B[k] = B[k] * B[k-1] / Bₒ\n B[k-1] = Bₒ\n\n bₒ = b[k] ; b[k] = b[k-1] ; b[k-1] = bₒ\n for j in 1:k-2\n tmp = μ[k,j] ; μ[k,j] = μ[k-1,j] ; μ[k-1,j] = tmp\n end\n for t in k+1:n\n tmp = μ[k,k-1] - μₒ * μ[t,k]\n μ[t,k-1] = μ[t,k] + μ[k,k-1] * tmp\n μ[t,k] = tmp\n end\n\n k = max(2, k-1)\n end\n end\n b\nend\n", "meta": {"hexsha": "305d473524abea64209bd87081a39f9969b8cd07", "size": 1985, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/lll.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/RiemannTheta.jl-803aa692-84f0-5eed-9cc1-fbb46710dded", "max_stars_repo_head_hexsha": "af371a543767493c495f535ada2ce896a27644e0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/lll.jl", "max_issues_repo_name": "UnofficialJuliaMirrorSnapshots/RiemannTheta.jl-803aa692-84f0-5eed-9cc1-fbb46710dded", "max_issues_repo_head_hexsha": "af371a543767493c495f535ada2ce896a27644e0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/lll.jl", "max_forks_repo_name": "UnofficialJuliaMirrorSnapshots/RiemannTheta.jl-803aa692-84f0-5eed-9cc1-fbb46710dded", "max_forks_repo_head_hexsha": "af371a543767493c495f535ada2ce896a27644e0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.1265822785, "max_line_length": 80, "alphanum_fraction": 0.3964735516, "num_tokens": 690, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133430934989, "lm_q2_score": 0.8633916152464016, "lm_q1q2_score": 0.8115132994851413}} {"text": "### A Pluto.jl notebook ###\n# v0.17.7\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ 64bba790-89ad-11ec-272d-af9bb791e63d\nbegin\n\tusing Turing, MCMCChains\n\tusing Distributions, StatsPlots, Random\nend\n\n# ╔═╡ dc1177eb-66ec-4b82-ad0d-5e76539faa24\nmd\"\"\"\n### Gaussian Mixture Model\n\nIn this post we see an example of gaussian mixture model, which is a clustering model of unsupervised learning.\nAs stated, gaussian mixture model has several normal distributions. The aim of this model is to find optimal value of parameters(μ, Σ, π) that describe clusters.\n\nLet's say we have n data and assumes K clusters.\n- θ parameters\n- Z latent variables(hidden variables)\n- X data\n\nWhat are the parameters here?\n- θ = {μ(μ₁ . . . μₖ), Σ(Σ₁ . . . Σₖ), π(π₁ . . . πₖ)}\n- μₖ is mean of kᵗʰ cluster\n- Σₖ is covariance matrix of kᵗʰ cluster\n- πₖ represent for probability of selecting kᵗʰ cluster\n\nWe assume that we have n data.\n- {(X₁, Z₁) . . . (Xₙ, Zₙ)}\n- Z can be interpreted as one-hot vector with only 1 in kᵗʰ position and 0 others.\n\n\"\"\"\n\n# ╔═╡ fc380f54-24c7-440a-9cfd-04d3a8ea3184\n# Set a random seed.\nRandom.seed!(3)\n\n# ╔═╡ 38643e87-ee2d-4aae-83f7-abc6199f54aa\n# Construct 30 data points for each cluster.\nN = 30\n\n# ╔═╡ b7acb16e-b6d1-4e53-80c4-0819259a6130\n# Parameters for each cluster, we assume that each cluster is Gaussian distributed in the example.\nμs = [4.0, 1.0]\n\n# ╔═╡ 24553bda-3ce4-4156-bf8a-0a1fe3df2afb\n# Construct the data points.\nx = mapreduce(c -> rand(MvNormal([μs[c], μs[c]], 1.), N), hcat, 1:2)\n\n# ╔═╡ a0871b5d-02f4-4f16-9bcd-2e85264b0f41\ngr()\n\n# ╔═╡ b5f72bd8-97b7-435f-8940-950af86aed91\n# Visualization.\nscatter(x[1,:], x[2,:], legend = false, title = \"Dataset\", fmt=\"png\")\n\n# ╔═╡ 00f15e65-5133-4453-88af-c9479c8e3f9e\n@model function GaussianMixtureModel(x)\n\n D, N = size(x)\n\n # Draw the parameters for cluster 1.\n μ1 ~ Normal()\n\n # Draw the parameters for cluster 2.\n μ2 ~ Normal()\n\n μ = [μ1, μ2]\n\t\t\n # draw the weights for the K clusters from a Dirichlet distribution.\n\n α = 1.0\n\tπ ~ Dirichlet(D, α)\n\n # Draw assignments for each datum and generate it from a multivariate normal.\n k = Vector{Int}(undef, N)\n for i in 1:N\n k[i] ~ Categorical(π)\n x[:,i] ~ MvNormal([μ[k[i]], μ[k[i]]], 1.)\n end\n return k\nend;\n\n\n# ╔═╡ a45b70a7-acb0-48f1-8ae5-31c63992cbfa\ngmm_model = GaussianMixtureModel(x);\n\n# ╔═╡ 7fe105ae-7d5c-4dd8-8104-fdaa9a18ea25\ngmm_sampler = Gibbs(PG(100, :k), HMC(0.05, 10, :μ1, :μ2))\n\n# ╔═╡ c30df2ab-45c2-42c3-9e72-2c4759b9b640\ntchain = sample(gmm_model, gmm_sampler, MCMCThreads(), 200, 3)\n\n# ╔═╡ 9f18a980-5277-4419-912b-e261451427b0\nsummarystats(tchain)\n\n# ╔═╡ c726de5c-e241-4fa5-984d-c4184a626e52\nnames(tchain)\n\n# ╔═╡ 166d94ed-1e70-4ef6-8b78-f5b566216246\nids = findall(map(name -> occursin(\"μ\", string(name)), names(tchain)))\n\n# ╔═╡ 29c24b11-fad4-4a2f-afbe-9f6afaab82f1\np = plot(tchain[:, ids, :]; legend=true, labels=[\"Mu 1\" \"Mu 2\"], colordim=:parameter)\n\n# ╔═╡ 40214128-1ed7-4b37-b19e-78fafdebbb0b\ntchain2 = tchain[:, :, 3];\n\n# ╔═╡ 2c2ea945-795a-4e0a-bc16-c49fa15bee1c\n# Helper function for plotting\nfunction predict(x, y, π, μ)\n \n return Turing.logaddexp(\n log(π[1]) + logpdf(MvNormal([μ[1], μ[1]], 1.), [x, y]),\n log(π[2]) + logpdf(MvNormal([μ[2], μ[2]], 1.), [x, y])\n )\nend\n\n# ╔═╡ 60d2ae42-7e56-418e-88a9-2ad16e84555e\n# plotting contour plot\nbegin\n\tcontour(range(-2, stop = 8), range(-2, stop = 8),\n\t (x, y) -> predict(x, y, [mean(tchain2[:\"π[1]\"]), mean(tchain2[:\"π[2]\"])], \t[mean(tchain2[:μ1]), mean(tchain2[:μ2])])\n\t)\n\tscatter!(x[1,:], x[2,:]; legend=false, title=\"Dataset\")\n\t\nend\n\n# ╔═╡ 88c6f515-b4d9-4e85-9157-2fc1e81483a2\nwireframe(range(-2, stop = 8), range(-2, stop = 8),\n\t (x, y) -> predict(x, y, [mean(tchain2[:\"π[1]\"]), mean(tchain2[:\"π[2]\"])], \t[mean(tchain2[:μ1]), mean(tchain2[:μ2])]))\n\n# ╔═╡ 63703e6a-7adb-43c3-82aa-34f5aa746017\nmean(MCMCChains.group(tchain2, :k))\n\n# ╔═╡ b81eb0e7-d3cd-49aa-ae14-b8436dc515a4\nassignments = round.(mean(MCMCChains.group(tchain2, :k)).nt.mean)\n\n# ╔═╡ 60b70f36-ffac-45c3-a3d4-78991ab57bab\nscatter(x[1,:], x[2,:]; legend=false,\n title=\"Assignments on Dataset\", zcolor=assignments)\n\n# ╔═╡ 00000000-0000-0000-0000-000000000001\nPLUTO_PROJECT_TOML_CONTENTS = \"\"\"\n[deps]\nDistributions = \"31c24e10-a181-5473-b8eb-7969acd0382f\"\nMCMCChains = \"c7f686f2-ff18-58e9-bc7b-31028e88f75d\"\nRandom = \"9a3f8284-a2c9-5f02-9a11-845980a1fd5c\"\nStatsPlots = \"f3b207a7-027a-5e70-b257-86293d7955fd\"\nTuring = \"fce5fe82-541a-59a6-adf8-730c64b5f9a0\"\n\n[compat]\nDistributions = \"~0.25.47\"\nMCMCChains = \"~5.0.3\"\nStatsPlots = \"~0.14.33\"\nTuring = \"~0.20.1\"\n\"\"\"\n\n# ╔═╡ 00000000-0000-0000-0000-000000000002\nPLUTO_MANIFEST_TOML_CONTENTS = \"\"\"\n# This file is machine-generated - editing it directly is not advised\n\n[[AbstractFFTs]]\ndeps = [\"ChainRulesCore\", \"LinearAlgebra\"]\ngit-tree-sha1 = \"6f1d9bc1c08f9f4a8fa92e3ea3cb50153a1b40d4\"\nuuid = \"621f4979-c628-5d54-868e-fcf4e3e8185c\"\nversion = \"1.1.0\"\n\n[[AbstractMCMC]]\ndeps = [\"BangBang\", \"ConsoleProgressMonitor\", \"Distributed\", \"Logging\", \"LoggingExtras\", \"ProgressLogging\", \"Random\", \"StatsBase\", \"TerminalLoggers\", \"Transducers\"]\ngit-tree-sha1 = \"db0a7ff3fbd987055c43b4e12d2fa30aaae8749c\"\nuuid = \"80f14c24-f653-4e6a-9b94-39d6b0f70001\"\nversion = \"3.2.1\"\n\n[[AbstractPPL]]\ndeps = [\"AbstractMCMC\", \"DensityInterface\", \"Setfield\"]\ngit-tree-sha1 = \"ca54027a17ca3133b36166191b5faa8a404e92d3\"\nuuid = \"7a57a42e-76ec-4ea3-a279-07e840d6d9cf\"\nversion = \"0.4.0\"\n\n[[AbstractTrees]]\ngit-tree-sha1 = \"03e0550477d86222521d254b741d470ba17ea0b5\"\nuuid = \"1520ce14-60c1-5f80-bbc7-55ef81b5835c\"\nversion = \"0.3.4\"\n\n[[Adapt]]\ndeps = [\"LinearAlgebra\"]\ngit-tree-sha1 = \"af92965fb30777147966f58acb05da51c5616b5f\"\nuuid = \"79e6a3ab-5dfb-504d-930d-738a2a938a0e\"\nversion = \"3.3.3\"\n\n[[AdvancedHMC]]\ndeps = [\"AbstractMCMC\", \"ArgCheck\", \"DocStringExtensions\", \"InplaceOps\", \"LinearAlgebra\", \"ProgressMeter\", \"Random\", \"Requires\", \"Setfield\", \"Statistics\", \"StatsBase\", \"StatsFuns\", \"UnPack\"]\ngit-tree-sha1 = \"189473a73d664fe2496675775b6c8a732b8dfe26\"\nuuid = \"0bf59076-c3b1-5ca4-86bd-e02cd72cde3d\"\nversion = \"0.3.3\"\n\n[[AdvancedMH]]\ndeps = [\"AbstractMCMC\", \"Distributions\", \"Random\", \"Requires\"]\ngit-tree-sha1 = \"8ad8bfddf8bb627d689ecb91599c349cbf15e971\"\nuuid = \"5b7e9947-ddc0-4b3f-9b55-0d8042f74170\"\nversion = \"0.6.6\"\n\n[[AdvancedPS]]\ndeps = [\"AbstractMCMC\", \"Distributions\", \"Libtask\", \"Random\", \"StatsFuns\"]\ngit-tree-sha1 = \"59c47e9525d5a807a950d8b79b5d6e60b2f6de82\"\nuuid = \"576499cb-2369-40b2-a588-c64705576edc\"\nversion = \"0.3.3\"\n\n[[AdvancedVI]]\ndeps = [\"Bijectors\", \"Distributions\", \"DistributionsAD\", \"DocStringExtensions\", \"ForwardDiff\", \"LinearAlgebra\", \"ProgressMeter\", \"Random\", \"Requires\", \"StatsBase\", \"StatsFuns\", \"Tracker\"]\ngit-tree-sha1 = \"130d6b17a3a9d420d9a6b37412cae03ffd6a64ff\"\nuuid = \"b5ca4192-6429-45e5-a2d9-87aec30a685c\"\nversion = \"0.1.3\"\n\n[[ArgCheck]]\ngit-tree-sha1 = \"a3a402a35a2f7e0b87828ccabbd5ebfbebe356b4\"\nuuid = \"dce04be8-c92d-5529-be00-80e4d2c0e197\"\nversion = \"2.3.0\"\n\n[[ArgTools]]\nuuid = \"0dad84c5-d112-42e6-8d28-ef12dabb789f\"\n\n[[Arpack]]\ndeps = [\"Arpack_jll\", \"Libdl\", \"LinearAlgebra\", \"Logging\"]\ngit-tree-sha1 = \"91ca22c4b8437da89b030f08d71db55a379ce958\"\nuuid = \"7d9fca2a-8960-54d3-9f78-7d1dccf2cb97\"\nversion = \"0.5.3\"\n\n[[Arpack_jll]]\ndeps = [\"Libdl\", \"OpenBLAS_jll\", \"Pkg\"]\ngit-tree-sha1 = \"e214a9b9bd1b4e1b4f15b22c0994862b66af7ff7\"\nuuid = \"68821587-b530-5797-8361-c406ea357684\"\nversion = \"3.5.0+3\"\n\n[[ArrayInterface]]\ndeps = [\"Compat\", \"IfElse\", \"LinearAlgebra\", \"Requires\", \"SparseArrays\", \"Static\"]\ngit-tree-sha1 = \"745233d77146ad221629590b6d82fe7f1ddb478f\"\nuuid = \"4fba245c-0d91-5ea0-9b3e-6abc04ee57a9\"\nversion = \"4.0.3\"\n\n[[Artifacts]]\nuuid = \"56f22d72-fd6d-98f1-02f0-08ddc0907c33\"\n\n[[AxisAlgorithms]]\ndeps = [\"LinearAlgebra\", \"Random\", \"SparseArrays\", \"WoodburyMatrices\"]\ngit-tree-sha1 = \"66771c8d21c8ff5e3a93379480a2307ac36863f7\"\nuuid = \"13072b0f-2c55-5437-9ae7-d433b7a33950\"\nversion = \"1.0.1\"\n\n[[AxisArrays]]\ndeps = [\"Dates\", \"IntervalSets\", \"IterTools\", \"RangeArrays\"]\ngit-tree-sha1 = \"d127d5e4d86c7680b20c35d40b503c74b9a39b5e\"\nuuid = \"39de3d68-74b9-583c-8d2d-e117c070f3a9\"\nversion = \"0.4.4\"\n\n[[BangBang]]\ndeps = [\"Compat\", \"ConstructionBase\", \"Future\", \"InitialValues\", \"LinearAlgebra\", \"Requires\", \"Setfield\", \"Tables\", \"ZygoteRules\"]\ngit-tree-sha1 = \"d648adb5e01b77358511fb95ea2e4d384109fac9\"\nuuid = \"198e06fe-97b7-11e9-32a5-e1d131e6ad66\"\nversion = \"0.3.35\"\n\n[[Base64]]\nuuid = \"2a0f44e3-6c83-55bd-87e4-b1978d98bd5f\"\n\n[[Baselet]]\ngit-tree-sha1 = \"aebf55e6d7795e02ca500a689d326ac979aaf89e\"\nuuid = \"9718e550-a3fa-408a-8086-8db961cd8217\"\nversion = \"0.1.1\"\n\n[[Bijectors]]\ndeps = [\"ArgCheck\", \"ChainRulesCore\", \"Compat\", \"Distributions\", \"Functors\", \"IrrationalConstants\", \"LinearAlgebra\", \"LogExpFunctions\", \"MappedArrays\", \"Random\", \"Reexport\", \"Requires\", \"Roots\", \"SparseArrays\", \"Statistics\"]\ngit-tree-sha1 = \"369af32fcb9be65d496dc43ad0bb713705d4e859\"\nuuid = \"76274a88-744f-5084-9051-94815aaf08c4\"\nversion = \"0.9.11\"\n\n[[Bzip2_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"19a35467a82e236ff51bc17a3a44b69ef35185a2\"\nuuid = \"6e34b625-4abd-537c-b88f-471c36dfa7a0\"\nversion = \"1.0.8+0\"\n\n[[Cairo]]\ndeps = [\"Cairo_jll\", \"Colors\", \"Glib_jll\", \"Graphics\", \"Libdl\", \"Pango_jll\"]\ngit-tree-sha1 = \"d0b3f8b4ad16cb0a2988c6788646a5e6a17b6b1b\"\nuuid = \"159f3aea-2a34-519c-b102-8c37f9878175\"\nversion = \"1.0.5\"\n\n[[Cairo_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"Fontconfig_jll\", \"FreeType2_jll\", \"Glib_jll\", \"JLLWrappers\", \"LZO_jll\", \"Libdl\", \"Pixman_jll\", \"Pkg\", \"Xorg_libXext_jll\", \"Xorg_libXrender_jll\", \"Zlib_jll\", \"libpng_jll\"]\ngit-tree-sha1 = \"4b859a208b2397a7a623a03449e4636bdb17bcf2\"\nuuid = \"83423d85-b0ee-5818-9007-b63ccbeb887a\"\nversion = \"1.16.1+1\"\n\n[[ChainRules]]\ndeps = [\"ChainRulesCore\", \"Compat\", \"IrrationalConstants\", \"LinearAlgebra\", \"Random\", \"RealDot\", \"SparseArrays\", \"Statistics\"]\ngit-tree-sha1 = \"849d4cb467ea3ecbbd3efe68dacd36f9429b543c\"\nuuid = \"082447d4-558c-5d27-93f4-14fc19e9eca2\"\nversion = \"1.26.0\"\n\n[[ChainRulesCore]]\ndeps = [\"Compat\", \"LinearAlgebra\", \"SparseArrays\"]\ngit-tree-sha1 = \"f9982ef575e19b0e5c7a98c6e75ee496c0f73a93\"\nuuid = \"d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4\"\nversion = \"1.12.0\"\n\n[[ChangesOfVariables]]\ndeps = [\"ChainRulesCore\", \"LinearAlgebra\", \"Test\"]\ngit-tree-sha1 = \"bf98fa45a0a4cee295de98d4c1462be26345b9a1\"\nuuid = \"9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0\"\nversion = \"0.1.2\"\n\n[[Clustering]]\ndeps = [\"Distances\", \"LinearAlgebra\", \"NearestNeighbors\", \"Printf\", \"SparseArrays\", \"Statistics\", \"StatsBase\"]\ngit-tree-sha1 = \"75479b7df4167267d75294d14b58244695beb2ac\"\nuuid = \"aaaa29a8-35af-508c-8bc3-b662a17a0fe5\"\nversion = \"0.14.2\"\n\n[[ColorSchemes]]\ndeps = [\"ColorTypes\", \"Colors\", \"FixedPointNumbers\", \"Luxor\", \"Random\"]\ngit-tree-sha1 = \"5b7d2a8b53c68dfdbce545e957a3b5cc4da80b01\"\nuuid = \"35d6a980-a343-548e-a6ea-1d62b119f2f4\"\nversion = \"3.17.0\"\n\n[[ColorTypes]]\ndeps = [\"FixedPointNumbers\", \"Random\"]\ngit-tree-sha1 = \"024fe24d83e4a5bf5fc80501a314ce0d1aa35597\"\nuuid = \"3da002f7-5984-5a60-b8a6-cbb66c0b333f\"\nversion = \"0.11.0\"\n\n[[Colors]]\ndeps = [\"ColorTypes\", \"FixedPointNumbers\", \"Reexport\"]\ngit-tree-sha1 = \"417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40\"\nuuid = \"5ae59095-9a9b-59fe-a467-6f913c188581\"\nversion = \"0.12.8\"\n\n[[Combinatorics]]\ngit-tree-sha1 = \"08c8b6831dc00bfea825826be0bc8336fc369860\"\nuuid = \"861a8166-3701-5b0c-9a16-15d98fcdc6aa\"\nversion = \"1.0.2\"\n\n[[CommonSolve]]\ngit-tree-sha1 = \"68a0743f578349ada8bc911a5cbd5a2ef6ed6d1f\"\nuuid = \"38540f10-b2f7-11e9-35d8-d573e4eb0ff2\"\nversion = \"0.2.0\"\n\n[[CommonSubexpressions]]\ndeps = [\"MacroTools\", \"Test\"]\ngit-tree-sha1 = \"7b8a93dba8af7e3b42fecabf646260105ac373f7\"\nuuid = \"bbf7d656-a473-5ed7-a52c-81e309532950\"\nversion = \"0.3.0\"\n\n[[Compat]]\ndeps = [\"Base64\", \"Dates\", \"DelimitedFiles\", \"Distributed\", \"InteractiveUtils\", \"LibGit2\", \"Libdl\", \"LinearAlgebra\", \"Markdown\", \"Mmap\", \"Pkg\", \"Printf\", \"REPL\", \"Random\", \"SHA\", \"Serialization\", \"SharedArrays\", \"Sockets\", \"SparseArrays\", \"Statistics\", \"Test\", \"UUIDs\", \"Unicode\"]\ngit-tree-sha1 = \"44c37b4636bc54afac5c574d2d02b625349d6582\"\nuuid = \"34da2185-b29b-5c13-b0c7-acf172513d20\"\nversion = \"3.41.0\"\n\n[[CompilerSupportLibraries_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"e66e0078-7015-5450-92f7-15fbd957f2ae\"\n\n[[CompositionsBase]]\ngit-tree-sha1 = \"455419f7e328a1a2493cabc6428d79e951349769\"\nuuid = \"a33af91c-f02d-484b-be07-31d278c5ca2b\"\nversion = \"0.1.1\"\n\n[[ConsoleProgressMonitor]]\ndeps = [\"Logging\", \"ProgressMeter\"]\ngit-tree-sha1 = \"3ab7b2136722890b9af903859afcf457fa3059e8\"\nuuid = \"88cd18e8-d9cc-4ea6-8889-5259c0d15c8b\"\nversion = \"0.1.2\"\n\n[[ConstructionBase]]\ndeps = [\"LinearAlgebra\"]\ngit-tree-sha1 = \"f74e9d5388b8620b4cee35d4c5a618dd4dc547f4\"\nuuid = \"187b0558-2788-49d3-abe0-74a17ed4e7c9\"\nversion = \"1.3.0\"\n\n[[Contour]]\ndeps = [\"StaticArrays\"]\ngit-tree-sha1 = \"9f02045d934dc030edad45944ea80dbd1f0ebea7\"\nuuid = \"d38c429a-6771-53c6-b99e-75d170b6e991\"\nversion = \"0.5.7\"\n\n[[Crayons]]\ngit-tree-sha1 = \"249fe38abf76d48563e2f4556bebd215aa317e15\"\nuuid = \"a8cc5b0e-0ffa-5ad4-8c14-923d3ee1735f\"\nversion = \"4.1.1\"\n\n[[DataAPI]]\ngit-tree-sha1 = \"cc70b17275652eb47bc9e5f81635981f13cea5c8\"\nuuid = \"9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a\"\nversion = \"1.9.0\"\n\n[[DataStructures]]\ndeps = [\"Compat\", \"InteractiveUtils\", \"OrderedCollections\"]\ngit-tree-sha1 = \"3daef5523dd2e769dad2365274f760ff5f282c7d\"\nuuid = \"864edb3b-99cc-5e75-8d2d-829cb0a9cfe8\"\nversion = \"0.18.11\"\n\n[[DataValueInterfaces]]\ngit-tree-sha1 = \"bfc1187b79289637fa0ef6d4436ebdfe6905cbd6\"\nuuid = \"e2d170a0-9d28-54be-80f0-106bbe20a464\"\nversion = \"1.0.0\"\n\n[[DataValues]]\ndeps = [\"DataValueInterfaces\", \"Dates\"]\ngit-tree-sha1 = \"d88a19299eba280a6d062e135a43f00323ae70bf\"\nuuid = \"e7dc6d0d-1eca-5fa6-8ad6-5aecde8b7ea5\"\nversion = \"0.4.13\"\n\n[[Dates]]\ndeps = [\"Printf\"]\nuuid = \"ade2ca70-3891-5945-98fb-dc099432e06a\"\n\n[[DefineSingletons]]\ngit-tree-sha1 = \"0fba8b706d0178b4dc7fd44a96a92382c9065c2c\"\nuuid = \"244e2a9f-e319-4986-a169-4d1fe445cd52\"\nversion = \"0.1.2\"\n\n[[DelimitedFiles]]\ndeps = [\"Mmap\"]\nuuid = \"8bb1440f-4735-579b-a4ab-409b98df4dab\"\n\n[[DensityInterface]]\ndeps = [\"InverseFunctions\", \"Test\"]\ngit-tree-sha1 = \"80c3e8639e3353e5d2912fb3a1916b8455e2494b\"\nuuid = \"b429d917-457f-4dbc-8f4c-0cc954292b1d\"\nversion = \"0.4.0\"\n\n[[DiffResults]]\ndeps = [\"StaticArrays\"]\ngit-tree-sha1 = \"c18e98cba888c6c25d1c3b048e4b3380ca956805\"\nuuid = \"163ba53b-c6d8-5494-b064-1a9d43ac40c5\"\nversion = \"1.0.3\"\n\n[[DiffRules]]\ndeps = [\"IrrationalConstants\", \"LogExpFunctions\", \"NaNMath\", \"Random\", \"SpecialFunctions\"]\ngit-tree-sha1 = \"84083a5136b6abf426174a58325ffd159dd6d94f\"\nuuid = \"b552c78f-8df3-52c6-915a-8e097449b14b\"\nversion = \"1.9.1\"\n\n[[Distances]]\ndeps = [\"LinearAlgebra\", \"SparseArrays\", \"Statistics\", \"StatsAPI\"]\ngit-tree-sha1 = \"3258d0659f812acde79e8a74b11f17ac06d0ca04\"\nuuid = \"b4f34e82-e78d-54a5-968a-f98e89d6e8f7\"\nversion = \"0.10.7\"\n\n[[Distributed]]\ndeps = [\"Random\", \"Serialization\", \"Sockets\"]\nuuid = \"8ba89e20-285c-5b6f-9357-94700520ee1b\"\n\n[[Distributions]]\ndeps = [\"ChainRulesCore\", \"DensityInterface\", \"FillArrays\", \"LinearAlgebra\", \"PDMats\", \"Printf\", \"QuadGK\", \"Random\", \"SparseArrays\", \"SpecialFunctions\", \"Statistics\", \"StatsBase\", \"StatsFuns\", \"Test\"]\ngit-tree-sha1 = \"d6dc7e55eb64d9d2326436df15df9c8e4d8f1cd4\"\nuuid = \"31c24e10-a181-5473-b8eb-7969acd0382f\"\nversion = \"0.25.47\"\n\n[[DistributionsAD]]\ndeps = [\"Adapt\", \"ChainRules\", \"ChainRulesCore\", \"Compat\", \"DiffRules\", \"Distributions\", \"FillArrays\", \"LinearAlgebra\", \"NaNMath\", \"PDMats\", \"Random\", \"Requires\", \"SpecialFunctions\", \"StaticArrays\", \"StatsBase\", \"StatsFuns\", \"ZygoteRules\"]\ngit-tree-sha1 = \"61805bf57113a52435a13ca0bb588daf8848784d\"\nuuid = \"ced4e74d-a319-5a8a-b0ac-84af2272839c\"\nversion = 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\"0d47668e-0667-5a69-a72c-f761630bfb7e\"\nversion = \"0.3.9+1\"\n\n[[Xorg_xcb_util_wm_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xcb_util_jll\"]\ngit-tree-sha1 = \"e78d10aab01a4a154142c5006ed44fd9e8e31b67\"\nuuid = \"c22f9ab0-d5fe-5066-847c-f4bb1cd4e361\"\nversion = \"0.4.1+1\"\n\n[[Xorg_xkbcomp_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libxkbfile_jll\"]\ngit-tree-sha1 = \"4bcbf660f6c2e714f87e960a171b119d06ee163b\"\nuuid = \"35661453-b289-5fab-8a00-3d9160c6a3a4\"\nversion = \"1.4.2+4\"\n\n[[Xorg_xkeyboard_config_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xkbcomp_jll\"]\ngit-tree-sha1 = \"5c8424f8a67c3f2209646d4425f3d415fee5931d\"\nuuid = \"33bec58e-1273-512f-9401-5d533626f822\"\nversion = \"2.27.0+4\"\n\n[[Xorg_xtrans_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"79c31e7844f6ecf779705fbc12146eb190b7d845\"\nuuid = \"c5fb5394-a638-5e4d-96e5-b29de1b5cf10\"\nversion = \"1.4.0+3\"\n\n[[Zlib_jll]]\ndeps = [\"Libdl\"]\nuuid = \"83775a58-1f1d-513f-b197-d71354ab007a\"\n\n[[Zstd_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"e45044cd873ded54b6a5bac0eb5c971392cf1927\"\nuuid = \"3161d3a3-bdf6-5164-811a-617609db77b4\"\nversion = \"1.5.2+0\"\n\n[[ZygoteRules]]\ndeps = [\"MacroTools\"]\ngit-tree-sha1 = \"8c1a8e4dfacb1fd631745552c8db35d0deb09ea0\"\nuuid = \"700de1a5-db45-46bc-99cf-38207098b444\"\nversion = \"0.2.2\"\n\n[[gdk_pixbuf_jll]]\ndeps = [\"Artifacts\", \"Glib_jll\", \"JLLWrappers\", \"JpegTurbo_jll\", \"Libdl\", \"Libtiff_jll\", \"Pkg\", \"Xorg_libX11_jll\", \"libpng_jll\"]\ngit-tree-sha1 = \"c23323cd30d60941f8c68419a70905d9bdd92808\"\nuuid = \"da03df04-f53b-5353-a52f-6a8b0620ced0\"\nversion = \"2.42.6+1\"\n\n[[libass_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"FreeType2_jll\", \"FriBidi_jll\", \"HarfBuzz_jll\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"5982a94fcba20f02f42ace44b9894ee2b140fe47\"\nuuid = \"0ac62f75-1d6f-5e53-bd7c-93b484bb37c0\"\nversion = \"0.15.1+0\"\n\n[[libfdk_aac_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"daacc84a041563f965be61859a36e17c4e4fcd55\"\nuuid = \"f638f0a6-7fb0-5443-88ba-1cc74229b280\"\nversion = \"2.0.2+0\"\n\n[[libpng_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"94d180a6d2b5e55e447e2d27a29ed04fe79eb30c\"\nuuid = \"b53b4c65-9356-5827-b1ea-8c7a1a84506f\"\nversion = \"1.6.38+0\"\n\n[[libvorbis_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Ogg_jll\", \"Pkg\"]\ngit-tree-sha1 = \"b910cb81ef3fe6e78bf6acee440bda86fd6ae00c\"\nuuid = \"f27f6e37-5d2b-51aa-960f-b287f2bc3b7a\"\nversion = \"1.3.7+1\"\n\n[[nghttp2_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"8e850ede-7688-5339-a07c-302acd2aaf8d\"\n\n[[p7zip_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"3f19e933-33d8-53b3-aaab-bd5110c3b7a0\"\n\n[[x264_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"4fea590b89e6ec504593146bf8b988b2c00922b2\"\nuuid = \"1270edf5-f2f9-52d2-97e9-ab00b5d0237a\"\nversion = \"2021.5.5+0\"\n\n[[x265_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"ee567a171cce03570d77ad3a43e90218e38937a9\"\nuuid = \"dfaa095f-4041-5dcd-9319-2fabd8486b76\"\nversion = \"3.5.0+0\"\n\n[[xkbcommon_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Wayland_jll\", \"Wayland_protocols_jll\", \"Xorg_libxcb_jll\", \"Xorg_xkeyboard_config_jll\"]\ngit-tree-sha1 = \"ece2350174195bb31de1a63bea3a41ae1aa593b6\"\nuuid = \"d8fb68d0-12a3-5cfd-a85a-d49703b185fd\"\nversion = \"0.9.1+5\"\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╟─dc1177eb-66ec-4b82-ad0d-5e76539faa24\n# ╟─64bba790-89ad-11ec-272d-af9bb791e63d\n# ╠═fc380f54-24c7-440a-9cfd-04d3a8ea3184\n# ╠═38643e87-ee2d-4aae-83f7-abc6199f54aa\n# ╠═b7acb16e-b6d1-4e53-80c4-0819259a6130\n# ╠═24553bda-3ce4-4156-bf8a-0a1fe3df2afb\n# ╟─a0871b5d-02f4-4f16-9bcd-2e85264b0f41\n# ╠═b5f72bd8-97b7-435f-8940-950af86aed91\n# ╠═00f15e65-5133-4453-88af-c9479c8e3f9e\n# ╠═a45b70a7-acb0-48f1-8ae5-31c63992cbfa\n# ╠═7fe105ae-7d5c-4dd8-8104-fdaa9a18ea25\n# ╠═c30df2ab-45c2-42c3-9e72-2c4759b9b640\n# ╠═9f18a980-5277-4419-912b-e261451427b0\n# ╠═c726de5c-e241-4fa5-984d-c4184a626e52\n# ╠═166d94ed-1e70-4ef6-8b78-f5b566216246\n# ╠═29c24b11-fad4-4a2f-afbe-9f6afaab82f1\n# ╠═40214128-1ed7-4b37-b19e-78fafdebbb0b\n# ╠═2c2ea945-795a-4e0a-bc16-c49fa15bee1c\n# ╠═60d2ae42-7e56-418e-88a9-2ad16e84555e\n# ╠═88c6f515-b4d9-4e85-9157-2fc1e81483a2\n# ╠═63703e6a-7adb-43c3-82aa-34f5aa746017\n# ╠═b81eb0e7-d3cd-49aa-ae14-b8436dc515a4\n# ╠═60b70f36-ffac-45c3-a3d4-78991ab57bab\n# ╟─00000000-0000-0000-0000-000000000001\n# ╟─00000000-0000-0000-0000-000000000002\n", "meta": {"hexsha": "827cfed948239352dd36491db6e902a5822ea9b9", "size": 59144, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "GMM.jl", "max_stars_repo_name": "delphinH/imagetoy", "max_stars_repo_head_hexsha": "a74e902aeaf8363aedef547de5aa3aa569e926bf", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "GMM.jl", "max_issues_repo_name": "delphinH/imagetoy", "max_issues_repo_head_hexsha": "a74e902aeaf8363aedef547de5aa3aa569e926bf", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "GMM.jl", "max_forks_repo_name": "delphinH/imagetoy", "max_forks_repo_head_hexsha": "a74e902aeaf8363aedef547de5aa3aa569e926bf", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.4060500291, "max_line_length": 438, "alphanum_fraction": 0.735493034, "num_tokens": 26939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133498259924, "lm_q2_score": 0.863391599428538, "lm_q1q2_score": 0.8115132904304986}} {"text": "\r\n\r\nfunction triangular_solve_upper_rowwise(U, b)\r\n\r\n n_row = size(U, 1)\r\n x = zeros(eltype(b), n_row)\r\n\r\n @inbounds for ii = n_row:-1:1\r\n internal_sum = zero(eltype(b))\r\n @inbounds @simd for jj = n_row:-1:(ii+1)\r\n @fastmath internal_sum += U[ii, jj] * x[jj]\r\n end\r\n @fastmath x[ii] = (b[ii] - internal_sum) / U[ii, ii]\r\n end\r\n\r\n return x\r\nend\r\n\r\nfunction triangular_solve_upper_colwise(U, b)\r\n# function triangular_solve_upper_colwise(U{T}, b::Array{T}) where {T}\r\n\r\n n_col = size(U, 2)\r\n x = copy(b)\r\n\r\n @inbounds for jj = n_col:-1:1\r\n @fastmath x[jj] = x[jj] / U[jj, jj]\r\n # @view( x[1:(jj-1)] ) = @view( x[1:(jj-1)] ) - @view( U[1:(jj-1), jj] ) .* x[jj]\r\n @inbounds @simd for ii = 1:(jj-1)\r\n @fastmath x[ii] -= U[ii, jj] .* x[jj]\r\n end\r\n end\r\n\r\n return x\r\nend\r\n\r\nfunction triangular_solve_upper_colwise_transposed(L, b)\r\n # the input matrix L is a lower triangular matrix\r\n # but it solved as it was U = L', L' * x = b\r\n\r\n #=\r\n n_row = size(L, 1)\r\n x = zeros(eltype(b), n_row)\r\n\r\n @inbounds for ii = n_row:-1:1\r\n internal_sum = zero(eltype(b))\r\n @inbounds @simd for jj = n_row:-1:(ii+1)\r\n @fastmath internal_sum += L[jj, ii] * x[jj]\r\n end\r\n @inbounds @fastmath x[ii] = (b[ii] - internal_sum) / L[ii, ii]\r\n end\r\n\r\n return x\r\n =#\r\n return triangular_solve_upper_colwise(L', b)\r\nend\r\n\r\nfunction triangular_solve_lower_rowwise(L, b)\r\n# function triangular_solve_lower_rowwise(L{T}, b::Array{T}) where {T}\r\n\r\n n_row = size(L, 1)\r\n x = zeros(eltype(b), n_row)\r\n \r\n for ii = 1:n_row\r\n internal_sum = zero(eltype(b))\r\n @inbounds @simd for jj = 1:(ii-1)\r\n @fastmath internal_sum += L[ii, jj] * x[jj]\r\n end\r\n @fastmath x[ii] = (b[ii] - internal_sum) / L[ii, ii];\r\n end\r\n \r\n return x\r\nend\r\n\r\n\r\nfunction triangular_solve_lower_colwise(L, b)\r\n# function triangular_solve_lower_colwise(L{T}, b::Array{T}) where {T}\r\n\r\n (n_row, n_col) = size(L)\r\n x = copy(b)\r\n\r\n # Single threaded\r\n @inbounds for jj = 1:n_col\r\n @fastmath x[jj] = x[jj] / L[jj, jj]\r\n @inbounds @simd for rr = (jj+1):n_row\r\n @fastmath x[rr] -= L[rr, jj] .* x[jj]\r\n end\r\n end\r\n\r\n return x\r\nend\r\n\r\n\r\nfunction triangular_solve_lower_sparse(spmat::SparseMatrixCSC, b)\r\n # forward substitution\r\n\r\n n_row = spmat.n_row\r\n n_col = spmat.n_col\r\n \r\n x = copy(b)\r\n L11 = spmat[1] # 1,1 location\r\n x[1] = x[1] / L11\r\n\r\n L::Int = length(spmat)\r\n idx_end::Int = 0\r\n\r\n for cc = 1:spmat.n_col\r\n \r\n idx_start = spmat.col_begins_index[cc]\r\n if idx_start == 0\r\n error(\"Singular matrix cannot be solved\")\r\n end\r\n\r\n # find next nonzero index\r\n idx_end = find_next_nonempty_col_begin(spmat, cc)\r\n\r\n for rr = idx_start:(idx_end-1)\r\n a_ij = spmat.data[rr]\r\n ii = spmat.row_index[rr]\r\n x[ii] /= a_ij\r\n end\r\n\r\n end\r\n\r\n return x\r\nend\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n", "meta": {"hexsha": "625bafa4f4d5f8419260c12b184261f403d2cc3a", "size": 3099, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "__lib__/math/linalg/src/triangular.jl", "max_stars_repo_name": "HomoModelicus/julia", "max_stars_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "__lib__/math/linalg/src/triangular.jl", "max_issues_repo_name": "HomoModelicus/julia", "max_issues_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "__lib__/math/linalg/src/triangular.jl", "max_forks_repo_name": "HomoModelicus/julia", "max_forks_repo_head_hexsha": "26be81348032ccd2728046193ce627c823a3804b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.3007518797, "max_line_length": 90, "alphanum_fraction": 0.5359793482, "num_tokens": 943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9399133447766225, "lm_q2_score": 0.863391599428538, "lm_q1q2_score": 0.811513286070915}} {"text": "export tukey\n\"\"\"\n tukey(N::Int, r::AbstractArray) -> AbstractArray\n\nTukey window is a rectangular window with the first and\nlast r/2 percent of the samples equal to parts of a cosine.\n\"\"\"\nfunction tukey(N::Int, r::AbstractFloat=0.5)\n n = (0:N-1)/(N-1)\n w = zeros(N)\n for i = 1:N\n x = n[i]\n if 0 <= x <= r/2\n w[i] = 0.5 + 0.5*cos(2*pi/r*(x-r/2))\n elseif r/2 <= x <= 1-r/2\n w[i] = 1.0\n else\n w[i] = 0.5 + 0.5*cos(2*pi/r*(x-1+r/2))\n end\n end\n return w\nend\n\n\nexport gauss\n\"\"\"\n gauss(N::Int, a::AbstractFloat=0.5)\n\na is proportional to the standard deviation\n\"\"\"\nfunction gauss(N::Int, a::AbstractFloat=0.4)\n n = 0:N-1\n m = (N-1)/2\n return @. exp(-((n - m)/m/a)^2/2)\nend\n", "meta": {"hexsha": "2a014df884980819f54e6acaad522cae598fd409", "size": 764, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/2-binary.jl", "max_stars_repo_name": "sonosole/WindowFunctions.jl", "max_stars_repo_head_hexsha": "82c67745ad4a6ef1d751f45d3abd6239ca5afd7a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-08-13T16:25:11.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-13T16:25:11.000Z", "max_issues_repo_path": "src/2-binary.jl", "max_issues_repo_name": "sonosole/WindowFunctions.jl", "max_issues_repo_head_hexsha": "82c67745ad4a6ef1d751f45d3abd6239ca5afd7a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/2-binary.jl", "max_forks_repo_name": "sonosole/WindowFunctions.jl", "max_forks_repo_head_hexsha": "82c67745ad4a6ef1d751f45d3abd6239ca5afd7a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.2222222222, "max_line_length": 59, "alphanum_fraction": 0.5248691099, "num_tokens": 282, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9284088084787997, "lm_q2_score": 0.8740772482857833, "lm_q1q2_score": 0.811501016599432}} {"text": "\n\"\"\"\n\tmeanconfid(x;sigma)\nCompute the mean with confidence interval assuming normal\ndistribution (or large sample size => central limit theorem)\n\n**Input**\n* x: vector of samples (NaNs will be removed)\n* sigma: optional input in case sigma (standard deviation, not variance) is known\n* alpha: (1-alpha) gives the conficende interval for x_mean\n\n**Output**\n* x_mean: estimated mean\n* x_confid: tuple with (lower,upper) confidence bounds defined by alpha\n\n**Example**\n```\n# known standard deviation\nx = [2.,3.,5.,6.,9.]\nx_mean,x_confid = meanconfid(x,sigma=2.5)\n\n# including NaN and Unknown sigma\npush!(x,NaN);\nx_mean,x_confid = meanconfid(x)\n```\n\"\"\"\nfunction meanconfid(x::Vector{Float64};alpha::Float64=0.05,sigma::Float64=NaN)\n\tx_use = HypoTest.prepdata(x);\n\tx_mean = Distributions.mean(x_use);\n\ts = isnan(sigma) ? Distributions.std(x_use) : sigma;\n\tif isnan(sigma) && length(x_use) < 100\n\t\td = Distributions.TDist(length(x_use)-1);\n\telse\n\t\td = Distributions.Normal();\n\tend\n\tq = Distributions.quantile(d,1-alpha/2);\n\treturn x_mean,(x_mean-q*s/sqrt(length(x_use)),x_mean+q*s/sqrt(length(x_use)))\nend\n\n\"\"\"\n\tstdconfid(x;x_mean)\nCompute the standard deviation with confidence interval assuming normal\ndistribution\n\n**Input**\n* x: vector of samples (NaNs will be removed)\n* x_mean: optional input in case true mean is known (=>n instead n-1)\n* alpha: (1-alpha) gives the conficende interval for sigma\n\n**Output**\n* sigma: estimated standard deviation (not variance)\n* x_confid: tuple with (lower,upper) confidence bounds defined by alpha\n\n**Example**\n```\nx = [1.95,1.96,1.98,1.99,2.00,2.00,2.01,2.01,2.01,2.01,\n\t2.02,2.02,2.03,2.03,2.04,2.04,2.05,2.06,2.08,2.10];\n# unknown mean value\ns,s_interval = stdconfid(x,alpha=0.05)\n\n# known mean\nx = [19.01,19.02,18.99,19.00,19.05,19.05,19.0,18.98,18.99,19.0];\ns,s_interval = stdconfid(x,alpha=0.05,x_mean=19.)\n```\n\"\"\"\nfunction stdconfid(x::Vector{Float64};alpha::Float64=0.05,x_mean::Float64=NaN)\n\tx_use = HypoTest.prepdata(x);\n\tsigma = Distributions.std(x_use, mean = isnan(x_mean) ? nothing : x_mean,\n\t\t\t\t\t\t\t\tcorrected = isnan(x_mean) ? true : false);\n\tk = isnan(x_mean) ? length(x_use)-1 : length(x_use);\n\td = Distributions.Chisq(k);\n\tqlow,qhigh = Distributions.quantile(d,1-alpha/2),Distributions.quantile(d,alpha/2);\n\treturn sigma,(sqrt(k*sigma^2/qlow), sqrt(k*sigma^2/qhigh))\nend\n", "meta": {"hexsha": "fb424cee9b947666a3d9c34a3dfb124f01f4471d", "size": 2326, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/momentestim.jl", "max_stars_repo_name": "emenems/HypoTest", "max_stars_repo_head_hexsha": "0cdbecb705825fc8bef718565400796aae44db84", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/momentestim.jl", "max_issues_repo_name": "emenems/HypoTest", "max_issues_repo_head_hexsha": "0cdbecb705825fc8bef718565400796aae44db84", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/momentestim.jl", "max_forks_repo_name": "emenems/HypoTest", "max_forks_repo_head_hexsha": "0cdbecb705825fc8bef718565400796aae44db84", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.0133333333, "max_line_length": 84, "alphanum_fraction": 0.7115219261, "num_tokens": 748, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9284087985746093, "lm_q2_score": 0.8740772368049823, "lm_q1q2_score": 0.8115009972835279}} {"text": "using FrankWolfe\nusing LinearAlgebra\nusing LaTeXStrings\nusing Plots\n\n# # FrankWolfe for scaled, shifted ``\\ell^1`` and ``\\ell^{\\infty}`` norm balls\n\n# In this example, we run the vanilla FrankWolfe algorithm on a scaled and shifted ``\\ell^1`` and ``\\ell^{\\infty}`` norm ball, using the `ScaledBoundL1NormBall`\n# and `ScaledBoundLInfNormBall` LMOs. We shift both onto the point ``(1,0)`` and then scale them by a factor of ``2`` along the x-axis. We project the point ``(2,1)`` onto the polytopes.\n\nn = 2\n\nk = 1000\n\nxp = [2.0,1.0]\n\nf(x) = norm(x-xp)^2\n\nfunction grad!(storage,x)\n @. storage = 2 * (x - xp)\n return nothing\nend\n\nlower = [-1.0,-1.0]\nupper = [3.0,1.0]\n\nl1 = FrankWolfe.ScaledBoundL1NormBall(lower, upper)\n\nlinf = FrankWolfe.ScaledBoundLInfNormBall(lower, upper)\n\nx1 = FrankWolfe.compute_extreme_point(l1, zeros(n))\ngradient = collect(x1)\n\nx_l1, v_1, primal_1, dual_gap_1, trajectory_1 = FrankWolfe.frank_wolfe(\n f,\n grad!,\n l1,\n collect(copy(x1)),\n max_iteration=k,\n line_search=FrankWolfe.Shortstep(),\n L=2,\n print_iter=50,\n emphasis=FrankWolfe.memory,\n verbose=true,\n trajectory=true,\n);\n\nprintln(\"\\nFinal solution: \", x_l1)\n\nx2 = FrankWolfe.compute_extreme_point(linf, zeros(n))\ngradient = collect(x2)\n\nx_linf, v_2, primal_2, dual_gap_2, trajectory_2 = FrankWolfe.frank_wolfe(\n f,\n grad!,\n linf,\n collect(copy(x2)),\n max_iteration=k,\n line_search=FrankWolfe.Shortstep(),\n L=2,\n print_iter=50,\n emphasis=FrankWolfe.memory,\n verbose=true,\n trajectory=true,\n);\n\nprintln(\"\\nFinal solution: \", x_linf)\n\n\n# We plot the polytopes alongside the solutions from above:\n\nxcoord1 = [1,3,1,-1,1]\nycoord1 = [-1,0,1,0,-1]\n\nxcoord2 = [3,3,-1,-1,3]\nycoord2 = [-1,1,1,-1,-1]\n\nplot(xcoord1, ycoord1, title = \"Visualization of scaled shifted norm balls\", lw = 2, label = L\"\\ell^1 \\textrm{ norm}\")\nplot!(xcoord2, ycoord2, lw = 2, label = L\"\\ell^{\\infty} \\textrm{ norm}\")\nplot!([x_l1[1]], [x_l1[2]], seriestype = :scatter, lw = 5, color = \"blue\", label = L\"\\ell^1 \\textrm{ solution}\")\nplot!([x_linf[1]], [x_linf[2]], seriestype = :scatter, lw = 5, color = \"orange\", label = L\"\\ell^{\\infty} \\textrm{ solution}\", legend = :bottomleft)\n", "meta": {"hexsha": "d25e20b751edb33693009a9d2d16aa55514ebf4d", "size": 2198, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/docs_7_shifted_norm_polytopes.jl", "max_stars_repo_name": "gdalle/FrankWolfe.jl-2", "max_stars_repo_head_hexsha": "c3b3903c4808e24aa9e0f655aa2f8de0f2c1571c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 48, "max_stars_repo_stars_event_min_datetime": "2021-03-27T15:50:25.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-18T15:11:21.000Z", "max_issues_repo_path": "examples/docs_7_shifted_norm_polytopes.jl", "max_issues_repo_name": "gdalle/FrankWolfe.jl-2", "max_issues_repo_head_hexsha": "c3b3903c4808e24aa9e0f655aa2f8de0f2c1571c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 132, "max_issues_repo_issues_event_min_datetime": "2021-03-29T18:49:01.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-30T01:33:22.000Z", "max_forks_repo_path": "docs/src/examples/6_shifted_norm_polytopes.jl", "max_forks_repo_name": "dviladrich95/FrankWolfe.jl", "max_forks_repo_head_hexsha": "bcb441e52918bd1103f13296082cd7a8bc22607b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 11, "max_forks_repo_forks_event_min_datetime": "2021-06-02T13:38:48.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-17T10:04:42.000Z", "avg_line_length": 26.8048780488, "max_line_length": 186, "alphanum_fraction": 0.6665150136, "num_tokens": 768, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9284088005554476, "lm_q2_score": 0.874077222043951, "lm_q1q2_score": 0.8115009853106622}} {"text": "# Word Problems Logical Reasoning Tests\n# http://www.graduatewings.co.uk/word-problems-logical-tests\nusing AbstractLogic\n\n# Question 1\n# Sam is shorter than Hannah. William is shorter than Michael who is shorter than\n# Hannah. Hannah and William are shorter than Fiona. Who is the tallest?\nlogicset = logicalparse([\"Sam, Hannah, William, Michael, Fiona ∈ 1:5\"])\nlogicset = logicalparse([\"Sam < Hannah; William < Michael; Michael < Hannah\"], logicset)\nlogicset = logicalparse([\"Hannah < Fiona; William < Fiona\"], logicset)\n\n# A) Sam B) Hannah C) William D) Fiona E) Cannot Say\nsearch(\"{{i}} > {{!i}}\", logicset)\n# :Fiona is a match with 13 feasible combinations out of 13.\n# Therefore we choose D) Fiona\n\n\n# Question 2\n# Ben is taller than Sharon who is taller than Ralph. Ben is also taller than Mike.\nlogicset = logicalparse([\"Ben, Sharon, Ralph, Mike ∈ 1:4\"])\nlogicset = logicalparse([\"Ben > Sharon; Sharon > Ralph; Ben > Mike\"], logicset)\n\n# Who is the shortest?\n# A) Ben B) Sharon C) Ralph D) Mike E) Cannot Say\nsearch(\"{{i}} < {{!i}}\", logicset)\n# :Ralph is a possible match with 6 feasible combinations out of 11.\n# :Mike is a possible match with 1 feasible combinations out of 11.\n# Therefore we choose E) Cannot Say\n", "meta": {"hexsha": "fb93ce40f52955561fa0c8d2e0856aa1f77d66d8", "size": 1248, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/juliashell/OrderedReasoningProblems.jl", "max_stars_repo_name": "UnofficialJuliaMirrorSnapshots/AbstractLogic.jl-bd85187e-0531-4a3e-9fea-713204a818a2", "max_stars_repo_head_hexsha": "1b8adac10854471ec7ce83b9039cdeb1e43c0208", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2019-09-24T23:44:22.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-01T18:03:25.000Z", "max_issues_repo_path": "examples/juliashell/OrderedReasoningProblems.jl", "max_issues_repo_name": "UnofficialJuliaMirrorSnapshots/AbstractLogic.jl-bd85187e-0531-4a3e-9fea-713204a818a2", "max_issues_repo_head_hexsha": "1b8adac10854471ec7ce83b9039cdeb1e43c0208", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 12, "max_issues_repo_issues_event_min_datetime": "2019-08-08T19:13:18.000Z", "max_issues_repo_issues_event_max_datetime": "2019-08-10T01:13:59.000Z", "max_forks_repo_path": "examples/juliashell/OrderedReasoningProblems.jl", "max_forks_repo_name": "UnofficialJuliaMirrorSnapshots/AbstractLogic.jl-bd85187e-0531-4a3e-9fea-713204a818a2", "max_forks_repo_head_hexsha": "1b8adac10854471ec7ce83b9039cdeb1e43c0208", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-02-22T01:12:36.000Z", "max_forks_repo_forks_event_max_datetime": "2021-02-22T01:12:36.000Z", "avg_line_length": 43.0344827586, "max_line_length": 88, "alphanum_fraction": 0.703525641, "num_tokens": 339, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545348152282, "lm_q2_score": 0.8558511451289037, "lm_q1q2_score": 0.8114791443807761}} {"text": "module problem116\n\ndescription = \"\"\"\nRed, green or blue tiles\nA row of five black square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four).\nIf red tiles are chosen there are exactly seven ways this can be done.\nIf green tiles are chosen there are three ways.\nAnd if blue tiles are chosen there are two ways.\nAssuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways of replacing the black tiles in a row measuring five units in length.\n\nHow many different ways can the black tiles in a row measuring fifty units in length be replaced if colours cannot be mixed and at least one coloured tile must be used?\n\"\"\"\n\n# Fill function, based on problems 114, 115\nfunction F(m::Integer, n :: Integer, memo = zeros(Int, n))\n # Base case: no space remaining in the row\n if n < m return 0 end\n\n # Recursive case, count 1 for each placement of block at positions 1:n-m+1, and all ways of\n # filling the remaining space at the end of the row after each placement, memoizing results.\n if memo[n] == 0\n total = 0\n for blockpos = 1:(n-m+1)\n total += 1 + F(m, n - blockpos - m + 1, memo)\n end\n memo[n] = total\n end\n return memo[n]\nend\n\n# total ways to fill a row with only red,blue or green tiles.\nfunction homogenous_replacements(n :: Integer)\n return F(2, n) + F(3, n) + F(4, n)\nend\n\nusing Base.Test\n\n@test F(2, 5) == 7\n@test F(3, 5) == 3\n@test F(4, 5) == 2\n@test homogenous_replacements(5) == (7+3+2)\n\nend", "meta": {"hexsha": "e892f209b9e4922bc927061b91981bc101e6e473", "size": 1524, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "julia/problem116.jl", "max_stars_repo_name": "mbuhot/mbuhot-euler-solutions", "max_stars_repo_head_hexsha": "30066543cfd2d84976beb0605839750b64f4b8ef", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2015-12-18T13:25:41.000Z", "max_stars_repo_stars_event_max_datetime": "2015-12-18T13:25:41.000Z", "max_issues_repo_path": "julia/problem116.jl", "max_issues_repo_name": "mbuhot/mbuhot-euler-solutions", "max_issues_repo_head_hexsha": "30066543cfd2d84976beb0605839750b64f4b8ef", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "julia/problem116.jl", "max_forks_repo_name": "mbuhot/mbuhot-euler-solutions", "max_forks_repo_head_hexsha": "30066543cfd2d84976beb0605839750b64f4b8ef", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.4418604651, "max_line_length": 176, "alphanum_fraction": 0.7027559055, "num_tokens": 432, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9615338057771058, "lm_q2_score": 0.843895106480586, "lm_q1q2_score": 0.8114336734109537}} {"text": "#solve eq x=cos(x)\nfunction main()\n x=0.0\n xnext = cos(x)\n iter=0\n eps=1e-7\n iteration=1000000\n is_convergence=false\n\n for n in 1:iteration\n xnext = cos(x)\n if(abs(xnext-x)
![\"wedge](\"https://raw.githubusercontent.com/UCLAMAEThreads/MAE150A/master/notebook/WedgeDiagram.svg\")
\n=#\n\n#=\nFor the flow past a wedge with half-angle $\\beta/2$, potential flow theory predicts\nthat the external velocity has the form,\n$$U_e(x) = A x^{m}$$\nwhere $A$ is a positive constant and\n$$ m = \\frac{\\beta}{2-\\beta}$$\n\nOur focus here is on the boundary layer that develops along the wall of the wedge.\nThe boundary-layer flow has a **self-similar form** as it develops along the $x$ direction. Just as the Rayleigh flow develops with the same profile shape in time, with a time-varying thickness $\\delta(t) = (\\nu t)^{1/2}$, the boundary layer's profile has the same shape at every $x$ location, but with **spatially-varying thickness** proportional to\n$$\\delta(x) = \\left( \\frac{\\nu x}{U_e(x)}\\right)^{1/2}$$\n\nThis means that the velocity tangent to the wall can be written in the form\n$$ u(x,y) = U_e(x) F'(\\eta) $$\nwhere\n$$\\eta = \\dfrac{y}{\\delta(x)}$$\n\nWith this form, the boundary layer equations reduce to an ordinary differential\nequation for $F$ in this case,\n\n$$ F''' + \\frac{1}{2} (m+1) F F'' + m (1-F'^2) = 0$$\n\nThis is called the **Falkner-Skan equation**. The equation is non-linear, however,\nand must be solved computationally. The tools to solve it are provided for you here.\n=#\n\n#=\n#### Boundary conditions\nThe boundary conditions on the flow are no-slip ($u = 0$ at $y = 0$), no-flow-through\n($v = 0$ at $y = 0$). Also, the flow must approach the external velocity outside\nthe boundary layer. This means $u \\rightarrow U_e$ as $y \\rightarrow \\infty$.\n=#\n\n#=\n#### Special cases\nThe case $\\beta = 0$ (so that $m = 0$) corresponds to a flat plate, parallel to\nfree stream, and $U_e(x) = U_\\infty$. This is called a **Blasius boundary layer**.\n\nWhen $\\beta > 0$, the external flow is accelerating, and the pressure gradient\nis said to be **favorable**.\n\nWhen $\\beta < 0$, the external flow is decelerating, and the pressure gradient\nis adverse. The lowest admissible value of $\\beta$ is approximately $-0.1999$.\nBelow this, the boundary layer equations do not hold.\n\n#### Other boundary-layer quantities\n*Every* measure of thickness of the boundary layer is proportional to $\\delta(x)$.\nHowever, we need the solution of the Falkner-Skan equation to determine what\nthe constant of proportionality is.\n\nFor example, the 99% thickness is\n\n$$\\delta_{99}(x) = C_{1}\\delta(x)$$\n\nthe displacement thickness is\n\n$$\\delta^{*}(x) = C_{2} \\delta(x)$$\n\nand the momentum thickness is\n\n$$\\theta(x) = C_{3} \\delta(x)$$\n\nThese constants $C_1$, $C_2$ and $C_3$ depend on $\\beta$, and they can only be\nfound by solving the Falkner-Skan problem.\n\nAnother quantity of interest is the **skin friction coefficient**,\n\n$$c_{f} = \\dfrac{\\tau_{w}}{\\frac{1}{2} \\rho U_e^2} $$\n\nIt can be shown that the skin friction coefficient of a Falkner-Skan boundary layer must be of the form\n\n$$c_{f} = \\dfrac{C_4}{(U_e(x) x/\\nu)^{1/2}} $$\n\nAs with the other constants, $C_4$ can only be found by solving the Falkner-Skan\nequation.\n=#\n\n# ### Set up the module\nusing MAE250B\n#-\nusing Plots\n\n#=\n### Generate the velocity profile and other solution features\nHere, we will compute the self-similar velocity profile and compute other\nboundary-layer quantities. The only required input for this `falknerskan` function\nis $\\beta$. It returns the all of the aspects of the solution described above.\n=#\n\nβ = 0.0\nsol = falknerskan(β);\n\n#=\nFor example, the proportionality constant on the 99 percent thickness is\n=#\nd99(sol)\n\n#=\nwhich means that, for $\\beta = 0$,\n\n$$\\delta_{99}(x) = 4.91 \\sqrt{\\dfrac{\\nu x}{U_e}} $$\n\nor, in other words,\n\n$$\\dfrac{\\delta_{99}(x)}{x} = \\dfrac{4.91}{\\sqrt{Re_x}} $$\n\nSimilarly, the momentum thickness:\n=#\ntheta(sol)\n\n#=\nso that\n\n$$\\dfrac{\\theta(x)}{x} = \\dfrac{0.664}{\\sqrt{Re_x}} $$\n=#\n\n#=\n#### Let's plot the profile\n=#\nplot(u(sol),η(sol),xlim=(0,2),ylim=(0,Inf),xlabel=L\"F'(\\eta) = u/U_e\",ylabel=L\"\\eta = y/\\delta(x)\")\n\n#=\nThe vertical velocity profile is\n=#\nplot(v(sol),η(sol),xlim=(0,2),ylim=(0,Inf),xlabel=L\"v/U_e/Re_x^{1/2}\",ylabel=L\"\\eta = y/\\delta(x)\")\n\n\n#=\n#### Drag force\nWhen $\\beta = 0$ (Blasius), the drag coefficient on a plate of length $L$ is equal to\n\n$$C_D = \\dfrac{4\\theta(L)}{L} = \\dfrac{2.656}{\\sqrt{Re_L}}$$\n=#\n\n#=\nIf you wish to specify a wall velocity, you can use the optional argument `Vw = `,\ne.g.,\n=#\nsol = falknerskan(β,Vw=0.2)\nplot(u(sol),η(sol),xlim=(0,2),ylim=(0,Inf),xlabel=L\"F'(\\eta) = u/U_e\",ylabel=L\"\\eta = y/\\delta(x)\")\n", "meta": {"hexsha": "010be7428b75289c3ff5410ddd32ebb554313d44", "size": 4766, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test/literate/3.2-BlasiusAndFalknerSkan.jl", "max_stars_repo_name": "UCLAMAEThreads/MAE250B", "max_stars_repo_head_hexsha": "e5624d302eb216401827130e74ff95d9886adc90", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/literate/3.2-BlasiusAndFalknerSkan.jl", "max_issues_repo_name": "UCLAMAEThreads/MAE250B", "max_issues_repo_head_hexsha": "e5624d302eb216401827130e74ff95d9886adc90", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/literate/3.2-BlasiusAndFalknerSkan.jl", "max_forks_repo_name": "UCLAMAEThreads/MAE250B", "max_forks_repo_head_hexsha": "e5624d302eb216401827130e74ff95d9886adc90", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.3552631579, "max_line_length": 350, "alphanum_fraction": 0.6898866974, "num_tokens": 1460, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.920789673717312, "lm_q2_score": 0.8774767874818409, "lm_q1q2_score": 0.8079715648399194}}
{"text": "\"\"\"\nA palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.\n\nFind the largest palindrome made from the product of two 3-digit numbers.\n\"\"\"\n\nispalindrome(a) = string(a) == reverse(string(a))\n\n## Bruteforce Method\nn = 100:999\npalindromes = []\n\nfor i = n\n for j = n\n ispalindrome(i*j) && push!(palindromes,i*j)\n end\nend\n\nfindmax(palindromes)", "meta": {"hexsha": "11e40fccf3dc1e8306359ac8900ce8ee4819df66", "size": 430, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/P4_largest_palindrome_product.jl", "max_stars_repo_name": "rojesh-shikhrakar/ProjectEuler.jl", "max_stars_repo_head_hexsha": "ee16f625e11d4076e4bff1b02d71913468fb3555", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/P4_largest_palindrome_product.jl", "max_issues_repo_name": "rojesh-shikhrakar/ProjectEuler.jl", "max_issues_repo_head_hexsha": "ee16f625e11d4076e4bff1b02d71913468fb3555", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/P4_largest_palindrome_product.jl", "max_forks_repo_name": "rojesh-shikhrakar/ProjectEuler.jl", "max_forks_repo_head_hexsha": "ee16f625e11d4076e4bff1b02d71913468fb3555", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.6315789474, "max_line_length": 133, "alphanum_fraction": 0.6906976744, "num_tokens": 128, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026550642018, "lm_q2_score": 0.8807970654616711, "lm_q1q2_score": 0.8079574867207485}}
{"text": "\nfunction normal_approximation_to_binomial(n, p)\n mu = n * p\n sigma = sqrt(n * (1-p) * p)\n return mu, sigma \nend\n\nmu, sigma = normal_approximation_to_binomial(1000, 0.5)\nprintln(mu)\nprintln(sigma)\n\n@assert mu == 500 \n@assert 15.8 < sigma < 15.9", "meta": {"hexsha": "c5c40961eb9986e6daa5e1ca402bc34b90be8f45", "size": 253, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "scratch/julia/e1-binomial-mu-sigma.jl", "max_stars_repo_name": "vettukal/data-science-from-scratch.jl", "max_stars_repo_head_hexsha": "5404d308ab9aee1fcbe9d7b4bbeb347ef636b234", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "scratch/julia/e1-binomial-mu-sigma.jl", "max_issues_repo_name": "vettukal/data-science-from-scratch.jl", "max_issues_repo_head_hexsha": "5404d308ab9aee1fcbe9d7b4bbeb347ef636b234", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scratch/julia/e1-binomial-mu-sigma.jl", "max_forks_repo_name": "vettukal/data-science-from-scratch.jl", "max_forks_repo_head_hexsha": "5404d308ab9aee1fcbe9d7b4bbeb347ef636b234", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.4615384615, "max_line_length": 55, "alphanum_fraction": 0.6679841897, "num_tokens": 85, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9621075777163567, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.807914409704442}}
{"text": "### A Pluto.jl notebook ###\n# v0.14.0\n\nusing Markdown\nusing InteractiveUtils\n\n# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).\nmacro bind(def, element)\n quote\n local el = $(esc(element))\n global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing\n el\n end\nend\n\n# ╔═╡ 7cd44a22-91f0-11eb-056d-699740ca7da9\n#Set up the working environment\nbegin\n\tusing Pkg\n\tPkg.activate(\"..\")\n\tusing Plots, PlutoUI,LaTeXStrings, DifferentialEquations\nend\n\n# ╔═╡ 3199bb4e-91ed-11eb-37fa-abed99588e6b\nmd\"\"\"\n# Epidemics - The SIR Model\n- developed by *Kermack* and *McKendric* in the 1930s to describe the spread of malaria\n- describes the spread of a disease from a population of infected individuals over time\n- The total population is constant and is divided into three groups: *susceptible (S), infected (I) and recovered (R)*\n- immune and deceased individuals are \"lumped\" together into group R\n- only susceptible individuals can get the disease\n- the incubation period is negligible - infected individuals are infectious right away\n\n$$S \\xrightarrow{r} I \\xrightarrow{a} R$$\n\nThis gives rise to the following set of differential equations:\n\n$$\\begin{aligned}\n\t\\frac{dS}{dt} = -rSI\\\\\n\t\\frac{dI}{dt} = rSI - aI\\\\\n\t\\frac{dR}{dt} = aI\n\\end{aligned}$$\n\nS, I & R represent parts of a constant total population, hence the following two relations hold\n\n$$S(t) + I(t) + R(t) = N$$\n\n$$\\frac{dS}{dt} + \\frac{dI}{dt} + \\frac{dR}{dt} = 0$$\n\n> If the latter relationship does not immediately make sense to you, pause for a second to figure it out.\n\nIn order to make things a bit more comparable, we'll consider the total population to be normalised. Hence $N=1$, and values for $S$, $I$, and $R$ correspond to fractions of the population.\n\n## Finding the right parameters\nOnce we have formulated a model - establishing a way how things depend on each other - we need to find the right parameters, and thereby establish how strong these links are.\n\nThe best way to do this is to look at real data. Let us try to parametrise the model to roughly mimic the dynamics of COVID-19. We know, that the base reproduction number, i.e. the number of individuals infected by a single infected person, is on the order of $3$, if no interventions (masks, lockdowns, etc.) are in place. This base reproduction number falls right out of our model. (In fact it was first defined based on a similar model). It is given by the following relationship:\n\n$$R_0 = \\frac{rS_0}{a}$$\n\nIf we have a rough idea of the fractions of the population that is either infected ($I_0$) or resistant ($R_0$), we can easily determine the fraction of susceptible individuals ($S_0$).\n\nThe parameters $r$ and $a$ are harder to determine. However, if we have an idea of the amount of time infected individuals remain infectious on average, we can estimate the recovery rate $a$ using the following relationship:\n\n$$t_{infectious} = \\frac{1}{a}$$\n\n> By thinking about what the equations in the model mean, convince yourself that the relationship between our parameter $a$ and the average infectious time holds\n\"\"\"\n\n# ╔═╡ 1fc5f952-91ff-11eb-3366-9f477c640915\nmd\"\"\"\nThe code below implements the SIR-model and plots the output. Parameters can be adjusted using the sliders above the graphs. The simulation, as well as values for $R_0$ and $t_{infectious}$ are updated accordingly.\n\n> **Exercise 1 - Understanding the model:** Try to paramterise the model to capture the dynamics of COVID-19 without any interventions. Start by changing $a$ and $r$ and observe the effect on the overall dynamics (as shown in the timecourse plot), as well as on the base reproduction rate and the infectious period. - Use an average infectious period of 10 days and a base reproduction rate of approximately 3.5 to guide your parametrisation.\n>\n> - How strong is the effect of $r$ and $a$ on the base reproduction number?\n> - What is the effect of a higher proportion of infected individuals at the start on the dynamics of the outbreak and $R_0$?\n> - Assuming a starting population of infectious individuals of 1%, how long does it take until the outbreak peaks? Roughly how large is the fraction of the population that has recovered at the end?\n> - Assuming a case mortality (percentage of people with the disease that die) of 2%, what would be your estimate of the total number of fatalities at the end of the outbreak?\n> - Research online to try to come up with reasonable estimates for the fraction of people currently infected and the number of recovered people (including vaccinations) to be used as $I(0)$ and $R(0)$ respectively\n\"\"\"\n\n# ╔═╡ bfe81088-9203-11eb-162b-f373131bc105\nmd\"\"\"\n> **Exercise 2 - Lowering $R_0$:** The current base reproduction rate in Austria is estimated to be slightly above 1. Change the transmission rate accordingly.\n>\n> - How has the change affected the dynamics of the outbreak (time to peak, fraction of population that has recovered, fraction of the population infected at peak, estimated number of fatalities)?\n\"\"\"\n\n# ╔═╡ e574a398-9203-11eb-0006-c38430911ee1\nmd\"\"\"\n>**Exercise 3 - Intensive Care:** Roughly 1.5% of cases need to be treated in intensive care. Roughly 2500 beds in intensive care units are available, of which roughly 1000 beds can be made available to treat COVID-19 patients. This translates to approximately 0.0001 ICU beds per capita. \n>\n> - Assuming that the recovery rate of ICU patients is unchanged, what will be the demand for ICU beds at the peak of the outbreak with a base reproduction number of 1.17? How does this compare to the ICU bed capacity estimated above?\n> - How low would $R_0$ need to be in order not to overwhelm ICU capacity?\n> - Give five reasons why this scenario might be an oversimplification? Make an educated guess, as to whether you're likely to over- or underestimate the number of patients needing ICU-care.\n\"\"\"\n\n# ╔═╡ ac0c0230-91f0-11eb-057d-2103908afe51\nmd\"\"\" ## Simulation\n*Change the sliders to adjust the corresponding values*:\n\n| Parameter | Value |\n|-------------------|------------------------|\n transmission rate $r$ | $(@bind r Slider(0:0.005:1, default=0.125, show_value=true)) |\n| recovery rate $a$ | $(@bind a Slider(0:0.001:0.25, default=0.1, show_value=true)) |\n| initial number of infected $I(0)$ | $(@bind I0 Slider(0.0001:0.001:0.1, default=0.01, show_value=true)) |\n| initial number of resistant $R(0)$ | $(@bind R0 Slider(0:0.001:0.1, default=0,show_value=true)) |\n| timespan $t_{max}$ | $(@bind tmax Slider(10.0:10.0:2000.0, default=100.0,show_value=true)) days |\n| zoom y-axis | $(@bind ymax Slider(0.05:0.05:1.0, default=1.0,show_value=true)) |\n| Population size | $(@bind popsize NumberField(500000:10000000000, default=9044650)) |\n\"\"\"\n\n# ╔═╡ a5f6c71c-920a-11eb-09e9-098f52c01f11\nmd\"\"\"\n!!! danger \"Beware!\"\n\n The blocks below are used to specify the model. Feel free to play around with them, but proceed with care!\n\"\"\"\n\n# ╔═╡ 618d69dc-91f1-11eb-211f-d7548bf4a306\nfunction sir!(du,u,p,t)\n\tdu[1] = -p[1] * u[1] * u[2]\n\tdu[2] = p[1] * u[1] * u[2] - p[2] * u[2]\n\tdu[3] = p[2] * u[2]\nend\n\n# ╔═╡ 9a4c2f80-91f1-11eb-2b14-1784007ca92c\np = [r, a]\n\n# ╔═╡ a6c393c8-91f1-11eb-0be0-99d47c2b207f\nu0 = [1.0 - I0 - R0, I0, R0]\n\n# ╔═╡ eccbab9e-91f1-11eb-19b2-d733609880d0\ntspan = (0.0, 2000)\n\n# ╔═╡ f2abe15a-91f1-11eb-17c2-e99b8b67502f\nprob = ODEProblem(sir!,u0,tspan,p)\n\n# ╔═╡ 4c7f9b82-91f1-11eb-0064-e70bb182dae0\nbegin\n\tsol = solve(prob, dt=1.0, dense = true)\n\t\n\tsimplot = plot(sol.t, [sol[3,:] + sol[2,:] + sol[1,:] sol[3,:] + sol[2,:] sol[2,:]],\n\t\tfill = true, \n\t\tcolor = [\"gray89\" \"paleturquoise3\" \"lightcoral\"], \n\t\tlabel = [\"S(t)\" \"R(t)\" \"I(t)\"], \n\t\txlabel = \"t / days\", \n\t\ttitle = \"Timecourse\", \n\t\txlims = (0, tmax), \n\t\tylims = (0, ymax))\n\t\n\tphaseplot1 = plot(sol, vars = (1,2), \n\t\txlabel = \"S(t)\", \n\t\tylabel = \"I(t)\", \n\t\tlabel = \"\", \n\t\ttitle = \"Phaseplanes\", \n\t\txlims = (0,ymax), \n\t\tylims = (0,ymax), \n\t\tcolor = \"skyblue4\")\n\t\n\tphaseplot2 = plot(sol, vars = (3,2), \n\t\txlabel = \"R(t)\", \n\t\tylabel = \"I(t)\", \n\t\tlabel = \"\", \n\t\txlims = (0,ymax), \n\t\tylims = (0,ymax), \n\t\tcolor = \"skyblue4\")\n\t\n\tphaseplot = plot(phaseplot1, phaseplot2, layout = (2,1))\n\t\n\t#plot(simplot, phaseplot, layout = (1, 2),fontfamily=\"Computer Modern\")\n\tplot(simplot,fontfamily=\"Computer Modern\")\nend\n\n# ╔═╡ 0a8e0895-2de8-435e-827c-fb216ab07f10\nfunction getres(solution, i, t, p)\n\tv = solution(t)[i]*p\n\treturn round(maximum([0.0 v]))\nend\n\n# ╔═╡ 0b66bc58-91fa-11eb-24f1-7f9f839a6cbf\nmd\"\"\"\nBase reprodution rate = $(round((r*(1-I0-R0)/a), sigdigits = 3)), \nInfectious period = $(round(1/a, sigdigits = 3))\n\n| t(days) | I(t) | R(t) |\n|-----------|--------|---------|\n| 0 | $(getres(sol, 2, 0, popsize)) | $(getres(sol, 3, 0, popsize)) |\n| $tmax | $(getres(sol, 2, tmax, popsize)) | $(getres(sol, 3, tmax, popsize)) |\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╟─7cd44a22-91f0-11eb-056d-699740ca7da9\n# ╟─3199bb4e-91ed-11eb-37fa-abed99588e6b\n# ╟─1fc5f952-91ff-11eb-3366-9f477c640915\n# ╟─bfe81088-9203-11eb-162b-f373131bc105\n# ╟─e574a398-9203-11eb-0006-c38430911ee1\n# ╟─ac0c0230-91f0-11eb-057d-2103908afe51\n# ╟─0b66bc58-91fa-11eb-24f1-7f9f839a6cbf\n# ╟─4c7f9b82-91f1-11eb-0064-e70bb182dae0\n# ╟─a5f6c71c-920a-11eb-09e9-098f52c01f11\n# ╟─618d69dc-91f1-11eb-211f-d7548bf4a306\n# ╟─9a4c2f80-91f1-11eb-2b14-1784007ca92c\n# ╟─a6c393c8-91f1-11eb-0be0-99d47c2b207f\n# ╟─eccbab9e-91f1-11eb-19b2-d733609880d0\n# ╟─f2abe15a-91f1-11eb-17c2-e99b8b67502f\n# ╟─0a8e0895-2de8-435e-827c-fb216ab07f10\n", "meta": {"hexsha": "c8997c9731a4c1b978c44e4e204dbbac09533513", "size": 9582, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "notebooks/03_modelling-basics_sir.jl", "max_stars_repo_name": "el-uhu/intro-to-modelling", "max_stars_repo_head_hexsha": "b86d558619c208184fc813e01c39de3d3b2e3f47", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notebooks/03_modelling-basics_sir.jl", "max_issues_repo_name": "el-uhu/intro-to-modelling", "max_issues_repo_head_hexsha": "b86d558619c208184fc813e01c39de3d3b2e3f47", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notebooks/03_modelling-basics_sir.jl", "max_forks_repo_name": "el-uhu/intro-to-modelling", "max_forks_repo_head_hexsha": "b86d558619c208184fc813e01c39de3d3b2e3f47", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.8468899522, "max_line_length": 483, "alphanum_fraction": 0.7032978501, "num_tokens": 3215, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9621075701109193, "lm_q2_score": 0.83973396967765, "lm_q1q2_score": 0.8079144091061602}}
{"text": "\"\"\"\n div_rem(x, y)\n\nComputes `quotient = fl(x/y)` and `remainder = x - quotient*y` such that `(x-remainder) / y = quotient`.\n\"\"\"\n@inline function div_rem(x::T, y::T) where {T}\n quotient = x / y\n remainder = fma(-y, quotient, x)\n return quotient, remainder\nend\n\n\"\"\"\n sqrt_rem(x)\n\nComputes `root = fl(sqrt(x))` and `remainder = x - root*root` such that `sqrt(x-remainder) = root`.\n\"\"\"\n@inline function sqrt_rem(x::T) where {T}\n root = sqrt(x)\n remainder = fma(-root, root, x)\n return root, remainder\nend\n\n#=\n Those remainders are exact.\n For more informations, applications and additional references, see the introduction of\n Sylvie Boldo and Jean-Michel Muller\n Some Functions Computable with a Fused-mac\n=#\n", "meta": {"hexsha": "13b8729472fb1e9aa0cd1855067cf4a756eccca3", "size": 733, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/remainder.jl", "max_stars_repo_name": "nestordemeure/ErrorfreeArithmetic.jl", "max_stars_repo_head_hexsha": "af69f112260be0c99f21ce7f3e71d83ae891dc18", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2018-08-07T17:03:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-17T11:37:33.000Z", "max_issues_repo_path": "src/remainder.jl", "max_issues_repo_name": "nestordemeure/ErrorfreeArithmetic.jl", "max_issues_repo_head_hexsha": "af69f112260be0c99f21ce7f3e71d83ae891dc18", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2017-04-07T15:19:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-15T03:34:13.000Z", "max_forks_repo_path": "src/remainder.jl", "max_forks_repo_name": "nestordemeure/ErrorfreeArithmetic.jl", "max_forks_repo_head_hexsha": "af69f112260be0c99f21ce7f3e71d83ae891dc18", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2017-04-10T00:18:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-03T12:54:51.000Z", "avg_line_length": 25.275862069, "max_line_length": 104, "alphanum_fraction": 0.6643929059, "num_tokens": 211, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9572778024535095, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.8078420436372201}}
{"text": "# p25.jl - stability regions for ODE formulas\n\n# Adams-Bashforth:\nclf(); subplot(221);\nzplot(z) = plot(real(z),imag(z));\nplot([-8,8],[0,0],\"k-\"); plot([0,0],[-8,8],\"k-\");\nz = exp.(1im*pi*(0:200)/100); r = z-1;\ns = 1; zplot(r./s); # order 1\ns = (3-1./z)/2; zplot(r./s); # order 2\ns = (23-16./z+5./z.^2)/12; zplot(r./s); # order 3\naxis(\"square\"); axis([-2.5,.5,-1.5,1.5]); grid(true)\ntitle(\"Adams—Bashforth\");\n\n# Adams-Moulton:\nsubplot(222);\nplot([-8,8],[0,0],\"k-\"); plot([0,0],[-8,8],\"k-\");\ns = (5*z+8-1./z)/12; zplot(r./s); # order 3\ns = (9*z+19-5./z+1./z.^2)/24; zplot(r./s); # order 4\ns = (251*z+646-264./z+106./z.^2-19./z.^3)/720; zplot(r./s) ; # 5\nd = 1-1./z;\ns = 1-d/2-d.^2/12-d.^3/24-19*d.^4/720-3*d.^5/160; zplot(d./s); # 6\naxis(\"square\"); axis([-7,1,-4,4]); grid(true);\ntitle(\"Adams—Moulton\");\n\n# Backward differentiation:\nsubplot(223);\nplot([-40,40],[0,0],\"k-\"); plot([0,0],[-40,40],\"k-\");\nr = 0;\nfor i = 1:6 # orders 1-6\n r += (d.^i)/i;\n zplot(r);\nend\naxis(\"square\"); axis([-15,35,-25,25]); grid(true);\ntitle(\"backward differentiation\")\n\n# Runge-Kutta:\nsubplot(224);\nplot([-8,8],[0,0],\"k-\"); plot([0,0],[-8,8],\"k-\");\nw = zeros(4)+0im; W = w.';\nfor i = 2:length(z)\n # orders 1-4\n w[1] -= (1+w[1]-z[i]);\n w[2] -= (1+w[2]+.5*w[2]^2-z[i]^2)/(1+w[2]);\n w[3] -= (1+w[3]+.5*w[3]^2+w[3]^3/6-z[i]^3)/(1+w[3]+w[3]^2/2);\n w[4] -= (1+w[4]+.5*w[4]^2+w[4]^3/6+w[4].^4/24-z[i]^4)/(1+w[4]+w[4]^2/2+w[4]^3/6);\n W = [W; w.'];\nend\nzplot(W); axis(\"square\"); axis([-5,2,-3.5,3.5]); grid(true);\ntitle(\"Runge—Kutta\");\n", "meta": {"hexsha": "4ebe095b455f9945df6fc6e480f4314af3930906", "size": 1630, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/scripts/p25.jl", "max_stars_repo_name": "tobydriscoll/SpectralMethodsTrefethen.jl", "max_stars_repo_head_hexsha": "633da18bcbaa169c2e23401e5a7536c18bdc5511", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-06-06T19:36:29.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-31T15:07:11.000Z", "max_issues_repo_path": "src/scripts/p25.jl", "max_issues_repo_name": "tobydriscoll/SpectralMethodsTrefethen.jl", "max_issues_repo_head_hexsha": "633da18bcbaa169c2e23401e5a7536c18bdc5511", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/scripts/p25.jl", "max_forks_repo_name": "tobydriscoll/SpectralMethodsTrefethen.jl", "max_forks_repo_head_hexsha": "633da18bcbaa169c2e23401e5a7536c18bdc5511", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.6, "max_line_length": 85, "alphanum_fraction": 0.4699386503, "num_tokens": 743, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545362802363, "lm_q2_score": 0.8519528038477824, "lm_q1q2_score": 0.8077829156649412}}
{"text": "# # m2.1t.jl\n\n# Load Julia packages (libraries) needed\n\nusing TuringModels\n\n# ### snippet 2.8t\n\n# Define the data\n\nk = 6; n = 9;\n\n# Define the model\n\n@model globe_toss(n, k) = begin\n theta ~ Beta(1, 1) # prior\n k ~ Binomial(n, theta) # model\n return k, theta\nend;\n\n# Use Turing mcmc\n\nchns = sample(globe_toss(n, k), NUTS(0.65), 1000)\n\n# Look at the proper draws (in corrected chn2)\n\nchns |> display\n\n# Show the hpd region\n\nhpd(chns, alpha=0.055) |> display\n\n# End of `02/m2.1t.jl`\n", "meta": {"hexsha": "9b16eac754df08536af88e917fe51bd69fe5d3b5", "size": 484, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "scripts/02/m2.1t.jl", "max_stars_repo_name": "yiyuezhuo/TuringModels.jl", "max_stars_repo_head_hexsha": "88755d21f128de893873820f28f5019856ff8d24", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "scripts/02/m2.1t.jl", "max_issues_repo_name": "yiyuezhuo/TuringModels.jl", "max_issues_repo_head_hexsha": "88755d21f128de893873820f28f5019856ff8d24", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "scripts/02/m2.1t.jl", "max_forks_repo_name": "yiyuezhuo/TuringModels.jl", "max_forks_repo_head_hexsha": "88755d21f128de893873820f28f5019856ff8d24", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 14.2352941176, "max_line_length": 49, "alphanum_fraction": 0.6466942149, "num_tokens": 180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.948154531885212, "lm_q2_score": 0.8519528000888386, "lm_q1q2_score": 0.8077829083565283}}
{"text": "module Copeland\n\nfunction level_of_dominance(v1::Array{Int,1}, v2::Array{Int,1})::Int64\n lod = zero(Int64)\n n = length(v1)\n for i = 1:n\n if v1[i] < v2[i]\n lod += 1\n end\n end\n return lod\nend\n\nfunction dominance_scores(ordering_mat::Array{Int, 2})::Array{Int, 2}\n n, _ = size(ordering_mat)\n lev_dom = zeros(Int64, n, n)\n for i in 1:n\n for j in 1:n\n lev_dom[i, j] = level_of_dominance(ordering_mat[i,:], ordering_mat[j,:])\n end\n end\n return lev_dom\nend\n\nfunction winloss_scores(dommat::Array{Int, 2})::Array{Int, 2}\n n, _ = size(dommat)\n winlossmat = zeros(Int64, n, n)\n for i in 1:n\n for j in 1:n \n winlossmat[i, j] = Int(sign(dommat[i, j] - dommat[j, i]))\n end\n end\n return winlossmat\nend\n\nfunction copeland(ordering_mat::Array{Int, 2})::Array{Int, 1}\n winlosses = ordering_mat |> dominance_scores |> winloss_scores\n n, _ = size(winlosses)\n scores = map(i -> Int(sum(winlosses[i, :])), 1:n)\n return scores\nend\n\nend # end of module", "meta": {"hexsha": "def1f8c55ee7c493d1afd25e903753fea99445df", "size": 1061, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/copeland.jl", "max_stars_repo_name": "jbytecode/jMCDM", "max_stars_repo_head_hexsha": "04fd374fd8cff2ba359e2f51568c65ea1675b93d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2021-02-09T17:42:34.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T22:21:26.000Z", "max_issues_repo_path": "src/copeland.jl", "max_issues_repo_name": "jbytecode/jMCDM", "max_issues_repo_head_hexsha": "04fd374fd8cff2ba359e2f51568c65ea1675b93d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 14, "max_issues_repo_issues_event_min_datetime": "2021-02-09T16:56:15.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-27T18:11:34.000Z", "max_forks_repo_path": "src/copeland.jl", "max_forks_repo_name": "jbytecode/jMCDM", "max_forks_repo_head_hexsha": "04fd374fd8cff2ba359e2f51568c65ea1675b93d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 24, "max_forks_repo_forks_event_min_datetime": "2021-02-12T16:01:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-24T22:02:44.000Z", "avg_line_length": 24.6744186047, "max_line_length": 84, "alphanum_fraction": 0.5947219604, "num_tokens": 356, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9481545289551957, "lm_q2_score": 0.8519528019683106, "lm_q1q2_score": 0.8077829076423226}}
{"text": "### A Pluto.jl notebook ###\n# v0.17.4\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ efff10ac-824b-422b-b2f9-4074ea3437fb\nbegin\n\tusing Pkg; Pkg.activate(@__DIR__); Pkg.instantiate()\n\tPkg.precompile()\n\n\tusing PlutoUI\n\tusing UMAP\n\tusing Makie\n\tusing GLMakie\n\tusing XLSX\n\tusing VegaDatasets\n\tusing DataFrames\n\tusing MultivariateStats\n\tusing RDatasets\n\tusing StatsBase\n\tusing Statistics\n\tusing LinearAlgebra\n\tusing Plots\n\tusing ScikitLearn\n\tusing MLBase\n\tusing Distances\nend;\n\n# ╔═╡ 85e2d832-542c-49b0-a26d-b57a7a3d4404\nPlutoUI.TableOfContents(aside=true, indent=true, depth=3)\n\n# ╔═╡ 6e8c97d2-4d9a-4f00-99e1-69054903635c\nhtml\"\"\"\n\n\t![\"Course-Logo\"](\"https://github.com/JuliaAcademy/DataScience/blob/main/datascience.png?raw=true\")
\n
![\"Flow](\"img/g1.svg\")
\r\n\r\nG = [\r\n0 100 30 0 0;\r\n0 0 20 0 0; \r\n0 0 0 10 60;\r\n0 15 0 0 50;\r\n0 0 0 0 0\r\n]\r\n\r\nn = size(G)[1]\r\n\r\nshortest_path = Model(with_optimizer(GLPK.Optimizer))\r\n\r\n@variable(shortest_path, x[1:n,1:n], Bin)\r\n@constraint(shortest_path, [i = 1:n, j = 1:n; G[i,j] == 0], x[i,j] == 0) # Arcs with zero cost are not a part of the path as they do no exist\r\n@constraint(shortest_path, [i = 1:n; i != 1 && i != 2], sum(x[i,:]) == sum(x[:,i])) # Flow conservation constraint\r\n@constraint(shortest_path, sum(x[1,:]) - sum(x[:,1]) == 1) # Flow coming out of source = 1\r\n@constraint(shortest_path, sum(x[2,:]) - sum(x[:,2]) == -1) # Flowing coming out of destination = -1 i.e. Flow entering destination = 1 \r\n@objective(shortest_path, Min, sum(G .* x))\r\n\r\noptimize!(shortest_path)\r\n@show objective_value(shortest_path);\r\n@show value.(x);\r\n\r\n#' ## The Assignment Problem\r\n#' Suppose that there are $n$ persons and $n$ objects that we have to match on a one-to-one basis. \r\n#' There is a benefit or value $a_{i,j}$ for matching person $i$ with object $j$, and \r\n#' we want to assign persons to objects so as to maximize the total benefit. \r\n#' There is also a restriction that person $i$ can be assigned to object $j$ only if $(i, j)$ belongs to a given set of pairs $A$. \r\n#' Mathematically, we want to find a set of person-object pairs $(1, j_{1}),..., (n, j_{n})$ from $A$ such that \r\n#' the objects $j_{1},...,j_{n}$ are all distinct, and the total benefit $\\sum_{i=1}^{y} a_{ij_{i}}$ is maximized.\r\n\r\n#' $$\r\n#' \\begin{align*}\r\n#' \\max && \\sum_{(i,j) \\in A} a_{i,j} \\times y_{i,j} \\\\\r\n#' s.t. && \\sum_{\\{j|(i,j) \\in A\\}} y_{i,j} = 1 && \\forall i = \\{1,2....n\\} \\\\\r\n#' && \\sum_{\\{i|(i,j) \\in A\\}} y_{i,j} = 1 && \\forall j = \\{1,2....n\\} \\\\\r\n#' && y_{i,j} \\in \\{0,1\\} && \\forall (i,j) \\in \\{1,2...k\\}\r\n#' \\end{align*}\r\n#' $$\r\n\r\n#' ![\"Flow](\"img/g2.svg\")
\r\n\r\nG = [\r\n6 4 5 0;\r\n0 3 6 0;\r\n5 0 4 3;\r\n7 5 5 5;\r\n]\r\n\r\nn = size(G)[1]\r\n\r\nassignment = Model(with_optimizer(GLPK.Optimizer))\r\n@variable(assignment, y[1:n,1:n], Bin)\r\n@constraint(assignment, [i = 1:n], sum(y[:,i]) == 1) # One person can only be assigned to one object\r\n@constraint(assignment, [j = 1:n], sum(y[j,:]) == 1) # One object can only be assigned to one person\r\n@objective(assignment, Max, sum(G .* y))\r\n\r\noptimize!(assignment)\r\n@show objective_value(assignment);\r\n@show value.(y);\r\n\r\n#' ## The Max-Flow Problem\r\n#' In the max-flow problem, we have a graph with two special nodes: the $source$, denoted by $s$, and the $sink$, denoted by $t$. \r\n#' The objective is to move as much flow as possible from $s$ into $t$ while observing the capacity constraints.\r\n\r\n#' $$\r\n#' \\begin{align*}\r\n#' \\max && \\sum_{v:(s,v) \\in E} f(s,v) \\\\\r\n#' s.t. && \\sum_{u:(u,v) \\in E} f(u,v) = \\sum_{w:(v,w) \\in E} f(v,w) && \\forall v \\in V - \\{s,t\\} \\\\\r\n#' && f(u,v) \\leq c(u,v) && \\forall (u,v) \\in E \\\\\r\n#' && f(u,v) \\geq 0 && \\forall (u,v) \\in E \r\n#' \\end{align*}\r\n#' $$\r\n\r\n#' ![\"Flow](\"img/g3.svg\")
\r\n\r\nG = [\r\n0 3 2 2 0 0 0 0 \r\n0 0 0 0 5 1 0 0 \r\n0 0 0 0 1 3 1 0 \r\n0 0 0 0 0 1 0 0 \r\n0 0 0 0 0 0 0 4 \r\n0 0 0 0 0 0 0 2 \r\n0 0 0 0 0 0 0 4 \r\n0 0 0 0 0 0 0 0 \r\n]\r\n\r\nn = size(G)[1]\r\n\r\nmax_flow = Model(with_optimizer(GLPK.Optimizer))\r\n\r\n@variable(max_flow, f[1:n,1:n] >= 0)\r\n@constraint(max_flow, [i = 1:n, j = 1:n], f[i,j] <= G[i,j]) # Capacity constraints\r\n@constraint(max_flow, [i = 1:n; i != 1 && i != 8], sum(f[i,:]) == sum(f[:,i])) # Flow conservation contraints\r\n@objective(max_flow, Max, sum(f[1, :]))\r\n\r\noptimize!(max_flow)\r\n@show objective_value(max_flow);\r\n@show value.(f);", "meta": {"hexsha": "c0445120a8a2c69627778ee2f4e618542e87424a", "size": 5341, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "script/modelling/network_flows.jl", "max_stars_repo_name": "carlosal1015/JuMPTutorials.jl", "max_stars_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "script/modelling/network_flows.jl", "max_issues_repo_name": "carlosal1015/JuMPTutorials.jl", "max_issues_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "script/modelling/network_flows.jl", "max_forks_repo_name": "carlosal1015/JuMPTutorials.jl", "max_forks_repo_head_hexsha": "4d9a86ea310ecc7a22de7f14b783dbd218e4b612", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.8582089552, "max_line_length": 142, "alphanum_fraction": 0.6098108968, "num_tokens": 1845, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109798251322, "lm_q2_score": 0.8479677622198946, "lm_q1q2_score": 0.8059178717515347}}
{"text": "\"\"\"Returns a matrix of water depth for the whole domain that contains a\nGaussian seamount in the middle. Water depth, heigth and width of the\nseamount are adjusted with the constants water_depth, topofeat_height and topofeat_width.\"\"\"\nfunction seamount()\n xx_T,yy_T = meshgrid(x_T_halo,y_T_halo)\n bumpx = exp.(-((xx_T .- Lx/2).^2)/(2*topofeat_width^2))\n bumpy = exp.(-((yy_T .- Ly/2).^2)/(2*topofeat_width^2))\n\n H = water_depth .- topofeat_height*bumpx.*bumpy\n return Numtype.(H)\nend\n\n\"\"\"Returns a matrix of water depth for the whole domain that contains a\nmeridional Gaussian ridge in the middle. Water depth, heigth and width of the\nridge are adjusted with the constants water_depth, topofeat_height and topofeat_width.\"\"\"\nfunction ridge()\n xx_T,yy_T = meshgrid(x_T_halo,y_T_halo)\n bumpx = exp.(-((xx_T .- Lx/2).^2)/(2*topofeat_width^2))\n\n H = water_depth .- topofeat_height*bumpx\n return Numtype.(H)\nend\n\n\"\"\"Same as ridge() but for 3 ridges at 1/4,1/2,3/4 of the domain.\"\"\"\nfunction ridges()\n xx_T,yy_T = meshgrid(x_T_halo,y_T_halo)\n\n # bumps in x direction\n # shift slightly left/right to avoid a symmetric solution\n b0x = exp.(-(xx_T.^2)/(2*topofeat_width^2))\n b1x = exp.(-((xx_T .- 0.99*Lx/4).^2)/(2*topofeat_width^2))\n b2x = exp.(-((xx_T .- 1.01*Lx/2).^2)/(2*topofeat_width^2))\n b3x = exp.(-((xx_T .- 0.99*3*Lx/4).^2)/(2*topofeat_width^2))\n b4x = exp.(-((xx_T .- Lx).^2)/(2*topofeat_width^2))\n\n th = topofeat_height # for convenience\n H = water_depth .- th*b0x .- th*b1x .- th*b2x .- th*b3x .- th*b4x\n return Numtype.(H)\nend\n\n\"\"\"Returns a matrix of constant water depth specified by the constant water_depth.\"\"\"\nfunction flat_bottom()\n H = fill(water_depth,(nx+2*haloη,ny+2*haloη))\n return Numtype.(H)\nend\n\n# rename for convenience\nif topography_feature == \"ridge\"\n topography = ridge\nelseif topography_feature == \"ridges\"\n topography = ridges\nelseif topography_feature == \"seamount\"\n topography = seamount\nelseif topography_feature == \"flat\"\n topography = flat_bottom\nelse\n throw(error(\"Topography feature not correctly declared.\"))\nend\n", "meta": {"hexsha": "b5796e2fb2bfda8672de605da5d334c01687a226", "size": 2129, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/bottom_topography.jl", "max_stars_repo_name": "eschnett/juls", "max_stars_repo_head_hexsha": "2921372578c880c52c07f5449afc234c69433148", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/bottom_topography.jl", "max_issues_repo_name": "eschnett/juls", "max_issues_repo_head_hexsha": "2921372578c880c52c07f5449afc234c69433148", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/bottom_topography.jl", "max_forks_repo_name": "eschnett/juls", "max_forks_repo_head_hexsha": "2921372578c880c52c07f5449afc234c69433148", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.0847457627, "max_line_length": 92, "alphanum_fraction": 0.6876467825, "num_tokens": 693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109742068042, "lm_q2_score": 0.8479677660619633, "lm_q1q2_score": 0.8059178706389181}}
{"text": "\n\"\"\"\n (uvels, vvels, wvels) = streamfunction_xz()\n\nDefine a two-dimensional flow field in the x-z directions from a simple streamfunction.\n\"\"\"\nfunction streamfunction_xz()\n scal = 2e-1\n f(x, y, z) = scal*sin(x*2π/128)*sin(z*π/128) #stream function\n\n ϕcorners=[f(x,0.,z) for x in 0:128, z in -128:0]\n ϕcenters=[f(x,0.,z) for x in 0.5:128, z in -128:-0.5]\n\n uu=-diff(ϕcorners,dims=2)[1:end-1,:]\n ww=diff(ϕcorners,dims=1)\n uu=reshape(uu,(128,1,128))\n ww=reshape(ww,(128,1,129))\n uu = reverse(uu, dims=3)\n ww = reverse(ww, dims=3)\n\n uvels = fill(uu, 2)\n vvels = fill(0*uu, 2)\n wvels = fill(ww, 2)\n uvels = cat(uvels..., dims=4)\n vvels = cat(vvels..., dims=4)\n wvels = cat(wvels..., dims=4)\n\n return uvels, vvels, wvels, ϕcenters\nend\n\n\n\"\"\"\n (uvels, vvels, wvels) = streamfunction_xy()\n\nDefine a two-dimensional flow field in the x-y directions from a simple streamfunction\nas explained at https://shaddenlab.berkeley.edu/uploads/LCS-tutorial/examples.html#Sec7.1\n\"\"\"\nfunction streamfunction_xy()\n scal = 3e-1\n f(x, y, z) = scal*(0.3*sin(x*1π/128)*sin(y*2π/128)+0.7*sin(x*2π/128)*sin(y*1π/128)) #stream function\n\n ϕcorners=[f(x,y,0.) for x in 0:128, y in 0:128]\n ϕcenters=[f(x,y,0.) for x in 0.5:128, y in 0.5:128]\n\n uu=-diff(ϕcorners,dims=2)[1:end-1,:]\n vv=diff(ϕcorners,dims=1)[:,1:end-1]\n uu=reshape(uu,(128,128,1))\n vv=reshape(vv,(128,128,1))\n ww=zeros(128,128,2)\n\n uvels = fill(uu, 2)\n vvels = fill(vv, 2)\n wvels = fill(ww, 2)\n uvels = cat(uvels..., dims=4)\n vvels = cat(vvels..., dims=4)\n wvels = cat(wvels..., dims=4)\n\n return uvels, vvels, wvels, ϕcenters\nend", "meta": {"hexsha": "68e7e05bc04269f25d3caee539c0d892133e85ca", "size": 1666, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/helper_functions.jl", "max_stars_repo_name": "JuliaOcean/PlanktonIndividuals.jl", "max_stars_repo_head_hexsha": "5ac96b3ef90a21f377c119557665997fabe0944d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2020-11-21T21:13:52.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-15T10:30:39.000Z", "max_issues_repo_path": "examples/helper_functions.jl", "max_issues_repo_name": "gaelforget/PlanktonIndividuals.jl", "max_issues_repo_head_hexsha": "f7a130b0d9ece421da42c737c080f53411db5821", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 20, "max_issues_repo_issues_event_min_datetime": "2020-07-08T18:13:44.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-14T17:44:10.000Z", "max_forks_repo_path": "examples/helper_functions.jl", "max_forks_repo_name": "gaelforget/PlanktonIndividuals.jl", "max_forks_repo_head_hexsha": "f7a130b0d9ece421da42c737c080f53411db5821", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-22T22:23:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-22T22:23:33.000Z", "avg_line_length": 28.2372881356, "max_line_length": 104, "alphanum_fraction": 0.6128451381, "num_tokens": 685, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109742068042, "lm_q2_score": 0.847967764140929, "lm_q1q2_score": 0.8059178688131459}}
{"text": "function init_CN(model)\n nx, ny = model.grid.nx, model.grid.ny\n Δx, Δy = model.grid.Δx, model.grid.Δy\n ν, Δt = model.parameters.ν, model.Δt\n\n Nx = (nx-2)\n Ny = (ny-2)\n Nxy = Nx*Ny\n βx = 0.5*ν*Δt/(Δx*Δx)\n βy = 0.5*ν*Δt/(Δy*Δy)\n\n # New a1\n a1 = spzeros(Nxy,Nxy)\n ij_ind = LinearIndices((1:Ny, 1:Nx))\n for ij in ij_ind\n a1[ij,ij] = 1 + 2*βx + 2*βy\n end\n for ij in 1:Nx*(Ny-1)\n a1[view(ij_ind,2:Ny,:)[ij],view(ij_ind,1:Ny-1,:)[ij]] = -βy # Up\n a1[view(ij_ind,1:Ny-1,:)[ij],view(ij_ind,2:Ny,:)[ij]] = -βy # Down\n end\n for ij in 1:(Nx-1)*Ny\n a1[view(ij_ind,:,2:Nx)[ij],view(ij_ind,:,1:Nx-1)[ij]] = -βx # Right\n a1[view(ij_ind,:,1:Nx-1)[ij],view(ij_ind,:,2:Nx)[ij]] = -βx # Left\n end\n\n # New a2\n a2 = spzeros(Nxy,Nxy)\n ij_ind = LinearIndices((1:Ny, 1:Nx))\n for ij in ij_ind\n a2[ij,ij] = 1 - 2*βx - 2*βy\n end\n for ij in 1:Nx*(Ny-1)\n a2[view(ij_ind,2:Ny,:)[ij],view(ij_ind,1:Ny-1,:)[ij]] = βy # Up\n a2[view(ij_ind,1:Ny-1,:)[ij],view(ij_ind,2:Ny,:)[ij]] = βy # Down\n end\n for ij in 1:(Nx-1)*Ny\n a2[view(ij_ind,:,2:Nx)[ij],view(ij_ind,:,1:Nx-1)[ij]] = βx # Right\n a2[view(ij_ind,:,1:Nx-1)[ij],view(ij_ind,:,2:Nx)[ij]] = βx # Left\n end\n\n a1_lu = factorize(a1)\n return a1_lu, a2\nend\nfunction init_ADI(model)\n nx, ny = model.grid.nx, model.grid.ny\n Δx, Δy = model.grid.Δx, model.grid.Δy\n ν, Δt = model.parameters.ν, model.Δt\n\n βx = 0.5*ν*Δt/(Δx*Δx)\n βy = 0.5*ν*Δt/(Δy*Δy)\n d1x = ones(nx-2) .+ 2 * βx\n e1x = -βx * ones(nx-3)\n d1y = ones(ny-2) .+ 2 * βy\n e1y = -βy * ones(ny-3)\n # matrix ( I - 0.5 nu dt delta_x^2 )\n a1x = SymTridiagonal(d1x,e1x)\n a1y = SymTridiagonal(d1y,e1y)\n\n d2x = ones(nx-2) .- 2 * βx\n e2x = -βx * ones(nx-3)\n d2y = ones(ny-2) .- 2 * βy\n e2y = -βy * ones(ny-3)\n # matrix ( I + 0.5 nu dt delta_x^2)\n a2x = SymTridiagonal(d2x,-e2x)\n a2y = SymTridiagonal(d2y,-e2y)\n\n #LDLt factorize\n a1x_ldlt = factorize(a1x)\n a1y_ldlt = factorize(a1y)\n return a1x_ldlt, a1y_ldlt, a2x, a2y\nend\n", "meta": {"hexsha": "ab75d56aeae92ce7e00fbc057dc748065a047dd2", "size": 2106, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/matrix_solvers.jl", "max_stars_repo_name": "williamjsdavis/NavierStokes2D.jl", "max_stars_repo_head_hexsha": "bc4e15beb0b32ee65106b0795381f96c7ac25122", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-16T00:15:48.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-16T00:15:48.000Z", "max_issues_repo_path": "src/matrix_solvers.jl", "max_issues_repo_name": "williamjsdavis/NavierStokes2D.jl", "max_issues_repo_head_hexsha": "bc4e15beb0b32ee65106b0795381f96c7ac25122", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/matrix_solvers.jl", "max_forks_repo_name": "williamjsdavis/NavierStokes2D.jl", "max_forks_repo_head_hexsha": "bc4e15beb0b32ee65106b0795381f96c7ac25122", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.8493150685, "max_line_length": 75, "alphanum_fraction": 0.5365622032, "num_tokens": 1027, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109770159682, "lm_q2_score": 0.8479677602988602, "lm_q1q2_score": 0.8059178675436821}}
{"text": "\"\"\"\n poisson_eigenvalues(N, L, dim, ::Periodic)\n\nReturn the eigenvalues satisfying the discrete form of Poisson's equation\nwith periodic boundary conditions along the dimension `dim` with `N` grid\npoints and domain extent `L`.\n\"\"\"\nfunction poisson_eigenvalues(N, L, dim, ::Periodic)\n inds = reshape(1:N, reshaped_size(N, dim)...)\n return @. (2sin((inds - 1) * π / N) / (L / N))^2\nend\n\n\"\"\"\n poisson_eigenvalues(N, L, dim, ::Bounded)\n\nReturn the eigenvalues satisfying the discrete form of Poisson's equation\nwith staggered Neumann boundary conditions along the dimension `dim` with\n`N` grid points and domain extent `L`.\n\"\"\"\nfunction poisson_eigenvalues(N, L, dim, ::Bounded)\n inds = reshape(1:N, reshaped_size(N, dim)...)\n return @. (2sin((inds - 1) * π / 2N) / (L / N))^2\nend\n\n\"\"\"\n poisson_eigenvalues(N, L, dim, ::Flat)\n\nNo need to do any transforms along `Flat` dimensions so just return zeros\nsince they will be added to the eigenvalues for other dimensions.\n\"\"\"\npoisson_eigenvalues(N, L, dim, ::Flat) = zeros(1, 1, 1)\n", "meta": {"hexsha": "c2cc5ebd22c522f59a33a7a5f3bdce1d84efe511", "size": 1044, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/Solvers/poisson_eigenvalues.jl", "max_stars_repo_name": "charleskawczynski/Oceananigans.jl", "max_stars_repo_head_hexsha": "c34e6cd2166bbaa057186ffa795d348c1802485f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 393, "max_stars_repo_stars_event_min_datetime": "2020-05-07T19:20:35.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T02:09:52.000Z", "max_issues_repo_path": "src/Solvers/poisson_eigenvalues.jl", "max_issues_repo_name": "charleskawczynski/Oceananigans.jl", "max_issues_repo_head_hexsha": "c34e6cd2166bbaa057186ffa795d348c1802485f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1408, "max_issues_repo_issues_event_min_datetime": "2020-05-09T11:39:27.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T23:52:36.000Z", "max_forks_repo_path": "src/Solvers/poisson_eigenvalues.jl", "max_forks_repo_name": "charleskawczynski/Oceananigans.jl", "max_forks_repo_head_hexsha": "c34e6cd2166bbaa057186ffa795d348c1802485f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 99, "max_forks_repo_forks_event_min_datetime": "2020-05-10T02:59:08.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-20T21:50:55.000Z", "avg_line_length": 32.625, "max_line_length": 73, "alphanum_fraction": 0.6915708812, "num_tokens": 306, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9504109812297141, "lm_q2_score": 0.8479677545357568, "lm_q1q2_score": 0.805917865639486}}
{"text": "## In this notebook, we'll look at stochastic gradient descent on a least squares\n## system\n\n##\nusing Plots\npyplot()\ntheme(:dark)\n##\n# https://vincentarelbundock.github.io/Rdatasets/csv/carData/Davis.csv\nusing CSV\ndf = CSV.read(download(\"https://vincentarelbundock.github.io/Rdatasets/csv/carData/Davis.csv\"))\nX = Float64.([df[:weight] df[:height]])\n# Filter the data\ngoodpts = X[:,1] .<= 115\nXf = X[goodpts,:]\n##\nscatter(Xf[:,1],Xf[:,2],label=\"\",xlabel=\"weight (kg)\", ylabel=\"height (cm)\")\n##\nx = Xf[:,1]\nA = [x ones(length(x))]\nb = Xf[:,2]\nab = A\\b # solve the least squares problem\n\n\n##\nusing Printf\nusing LinearAlgebra\n\"\"\" Run stochastic gradient descent to solve a least squares problem. \"\"\"\nfunction stochastic_gradient_descent_leastsquares(A,b;\n step=0.0001, maxiter=10*size(A,1),tol=1e-6)\n x = zeros(size(A,2)) # start at 0\n for i=1:1000000\n si = rand(1:size(A,1))\n ai = A[si,:]\n bi = b[si]\n g = 2.0.*(ai'*x - bi)*ai\n x -= step*g\n end\n return x\nend\nstochastic_gradient_descent_leastsquares(A,b)\n## Animate the algorithm\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)/size(A,1)\nezsurf(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot(x, y, Z, st=:surface)\nend\nezsurf(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A,b))\nplot!(colorbar=false,size=(450,450),dpi=300)\nplot!(camera=(170,30))\nx = zeros(size(A,2)) # initialize\np = plot3d!(1, label=\"\", marker = 2)\nstep = 0.0001\nanim = Animation()\nfor i=0:1000000\n global x\n g = (A'*A*x - A'*b)/size(A,1)\n x -= step*g\n if ((i <= 10) || ((i <= 30) & (rem(i,2) == 0)) || ((i <= 100) && rem(i,5) == 0) ||\n ((i <= 1000) && rem(i,10) == 0) || ((i <=10000) && rem(i,100)==0) ||\n ((i <= 100000) && rem(i,1000)==0) || rem(i,10000)==0)\n push!(p[1][2], x[1], x[2], quadratic_leastsq(A,b)(x))\n title!(@sprintf(\"Iteration %3i Residual = %.3e\", i, norm(g)))\n @printf(\"Iteration %3i Residual = %.3e\\n\", i, norm(g))\n frame(anim)\n end\nend\ngif(anim, \"stochatic-grad-descent-1.gif\",fps=30)\n\n##\n## Animate the stochastic algorithm\nusing Random\nRandom.seed!(0)\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)/size(A,1)\nezsurf(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot(x, y, Z, st=:surface)\nend\nezsurf(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A,b))\nplot!(colorbar=false,size=(450,450),dpi=300)\nplot!(camera=(170,30))\nx = zeros(size(A,2)) # initialize\nxgd = zeros(size(A,2)) # initialize\np = plot3d!(1, label=\"\", color=3)\np2 = plot3d!(1, label=\"\", color=2)\nstep = 0.0001\nanim = Animation()\nfor i=0:1000000\n global x, xgd\n si = rand(1:size(A,1))\n ai = A[si,:]\n bi = b[si]\n g = (ai'*x - bi)*ai\n x -= step*g\n\n xgd -= step*(A'*A*xgd - A'*b)/size(A,1)\n if ((i <= 10) || ((i <= 30) & (rem(i,2) == 0)) || ((i <= 100) && rem(i,5) == 0) ||\n ((i <= 1000) && rem(i,10) == 0) || ((i <=10000) && rem(i,100)==0) ||\n ((i <= 100000) && rem(i,1000)==0) || rem(i,10000)==0)\n push!(p[1][2], x[1], x[2], quadratic_leastsq(A,b)(x))\n push!(p[1][3], xgd[1], xgd[2], quadratic_leastsq(A,b)(xgd))\n title!(@sprintf(\"Iteration %3i Residual = %.3e\", i, norm(g)))\n @printf(\"Iteration %3i Residual = %.3e\\n\", i, norm(g))\n frame(anim)\n end\nend\ngif(anim, \"stochatic-grad-descent-2.gif\",fps=30)\n\n\n## Show the quadratic surface\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)\nezsurf(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot(x, y, Z, st=:surface)\nend\nezsurf(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A,b))\n##\nplot!(colorbar=false,size=(300,300),dpi=300)\nanim = @animate for i=1:360\n plot!(camera=(i,30))\nend\ngif(anim, \"leastsq-quadratic.gif\")\n\n##\n## Show the quadratic surface with alpha-blending\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)\nezsurf(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot()\n plot!(x, y, Z, st=:surface)\nend\nezsurf(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A,b))\n\n##\n## Show k random samples of the quadratic surface\nplotlyjs()\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)/size(A,1)\nezsurf!(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot!(x, y, Z, st=:surface, alpha=0.1)\nend\norig = ezsurf(-3:0.25:3,-50:1:250,\n quadratic_leastsq(A,b))\nnsamp = 25\nk = 25\nfor i=1:k\n subset = randperm(size(A,1))[1:25] # choose 25 of 200 samples.\n ezsurf!(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A[subset,:],b[subset,:]))\nend\nplot!() # get the plot to show\n## Show k random samples of the quadratic surface\nusing Random\nusing LinearAlgebra\nplotlyjs()\nquadratic_leastsq(A,b) = x-> (0.5*norm(A*x - b)^2)/size(A,1)\nezsurf!(x, y, f) = begin\n X = repeat(x', length(y), 1)\n Y = repeat(y, 1, length(x))\n # Evaluate each f(x, y)\n Z = map((x,y) -> f([x,y]), X, Y)\n plot!(x, y, Z, st=:surface, alpha=0.4)\nend\nplot(framestyle=:none,colorbar=false,size=(300,300),dpi=300)\nnsamp = 10\nk = 25\nfor i=1:k\n subset = randperm(size(A,1))[1:nsamp] # choose 25 of 200 samples.\n ezsurf!(-3:0.25:3,-50:2:250,\n quadratic_leastsq(A[subset,:],b[subset,:]))\nend\nplot!() # get the plot to show\n##\n#plot!(colorbar=false,size=(300,300),dpi=300)\nplot!(colorbar=false,size=(900,900))\nanim = @animate for i=2:2:360\n plot!(camera=(i,30))\nend\ngif(anim, \"leastsq-quadratic-samples.gif\")\n\n\n## Show the same thing with a log10\nusing Random\nusing LinearAlgebra\nplotlyjs()\nquadratic_leastsq(A,b) = x-> log10(0.5*norm(A*x - b)^2)/size(A,1)\nplot(framestyle=:none,colorbar=false,size=(300,300),dpi=300)\nnsamp = 10\nk = 25\nfor i=1:k\n subset = randperm(size(A,1))[1:nsamp] # choose 25 of 200 samples.\n ezsurf!(-3:0.1:3,-50:0.5:250,\n quadratic_leastsq(A[subset,:],b[subset,:]))\nend\nplot!() # get the plot to show\n##\n#plot!(colorbar=false,size=(300,300),dpi=300)\nplot!(colorbar=false,size=(900,900))\nanim = @animate for i=2:2:360\n plot!(camera=(i,30))\nend\ngif(anim, \"leastsq-quadratic-samples-log.gif\")\n##\n", "meta": {"hexsha": "73d845f876578c4c9bb5c37b686cd8cc68d2e63a", "size": 6260, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "4-unit-3-demos/stochastic-gradient-descent.jl", "max_stars_repo_name": "dgleich/cs590-ncds", "max_stars_repo_head_hexsha": "bd28821e2f805c2af1a1ae3cb7879e4e6e07bc56", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2019-04-07T15:19:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-07T04:43:33.000Z", "max_issues_repo_path": "4-unit-3-demos/stochastic-gradient-descent.jl", "max_issues_repo_name": "dgleich/cs590-ncds", "max_issues_repo_head_hexsha": "bd28821e2f805c2af1a1ae3cb7879e4e6e07bc56", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "4-unit-3-demos/stochastic-gradient-descent.jl", "max_forks_repo_name": "dgleich/cs590-ncds", "max_forks_repo_head_hexsha": "bd28821e2f805c2af1a1ae3cb7879e4e6e07bc56", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2019-07-13T03:13:53.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-17T01:37:03.000Z", "avg_line_length": 28.0717488789, "max_line_length": 95, "alphanum_fraction": 0.5945686901, "num_tokens": 2424, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9334308091776496, "lm_q2_score": 0.8633916082162403, "lm_q1q2_score": 0.8059163274944774}}
{"text": "\n\"\"\"\n Exponential(lnℓ::AbstractFloat) <: RadialKernel{Euclidean}\n\nThe exponential kernel with length scale \n``\\\\exp(\\\\ln \\\\ell) = \\\\ell > 0``.\n\n```math\nk_{\\\\ell}(x, y) = k_{\\\\ell}(\\\\lVert x - y\\\\rVert_2) = \n\\\\exp\\\\bigg\\\\lbrace -\\\\frac{\\\\lVert x - y\\\\rVert_2}{\\\\ell}\\\\bigg\\\\rbrace\n```\n\"\"\"\nmutable struct Exponential{F<:AbstractFloat} <: RadialKernel{Euclidean}\n dist::Euclidean\n lnℓ::F\n ℓ::F\n function Exponential(dist::Euclidean, lnℓ::AbstractFloat)\n new{typeof(lnℓ)}(dist, lnℓ, exp(lnℓ))\n end\nend\n\nExponential(lnℓ::AbstractFloat) = Exponential(Euclidean(), lnℓ)\n\n@inline (k::Exponential)(τ::AbstractFloat) = exp(-τ / k.ℓ)\n\nnumparams(::Exponential) = (1,)\nparamtypes(::Exponential{F}) where F = (F,)\nparams(k::Exponential) = (lnℓ = k.lnℓ,)\nfunction setparams!(k::Exponential{F}, lnℓ::F) where F\n k.lnℓ = lnℓ\n k.ℓ = exp(lnℓ)\nend\n\n\n", "meta": {"hexsha": "ea3c4332559a9ce1c2162e87c6d6037e5b1bd70a", "size": 861, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/kernels/radial/Exponential.jl", "max_stars_repo_name": "cameton/SKI.jl", "max_stars_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/kernels/radial/Exponential.jl", "max_issues_repo_name": "cameton/SKI.jl", "max_issues_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/kernels/radial/Exponential.jl", "max_forks_repo_name": "cameton/SKI.jl", "max_forks_repo_head_hexsha": "23b181da99b00b7f1a345ec1004c126624c508ed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.6, "max_line_length": 72, "alphanum_fraction": 0.6353077816, "num_tokens": 318, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9334308110294983, "lm_q2_score": 0.8633916064586998, "lm_q1q2_score": 0.8059163274528056}}
{"text": "function hypot(x::DoubleFloat{T}, y::DoubleFloat{T}) where {T<:IEEEFloat}\n ax = abs(x)\n ay = abs(y)\n ay, ax = minmax(ax, ay)\n\n r = ay\n if !iszero(ax)\n r /= ax\n end\n r *= r\n\n rr = ax * sqrt(one(DoubleFloat{T}) + r)\n\n # from Base\n # use type of rr to make sure that return type\n # is the same for all branches\n if isnan(ay)\n isinf(ax) && return oftype(rr, Inf)\n isinf(ay) && return oftype(rr, Inf)\n return oftype(rr, r)\n end\n return rr\n end\n\n\"\"\"\n normalize(x,y)\n\n x_normalized^2 + y_normalized^2 == one(promote_type(typeof(x),typeof(y)))\n\"\"\"\nfunction normalize(x::DoubleFloat{T}, y::DoubleFloat{T}) where {T<:IEEEFloat}\n hypotenuse = hypot(x, y)\n if !iszero(hypotenuse) && isfinite(hypotenuse)\n xnorm = x / hypotenuse\n ynorm = y / hypotenuse\n else\n # try for xnorm, ynorm such that xnorm + ynorm == 1\n xnorm = x/(x+y)\n ynorm = y/(x+y)\n end\n return xnorm, ynorm\nend\n", "meta": {"hexsha": "2b4537cfcf0e8a523e1f0a122bcaf6eeb0c3ee02", "size": 992, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/math/arithmetic/normalize_hypot.jl", "max_stars_repo_name": "UnofficialJuliaMirror/DoubleFloats.jl-497a8b3b-efae-58df-a0af-a86822472b78", "max_stars_repo_head_hexsha": "ccf0c6a690f81eec84caf080c99d58d11e72432d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 107, "max_stars_repo_stars_event_min_datetime": "2018-02-07T14:44:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T13:44:41.000Z", "max_issues_repo_path": "src/math/arithmetic/normalize_hypot.jl", "max_issues_repo_name": "UnofficialJuliaMirror/DoubleFloats.jl-497a8b3b-efae-58df-a0af-a86822472b78", "max_issues_repo_head_hexsha": "ccf0c6a690f81eec84caf080c99d58d11e72432d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 125, "max_issues_repo_issues_event_min_datetime": "2018-04-15T13:56:43.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T16:12:12.000Z", "max_forks_repo_path": "src/math/arithmetic/normalize_hypot.jl", "max_forks_repo_name": "UnofficialJuliaMirror/DoubleFloats.jl-497a8b3b-efae-58df-a0af-a86822472b78", "max_forks_repo_head_hexsha": "ccf0c6a690f81eec84caf080c99d58d11e72432d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 29, "max_forks_repo_forks_event_min_datetime": "2018-04-27T10:17:00.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T06:44:20.000Z", "avg_line_length": 23.619047619, "max_line_length": 77, "alphanum_fraction": 0.5675403226, "num_tokens": 317, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541659378681, "lm_q2_score": 0.8558511469672594, "lm_q1q2_score": 0.8059157979644224}}
{"text": "\n\"\"\"\n create_ols_approximation(y::Vector{<:Real}, x::Vector{<:Real}, base_x::Real = 0.0, degree::Integer = 1, intercept::Bool = true)\n create_ols_approximation(y::Vector{<:Real}, x::Union{Vector{DateTime},Vector{Date},Vector{Union{Date,DateTime}}}, base_x::Union{Date,DateTime} = global_base_date, degree::Integer = 1, intercept::Bool = true)\n\nAn approximation function calculated via OLS.\n\n### Inputs\n* `y` - A `Vector` with the y coordinates\n* `x` - A `Vector` with the x coordinates\n* `base_x` - A real that offsets the x. So a coordinate with x value of 2.0 will be converted to 1.8 if base_x is 0.2.\n* `degree` - What the highest power of x should be. So if this is 3 then the equation will have x, x^2, x^3 as predictors.\n* `intercept` - Should there be an x intercept.\n### Returns\n* A `Sum_Of_Functions` containing the approximation function.\n\"\"\"\nfunction create_ols_approximation(y::Vector{<:Real}, x::Vector{<:Real}, base_x::Real = 0.0, degree::Integer = 1, intercept::Bool = true)\n obs = length(y)\n if degree < 0\n error(\"Cannot approximate with OLS with a degree that is negative\")\n end\n x = x .- base_x\n if intercept\n X = ones(obs)\n for i in 1:degree\n X = hcat(X, (x .^ i))\n end\n else\n X = x\n for i in 2:degree\n X = hcat(X, (x .^ i))\n end\n end\n\n lm1 = fit(LinearModel, hcat(X), y)\n beta = lm1.pp.beta0\n func_array = Array{PE_Function,1}(undef,convert(Int, intercept) + degree)\n if intercept\n func_array[1] = PE_Function(beta[1], 0.0, base_x, 0)\n end\n for d in 1:degree\n func_array[d+Integer(intercept)] = PE_Function(beta[d+Integer(intercept)], 0.0, base_x, d)\n end\n return Sum_Of_Functions(func_array)\nend\n\nfunction create_ols_approximation(y::Vector{<:Real}, x::Union{Vector{DateTime},Vector{Date},Vector{Union{Date,DateTime}}}, base_x::Union{Date,DateTime} = global_base_date, degree::Integer = 1, intercept::Bool = true)\n base = years_from_global_base.(base_x)\n xx = years_from_global_base.(x)\n return create_ols_approximation(y, xx, base, degree, intercept)\nend\n\nfunction get_cholesky_coefficients(chebyshev::Sum_Of_Functions, y::Vector{<:Real}, unnormalised_nodes::Vector{<:Real})\n chebyshev_on_nodes = evaluate.(Ref(chebyshev), unnormalised_nodes)\n a = sum(y .* chebyshev_on_nodes) / sum(chebyshev_on_nodes .^ 2)\n return a\nend\n\n\"\"\"\n create_chebyshev_approximation(func::Function, nodes::Integer, degree::Integer, left::Real, right::Real)\n\nAn function that will approximate another function via Chebyshev polynomials.\n\n### Inputs\n* `func` - A function that you want to approximation\n* `nodes` - The number of approximation nodes\n* `degree` - The degree of the Chebyshev polynomials.\n* `left` - The left limit of the approximation\n* `right` - The right limit of the approximation.\n### Returns\n* A `Sum_Of_Functions` containing the approximation function.\n\"\"\"\nfunction create_chebyshev_approximation(func::Function, nodes::Integer, degree::Integer, left::Real, right::Real)\n # This is all after Algorithm 6.2 from Judd (1998) Numerical Methods in Economics.\n if nodes <= degree\n error(\"Need to have more nodes than degree to use a chebyshev approximation\")\n end\n k = 1:nodes\n unnormalised_nodes = -cos.( (((2 .* k) .- 1) ./ (2 * nodes)) .* pi )\n normalised_nodes = ((unnormalised_nodes .+ 1) .* ((right-left)/2)) .+ left\n y = func.(normalised_nodes)\n chebyshevs = get_chevyshevs_up_to(degree, true)\n a = get_cholesky_coefficients.(chebyshevs, Ref(y), Ref(unnormalised_nodes))\n transformed_chebyshevs = convert_to_linearly_rescale_inputs.(chebyshevs, (right-left)/2, +(right+left)/2)\n # Note that these alpha and beta parameters in the convert_to_linearly_rescale_inputs function differ from those in Judd because those did not work in this context.\n all_terms = a .* transformed_chebyshevs\n final_func = Sum_Of_Functions(all_terms)\n return final_func\nend\n", "meta": {"hexsha": "dbdb26cc0dd39b9aad922afa3ef027bc3a1cedf7", "size": 3982, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/7_regressions_and_approximation.jl", "max_stars_repo_name": "s-baumann/UnivariateFunctions.jl", "max_stars_repo_head_hexsha": "9e1a67e99ce55b109c850f51bef2491fee205c4a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-09-18T17:53:21.000Z", "max_stars_repo_stars_event_max_datetime": "2019-09-18T17:53:21.000Z", "max_issues_repo_path": "src/7_regressions_and_approximation.jl", "max_issues_repo_name": "s-baumann/UnivariateFunctions.jl", "max_issues_repo_head_hexsha": "9e1a67e99ce55b109c850f51bef2491fee205c4a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2018-10-15T19:14:20.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-21T00:04:18.000Z", "max_forks_repo_path": "src/7_regressions_and_approximation.jl", "max_forks_repo_name": "s-baumann/UnivariateFunctions.jl", "max_forks_repo_head_hexsha": "9e1a67e99ce55b109c850f51bef2491fee205c4a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.2444444444, "max_line_length": 216, "alphanum_fraction": 0.6948769463, "num_tokens": 1163, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541643004809, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.8059157931008721}}
{"text": "\"\"\"\n psMF(lnν)\n\nPress-Schechter halo multiplicity function.\n\n*Reference*: Press & Schechter, 187, 425 (1974)\n- See also Bond, Cole, Efstathiou & Kaiser, ApJ, 379, 440 (1991)\n- See Sheth & Tormen, MNRAS, 329, 61 (2002) for the notation used here. `psMF(lnν)` corresponds to νf(ν) in this reference.\n\n# Arguments\n- `lnν::Real`: natural logarithm of a threshold, ν, i.e., `lnν` = log(ν), defined by ν ≡ [δc/σ(R)]^2. Here, δc = 1.6865 and σ(R) is the r.m.s. mass fluctuation within a top-hat smoothing of scale R.\n\nPress & Schechter's multiplicity function is normalized, i.e,\n\n``∫_-∞^∞ dlnν psMF(lnν) = 1``\n\nThis function is based on [Cosmology Routine Library (CRL)](https://wwwmpa.mpa-garching.mpg.de/~komatsu/crl/).\n\"\"\"\nfunction psMF(lnν::Real)\n ν = exp(lnν)\n mf = exp(-ν / 2) * √(ν / 2π)\nend\n\n\"\"\"\n stMF(lnν)\n\nSheth-Tormen halo multiplicity function.\n\n*Reference*: Equation (2) of Sheth & Tormen, MNRAS, 329, 61 (2002)\n- See also Sheth, Mo & Tormen, MNRAS, 323, 1 (2001)\n- `stMF(lnν)` corresponds to νf(ν) in these references\n\n# Arguments\n- `lnν::Real`: natural logarithm of a threshold, ν, i.e., `lnν` = log(ν), defined by ν ≡ [δc/σ(R)]^2. Here, δc = 1.6865 and σ(R) is the r.m.s. mass fluctuation within a top-hat smoothing of scale R.\n\nSheth & Tormen's multiplicity function is normalized, i.e,\n\n``∫_-∞^∞ dlnν stMF(lnν) = 1``\n\nThis function is based on [Cosmology Routine Library (CRL)](https://wwwmpa.mpa-garching.mpg.de/~komatsu/crl/).\n\"\"\"\nfunction stMF(lnν::Real)\n A, p, q = 0.3222, 0.3, 0.707\n ν = exp(lnν)\n mf = A * (1 + (q * ν)^-p) * exp(-q * ν / 2) * √(q * ν / 2π)\nend\n\n\"\"\"\n jenkinsMF(lnν)\n\nJenkins et al.'s halo multiplicity function.\n\n*Reference*: Equation (B3) of Jenkins et al., MNRAS, 321, 372 (2001)\n- `jenkinsMF(lnν)` corresponds to f(σ)/2 in this reference, where σ=√(δc/ν).\n- We use the fitting function for Δm = 180. Here, Δm is an overdensity within a spherical region of radius R, whose mean density is equal to Δm times the mean **mass** density of the Universe.\n\n# Arguments\n- `lnν::Real`: natural logarithm of a threshold, ν, i.e., `lnν` = log(ν), defined by ν ≡ [δc/σ(R)]^2. Here, δc = 1.6865 and σ(R) is the r.m.s. mass fluctuation within a top-hat smoothing of scale R.\n\nJenkins et al.'s multiplicity function is NOT normalized, i.e,\n\n``∫_-∞^∞ dlnν jenkinsMF(lnν) ≠ 1``\n\nThis function is based on [Cosmology Routine Library (CRL)](https://wwwmpa.mpa-garching.mpg.de/~komatsu/crl/).\n\"\"\"\nfunction jenkinsMF(lnν::Real)\n ν = exp(lnν)\n σ = 1.6865 / √ν\n mf = 0.5 * 0.301 * exp(-abs(log(1 / σ) + 0.64)^3.82)\nend\n", "meta": {"hexsha": "8a397031d5c2b64f68cdf769e3d4bb932d1ef40b", "size": 2566, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/classicMF.jl", "max_stars_repo_name": "komatsu5147/HaloMF.jl", "max_stars_repo_head_hexsha": "70a2329082fa4b797f7b1e3ab9891d74c72218c9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/classicMF.jl", "max_issues_repo_name": "komatsu5147/HaloMF.jl", "max_issues_repo_head_hexsha": "70a2329082fa4b797f7b1e3ab9891d74c72218c9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/classicMF.jl", "max_forks_repo_name": "komatsu5147/HaloMF.jl", "max_forks_repo_head_hexsha": "70a2329082fa4b797f7b1e3ab9891d74c72218c9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.1408450704, "max_line_length": 198, "alphanum_fraction": 0.6554949337, "num_tokens": 951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541544761566, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.8059157898859988}}
{"text": "using Plots\n\nfunction bracket_minimum(f, x=0, s=1e-2, k=2.0)\n a, ya = x, f(x);\n b, yb = a + s, f(a + s);\n \n if yb > ya\n a, b = b, a;\n ya, yb = yb, ya;\n s = -s;\n end\n\n while true\n c, yc = b + s, f(b + s);\n if yc > yb\n return a < c ? (a, c) : (c, a);\n end\n\n a, ya, b, yb = b, yb, c, yc;\n s *= k;\n end\nend\n\n\n\n# test\n\nf1 = x -> (1 .+ x).^2;\nf2 = x -> (1 + x)^2;\nx = -3:0.1:3;\ny = f1(x);\n\na, b = bracket_minimum(f2, -2, 1e-2, 1.01);\n\nplot(x, y)\nvline!([a, b])\n", "meta": {"hexsha": "965346f287e14ef818396cd7f678ea958b25334b", "size": 543, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "bracketing/3.2-bracket_minimum.jl", "max_stars_repo_name": "tor4z/convex_optimization", "max_stars_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "bracketing/3.2-bracket_minimum.jl", "max_issues_repo_name": "tor4z/convex_optimization", "max_issues_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "bracketing/3.2-bracket_minimum.jl", "max_forks_repo_name": "tor4z/convex_optimization", "max_forks_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 14.6756756757, "max_line_length": 47, "alphanum_fraction": 0.3720073665, "num_tokens": 243, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541577509315, "lm_q2_score": 0.8558511414521922, "lm_q1q2_score": 0.8059157857643374}}
{"text": "\n\"\"\"\n angle(v::AbstractVector{<:Real}) -> Float\n\nAngle the vector `v` makes with the x axis, throws error if vector is not of size 2\n\"\"\"\nfunction angle(v::AbstractVector{<:Real})\n if length(v)!=2 error(\"Vector must be of length 2\") end\n atan(v[2],v[1])\nend\n\nangle(v::SVector{2, <:Real}) = atan(v[2],v[1])\n\n\"\"\"\n sqrt(z, branchcutangle::Real=π) -> typeof(z)\n\nReturn \\\\sqrt{x}, analytically continued from real line with the branch cut from 0 in a straight line along `branchcutangle`\n\"\"\"\nfunction sqrt(z, branchcutangle::Real)\n\n if branchcutangle > 0\n if angle(z) > branchcutangle return -sqrt(z) end\n else\n if angle(z) < branchcutangle return -sqrt(z) end\n end\n\n return sqrt(z)\nend\n\n\"\"\"\n round(num::Real, base::Real) -> Real\n\nNearest multiple of `base` to `num`, result will be promoted to num and base\n\"\"\"\nround(num::Real, base::Real) = round(num/base)*base\n\n\"\"\"\n ceil(num::Real, base::Real) -> Real\n\nSmallest multiple of `base` greater than `num`, result will be promoted to num and base\n\"\"\"\nceil(num::Real, base::Real) = ceil(num/base)*base\n\n\"\"\"\n ceil(num::Real, base::Real) -> Real\n\nLargest multiple of `base` smaller than `num`, result will be promoted to num and base\n\"\"\"\nfloor(num::Real, base::Real) = floor(num/base)*base\n\n", "meta": {"hexsha": "93e8f22e72a073b81d71d5aca7438c1851f95146", "size": 1278, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/maths.jl", "max_stars_repo_name": "jondea/JondeaUtils.jl", "max_stars_repo_head_hexsha": "09aec8840a4bbaaec3f5bfbd1a543ba60ab349a9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/maths.jl", "max_issues_repo_name": "jondea/JondeaUtils.jl", "max_issues_repo_head_hexsha": "09aec8840a4bbaaec3f5bfbd1a543ba60ab349a9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/maths.jl", "max_forks_repo_name": "jondea/JondeaUtils.jl", "max_forks_repo_head_hexsha": "09aec8840a4bbaaec3f5bfbd1a543ba60ab349a9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.0588235294, "max_line_length": 124, "alphanum_fraction": 0.6572769953, "num_tokens": 369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9572778073288128, "lm_q2_score": 0.8418256532040708, "lm_q1q2_score": 0.8058610154523385}}
{"text": "# Generates the dataset to be used in GMM example for EM course\n\nusing Plots, Distributions, Random, DataFrames, CSV\nRandom.seed!(123)\n\n# Set GMM parameters\nn = 200\nα = 0.4\nμ₀ = [-1, 2]\nμ₁ = [3.0, 7.0]\nΣ₀ = [2.0 1.0; 1.0 2.0]\nΣ₁ = [1.5 0; 0 0.5]\n𝒩₀ = MvNormal(μ₀, Σ₀)\n𝒩₁ = MvNormal(μ₁, Σ₁)\nX₁ = zeros(n)\nX₂ = zeros(n)\n\nfor i=1:n\n X₁[i],X₂[i] = rand() < α ? rand(𝒩₁) : rand(𝒩₀)\nend\n\n# Store in data frame\ndf = DataFrame(X1 = X₁, X2 = X₂)\n\n# Save to csv\nCSV.write(\"gmm_observations.csv\", df)\n", "meta": {"hexsha": "5b0ac913f9a82ffbb5e3f5c1b5a634294198ebfc", "size": 493, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_sample_generation.jl", "max_stars_repo_name": "seanrattana/courses", "max_stars_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2019-08-21T07:33:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-24T15:46:58.000Z", "max_issues_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_sample_generation.jl", "max_issues_repo_name": "seanrattana/courses", "max_issues_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-23T06:04:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-26T12:47:12.000Z", "max_forks_repo_path": "content/bayesian-inference-and-graphical-models/code/em_gmm_sample_generation.jl", "max_forks_repo_name": "seanrattana/courses", "max_forks_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-08-18T21:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T19:14:33.000Z", "avg_line_length": 18.2592592593, "max_line_length": 63, "alphanum_fraction": 0.6186612576, "num_tokens": 233, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9572778024535095, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.805860999951786}}
{"text": "# basisfunctions.jl\r\n\r\n\"\"\"\r\n fourier(k::Int)\r\n\r\nImplements the Fourier basis of functions ϕ_k, defined, if k is odd, by\r\nϕ_k(x)=sqrt(2)sin((k+1)π*x) and if k is even, by ϕ_k(x)=sqrt(2)cos(k*x).\r\nThis function is defined for k≥0 and ϕ_0≡1. \r\n\r\n# Examples\r\n\r\n```julia \r\nx = -1.0:0.01:2.0\r\ny = fourier(3).(x)\r\n```\r\n\"\"\"\r\nfunction fourier(k::Int)\r\n k ≥ 0 || throw(AssertionError(\"k should be non-negative\"))\r\n sqrttwo = sqrt(2.0)\r\n if k == 0\r\n return x -> 1.0\r\n elseif isodd(k)\r\n return x -> sqrttwo * sinpi(float(k+1)*x)\r\n else\r\n return x -> sqrttwo * cospi(float(k)*x)\r\n end\r\nend\r\n\r\n\"\"\"\r\n faberschauderone(x::Float64)\r\n\r\nImplements the first Faber-Schauder function defined by 1-2x for 0≤x≤1/2 and\r\n-1+2x for 0.5≤x≤1, and is 1-periodically extended to all x∈R.\r\n\r\n#See also: faberschauder\r\n\r\n#Warning\r\n\r\nNote the difference between faberschauderone and faberschauder. The first is a\r\nfunction that takes a Float64 and returns a Float64, the second takes (j,k) and\r\nreturns a anonymous function that takes a Float64 and returns a Float64.\r\n\r\n#Examples\r\n\r\n```julia\r\nx=-2.0:0.001:2.0\r\ny=faberschauderone.(x)\r\n```\r\n\"\"\"\r\nfunction faberschauderone(x::Float64)\r\n y = mod(x, 1.0)\r\n if 0≤y≤0.5\r\n return 1.0 - 2*y\r\n else\r\n return -1.0 + 2*y\r\n end\r\nend\r\n\r\n\"\"\"\r\n faberschauder(j::Int, k::Int)\r\n\r\nImplements the k-th Faber-Schauder function of level j. Here, j≥0 and 1≤k≤2^j.\r\nIt is a one periodic function and defined on [0,1] by 2^(j+1)(x-(k-1)2^(-j)) on\r\n(k-1)2^(-j)≤x≤(k-1/2)2^(-j) and 1 - 2^(j+1)(x-(k-1/2)2^(-j)) on\r\n[(k-1/2)2^(-j), k2^(-j)] and zero outside these intervals.\r\n\r\n# See also: faberschauderone.\r\n\r\n#Warning\r\n\r\nNote the difference between faberschauderone and faberschauder. The first is a\r\nfunction that takes a Float64 and returns a Float64, the second takes (j,k) and\r\nreturns a anonymous function that takes a Float64 and returns a Float64.\r\n\r\n#Example using PyPlot\r\n```julia\r\nusing PyPlot\r\nclf()\r\nj=3\r\nx=-2.0:0.001:2.0\r\nfor k in 1:2^j\r\n y = faberschauder(j,k).(x)\r\n plot(x,y)\r\nend\r\n```\r\n\"\"\"\r\nfunction faberschauder(j::Int,k::Int)\r\n j ≥ 0 || throw(AssertionError(\"j should be a nonnegative integer.\"))\r\n 1≤k≤2^j || throw(AssertionError(\"k should be an integer between 1 and 2^j.\"))\r\n twotothepowerminusj = 2.0^-j\r\n k_float = float(k)\r\n j_float = float(j)\r\n return function(x::Float64)\r\n y = mod(x,1.0)\r\n if y ≤ (k_float-1.0)*twotothepowerminusj || y ≥ k_float*twotothepowerminusj\r\n return 0.0\r\n elseif (k_float-1.0)*twotothepowerminusj < y ≤ (k_float-0.5)*twotothepowerminusj\r\n return 2.0^(j_float+1.0)*y-2*k_float+2.0\r\n else\r\n return 2.0*k_float - 2.0^(j_float+1.0) * y\r\n end\r\n end\r\nend\r\n", "meta": {"hexsha": "8ef4905b7c5e879cdc7453a75e5bff4623b08a83", "size": 2689, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/version0dot6/basisfunctions.jl", "max_stars_repo_name": "Jan-van-Waaij/BayesianNonparametricStatistics", "max_stars_repo_head_hexsha": "8ab5c9f995d83528688f061212b2580a53c226ff", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/version0dot6/basisfunctions.jl", "max_issues_repo_name": "Jan-van-Waaij/BayesianNonparametricStatistics", "max_issues_repo_head_hexsha": "8ab5c9f995d83528688f061212b2580a53c226ff", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/version0dot6/basisfunctions.jl", "max_forks_repo_name": "Jan-van-Waaij/BayesianNonparametricStatistics", "max_forks_repo_head_hexsha": "8ab5c9f995d83528688f061212b2580a53c226ff", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.8557692308, "max_line_length": 85, "alphanum_fraction": 0.6355522499, "num_tokens": 1005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9294403979493139, "lm_q2_score": 0.8670357580842941, "lm_q1q2_score": 0.8058580600301514}}
{"text": "### A Pluto.jl notebook ###\n# v0.12.18\n\nusing Markdown\nusing InteractiveUtils\n\n# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).\nmacro bind(def, element)\n quote\n local el = $(esc(element))\n global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing\n el\n end\nend\n\n# ╔═╡ 45f1331e-59b1-11eb-2bc0-c138b1421b55\nbegin\n\tusing Pkg\n\tPkg.activate(\".\")\n\tPkg.instantiate()\nend\n\n# ╔═╡ fc1a9e20-56aa-11eb-29b1-3be8043c6ede\nbegin\n\tusing Random\n\tRandom.seed!(0)\n\tusing Zygote\n\tusing LinearAlgebra\n\tusing Plots\n\tusing PlutoUI\n\tusing Images\n\tusing LaTeXStrings\nend\n\n# ╔═╡ 60a23536-59d0-11eb-368a-5588b71a6093\nmd\"# Curve fitting using AD\"\n\n# ╔═╡ 832e2c5a-59ce-11eb-32d7-85b05440f919\nmd\"## Problem Formulation\"\n\n# ╔═╡ e35400cc-56ad-11eb-2794-592020c6ac7e\nmd\"### The underlying mathematics of the problem\"\n\n# ╔═╡ 761fc420-57af-11eb-24b3-9728b06c482e\nmd\"We are trying to fit a function $f_{org}(x)$ to a polynomial function $f_{ML}(x)$\"\n\n# ╔═╡ 021f6506-57b1-11eb-2179-7b6f153fce2f\nmd\"$$f_{ML}(x_i) = \\displaystyle\\sum\\limits_{j=0}^nw_jx_i^j = w_0 + x_iw_1 + x_i^2w_2+\\ldots+x_i^nw_n$$\"\n\n# ╔═╡ df0e42cc-59be-11eb-367e-cf25ab5f105f\nmd\"so that $f_{org}(x_i) \\approx f_{ML}(x_i)$\"\n\n# ╔═╡ 2730398c-59bf-11eb-060e-276c0138395e\nmd\"We define $X, \\textbf{w}, \\textbf{y}$ for a more concise and all-round understanding of the problem,\"\n\n# ╔═╡ e50c3cc4-59b7-11eb-1479-8bba33d251fd\nmd\"$\\begin{gather}\nX=\n\\begin{bmatrix}\n1 & x_1 & \\ldots & x_1^n\\\\\n\\vdots & & \\ddots &\\vdots \\\\\n1 & x_m & & x_m^n\n\\end{bmatrix}\n\\quad\\quad\\quad\n\\textbf{w} = \\begin{bmatrix}\nw_o \\\\\nw_1 \\\\\n\\vdots\\\\\nw_n\n\\end{bmatrix}\n\\quad\\quad\\quad\n\\textbf{y} = \n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots\\\\\ny_m\n\\end{bmatrix}\n\\end{gather}$\"\n\n# ╔═╡ 51545508-59b9-11eb-3f94-59e743bfd1a8\nmd\" where,\n- The matrix $X$ is the data matrix. $X$ is a $(m\\times (n+1)) matrix$. $m$ is the number of data points and $n$ is the n is the degree of the plyonomial. The choice of this polynomial becomes clear in the next step.\n- The vector $\\textbf{w}$ represents the weights vector. $w_0$ the bias. $w_j$ is the weight corresponding to $x^j$.\n- The vector $\\textbf{y}$ represents the results vector. $y_i$ is the result of $x_i$ on applying the function $f$\"\n\n# ╔═╡ 70b0cea0-59ba-11eb-2ae9-439069c94467\nmd\"$\\begin{gather}\n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots\\\\\ny_m\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & x_1 & \\ldots & x_1^n\\\\\n\\vdots & & \\ddots &\\vdots \\\\\n1 & x_m & & x_m^n\n\\end{bmatrix}\n\\odot \n\\begin{bmatrix}\nw_o \\\\\nw_1 \\\\\n\\vdots\\\\\nw_n\n\\end{bmatrix}\n\\end{gather}$\"\n\n# ╔═╡ ea14bfa4-59ba-11eb-35a1-6548f64bed7e\nmd\"$\\begin{gather}\n\\quad \\quad \\\n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots\\\\\ny_m\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nw_o + x_1\\times w_1 &\\ldots &+x_1^n \\times w_n\\\\\n\\vdots \\\\\nw_0 + x_m\\times w_1 &\\ldots &+x_m^n\\times w_n\n\\end{bmatrix}\n\\end{gather}$\"\n\n# ╔═╡ 32bb7c2c-59bc-11eb-0914-d99847d081a3\nload(download(\"https://thehritikjain.files.wordpress.com/2020/05/1_e4_ptjctmaofsrpzczbv-g-1.jpeg?w=935\"))\n\n# ╔═╡ d2ea16ee-57b7-11eb-2d27-f3c04343dc59\nmd\"Before we approach the real problem of solving the approximate function, we need a predicting function and loss function to evaluate good our approximation is.\"\n\n# ╔═╡ fa7fe6b4-59c1-11eb-2486-d90418eaee4a\nmd\"### Predict and loss functions : \"\n\n# ╔═╡ e8864a02-59c1-11eb-12ff-c9a4ffde849d\nmd\"#### Predict function \"\n\n# ╔═╡ bf75963a-56ac-11eb-2b34-21b724a59f96\nfunction predict(x, ω::Vector)\n\tn = length(ω)\n\tx_vec = [x^i for i= 0:n-1]\n\treturn dot(x_vec, ω)\nend\n\n# ╔═╡ c8a25b68-59c1-11eb-363f-0772ec4edbf2\nmd\"**Usage Example :**\"\n\n# ╔═╡ 10be275c-59c1-11eb-10dc-4becfaf27f8d\nbegin\n\t# Try changing the values\n\tlocal ω = rand(5) # 4 polynomial weight vector\n\tlocal x = 3. # input\n\tpredict(x, ω)\nend\t\n\n# ╔═╡ d928b270-59c1-11eb-09d0-a9c3c878fe2d\nmd\"#### Square Loss function \"\n\n# ╔═╡ 0129d404-57b8-11eb-3dde-efa8a687c1d4\nfunction sq_loss(func, X, ω::Vector) \n\tn = length(ω)\n\ty = func.(X)\n\tŷ = [predict(x, ω) for x in X]\n\treturn sum((y-ŷ).^2)/n\nend\n\n# ╔═╡ bc758f7c-59c1-11eb-3f4a-5b987c7c6e0b\nmd\"**Usage Example :**\"\n\n# ╔═╡ 55bd943c-59c1-11eb-19d3-a7bba6fcbd30\nbegin\n\tlocal ω = rand(5) # 4 polynomial weight vector\n\tlocal X = [3.] # input\n\tlocal loss = sq_loss(sin, X, ω)\nend\n\n# ╔═╡ 00f8133c-57be-11eb-19d0-33006b7bbb06\nmd\"## Data Generation\"\n\n# ╔═╡ 6d407f0e-59ce-11eb-0ed6-33f817c914a3\nmd\"Choose the number of data points : \"\n\n# ╔═╡ 4b28aee8-56ac-11eb-3be7-0de4e9cb211b\n@bind n Slider(0:10, show_value = true, default=8)\n\n# ╔═╡ e6d69724-57bf-11eb-0337-db41600a8ebc\nmd\"We define a function that we want to approximate. **org\\_func** is the original function. We are trying to approximate *sin(x)* here. You can try your own function that you want to approximat inside org\\_func and chnages will be reflected in rest of the notebook.\"\n\n# ╔═╡ 9801c9ca-57bf-11eb-3f30-93055ccd6a0f\nfunction org_func(x)\n\treturn sin(x)\nend\n\n# ╔═╡ 2fcaeb98-56b1-11eb-0198-a3a5088dc007\nbegin\n\tX = [x for x in range(randn(), 12, length = n)]\n\ty = org_func.(X) +0.1* randn((n))\n\tplot(-0.5:0.01:12, org_func.(-0.5:0.01:12), label = L\"sin(x)\")\n\tscatter!(X, y, label = \"data points\", title = \"Our Data points\")\nend\n\n# ╔═╡ bd151b36-56b1-11eb-03d1-094237a3bafa\nmd\"## Approximation\"\n\n# ╔═╡ c7c2e3f2-57be-11eb-15c0-dbe72c95d654\nmd\"We will try to solve the problem of approximating the function by two differnt ways:\n- Linear Algebra Methods\n- Gradient Descent using AD\"\n\n# ╔═╡ 377b50e4-57bf-11eb-1741-a73fda781579\nmd\" ### Approximation by Linear Algebra\"\n\n# ╔═╡ 50c00e44-59c4-11eb-2726-27bee50e4fdd\nmd\"$\\begin{gather}\n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots\\\\\ny_m\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & x_1 & \\ldots & x_1^n\\\\\n\\vdots & & \\ddots &\\vdots \\\\\n1 & x_m & & x_m^n\n\\end{bmatrix}\n\\odot \n\\begin{bmatrix}\nw_o \\\\\nw_1 \\\\\n\\vdots\\\\\nw_n\n\\end{bmatrix}\n\\end{gather}$\"\n\n# ╔═╡ 586240fe-59c4-11eb-1427-312c47374f8a\nmd\"$\n\\ \\ \\ \\\n\\begin{gather}\n\\begin{bmatrix}\nw_o \\\\\nw_1 \\\\\n\\vdots\\\\\nw_n\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & x_1 & \\ldots & x_1^n\\\\\n\\vdots & & \\ddots &\\vdots \\\\\n1 & x_m & & x_m^n\n\\end{bmatrix}^{-1}\n\\odot \n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots\\\\\ny_m\n\\end{bmatrix}\n\n\\end{gather}$\"\n\n# ╔═╡ 4f5c808e-57bf-11eb-2a7a-3995d6f8f866\nmd\"This may seem like a very viable method, but try considering the case where $X$ is invertible. It is left as an exercise to the reader.\"\n\n# ╔═╡ 5dd40e02-57bf-11eb-2764-0f3f2b95517b\nmd\" ### Approximation by Gradient Descent\"\n\n# ╔═╡ ea205094-59c9-11eb-0f04-818f051b7d1e\nmd\"We will be using gradient descent algorithm to find $\\textbf{w}$ here. We are tired of mathematics by now and we somehow want the differentaition to happen automatically(almost magically). We will be using **Automatic Differentation**(or a more intuitive name : **Algorthimic Differentiation**). While how automatic differintation works is another day's agenda, we can have a little taste of magic of what AD does for us.\"\n\n# ╔═╡ b13969a4-59ca-11eb-15d8-0f9b5f225fed\nmd\"The gradient of a function $f(x)$ at $x_0$ is defined as $\\displaystyle\\frac{df(x)}{dx}\\Bigg|_{x=x_0}$\n\n$f^\\prime(x_0) = \\displaystyle\\frac{df(x)}{dx}\\Bigg|_{x=x_0}$\"\n\n# ╔═╡ e4f40500-59cb-11eb-1020-3b345469fb1f\nmd\"The Zygote package gives us a *gradient* API that we can use to find gradients\"\n\n# ╔═╡ 88002972-59cc-11eb-3d75-ab7e7bb45954\nmd\"$f(x) = x^2+4x+1$\n$f^\\prime(x) = 2x+4$\n$f^\\prime(5) = 2\\times 5 + 4 = 14$\"\n\n# ╔═╡ 3255da08-59cc-11eb-0bdc-61a05344ad50\ngradient(x->x^2+4x+1, 5)\n\n# ╔═╡ cd6a28f8-59cc-11eb-28f8-35c3a04414d8\nmd\"$f(x, y) = x^2+y^2+3xy$\n$\\displaystyle\\frac{\\partial f(x,y)}{\\partial x} = 2x+3y$\n$\\displaystyle\\frac{\\partial f(x,y)}{\\partial y} = 2y+3x$\"\n\n# ╔═╡ 4506b2d0-59cc-11eb-22e2-5f43abeb3ccd\ngradient( (x,y) -> x^2 + y^2 + 3x*y, 2,3)\n\n# ╔═╡ 608cc19c-59cd-11eb-092d-558d322966c6\nmd\"A train step function that returns the updated weights vector $\\textbf{w}$\"\n\n# ╔═╡ 84566232-57bf-11eb-390b-f33370fbd9a3\nfunction train_step(func, X, ω, η)::typeof(ω)\n\tgrad = gradient(ω->sq_loss(func, X, ω), ω)[1]\n\tω -= η * grad\n\treturn ω\nend\n\n# ╔═╡ de0aeb62-59cd-11eb-3e5f-a7cb176093c9\nmd\"Choose the degree of polynomial of $f_{ML}(x)$. You may need to choose different $\\eta$ and number of epochs to account for the model change\"\n\n# ╔═╡ ae1c0e18-59cd-11eb-2e61-fbe810bd5fa0\n@bind n_model Slider(1:1:7, default = 4, show_value = true)\n\n# ╔═╡ 5c85511c-5811-11eb-0f2e-2352309f2261\n## Intializing weights\nω = rand(n_model+1)\n\n# ╔═╡ 11337388-57c1-11eb-3418-3979d91e6af2\nfunction train(func, X, ω, η, epochs)::typeof(ω)\n\tfor i = 1:1:epochs\n\t\tω = train_step(func, X, ω, η)\n\t\tloss = sq_loss(func, X, ω)\n\t\tif isnan(loss) || isinf(loss) || iszero(loss)\n\t\t\tthrow(DomainError(\"Loss cannot take $(loss) value\"))\n\t\tend\n\tend\n\treturn ω\nend\n\n# ╔═╡ d722598c-57c2-11eb-3452-531fb6522e34\nω_trained = train(org_func, X, ω, 7e-9, 1e6)\n\n# ╔═╡ 11d38860-59c6-11eb-06e2-b10697f21f1d\nmd\"It is upto the user to decide how much accuracy he wants given the time constraints\"\n\n# ╔═╡ 8b6bc65e-57c7-11eb-372f-bd6f700363e1\nloss = sq_loss(org_func, X, ω_trained)\n\n# ╔═╡ 328c4584-57c4-11eb-3cf5-a57e033cf574\npred = [predict(x, ω_trained) for x in -0.5:0.01:13]\n\n# ╔═╡ a0dfe352-57c3-11eb-348f-8380ee6b3caf\nbegin\n\tplot(-0.5:0.01:13, org_func.(-0.5:0.01:13), label = L\"f_{org}(x)\")\n\tscatter!(X, y, label = \"Orginal data points\")\n\tŷ = [predict(x, ω_trained) for x in X]\n\tscatter!(X, ŷ, label = \"Predicted data points\")\n\tplot!(-0.5:0.01:13, pred, label = L\"f_{ML}(x)\")\nend\n\n# ╔═╡ Cell order:\n# ╟─45f1331e-59b1-11eb-2bc0-c138b1421b55\n# ╟─fc1a9e20-56aa-11eb-29b1-3be8043c6ede\n# ╟─60a23536-59d0-11eb-368a-5588b71a6093\n# ╟─832e2c5a-59ce-11eb-32d7-85b05440f919\n# ╟─e35400cc-56ad-11eb-2794-592020c6ac7e\n# ╟─761fc420-57af-11eb-24b3-9728b06c482e\n# ╟─021f6506-57b1-11eb-2179-7b6f153fce2f\n# ╟─df0e42cc-59be-11eb-367e-cf25ab5f105f\n# ╟─2730398c-59bf-11eb-060e-276c0138395e\n# ╟─e50c3cc4-59b7-11eb-1479-8bba33d251fd\n# ╟─51545508-59b9-11eb-3f94-59e743bfd1a8\n# ╟─70b0cea0-59ba-11eb-2ae9-439069c94467\n# ╟─ea14bfa4-59ba-11eb-35a1-6548f64bed7e\n# ╟─32bb7c2c-59bc-11eb-0914-d99847d081a3\n# ╟─d2ea16ee-57b7-11eb-2d27-f3c04343dc59\n# ╟─fa7fe6b4-59c1-11eb-2486-d90418eaee4a\n# ╟─e8864a02-59c1-11eb-12ff-c9a4ffde849d\n# ╠═bf75963a-56ac-11eb-2b34-21b724a59f96\n# ╟─c8a25b68-59c1-11eb-363f-0772ec4edbf2\n# ╠═10be275c-59c1-11eb-10dc-4becfaf27f8d\n# ╟─d928b270-59c1-11eb-09d0-a9c3c878fe2d\n# ╠═0129d404-57b8-11eb-3dde-efa8a687c1d4\n# ╟─bc758f7c-59c1-11eb-3f4a-5b987c7c6e0b\n# ╠═55bd943c-59c1-11eb-19d3-a7bba6fcbd30\n# ╟─00f8133c-57be-11eb-19d0-33006b7bbb06\n# ╟─6d407f0e-59ce-11eb-0ed6-33f817c914a3\n# ╟─4b28aee8-56ac-11eb-3be7-0de4e9cb211b\n# ╟─e6d69724-57bf-11eb-0337-db41600a8ebc\n# ╠═9801c9ca-57bf-11eb-3f30-93055ccd6a0f\n# ╠═2fcaeb98-56b1-11eb-0198-a3a5088dc007\n# ╟─bd151b36-56b1-11eb-03d1-094237a3bafa\n# ╟─c7c2e3f2-57be-11eb-15c0-dbe72c95d654\n# ╟─377b50e4-57bf-11eb-1741-a73fda781579\n# ╟─50c00e44-59c4-11eb-2726-27bee50e4fdd\n# ╟─586240fe-59c4-11eb-1427-312c47374f8a\n# ╟─4f5c808e-57bf-11eb-2a7a-3995d6f8f866\n# ╟─5dd40e02-57bf-11eb-2764-0f3f2b95517b\n# ╟─ea205094-59c9-11eb-0f04-818f051b7d1e\n# ╟─b13969a4-59ca-11eb-15d8-0f9b5f225fed\n# ╟─e4f40500-59cb-11eb-1020-3b345469fb1f\n# ╟─88002972-59cc-11eb-3d75-ab7e7bb45954\n# ╠═3255da08-59cc-11eb-0bdc-61a05344ad50\n# ╟─cd6a28f8-59cc-11eb-28f8-35c3a04414d8\n# ╠═4506b2d0-59cc-11eb-22e2-5f43abeb3ccd\n# ╟─608cc19c-59cd-11eb-092d-558d322966c6\n# ╠═84566232-57bf-11eb-390b-f33370fbd9a3\n# ╟─de0aeb62-59cd-11eb-3e5f-a7cb176093c9\n# ╟─ae1c0e18-59cd-11eb-2e61-fbe810bd5fa0\n# ╠═5c85511c-5811-11eb-0f2e-2352309f2261\n# ╠═11337388-57c1-11eb-3418-3979d91e6af2\n# ╠═d722598c-57c2-11eb-3452-531fb6522e34\n# ╟─11d38860-59c6-11eb-06e2-b10697f21f1d\n# ╠═8b6bc65e-57c7-11eb-372f-bd6f700363e1\n# ╠═328c4584-57c4-11eb-3cf5-a57e033cf574\n# ╟─a0dfe352-57c3-11eb-348f-8380ee6b3caf\n", "meta": {"hexsha": "6f9f25a63d19cf54c6cfc74f93f7e0756d71fc4d", "size": 11692, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "Study_Session_1/notebooks/Curve_Fitting.jl", "max_stars_repo_name": "khanfarhan10/Study-Group", 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{"text": "### A Pluto.jl notebook ###\n# v0.17.1\n\nusing Markdown\nusing InteractiveUtils\n\n# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).\nmacro bind(def, element)\n quote\n local iv = try Base.loaded_modules[Base.PkgId(Base.UUID(\"6e696c72-6542-2067-7265-42206c756150\"), \"AbstractPlutoDingetjes\")].Bonds.initial_value catch; b -> missing; end\n local el = $(esc(element))\n global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)\n el\n end\nend\n\n# ╔═╡ 1cdb4e08-99f7-48d9-924b-e105d1b266dc\nbegin\n using Plots; gr()\n\tusing Printf\n\tusing LaTeXStrings\n\tusing PlutoUI\n\tusing LinearAlgebra\n\timport Statistics: mean, cov\n\tusing MultivariateStats\n\tusing Random\n\t\n\tmd\"\"\"**Import julia libraries**\"\"\"\nend\n\n# ╔═╡ 721e5d79-4842-43cf-a2ff-4cf0fdb13f87\nbegin\n rng = MersenneTwister(220115)\n Base.show(io::IO, f::Float64) = @printf io \"%1.2f\" f\n\n\tmd\"\"\"Set the random seed and change the format of floating numbers for showing arrays.\"\"\"\nend\n\n# ╔═╡ 969a133a-9861-4248-92f0-757fd12f12fb\nmd\"# Principal Component Analysis (PCA): *A Physically Intuitive Mathematical Introduction*\n\nThe principal component analysis (PCA) involves rotating a cloud of data points in Euclidean space such that the variance is maximal along the first axis, the so-called first principal component. The principal axis theorem ensures that the data can be rotated in such away. In mathematical terms, the PCA involves finding an orthogonal linear coordinate transformation or, more generally, a new basis.\n\nThe mathematics behind the PCA is found again in the description of rotations of rigid bodies. This physical interpretation is instructive in understanding the PCA.\"\n\n# ╔═╡ 624fcb2a-8446-42c2-863f-e285f2c1fd86\nbegin\n\tN_slider = @bind N html\"\"\n\tR1_slider = @bind R1 html\"\"\n\tR2_slider = @bind R2 html\"\"\n\tangle_slider = @bind angle html\"\"\n\t\n\tmd\"\"\"## Define parameters to generate sample data\n\t\t\n\t\"\"\"\nend\n\n# ╔═╡ d8005539-13d5-4a3a-ad62-861c4f4bb9e0\nmd\"First, we will generate a cloud of $N$ randomly distributed data points in Euclidean space $\\mathbb{R}^n$ with coordinates $\\{ \\vec{x}^{(1)}, \\vec{x}^{(2)},\\dots, \\vec{x}^{(N)} \\} = X$. We will demonstrate the concept based on $3$-dimensional data, where $n=3$ and $X \\subset \\mathbb{R}^3$ with basis vectors $\\vec{e}_1$, $\\vec{e}_2$ and $\\vec{e}_3$ centered around the origin $(0, 0, 0)$. For simplicity, we will set all coordinates along the basis vector $\\vec{e}_3$ to zero. It will allow us to visualise the data in the $\\vec{e}_1$ and $\\vec{e}_2$ plane.\"\n\n# ╔═╡ 5619b1f9-38d1-42de-bd81-708567e2fa96\nmd\"\"\"\n**Define Parameters:**\n\n| | Min | Slider | Max |\n|-------|-----|:-------------:|-------|\n| N | 10 | $(N_slider) | 1000 |\n| R1 | 0.1 | $(R1_slider) | 10 |\n| R2 | 0.1 | $(R2_slider) | 10 |\n| angle | 0.0 |$(angle_slider)| 180 |\n\"\"\"\n\n# ╔═╡ 24f9bdd6-0337-4a23-9373-cfecabc2b029\nbegin\n\tRmax = max(R1, R2)\n\t\n\tmd\"\"\"\n\t\n\tN defines the number of points of the sample data. R1 and R2 define the\n\tvariance of the data and angle rotates the sample data in space.\n\n\t| Variable | N | R1 | R2 | angle | Rmax |\n\t|----------|----|-----|-----|-------|------|\n\t|**Values**| $(N) | $(R1) | $(R2) | $(round(angle, digits=2)) |$(Rmax) |\n\t\"\"\"\nend\n\n# ╔═╡ eacd1340-79af-4985-bf60-b8db8a380a20\nbegin\n\trot = [cos(deg2rad(angle)) sin(deg2rad(angle));\n\t\t -sin(deg2rad(angle)) cos(deg2rad(angle))]\n\t\n\tpts = randn(rng, N, 3)\n\tpts[:, 1] .*= R1\n\tpts[:, 2] .*= R2\n pts[:, 3] = zeros(N)\n\t\n\tpts[:, 1:2] = pts[:, 1:2] * rot\n\t\n\tmd\"\"\"Create $3$-dim array with sample data based on parameters defined in the table above.\n\t\n\t---\n\t\"\"\"\n\t\nend\n\n# ╔═╡ 6bcb1adc-633c-4730-b4e2-6b5b6f5237c4\nbegin\n\tlim = [-3 * Rmax, 3 * Rmax]\n\t\n\tu1 = [0.0 0.0; lim[2] lim[2]*tan(deg2rad(angle+90.0))]\n\tu1[2,:] = normalize!(u1[2,:])\n\tann1 = (u1[2,1].*2.5*Rmax, u1[2,2].*2.5*Rmax, Plots.text(L\"$\\mathbf{\\vec{u}_1}$\", :left))\n\t\n\tu2 = [0.0 0.0; lim[2] lim[2]*tan(deg2rad(angle))]\n\tu2[2,:] = normalize!(u2[2,:])\n\tann2 = (u2[2,1].*2.5*Rmax, u2[2,2].*2.5*Rmax, Plots.text(L\"$\\mathbf{\\vec{u}_2}$\", :left))\n\t\n\tl = @layout [ a{1.0w, 0.01h}\n\t [b{0.9w, 0.1h} _\n\t c{0.9w, 0.9h} d{0.1w, 0.9h}]]\n\n\ttitle = plot(title = \"Random Set of Points\",\n\t\t framestyle = :none,\n\t\t grid = false,\n\t\t showaxis = false,\n\t\t titlelocation = :center,\n\t\t bottom_margin = -40Plots.px)\n\t\n\th1 = histogram(pts[:,1],\n\t\t bins=range(lim[1], stop = lim[2], length = 20),\n\t\t normalize = true,\n\t\t fillcolor = \"#0044aa\",\n\t\t orientation = :vertical,\n\t\t framestyle = :none,\n\t title = L\"$a)$\",\n\t titlefontsize = 10,\n\t\t titlelocation=:left)\n\t\n\th2 = histogram(pts[:,2],\n\t\t bins=range(lim[1], stop = lim[2], length = 20),\n\t\t normalize = true,\n\t\t fillcolor = \"#0044aa\",\n\t\t orientation = :horizontal,\n\t\t framestyle = :none,\n\t title = L\"$c)$\",\n\t titlefontsize = 10,\n\t\t titlelocation=:right )\n\t\n\ts = scatter(pts[:,1], pts[:,2],\n\t\t markershape = :circle,\n\t\t markersize = 2,\n\t\t markercolor = \"#0044aa\",\n\t\t markerstrokecolor = \"#0044aa\",\n\t\t framestyle = :orign,\n\t\t xlabel = L\"\\vec{e}_1\",\n\t ylabel = L\"\\vec{e}_2\",\n\t\t xlims = (lim[1], lim[2]),\n\t\t ylims = (lim[1], lim[2]),\n\t\t aspect_ratio = :equal,\n\t\t title = L\"$b)$\",\n\t titlefontsize = 10,\n\t\t titlelocation=:left)\n\t\n\tplot(title, h1, s, h2,\n\t\t layout = l,\n\t\t legend = false,\n\t\t size = (600, 610))\n\n\tplot!(lim, lim*tan(deg2rad(angle)),\n\t\t color=\"#a6a6a6\",\n\t\t label=\"\",\n\t\t annotations = ann1,\n\t\t subplot = 3 )\n\t\n\tplot!(lim, lim*tan(deg2rad(angle+90.0)),\n\t\t color=\"#a6a6a6\",\n\t\t label=\"\",\n\t\t annotations = ann2,\n\t\t subplot = 3)\nend\n\n# ╔═╡ 075b42b8-476b-4436-a886-dcbb082b0ad7\nmd\"*Figure 1. a) Histogram of the random points distributed along $\\vec{e}_2$. b) Cloud of randomly distributed data points in Euclidean space $\\mathbb{R}^3$ with coordinates $\\{ \\vec{x}^{(1)}, \\vec{x}^{(2)},\\dots, \\vec{x}^{(N)} \\}$ with the basis vectors (axis in the plot b)) $\\vec{e}_1$ and $\\vec{e}_2$. The grey lines in plot b) indicate the principal axis $\\vec{u}_1$ and $\\vec{u}_2$. c) Histogram of the random points distributed along $\\vec{e}_1$. All values along $\\vec{e}_3$ are equal zero.*\n\n---\n\"\n\n# ╔═╡ 35f8cd6a-a87d-4af9-8590-3bcac79ef3ea\nmd\"## Moment of Inertia $\\boldsymbol{J}$\n\nThe moment of inertia $\\boldsymbol{J}$ of a rigid body, also called rotational inertia, determines the torque required for a desired angular acceleration around an axis of rotation. It depends on the mass distribution of the body and the selected axis of rotation. A body with a larger moment of inertia $\\boldsymbol{J}$ requires more torque to change the body's rate of rotation. For the same rigid body, different axes of rotation will have different moments of inertia associated with them. In other words, it depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate.\n\nAll moments of inertia of a rigid body can be summarised by a matrix. In general, it can be determined with respect to any point in space. For simplicity, we will calculate the moment of inertia with respect to the center of mass.\n\nThe principal axes of the body, also known as figure axes, and the principal moments of inertia can be found by rotating the cloud of point masses. In mathematical terms, the PCA involves finding an orthogonal linear coordinate transformation or, more generally, a new basis.\n\nThe figure axis corresponding to the largest principal moment of inertia is the area vector of the plane with the maximal spread of mass points.\n\n|  | \n|:--| \n| *Figure 2. The six cylinders have the same mass but different moments of inertia $\\boldsymbol{J}$. As they roll down the slope, cylinders with lower moments of inertia accelerate more quickly. ([Image taken from wikipedia](https://en.wikipedia.org/wiki/File:RollingVsInertia.gif))* |\n\"\n\n# ╔═╡ 9ee5e06b-318b-4750-82bb-9db2999f2d1e\nmd\"## Visual Comparison of the Principal Axis in PCA and the Moment of Inertia\n\nIn the following, we will interpret the set of random data points from above in two ways. First, we interpret the data points $X$ as statistically distributed data with Covariance matrix $\\boldsymbol{C}$. Second, $X$ represents a set of point masses representing a rigid body with the Moment of Inertia matrix $\\boldsymbol{J}$. \n\n---\"\n\n# ╔═╡ 5a400a58-4bfb-49fe-8286-2b8e56803710\nbegin\n\tl2 = @layout [c{0.5w, 1.0h} d{0.5w, 1.0h}]\n\n\ts2 = scatter(pts[:,1], pts[:,2],\n\t\t markershape = :circle,\n\t\t markersize = 2,\n\t\t markercolor = \"#0044aa\",\n\t\t markerstrokecolor = \"#0044aa\",\n\t\t framestyle = :orign,\n\t\t xlabel = L\"\\vec{e}_1\",\n\t ylabel = L\"\\vec{e}_2\",\n\t\t xlims = (lim[1], lim[2]),\n\t\t ylims = (lim[1], lim[2]),\n\t\t aspect_ratio = :equal)\n\t\n\tplot(s2, s2,\n\t\t layout = l2,\n\t\t legend = false,\n\t\t size = (600, 310))\n\n\tplot!(lim, lim*tan(deg2rad(angle+90.0)),\n\t\t color=\"#a6a6a6\",\n\t\t label=\"\",\n\t\t annotations = ann1,\n\t\t subplot = 1,\n\t\t title = \"a) PCA\",\n\t titlefontsize = 12,\n\t\t titlelocation=:left )\n\n\tplot!( u1[:,1].*Rmax,\n\t\t u1[:,2].*Rmax,\n\t\t subplot = 1,\n\t\t arrow=true,color=\"#6a0000\",linewidth=3,label=\"\")\n\t\n\t\n\tplot!(lim, lim*tan(deg2rad(angle)),\n\t\t color=\"#a6a6a6\",\n\t\t label=\"\",\n\t\t annotations = ann2,\n\t\t subplot = 2,\n\t\t title = \"b) Moment of Inertia\",\n\t titlefontsize = 12,\n\t\t titlelocation=:left)\n\n\tplot!( u2[:,1].*Rmax,\n\t\t u2[:,2].*Rmax,\n\t\t subplot = 2,\n\t\t arrow=true, color=\"#01493e\",linewidth=3,label=\"\")\n\t\nend\n\n# ╔═╡ 07371ed0-72e9-47d9-82d8-6f3cb0e16ec8\nmd\" *Figure 3. a) PCA: The first principal component $\\vec{u}_1$ is along the axis where the variance is maximally indicated by a red arrow b) Moment of Inertia: The figure axis $\\vec{u}_2$ corresponding to the (actually second) largest principal moment of inertia indicated by a green arrow.*\n\n---\n\"\n\n# ╔═╡ 2f20596e-efa9-4d00-9769-6e8689c99f05\nmd\"With the visual support of *Figure 1* and *3*, we expect that the principal axes of the PCA and Moment of Inertia are the same. However, the value of the largest principal component and principal moment of inertia will differ for most sets of data points.\n\n> *Note:* In physics, the moment of inertia is defined for a $3$-dimensional rigid body. For simplicity, we projected the data into the plane spanned by $\\vec{e}_1$ and $\\vec{e}_2$. In our example, the plane with the maximal spread of mass points is then spaned by $\\vec{e}_1 and \\vec{e}_2$. The principal axis corresponding to the largest moment is pointing out of the plane along $\\vec{e}_3$ and is orthogonal to $\\vec{e}_1$ and $\\vec{e}_2$.\n\nThe line along $\\vec{u}_1$ is equivalent to the direction where the variance is maximal. In the following, we will support our visual understanding by exploring the PCA and moment of inertia mathematically.\n\"\n\n# ╔═╡ 03664f5c-d45c-11ea-21b6-91cd647a07aa\nmd\"## Definition of the Moment of Inertia of a Rigid Body\n\nFor a rigid object of $N$ point masses $m_i$ in $\\mathbb{R}^n$, the moment of inertia $\\boldsymbol{J}$ is given by\n\n$$\\begin{equation*}\n \\boldsymbol{J} = \n \\begin{bmatrix}\n J_{1,1} & \\cdots & J_{1,n} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n J_{n,1} & \\cdots & J_{n,n} \n \\end{bmatrix}\n\\end{equation*}$$\n\nIts components are defined by Eq. (1) as\n\n$$\\begin{equation}\n J_{j,j'} =\n \\frac{1}{M}\n \\sum_{i=1}^{N} m_{i}\n \\left( ||\\vec{x}^{(i)}||^2 \\delta_{j,j'} - x_j^{(i)} x_{j'}^{(i)} \\right)\n\\end{equation}$$\n\nwhere $\\delta_{j,j'}$ is the Kronecker delta and $M = \\sum_{i}^{N} m_i$ is the total mass.\n\n> *Note:* Here, we normalize the moment of inertia by the total mass. In physics, the moment of inertia would not commonly be normalized like this.\n\nLooking closer, we see that $\\boldsymbol{J}$ is symmetric with $J_{j,j'} = J_{j',j}$. The spectral theorem tells us that $\\boldsymbol{J}$ has real eigenvalues $\\lambda$ and is diagonalisable by an orthogonal matrix (orthogonally diagonalizable).\n\"\n\n# ╔═╡ 2e1aa912-e7bd-4ffd-8060-05556a77bc8b\nmd\"## Definition of the Covariance Matrix\n\nThe covariance matrix $\\boldsymbol{C}$ for a cloud of points in Euclidean space centered around the mean is given by\n\n$$\\begin{equation*}\n \\boldsymbol{C} = \n \\begin{bmatrix}\n C_{1,1} & \\cdots & C_{1,n} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n C_{n,1} & \\cdots & C_{n,n} \n \\end{bmatrix}\n\\end{equation*}$$\n\nIts components are defined by Eq. (2) as\n\n$$\\begin{equation}\n C_{j,j'} =\n \\frac{1}{N}\n \\sum_{i=1}^{N}\n \\left( x_j^{(i)} x_{j'}^{(i)} \\right)\n\\end{equation}$$\"\n\n# ╔═╡ 8c8167ca-8679-4d2a-b15b-4c8f32cfb8b6\nmd\"## Solving the Eigenvalue Problem\n\nThe principal axes of the PCA and moment of inertia can be determined by rotating the data points in space. More precisely, the principal components and axes are calculated by solving an eigenvalue problem.\n\nA real symmetric matrix (like $\\boldsymbol{C}$ and $\\boldsymbol{J}$ ) has the eigendecomposition into the product of a rotation matrix $\\boldsymbol{R}$ and a diagonal matrix $\\boldsymbol{\\Lambda}$\n\n$$\\begin{equation*}\n \\boldsymbol{\\Lambda} = \n \\begin{bmatrix}\n \\lambda_{1} & \\cdots & 0 \\\\\n \\vdots & \\ddots & \\vdots \\\\\n 0 & \\cdots & \\lambda_{n} \n \\end{bmatrix}\n\\end{equation*}$$\n\ngiven by\n\n$\\boldsymbol{J} = \\boldsymbol{R} \\boldsymbol{\\Lambda } \\boldsymbol{R}^T$\n\nThe columns of the rotation matrix $\\boldsymbol{R}$ define the directions of the principal axes, and the constants $\\lambda_{1}, \\dots, \\lambda_{n}$ are the diagonal elements of the matrix $\\boldsymbol{\\Lambda}$ and\nare called the principal moments.\n\nThe structure of the matrices $\\boldsymbol{J}$ and $\\boldsymbol{C}$ is the same except for the sign of the off-diagonal elements. We will see in the following that the eigenvectors will be the same for $\\boldsymbol{C}$ and $\\boldsymbol{J}$. In addition, we will see how the eigenvalues $\\boldsymbol{\\Lambda}$ of $\\boldsymbol{C}$ relate to the eigenvalues of $\\boldsymbol{J}$. \n\"\n\n# ╔═╡ 1d1d7052-977e-4180-af75-f1171f1492b2\nmd\"## Showing the Equality of the Eigenvectors of $\\boldsymbol{C}$ and $\\boldsymbol{J}$\n\nLet's rewrite the moment of inertia matrix $\\boldsymbol{J}$ defined by Eq. (1) in terms of the covariance matrix $\\boldsymbol{C}$ in Eq. (2).\n\n$$\\begin{equation}\n \\boldsymbol{J} = tr(\\boldsymbol{C})\\boldsymbol{I} - \\boldsymbol{C} \\qquad \\text{Eq. (3)}\n\\end{equation}$$\n\nwhere $\\boldsymbol{I}$ is the identity matrix.\n\n$$\\begin{align}\nJ_{j,j'} & = tr(\\boldsymbol{C})\\boldsymbol{I_{j,j}} - \\boldsymbol{C_{j,j'}} \\\\\n\n& = \\frac{1}{N} \\sum_{i=1}^{N} \\sum_{j=1}^{n} \\left( x_j^{(i)} x_{j}^{(i)} \\right) - \\frac{1}{N} \\sum_{i=1}^{N} \\left( x_j^{(i)} x_{j'}^{(i)} \\right)\\\\\n\n& = \\frac{1}{N} \\sum_{i=1}^{N} \\left( \\sum_{j=1}^{n} x_j^{(i)} x_{j}^{(i)} - x_j^{(i)} x_{j'}^{(i)} \\right)\\\\\n\n& = \\frac{1}{N} \\sum_{i=1}^{N} \\left( ||\\vec{x}^{(i)}||^2 \\delta_{j,j'} - x_j^{(i)} x_{j'}^{(i)} \\right) \\\\\n\n& \\text{multiply the expression under the sum by } 1 \\\\\n\n& = \\frac{1}{N} \\sum_{i=1}^{N} 1 \\cdot \\left( ||\\vec{x}^{(i)}||^2 \\delta_{j,j'} - x_j^{(i)} x_{j'}^{(i)} \\right) \\\\\n\n& \\text{assume each data point has mass } m_{i} = 1 \\\\\n\n\n& = \\frac{1}{\\sum_{i}^{N} m_i} \\sum_{i=1}^{N} m_{i} \\left( ||\\vec{x}^{(i)}||^2 \\delta_{j,j'} - x_j^{(i)} x_{j'}^{(i)} \\right) \\\\\n\n& = \\frac{1}{M} \\sum_{i=1}^{N} m_{i} \\left( ||\\vec{x}^{(i)}||^2 \\delta_{j,j'} - x_j^{(i)} x_{j'}^{(i)} \\right) \\\\\n\n\\end{align}$$\n\"\n\n# ╔═╡ 48d9a159-5725-4c8d-8a3f-334c0f6e24cf\nmd\"To obtain the eigenvectors and eigenvalues of $\\boldsymbol{C}$, we solve the eigenvalue problem by decomposing $\\boldsymbol{C}$ into the product of a rotation matrix $\\boldsymbol{R}$ and a diagonal matrix $\\boldsymbol{\\Lambda}_{cov}$\n\n$\\boldsymbol{C} = \\boldsymbol{R} \\boldsymbol{\\Lambda}_{cov} \\boldsymbol{R}^T,$\n\nwhere $\\boldsymbol{R}$ is composed of the eigenvectors $\\vec{v}^{j}$. In the case of a $3$-dimensional space\n\n$\\boldsymbol{R} = \\left[ \\vec{v}^{1} \\; \\vec{v}^{2} \\; \\vec{v}^{3} \\right]$\n\nand\n\n$$\\begin{equation*}\n \\boldsymbol{\\boldsymbol{\\Lambda}_{cov}} = \n \\begin{bmatrix}\n \\lambda^{1}_{cov} & 0 & 0 \\\\\n 0 & \\lambda^{2}_{cov} & 0 \\\\\n 0 & 0 & \\lambda^{3}_{cov}\n \\end{bmatrix}\n\\end{equation*}$$\n\n\nThe $j^{th}$ eigenvector $\\vec{v}^{j}$ and $j^{th}$ eigenvalue $\\lambda^{j}_{cov}$ of the covariance matrix $\\boldsymbol{C}$ are given by\n\n$$\\begin{equation}\n \\boldsymbol{C}\\lambda^{j}_{cov} = \\lambda^{j}_{cov} \\vec{v}^{j}.\n\\end{equation}$$\n\nIn the following, we will drop the index $j$ and rewrite the formula from above as $\\boldsymbol{C}\\lambda_{cov} = \\lambda_{cov} \\vec{v}$. Multiplying equation Eq. (3) by the eigenvector $\\vec{v}$ from the right side, we find Eq. (4).\n\n$$\\begin{align}\n \\boldsymbol{J}\\vec{v} &= \\left( tr(\\boldsymbol{C})\\boldsymbol{I} - \\boldsymbol{C} \\right) \\vec{v} \\\\\n &= tr(\\boldsymbol{C})\\boldsymbol{I}\\vec{v} - \\boldsymbol{C} \\vec{v} \\\\\n\n &= tr(\\boldsymbol{C})\\boldsymbol{I}\\vec{v} - \\lambda_{cov} \\vec{v} \\\\\n\n &= tr(\\boldsymbol{C})\\vec{v} - \\lambda_{cov} \\vec{v} \\\\\n\n &= \\left( tr(\\boldsymbol{C}) - \\lambda_{cov} \\right) \\vec{v} \\\\\n\n &= \\lambda_{J} \\vec{v} \\qquad \\qquad \\qquad \\qquad \\text{Eq. (4)}\\\\\n\\end{align}$$\n\nEq. (4) implies that $\\boldsymbol{C}$ and $\\boldsymbol{J}$ have the same eigenvectors $\\vec{v}$.\n\"\n\n# ╔═╡ 9e8ce452-78bc-4f1c-95fa-37be90c1006c\nmd\"## Calculating the Eigenvalues of the Moment of Inertia Matrix\n\nWhile the eigenvectors of $\\boldsymbol{C}$ and $\\boldsymbol{J}$ are the same, the eigenvalues are not the same. To relate them we will need to note that\n\n$tr\\left( \\boldsymbol{C} \\right)= tr \\left( \\boldsymbol{R} \\boldsymbol{\\Lambda}_{cov} \\boldsymbol{R}^T \\right) = \\sum_{j=1}^{n} \\lambda^{j}_{cov} \\qquad \\quad \\text{Eq. (5)}$\n\nwhere we have used that the trace of a matrix is invariant under cyclic permutations, so\n\n$$\n\\begin{align}\n tr \\left( \\boldsymbol{R} \\boldsymbol{\\Lambda}_{cov} \\boldsymbol{R}^T \\right) &= \n tr \\left( \\boldsymbol{R}^T \\boldsymbol{R} \\boldsymbol{\\Lambda}_{cov} \\right)\\\\\n &= tr \\left( \\boldsymbol{I} \\boldsymbol{\\Lambda}_{cov} \\right) \\\\\n &= \\sum_{j=1}^{n} \\lambda^{j}_{cov}\n\\end{align}$$\n\n\"\n\n# ╔═╡ d3eb6336-9e2e-4503-8fa1-f49385b45b40\nmd\"Using Eq. (4) and Eq. (5), we can write the eigenvalues $\\lambda_{J}$ of the moment of inertia $\\boldsymbol{J}$ (Eq. (1)) as\n\n$$\\begin{align}\n \\lambda^{k}_{J} &= tr(\\boldsymbol{C}) - \\lambda^{k}_{cov} \\\\\n &= \\sum_{j=1}^{n} \\lambda^{j}_{cov} - \\lambda^{k}_{cov} \\qquad \\text{Eq. (6)}\n \\end{align}$$\n\nWe see that the $k^{th}$ eigenvalue $\\lambda^{k}_{J}$ can be expressed in terms of the eigenvalues $\\lambda_{cov}$.\n\"\n\n# ╔═╡ a3da380d-b7db-4b7d-a348-bf3ef9b59e04\nmd\"\"\"Physically we can gain an intuitive understanding for the relation between these two sets of eigenvalues by considering that the covariance eigenvalue along e.g. axis $\\vec{v}^{(1)}$ are determined by the component of the data points in $(1)$ direction, while the moment of inertia around for a rotation around the axis $(1)$ is determined by the euclidean distance of the data points from that axis.\"\"\"\n\n# ╔═╡ 96b8d373-3ce1-460a-a70c-3f9e026bb783\nmd\"## Calculating Eigenvalues and Eigenvectors Using the Above Dataset $X$\"\n\n# ╔═╡ 42c9d5b9-7939-4f1f-b148-c9347acb3b3d\nmd\"Next, we calculate covariance matrix elements, eigenvalues, and eigenvectors. The covariance matrix $\\boldsymbol{C}$ of the dataset $X$ is\"\n\n# ╔═╡ cbdf4dd7-5710-47d9-91aa-8dbae9712e26\nC = cov(pts; corrected=false)\n\n# ╔═╡ be55c1dd-0b85-4192-bbbb-2f397fae1d49\nmd\"For the cavariance matrix $\\boldsymbol{C}$, we find the eigenvalues $\\lambda_{cov}$ and the eigenvectors $v_{cov}$\"\n\n# ╔═╡ d283412a-a3a3-458c-aa8b-63e1c4f3bf15\nλ_cov, v_cov = eigen(C; sortby=-)\n\n# ╔═╡ 4b17638d-1465-4b21-a2a3-61f94cbd0bca\nmd\"The eigenvalues $\\lambda_{cov}$ are\"\n\n# ╔═╡ 19378277-61f3-4d59-b840-452cfdf58a3b\nλ_cov\n\n# ╔═╡ 30513f7e-d866-4078-95e2-ab05bfbcf547\nmd\"and the eigenvectors $\\vec{v}_{cov}$ are\"\n\n# ╔═╡ 32191a9f-14ed-4bf6-a8ea-6586b117e848\nv_cov[:,1], v_cov[:,2], v_cov[:,3]\n\n# ╔═╡ 11c37b73-ddfd-41cf-98f4-c461b265d1bf\nmd\"Using Eq. (6), we can calculate the eigenvalues $\\lambda_{J}$ of $\\boldsymbol{J}$\"\n\n# ╔═╡ b20fed20-a678-450b-a689-a51baa337153\nsum(λ_cov)-λ_cov[1], sum(λ_cov)-λ_cov[2], sum(λ_cov)-λ_cov[3]\n\n# ╔═╡ abbd82f5-31cc-4f45-9c7c-f401796174cb\nmd\"## Calculating the Eigenvalues $\\lambda_{J}$ and Eigenvectors $\\vec{v}$ of $\\boldsymbol{J}$\"\n\n# ╔═╡ eaa487b9-2728-49f8-8b5e-003926f4d544\nδ(i, j) = i == j ? 1 : 0\n\n# ╔═╡ 3e6282ce-f3ba-4591-976d-0c37d14755c7\n\"\"\"\n\tcalcJ(pts::AbstractMatrix)\n\nCalculates the moment of inertia matrix ``\\\\boldsymbol{J}`` given a set of points `pts`\n\"\"\"\nfunction calcJ(pts::AbstractMatrix)\n\tN, n = size(pts)\n\tJ = zeros(Float64, n, n)\n\t\n\tfor j1 in 1:n\n\t\tfor j2 in j1:n\n\t\t\tJ[j1, j2] = sum(1:N) do i\n\t\t\t\tnorm(pts[i, :])^2 * δ(j1, j2) - pts[i, j1] * pts[i, j2]\n\t\t\tend\n\n\t\t\tJ[j2, j1] = J[j1, j2]\n\t\tend\n\tend\n\n\treturn J ./ N\nend\n\n# ╔═╡ 6b58bfb2-be89-4cd2-9c67-85ca95d02267\nJ = calcJ(pts)\n\n# ╔═╡ 51f20ae2-e0b6-410d-8783-689f6de9a025\nmd\"For the moment of inertia matrix $\\boldsymbol{J}$, we find the eigenvalues $\\lambda$ and the eigenvectors $v$\"\n\n# ╔═╡ 4b47a6af-3c41-4575-a576-09145c535e22\nλ, v = eigen(J; sortby=-)\n\n# ╔═╡ 1ad77ac0-bd4c-407f-858f-6e435085d4d8\nmd\"In both interpretations of the cloud of data points -- first as a cloud of points centered around the mean and second as a rigid body rotating around the center of mass -- we obtain the same eigenvectors.\"\n\n# ╔═╡ e54b7b64-ebbe-426d-92fa-4959eebba826\nbegin\n p1 = plot(title = \"Dataset and the (Scaled up) Eigenvectors\",\n\t xlims = (-3 * Rmax, 3 * Rmax),\n\t ylims = (-3 * Rmax, 3 * Rmax) )\n\t\n scatter!(pts[:,1], pts[:,2],\n\t\t markershape = :circle,\n\t\t markersize = 2,\n\t\t markercolor = \"#0044aa\",\n\t\t markerstrokecolor = \"#0044aa\",\n\t\t framestyle = :zerolines,\n\t\t xlabel = L\"\\vec{e}_1\",\n\t ylabel = L\"\\vec{e}_2\",\n\t\t aspect_ratio = :equal,\n\t\t label=\"Points\")\n\n\tplot!( [0, v_cov[1,1] * sqrt(λ_cov[1])*2],\n\t\t [0, v_cov[2,1] * sqrt(λ_cov[1])*2],\n\t\t arrow=true,color=\"#6a0000\",linewidth=3,label=\"\")\n\t\n\tplot!( [0, v_cov[1,2] * sqrt(λ_cov[2])*2],\n\t\t [0, v_cov[2,2] * sqrt(λ_cov[2])*2],\n\t\t arrow=true, color=\"#01493e\",linewidth=3,label=\"\")\nend\n\n# ╔═╡ b6c98fa0-8b40-4654-aa0c-b0204ed9e291\nmd\"*Figure 4. Cloud of randomly distributed data points $X$. Overlayed are the scaled eigenvectors (shown as red and green arrows).*\n\n---\n\"\n\n# ╔═╡ 3c8c50dc-7b25-48d3-84f2-c2e1a9fb85dd\nmd\"Next, we use the eigenvectors and eigenvalues to rotate the data and represent it in the new basis $v = v_{cov} = [\\vec{u}_1, \\vec{u}_2, \\vec{u}_3]$.\"\n\n# ╔═╡ fd8ce478-87a8-48aa-b8d1-df80df93254b\npts_rot = pts * v_cov\n\n# ╔═╡ 8d9840d1-1e71-403b-8935-c6aeca8a0ee0\nmd\"\n---\n\"\n\n# ╔═╡ 2be79841-4bfd-47d1-9c50-fd1e33a766c5\nbegin\n p2 = plot(title = \"Rotated Dataset\",\n\t xlims = (-3 * Rmax, 3 * Rmax),\n\t ylims = (-3 * Rmax, 3 * Rmax) )\n\t\n scatter!(pts_rot[:,1], pts_rot[:,2],\n\t\t markershape = :circle,\n\t\t markersize = 2,\n\t\t markercolor = :steelblue,\n\t\t markerstrokecolor = :steelblue,\n\t\t xlabel = L\"\\vec{u}_1\",\n\t ylabel = L\"\\vec{u}_2\",\n\t\t aspect_ratio = :equal,\n\t\t label=\"Points\")\n\n\tplot!( [0, v_cov[1,1]*10.0],\n\t\t [0, v_cov[2,1]*10.0],\n\t\t arrow=true,color=:grey,linewidth=3,label=\"\")\n\t\n\tplot!( [0, v_cov[1,2]*10.0],\n\t\t [0, v_cov[2,2]*10.0],\n\t\t arrow=true,color=:grey,linewidth=3,label=\"\")\nend\n\n# ╔═╡ 9a90438b-67c2-4024-bf45-696d60b718df\nmd\"*Figure 5. Cloud of randomly distributed data points $X$ represented in the basis spanned by the vectors $[\\vec{u}_1 \\; \\vec{u}_2 \\; \\vec{u}_3]$. The grey arrows indicate the old basis vectors $\\vec{e}_1$ and $\\vec{e}_2$ in this new basis.*\n\n---\n\"\n\n# ╔═╡ 14bf83f9-9a19-42e0-ba41-c6a16c58a89c\nmd\"\"\"## Final Thoughts\nIn machine learning and data science, PCA is used for two reasons.\nFirst, the accuracy and numeric stability of some Machine Learning algorithms are sensitive to correlated input data. In particular, Machine Learning algorithms that perform an inversion of the covariance matrix may experience the singularity problem -- Gaussian Mixture Models come to mind. A different example is the application of random forest algorithms to detect interactions between different features, where large correlations can mask these interactions. Performing a PCA first allows us to tease out the effect of correlations, which can improve feature importance analysis.\n\n\nSecond, PCA is used to reduce the dimensionality of the dataset, e.g. for data compression. In our example, we used a 3-dimensional dataset $X$, but the $\\vec{e}_3$ component did not carry any information (by construction). We could have used a PCA to justify dropping the third dimension since a PCA would have shown that the variance is minimal in the $\\vec{e}_3$ direction. Using the projection of a higher dimensional dataset onto a lower-dimensional space like this is a powerful tool to handle high-dimensional data and deal with the [curse of dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality).\n\nBoth applications of PCA have been extensively discussed. See recommendations for further reading below.\n\n\"\"\"\n\n# ╔═╡ 3b409a8a-b03a-41e6-9e14-b9f73af1847b\nmd\"## Further Reading\n 1. **Bauckhage, Christian & Dong, Tiansi** A Tutorial on Principal Component Analysis Part 1: Motivation (2018).\n 2. **Bauckhage, Christian & Dong, Tiansi** A Tutorial on Principal Component Analysis Part 2: Principal Axis via Moment of Inertia (2018).\n 3. **Hong, Liang** [The Proof of Equivalence between \\\" Moment of Inertia Based Method\\\" and \\\"PCA Based Method\\\" in LOS Detection](10.4028/www.scientific.net/AMR.945-949.2071) Advanced Materials Research. 945-949. 2071-2074 (2014).\n 4. **Murphy, Kevin P.** [Machine Learning A Probabilistic Perspective](https://isbnsearch.org/isbn/9780262018029), The MIT Press, Chapter 12.2 (2012)\n 5. **Marsland, Stephen** [Machine Learning An Algorithmic Perspective](https://isbnsearch.org/isbn/9781466583283), 2nd Edition, Chapman and Hall/CRC (2014)\n 6. [Badr, Will](https://medium.com/@will.badr) [Why Feature Correlation Matters …. A Lot!](https://towardsdatascience.com/why-feature-correlation-matters-a-lot-847e8ba439c4) Blog TowardsDataScience (2019).\"\n\n# ╔═╡ 00000000-0000-0000-0000-000000000001\nPLUTO_PROJECT_TOML_CONTENTS = \"\"\"\n[deps]\nLaTeXStrings = \"b964fa9f-0449-5b57-a5c2-d3ea65f4040f\"\nLinearAlgebra = \"37e2e46d-f89d-539d-b4ee-838fcccc9c8e\"\nMultivariateStats = \"6f286f6a-111f-5878-ab1e-185364afe411\"\nPlots = \"91a5bcdd-55d7-5caf-9e0b-520d859cae80\"\nPlutoUI = \"7f904dfe-b85e-4ff6-b463-dae2292396a8\"\nPrintf = \"de0858da-6303-5e67-8744-51eddeeeb8d7\"\nRandom = \"9a3f8284-a2c9-5f02-9a11-845980a1fd5c\"\nStatistics = \"10745b16-79ce-11e8-11f9-7d13ad32a3b2\"\n\n[compat]\nLaTeXStrings = \"~1.3.0\"\nMultivariateStats = \"~0.8.0\"\nPlots = \"~1.23.6\"\nPlutoUI = \"~0.7.19\"\n\"\"\"\n\n# ╔═╡ 00000000-0000-0000-0000-000000000002\nPLUTO_MANIFEST_TOML_CONTENTS = \"\"\"\n# This file is machine-generated - editing it directly is not advised\n\n[[AbstractPlutoDingetjes]]\ndeps = [\"Pkg\"]\ngit-tree-sha1 = \"0bc60e3006ad95b4bb7497698dd7c6d649b9bc06\"\nuuid = \"6e696c72-6542-2067-7265-42206c756150\"\nversion = \"1.1.1\"\n\n[[Adapt]]\ndeps = [\"LinearAlgebra\"]\ngit-tree-sha1 = \"84918055d15b3114ede17ac6a7182f68870c16f7\"\nuuid = \"79e6a3ab-5dfb-504d-930d-738a2a938a0e\"\nversion = \"3.3.1\"\n\n[[ArgTools]]\nuuid = \"0dad84c5-d112-42e6-8d28-ef12dabb789f\"\n\n[[Arpack]]\ndeps = [\"Arpack_jll\", \"Libdl\", \"LinearAlgebra\"]\ngit-tree-sha1 = \"2ff92b71ba1747c5fdd541f8fc87736d82f40ec9\"\nuuid = \"7d9fca2a-8960-54d3-9f78-7d1dccf2cb97\"\nversion = \"0.4.0\"\n\n[[Arpack_jll]]\ndeps = [\"Libdl\", \"OpenBLAS_jll\", \"Pkg\"]\ngit-tree-sha1 = \"e214a9b9bd1b4e1b4f15b22c0994862b66af7ff7\"\nuuid = \"68821587-b530-5797-8361-c406ea357684\"\nversion = \"3.5.0+3\"\n\n[[Artifacts]]\nuuid = \"56f22d72-fd6d-98f1-02f0-08ddc0907c33\"\n\n[[Base64]]\nuuid = \"2a0f44e3-6c83-55bd-87e4-b1978d98bd5f\"\n\n[[Bzip2_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"19a35467a82e236ff51bc17a3a44b69ef35185a2\"\nuuid = \"6e34b625-4abd-537c-b88f-471c36dfa7a0\"\nversion = \"1.0.8+0\"\n\n[[Cairo_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"Fontconfig_jll\", \"FreeType2_jll\", \"Glib_jll\", \"JLLWrappers\", \"LZO_jll\", \"Libdl\", \"Pixman_jll\", \"Pkg\", \"Xorg_libXext_jll\", \"Xorg_libXrender_jll\", \"Zlib_jll\", \"libpng_jll\"]\ngit-tree-sha1 = \"4b859a208b2397a7a623a03449e4636bdb17bcf2\"\nuuid = \"83423d85-b0ee-5818-9007-b63ccbeb887a\"\nversion = \"1.16.1+1\"\n\n[[ChainRulesCore]]\ndeps = [\"Compat\", \"LinearAlgebra\", \"SparseArrays\"]\ngit-tree-sha1 = \"f885e7e7c124f8c92650d61b9477b9ac2ee607dd\"\nuuid = \"d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4\"\nversion = \"1.11.1\"\n\n[[ChangesOfVariables]]\ndeps = [\"LinearAlgebra\", \"Test\"]\ngit-tree-sha1 = \"9a1d594397670492219635b35a3d830b04730d62\"\nuuid = \"9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0\"\nversion = \"0.1.1\"\n\n[[ColorSchemes]]\ndeps = [\"ColorTypes\", \"Colors\", \"FixedPointNumbers\", \"Random\"]\ngit-tree-sha1 = \"a851fec56cb73cfdf43762999ec72eff5b86882a\"\nuuid = \"35d6a980-a343-548e-a6ea-1d62b119f2f4\"\nversion = \"3.15.0\"\n\n[[ColorTypes]]\ndeps = [\"FixedPointNumbers\", \"Random\"]\ngit-tree-sha1 = \"024fe24d83e4a5bf5fc80501a314ce0d1aa35597\"\nuuid = \"3da002f7-5984-5a60-b8a6-cbb66c0b333f\"\nversion = \"0.11.0\"\n\n[[Colors]]\ndeps = [\"ColorTypes\", \"FixedPointNumbers\", \"Reexport\"]\ngit-tree-sha1 = \"417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40\"\nuuid = \"5ae59095-9a9b-59fe-a467-6f913c188581\"\nversion = \"0.12.8\"\n\n[[Compat]]\ndeps = [\"Base64\", \"Dates\", \"DelimitedFiles\", \"Distributed\", \"InteractiveUtils\", \"LibGit2\", \"Libdl\", \"LinearAlgebra\", \"Markdown\", \"Mmap\", \"Pkg\", \"Printf\", \"REPL\", \"Random\", \"SHA\", \"Serialization\", \"SharedArrays\", \"Sockets\", \"SparseArrays\", \"Statistics\", \"Test\", \"UUIDs\", \"Unicode\"]\ngit-tree-sha1 = \"dce3e3fea680869eaa0b774b2e8343e9ff442313\"\nuuid = \"34da2185-b29b-5c13-b0c7-acf172513d20\"\nversion = \"3.40.0\"\n\n[[CompilerSupportLibraries_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"e66e0078-7015-5450-92f7-15fbd957f2ae\"\n\n[[Contour]]\ndeps = [\"StaticArrays\"]\ngit-tree-sha1 = \"9f02045d934dc030edad45944ea80dbd1f0ebea7\"\nuuid = \"d38c429a-6771-53c6-b99e-75d170b6e991\"\nversion = \"0.5.7\"\n\n[[DataAPI]]\ngit-tree-sha1 = \"cc70b17275652eb47bc9e5f81635981f13cea5c8\"\nuuid = \"9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a\"\nversion = \"1.9.0\"\n\n[[DataStructures]]\ndeps = [\"Compat\", \"InteractiveUtils\", \"OrderedCollections\"]\ngit-tree-sha1 = \"7d9d316f04214f7efdbb6398d545446e246eff02\"\nuuid = \"864edb3b-99cc-5e75-8d2d-829cb0a9cfe8\"\nversion = \"0.18.10\"\n\n[[DataValueInterfaces]]\ngit-tree-sha1 = \"bfc1187b79289637fa0ef6d4436ebdfe6905cbd6\"\nuuid = \"e2d170a0-9d28-54be-80f0-106bbe20a464\"\nversion = \"1.0.0\"\n\n[[Dates]]\ndeps = [\"Printf\"]\nuuid = \"ade2ca70-3891-5945-98fb-dc099432e06a\"\n\n[[DelimitedFiles]]\ndeps = [\"Mmap\"]\nuuid = \"8bb1440f-4735-579b-a4ab-409b98df4dab\"\n\n[[Distributed]]\ndeps = [\"Random\", \"Serialization\", \"Sockets\"]\nuuid = \"8ba89e20-285c-5b6f-9357-94700520ee1b\"\n\n[[DocStringExtensions]]\ndeps = [\"LibGit2\"]\ngit-tree-sha1 = \"b19534d1895d702889b219c382a6e18010797f0b\"\nuuid = \"ffbed154-4ef7-542d-bbb7-c09d3a79fcae\"\nversion = \"0.8.6\"\n\n[[Downloads]]\ndeps = [\"ArgTools\", \"LibCURL\", \"NetworkOptions\"]\nuuid = \"f43a241f-c20a-4ad4-852c-f6b1247861c6\"\n\n[[EarCut_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"3f3a2501fa7236e9b911e0f7a588c657e822bb6d\"\nuuid = \"5ae413db-bbd1-5e63-b57d-d24a61df00f5\"\nversion = \"2.2.3+0\"\n\n[[Expat_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"b3bfd02e98aedfa5cf885665493c5598c350cd2f\"\nuuid = \"2e619515-83b5-522b-bb60-26c02a35a201\"\nversion = \"2.2.10+0\"\n\n[[FFMPEG]]\ndeps = [\"FFMPEG_jll\"]\ngit-tree-sha1 = \"b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8\"\nuuid = \"c87230d0-a227-11e9-1b43-d7ebe4e7570a\"\nversion = \"0.4.1\"\n\n[[FFMPEG_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"FreeType2_jll\", \"FriBidi_jll\", \"JLLWrappers\", \"LAME_jll\", \"Libdl\", \"Ogg_jll\", \"OpenSSL_jll\", \"Opus_jll\", \"Pkg\", \"Zlib_jll\", \"libass_jll\", \"libfdk_aac_jll\", \"libvorbis_jll\", \"x264_jll\", \"x265_jll\"]\ngit-tree-sha1 = \"d8a578692e3077ac998b50c0217dfd67f21d1e5f\"\nuuid = \"b22a6f82-2f65-5046-a5b2-351ab43fb4e5\"\nversion = \"4.4.0+0\"\n\n[[FixedPointNumbers]]\ndeps = [\"Statistics\"]\ngit-tree-sha1 = \"335bfdceacc84c5cdf16aadc768aa5ddfc5383cc\"\nuuid = \"53c48c17-4a7d-5ca2-90c5-79b7896eea93\"\nversion = \"0.8.4\"\n\n[[Fontconfig_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"Expat_jll\", \"FreeType2_jll\", \"JLLWrappers\", \"Libdl\", \"Libuuid_jll\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"21efd19106a55620a188615da6d3d06cd7f6ee03\"\nuuid = \"a3f928ae-7b40-5064-980b-68af3947d34b\"\nversion = \"2.13.93+0\"\n\n[[Formatting]]\ndeps = [\"Printf\"]\ngit-tree-sha1 = \"8339d61043228fdd3eb658d86c926cb282ae72a8\"\nuuid = \"59287772-0a20-5a39-b81b-1366585eb4c0\"\nversion = \"0.4.2\"\n\n[[FreeType2_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"87eb71354d8ec1a96d4a7636bd57a7347dde3ef9\"\nuuid = \"d7e528f0-a631-5988-bf34-fe36492bcfd7\"\nversion = \"2.10.4+0\"\n\n[[FriBidi_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91\"\nuuid = \"559328eb-81f9-559d-9380-de523a88c83c\"\nversion = \"1.0.10+0\"\n\n[[GLFW_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Libglvnd_jll\", \"Pkg\", \"Xorg_libXcursor_jll\", \"Xorg_libXi_jll\", \"Xorg_libXinerama_jll\", \"Xorg_libXrandr_jll\"]\ngit-tree-sha1 = \"0c603255764a1fa0b61752d2bec14cfbd18f7fe8\"\nuuid = \"0656b61e-2033-5cc2-a64a-77c0f6c09b89\"\nversion = \"3.3.5+1\"\n\n[[GR]]\ndeps = [\"Base64\", \"DelimitedFiles\", \"GR_jll\", \"HTTP\", \"JSON\", \"Libdl\", \"LinearAlgebra\", \"Pkg\", \"Printf\", \"Random\", \"Serialization\", \"Sockets\", \"Test\", \"UUIDs\"]\ngit-tree-sha1 = \"30f2b340c2fff8410d89bfcdc9c0a6dd661ac5f7\"\nuuid = \"28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71\"\nversion = \"0.62.1\"\n\n[[GR_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"Cairo_jll\", \"FFMPEG_jll\", \"Fontconfig_jll\", \"GLFW_jll\", \"JLLWrappers\", \"JpegTurbo_jll\", \"Libdl\", \"Libtiff_jll\", \"Pixman_jll\", \"Pkg\", \"Qt5Base_jll\", \"Zlib_jll\", \"libpng_jll\"]\ngit-tree-sha1 = \"fd75fa3a2080109a2c0ec9864a6e14c60cca3866\"\nuuid = \"d2c73de3-f751-5644-a686-071e5b155ba9\"\nversion = \"0.62.0+0\"\n\n[[GeometryBasics]]\ndeps = [\"EarCut_jll\", \"IterTools\", \"LinearAlgebra\", \"StaticArrays\", \"StructArrays\", \"Tables\"]\ngit-tree-sha1 = \"58bcdf5ebc057b085e58d95c138725628dd7453c\"\nuuid = \"5c1252a2-5f33-56bf-86c9-59e7332b4326\"\nversion = \"0.4.1\"\n\n[[Gettext_jll]]\ndeps = [\"Artifacts\", \"CompilerSupportLibraries_jll\", \"JLLWrappers\", \"Libdl\", \"Libiconv_jll\", \"Pkg\", \"XML2_jll\"]\ngit-tree-sha1 = \"9b02998aba7bf074d14de89f9d37ca24a1a0b046\"\nuuid = \"78b55507-aeef-58d4-861c-77aaff3498b1\"\nversion = \"0.21.0+0\"\n\n[[Glib_jll]]\ndeps = [\"Artifacts\", \"Gettext_jll\", \"JLLWrappers\", \"Libdl\", \"Libffi_jll\", \"Libiconv_jll\", \"Libmount_jll\", \"PCRE_jll\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"a32d672ac2c967f3deb8a81d828afc739c838a06\"\nuuid = \"7746bdde-850d-59dc-9ae8-88ece973131d\"\nversion = \"2.68.3+2\"\n\n[[Graphite2_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"344bf40dcab1073aca04aa0df4fb092f920e4011\"\nuuid = \"3b182d85-2403-5c21-9c21-1e1f0cc25472\"\nversion = \"1.3.14+0\"\n\n[[Grisu]]\ngit-tree-sha1 = \"53bb909d1151e57e2484c3d1b53e19552b887fb2\"\nuuid = \"42e2da0e-8278-4e71-bc24-59509adca0fe\"\nversion = \"1.0.2\"\n\n[[HTTP]]\ndeps = [\"Base64\", \"Dates\", \"IniFile\", \"Logging\", \"MbedTLS\", \"NetworkOptions\", \"Sockets\", \"URIs\"]\ngit-tree-sha1 = \"14eece7a3308b4d8be910e265c724a6ba51a9798\"\nuuid = \"cd3eb016-35fb-5094-929b-558a96fad6f3\"\nversion = \"0.9.16\"\n\n[[HarfBuzz_jll]]\ndeps = [\"Artifacts\", \"Cairo_jll\", \"Fontconfig_jll\", \"FreeType2_jll\", \"Glib_jll\", \"Graphite2_jll\", \"JLLWrappers\", \"Libdl\", \"Libffi_jll\", \"Pkg\"]\ngit-tree-sha1 = \"129acf094d168394e80ee1dc4bc06ec835e510a3\"\nuuid = \"2e76f6c2-a576-52d4-95c1-20adfe4de566\"\nversion = \"2.8.1+1\"\n\n[[Hyperscript]]\ndeps = [\"Test\"]\ngit-tree-sha1 = \"8d511d5b81240fc8e6802386302675bdf47737b9\"\nuuid = \"47d2ed2b-36de-50cf-bf87-49c2cf4b8b91\"\nversion = \"0.0.4\"\n\n[[HypertextLiteral]]\ngit-tree-sha1 = \"2b078b5a615c6c0396c77810d92ee8c6f470d238\"\nuuid = \"ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2\"\nversion = \"0.9.3\"\n\n[[IOCapture]]\ndeps = [\"Logging\", \"Random\"]\ngit-tree-sha1 = \"f7be53659ab06ddc986428d3a9dcc95f6fa6705a\"\nuuid = \"b5f81e59-6552-4d32-b1f0-c071b021bf89\"\nversion = \"0.2.2\"\n\n[[IniFile]]\ndeps = [\"Test\"]\ngit-tree-sha1 = \"098e4d2c533924c921f9f9847274f2ad89e018b8\"\nuuid = \"83e8ac13-25f8-5344-8a64-a9f2b223428f\"\nversion = \"0.5.0\"\n\n[[InteractiveUtils]]\ndeps = [\"Markdown\"]\nuuid = \"b77e0a4c-d291-57a0-90e8-8db25a27a240\"\n\n[[InverseFunctions]]\ndeps = [\"Test\"]\ngit-tree-sha1 = \"a7254c0acd8e62f1ac75ad24d5db43f5f19f3c65\"\nuuid = \"3587e190-3f89-42d0-90ee-14403ec27112\"\nversion = \"0.1.2\"\n\n[[IrrationalConstants]]\ngit-tree-sha1 = \"7fd44fd4ff43fc60815f8e764c0f352b83c49151\"\nuuid = \"92d709cd-6900-40b7-9082-c6be49f344b6\"\nversion = \"0.1.1\"\n\n[[IterTools]]\ngit-tree-sha1 = \"05110a2ab1fc5f932622ffea2a003221f4782c18\"\nuuid = \"c8e1da08-722c-5040-9ed9-7db0dc04731e\"\nversion = \"1.3.0\"\n\n[[IteratorInterfaceExtensions]]\ngit-tree-sha1 = \"a3f24677c21f5bbe9d2a714f95dcd58337fb2856\"\nuuid = \"82899510-4779-5014-852e-03e436cf321d\"\nversion = \"1.0.0\"\n\n[[JLLWrappers]]\ndeps = [\"Preferences\"]\ngit-tree-sha1 = \"642a199af8b68253517b80bd3bfd17eb4e84df6e\"\nuuid = \"692b3bcd-3c85-4b1f-b108-f13ce0eb3210\"\nversion = \"1.3.0\"\n\n[[JSON]]\ndeps = [\"Dates\", \"Mmap\", \"Parsers\", \"Unicode\"]\ngit-tree-sha1 = \"8076680b162ada2a031f707ac7b4953e30667a37\"\nuuid = \"682c06a0-de6a-54ab-a142-c8b1cf79cde6\"\nversion = \"0.21.2\"\n\n[[JpegTurbo_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"d735490ac75c5cb9f1b00d8b5509c11984dc6943\"\nuuid = \"aacddb02-875f-59d6-b918-886e6ef4fbf8\"\nversion = \"2.1.0+0\"\n\n[[LAME_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"f6250b16881adf048549549fba48b1161acdac8c\"\nuuid = \"c1c5ebd0-6772-5130-a774-d5fcae4a789d\"\nversion = \"3.100.1+0\"\n\n[[LZO_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"e5b909bcf985c5e2605737d2ce278ed791b89be6\"\nuuid = \"dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac\"\nversion = \"2.10.1+0\"\n\n[[LaTeXStrings]]\ngit-tree-sha1 = 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\"26be8b1c342929259317d8b9f7b53bf2bb73b123\"\nuuid = \"d1454406-59df-5ea1-beac-c340f2130bc3\"\nversion = \"1.1.4+4\"\n\n[[Xorg_libXrandr_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libXext_jll\", \"Xorg_libXrender_jll\"]\ngit-tree-sha1 = \"34cea83cb726fb58f325887bf0612c6b3fb17631\"\nuuid = \"ec84b674-ba8e-5d96-8ba1-2a689ba10484\"\nversion = \"1.5.2+4\"\n\n[[Xorg_libXrender_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libX11_jll\"]\ngit-tree-sha1 = \"19560f30fd49f4d4efbe7002a1037f8c43d43b96\"\nuuid = \"ea2f1a96-1ddc-540d-b46f-429655e07cfa\"\nversion = \"0.9.10+4\"\n\n[[Xorg_libpthread_stubs_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"6783737e45d3c59a4a4c4091f5f88cdcf0908cbb\"\nuuid = \"14d82f49-176c-5ed1-bb49-ad3f5cbd8c74\"\nversion = \"0.1.0+3\"\n\n[[Xorg_libxcb_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"XSLT_jll\", \"Xorg_libXau_jll\", \"Xorg_libXdmcp_jll\", \"Xorg_libpthread_stubs_jll\"]\ngit-tree-sha1 = \"daf17f441228e7a3833846cd048892861cff16d6\"\nuuid = \"c7cfdc94-dc32-55de-ac96-5a1b8d977c5b\"\nversion = \"1.13.0+3\"\n\n[[Xorg_libxkbfile_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libX11_jll\"]\ngit-tree-sha1 = \"926af861744212db0eb001d9e40b5d16292080b2\"\nuuid = \"cc61e674-0454-545c-8b26-ed2c68acab7a\"\nversion = \"1.1.0+4\"\n\n[[Xorg_xcb_util_image_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xcb_util_jll\"]\ngit-tree-sha1 = \"0fab0a40349ba1cba2c1da699243396ff8e94b97\"\nuuid = \"12413925-8142-5f55-bb0e-6d7ca50bb09b\"\nversion = \"0.4.0+1\"\n\n[[Xorg_xcb_util_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libxcb_jll\"]\ngit-tree-sha1 = \"e7fd7b2881fa2eaa72717420894d3938177862d1\"\nuuid = \"2def613f-5ad1-5310-b15b-b15d46f528f5\"\nversion = \"0.4.0+1\"\n\n[[Xorg_xcb_util_keysyms_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xcb_util_jll\"]\ngit-tree-sha1 = \"d1151e2c45a544f32441a567d1690e701ec89b00\"\nuuid = \"975044d2-76e6-5fbe-bf08-97ce7c6574c7\"\nversion = \"0.4.0+1\"\n\n[[Xorg_xcb_util_renderutil_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xcb_util_jll\"]\ngit-tree-sha1 = \"dfd7a8f38d4613b6a575253b3174dd991ca6183e\"\nuuid = \"0d47668e-0667-5a69-a72c-f761630bfb7e\"\nversion = \"0.3.9+1\"\n\n[[Xorg_xcb_util_wm_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xcb_util_jll\"]\ngit-tree-sha1 = \"e78d10aab01a4a154142c5006ed44fd9e8e31b67\"\nuuid = \"c22f9ab0-d5fe-5066-847c-f4bb1cd4e361\"\nversion = \"0.4.1+1\"\n\n[[Xorg_xkbcomp_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_libxkbfile_jll\"]\ngit-tree-sha1 = \"4bcbf660f6c2e714f87e960a171b119d06ee163b\"\nuuid = \"35661453-b289-5fab-8a00-3d9160c6a3a4\"\nversion = \"1.4.2+4\"\n\n[[Xorg_xkeyboard_config_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Xorg_xkbcomp_jll\"]\ngit-tree-sha1 = \"5c8424f8a67c3f2209646d4425f3d415fee5931d\"\nuuid = \"33bec58e-1273-512f-9401-5d533626f822\"\nversion = \"2.27.0+4\"\n\n[[Xorg_xtrans_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"79c31e7844f6ecf779705fbc12146eb190b7d845\"\nuuid = \"c5fb5394-a638-5e4d-96e5-b29de1b5cf10\"\nversion = \"1.4.0+3\"\n\n[[Zlib_jll]]\ndeps = [\"Libdl\"]\nuuid = \"83775a58-1f1d-513f-b197-d71354ab007a\"\n\n[[Zstd_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6\"\nuuid = \"3161d3a3-bdf6-5164-811a-617609db77b4\"\nversion = \"1.5.0+0\"\n\n[[libass_jll]]\ndeps = [\"Artifacts\", \"Bzip2_jll\", \"FreeType2_jll\", \"FriBidi_jll\", \"HarfBuzz_jll\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"5982a94fcba20f02f42ace44b9894ee2b140fe47\"\nuuid = \"0ac62f75-1d6f-5e53-bd7c-93b484bb37c0\"\nversion = \"0.15.1+0\"\n\n[[libfdk_aac_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"daacc84a041563f965be61859a36e17c4e4fcd55\"\nuuid = \"f638f0a6-7fb0-5443-88ba-1cc74229b280\"\nversion = \"2.0.2+0\"\n\n[[libpng_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Zlib_jll\"]\ngit-tree-sha1 = \"94d180a6d2b5e55e447e2d27a29ed04fe79eb30c\"\nuuid = \"b53b4c65-9356-5827-b1ea-8c7a1a84506f\"\nversion = \"1.6.38+0\"\n\n[[libvorbis_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Ogg_jll\", \"Pkg\"]\ngit-tree-sha1 = \"c45f4e40e7aafe9d086379e5578947ec8b95a8fb\"\nuuid = \"f27f6e37-5d2b-51aa-960f-b287f2bc3b7a\"\nversion = \"1.3.7+0\"\n\n[[nghttp2_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"8e850ede-7688-5339-a07c-302acd2aaf8d\"\n\n[[p7zip_jll]]\ndeps = [\"Artifacts\", \"Libdl\"]\nuuid = \"3f19e933-33d8-53b3-aaab-bd5110c3b7a0\"\n\n[[x264_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"4fea590b89e6ec504593146bf8b988b2c00922b2\"\nuuid = \"1270edf5-f2f9-52d2-97e9-ab00b5d0237a\"\nversion = \"2021.5.5+0\"\n\n[[x265_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\"]\ngit-tree-sha1 = \"ee567a171cce03570d77ad3a43e90218e38937a9\"\nuuid = \"dfaa095f-4041-5dcd-9319-2fabd8486b76\"\nversion = \"3.5.0+0\"\n\n[[xkbcommon_jll]]\ndeps = [\"Artifacts\", \"JLLWrappers\", \"Libdl\", \"Pkg\", \"Wayland_jll\", \"Wayland_protocols_jll\", \"Xorg_libxcb_jll\", \"Xorg_xkeyboard_config_jll\"]\ngit-tree-sha1 = \"ece2350174195bb31de1a63bea3a41ae1aa593b6\"\nuuid = \"d8fb68d0-12a3-5cfd-a85a-d49703b185fd\"\nversion = \"0.9.1+5\"\n\"\"\"\n\n# ╔═╡ Cell order:\n# ╟─1cdb4e08-99f7-48d9-924b-e105d1b266dc\n# ╟─721e5d79-4842-43cf-a2ff-4cf0fdb13f87\n# ╟─969a133a-9861-4248-92f0-757fd12f12fb\n# ╟─624fcb2a-8446-42c2-863f-e285f2c1fd86\n# ╟─d8005539-13d5-4a3a-ad62-861c4f4bb9e0\n# ╟─5619b1f9-38d1-42de-bd81-708567e2fa96\n# ╟─24f9bdd6-0337-4a23-9373-cfecabc2b029\n# ╟─eacd1340-79af-4985-bf60-b8db8a380a20\n# ╟─6bcb1adc-633c-4730-b4e2-6b5b6f5237c4\n# 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╟─11c37b73-ddfd-41cf-98f4-c461b265d1bf\n# ╠═b20fed20-a678-450b-a689-a51baa337153\n# ╟─abbd82f5-31cc-4f45-9c7c-f401796174cb\n# ╟─eaa487b9-2728-49f8-8b5e-003926f4d544\n# ╟─3e6282ce-f3ba-4591-976d-0c37d14755c7\n# ╠═6b58bfb2-be89-4cd2-9c67-85ca95d02267\n# ╟─51f20ae2-e0b6-410d-8783-689f6de9a025\n# ╠═4b47a6af-3c41-4575-a576-09145c535e22\n# ╟─1ad77ac0-bd4c-407f-858f-6e435085d4d8\n# ╟─e54b7b64-ebbe-426d-92fa-4959eebba826\n# ╟─b6c98fa0-8b40-4654-aa0c-b0204ed9e291\n# ╟─3c8c50dc-7b25-48d3-84f2-c2e1a9fb85dd\n# ╠═fd8ce478-87a8-48aa-b8d1-df80df93254b\n# ╟─8d9840d1-1e71-403b-8935-c6aeca8a0ee0\n# ╟─2be79841-4bfd-47d1-9c50-fd1e33a766c5\n# ╟─9a90438b-67c2-4024-bf45-696d60b718df\n# ╟─14bf83f9-9a19-42e0-ba41-c6a16c58a89c\n# ╟─3b409a8a-b03a-41e6-9e14-b9f73af1847b\n# ╟─00000000-0000-0000-0000-000000000001\n# ╟─00000000-0000-0000-0000-000000000002\n", "meta": {"hexsha": "8eead89098e7af714d29a21276d5138bf1855602", "size": 59958, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "pca.jl", 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{"text": "function mean( img::AbstractArray{T,2} ) where {T<:Real}\n\n\tret = 0.0\n\tN = length( img );\n\n\t@simd for li in 1:N\n\t\t@inbounds ret += img[li]\n\tend\n\n\treturn ret/N\nend\n\nfunction variance( img::AbstractArray{T,2} ) where {T<:Real}\n\n\tmean = mean( img )\n\n\tret = 0.0\n\tN = length( img )\n\n\t@simd for li in 1:N\n\t\t@inbounds ret += ( img[li] - mean )*( img[li] - mean )\n\tend\n\n\treturn ret/N\nend\n\nfunction variance( img::AbstractArray{T,2}, mean ) where {T<:Real}\n\n\tret = 0.0\n\tN = length( img )\n\n\t@simd for li in 1:N\n\t\t@inbounds ret += ( img[li] - mean )*( img[li] - mean )\n\tend\n\n\treturn ret/N\nend\n\nfunction std( img::AbstractArray{T,2} ) where {T<:Real}\n\n\treturn sqrt( variance( img ) )\nend\n\nfunction std( img::AbstractArray{T,2}, mean ) where {T<:Real}\n\n\treturn sqrt( variance( img, mean ) )\nend\n\nfunction cov( img1::Array{T1,2}, img2::Array{T2,2}) where {T1<:Real,T2<:Real}\n\n\tlength( img1 ) !== length( img2 ) && ( error(\"Arrays have different lengths\") )\n\n\tm1 = mean( img1 )\n\ts1 = std( img1, m1 )\n\n\tm2 = mean( img2 )\n\ts2 = std( img2, m2 )\n\n\tret = 0.0\n\tN = length( img1 )\n\n\t@simd for li in 1:N\n\t\t@inbounds ret += ( img1[li] - m1 )*( img2[li] - m2 )\n\tend\n\n\treturn ret/N\nend\n\nfunction correlation( img1::Array{T1,2}, img2::Array{T2,2} ) where {T1<:Real,T2<:Real}\n\n\tlength( img1 ) !== length( img2 ) && ( error(\"Arrays have different lengths\") )\n\n\tm1 = mean( img1 )\n\ts1 = std( img1, m1 )\n\n\tm2 = mean( img2 )\n\ts2 = std( img2, m2 )\n\n\tret = 0.0\n\tN = length( img1 )\n\n\t@simd for li in 1:N\n\t\t@inbounds ret += ( img1[li] - m1 )*( img2[li] - m2 )\n\tend\n\n\treturn ret/(N*s1*s2)\nend\n\nfunction crossCorrelationZNCC( f::Array{T,2}, g::Array{T,2} ) where {T<:Real}\n\n\t\tlN = div( length( f ), 16 );\n\n\t\t# statstics for ZNCC\n\t\t#=\n\t\tmeanf = mean( f );\n\t\tmeanf2 = meanf * meanf;\n\n\t\tmeang = mean( g );\n\t\tmeang2 = meang * meang;\n\t\t=#\n\n\t\t# integral Arrays to compute stdf at each possible translation\n\t\tsumf = integralArray( f );\n\t\tsumf2 = integralArray( f, fun=(x)->(x*x) );\n\t\tsumg = integralArray( g );\n\t\tsumg2 = integralArray( g, fun=(x)->(x*x) );\n\n\t # correlation size = N + M - 1\n\t csize = size(f) .+ size(g) .- 1;\n\t h, w = csize;\n\t\tnum = zeros( Float32, csize ); # we compute the numerator with FFT for each cross-correlation element\n\t\tcorr = zeros( Float32, csize );\n\n\t\t#=\n\t\tpadf = zeros( Complex{T}, csize );\n\t\tpadf[ 1:size(f,1), 1:size(f,2) ] .= complex.( f .- meanf, 0.0 );\n\n\t\tpadg = zeros( Complex{T}, csize );\n\t\tpadg[ 1:size(g,1), 1:size(g,2) ] .= complex.( g .- meang, 0.0 );\n\n\t\tplan = plan_fft!( padf );\n\t\tiplan = plan_ifft!( padf );\n\n\t\t# cross-correlation ( f - meanf ) and ( g - meang ) with FFT\n\t\tplan * padf;\n\t\tplan * padg;\n\t\t@inbounds @simd for e in 1:length(padf)\n\t\t\tpadf[e] = conj( padf[e] ) * padg[e];\n\t\tend\n\n\t\tiplan * padf;\n\t\t@inbounds @simd for e in 1:length(padf)\n\t\t\tcorr[e] = real( padf[e] )\n\t\tend\n\n\t\tshifts = div.( csize, 2 ).- 1\n\t Base.circshift!( num, corr, shifts )\n\t\t=#\n\n\t\t# computing the denominator\n\t\t# Computing denominator through summed-area tables.\n\t @inbounds for col in 1:csize[2]\n\n\t\t\tminCol_g = 1 + max( 0, col - size(f,2) )\n\t\t\tmaxCol_g = min( col, size(g,2) )\n\n\t\t\t#\n\t\t\tminCol_f = max( 1, 1 + size(f,2) - col )\n\t maxCol_f = size(f,2) - max( 0, col - size(g,2) )\n\n\t\t\tlenCol_g = length( minCol_g:maxCol_g );\n\n\t for row in 1:csize[1]\n\n\t\t\t\tminRow_g = 1 + max( 0, row - size(f,1) );\n \tmaxRow_g = min( row, size(g,1) );\n\n\t\t\t\t#\n\t\t\t\tminRow_f = max( 1, 1 + size(f,1) - row )\n\t maxRow_f = size(f,1) - max( 0, row - size(g,1) )\n\n\t\t\t\tlenRow_g = length( minRow_g:maxRow_g )\n\t\t\t\tN = lenRow_g*lenCol_g\n\n\t\t\t\t#\n\t\t\t\tF, F2 = integralAreaZNCC( sumf, sumf2, row, col, size(f), size(g) )\n\t\t\t\tG, G2 = integralAreaZNCC( sumg, sumg2, row, col, size(g), size(g) )\n\n\t\t\t\t#\n\t\t\t\tmeanf = F/N;\n\t\t\t\tmeanf2 = meanf * meanf;\n\t\t\t\tstdf = sqrt( F2 + N*meanf2 - 2*F*meanf );\n\t\t\t\tmeang = G/N;\n\t\t\t\tmeang2 = meang * meang;\n\t\t\t\tstdg = sqrt( G2 + N*meang2 - 2*G*meang );\n\n\t\t\t\t#\n\t\t\t\tsubg = view( g, minRow_g:maxRow_g, minCol_g:maxCol_g )\n\t\t\t\tsubf = view( f, minRow_f:maxRow_f, minCol_f:maxCol_f )\n\n\t\t\t\t#\n\t\t\t\tzncc = 0.0\n\t\t\t\tfor e in 1:length(subg)\n\t\t\t\t\tzncc += ( subf[e] - meanf )*( subg[e] - meang )\n\t\t\t\tend\n\n\t\t\t\t#\n\t\t\t\tcorr[ row, col ] = zncc/(stdg*stdf)\n\n\t\t\t\t#=\n\t\t\t\tif N < lN\n\t\t\t\t\tcorr[ row, col ] = 0.0\n\t\t\t\telse\n\t\t F, F2 = integralAreaZNCC( sumf, sumf2, row, col, size(f), size(g) )\n\t\t\t\t\tG, G2 = integralAreaZNCC( sumg, sumg2, row, col, size(g), size(g) )\n\n\t\t corr[ row, col ] = num[ row, col ]/( sqrt( F2 + N*meanf2 - 2*F*meanf )*sqrt( G2 + N*meang2 - 2*G*meang ) );\n\t\t\t\tend\n\t\t\t\t=#\n\t end\n\t end\n\n\t\treturn corr;\nend\n", "meta": {"hexsha": "513eabc3b2b412cb8a63b5fab009c212814775fb", "size": 4574, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/stats.jl", "max_stars_repo_name": "Marc-3d/ImageAnalysis", "max_stars_repo_head_hexsha": "0e99582d05adbc905e6c217ea90b6ac171d41e18", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/stats.jl", "max_issues_repo_name": "Marc-3d/ImageAnalysis", "max_issues_repo_head_hexsha": "0e99582d05adbc905e6c217ea90b6ac171d41e18", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/stats.jl", "max_forks_repo_name": "Marc-3d/ImageAnalysis", "max_forks_repo_head_hexsha": "0e99582d05adbc905e6c217ea90b6ac171d41e18", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.312195122, "max_line_length": 123, "alphanum_fraction": 0.5636204635, "num_tokens": 1783, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "# # Helmholtz decomposition\n\n#md # ```@meta\n#md # CurrentModule = ImmersedLayers\n#md # ```\n\n#=\nFor vector fields on the grid, we often prefer to work with potentials\nof that field and compose the vector field from these potentials. This\nis the so-called Helmholtz decomposition\n\n$$\\mathbf{v} = \\nabla\\phi + \\nabla\\times\\psi$$\n\nwhere $\\psi$ is a vector potential (or streamfunction in two dimensions)\nand $\\phi$ is a scalar potential. Note that, in an unbounded domain, $\\phi$\nsatisfies a Poisson equation:\n\n$$\\nabla^2\\phi = \\nabla\\cdot\\mathbf{v}$$\n\nand, assuming that $\\psi$ has zero 'gauge' (i.e., its divergence is zero), then\n$\\psi$ also satisfies a Poisson equation:\n\n$$\\nabla^2\\psi = -\\nabla\\times\\mathbf{v}$$\n\nSo we can recover the velocity field from the divergence and curl of the\nvector field (in fluid dynamics, these are the rate of dilatation and the vorticity,\nrespectively) by solving these equations for the potentials and then\nevaluating their gradient and curl.\n\nWith immersed layers, these are extendable to domains with surfaces\nand their associated jump in the vector field $[\\mathbf{v}] = \\mathbf{v}^+ - \\mathbf{v}^-$.\nThe crucial relationships are\n\n$$\\nabla\\cdot{\\overline{\\mathbf{v}}} = \\overline{\\nabla\\cdot\\mathbf{v}} + \\delta(\\chi)\\mathbf{n}\\cdot[\\mathbf{v}]$$\n\nand\n\n$$\\nabla\\times{\\overline{\\mathbf{v}}} = \\overline{\\nabla\\times\\mathbf{v}} + \\delta(\\chi)\\mathbf{n}\\times[\\mathbf{v}]$$\n\nThese are, respectively, the divergence and curl of the masked vector field, $\\overline{\\mathbf{v}}$.\nIn `ImmersedLayers.jl`, we are always working with the masked form of a field,\nwhether we acknowledge it or not. In contrast, the first terms on the right-hand sides of these are the masked divergence and curl\nof the vector field in each side of the surface. The difference between these is\na type of single-layer surface term---a 'source sheet' or a 'vortex sheet', respectively---\ndue to the jump in vector field. If we omit the overbar notation, then the Helmholtz\ndecomposition and the Poisson equations all still hold.\n\nThis package has a number of routines that can be used to perform the\noperations described here. They rely on two caches, [`ScalarPotentialCache`](@ref)\nand [`VectorPotentialCache`](@ref), each of which supports one of the\nrespective potentials. The function [`vecfield_helmholtz!`](@ref) carries\nout all of these operations in order to assemble the vector field $\\mathbf{v}$\nfrom the masked divergence and curl fields and the jump in the vector field.\n(We can also add an additional irrotational, divergence-free vector field during\nthis assembly.\n=#\n\n\n#md # ## Helmholtz decomposition functions\n\n#md # ```@docs\n#md # ScalarPotentialCache\n#md # VectorPotentialCache\n#md # vectorpotential_from_masked_curlv!\n#md # scalarpotential_from_masked_divv!\n#md # vectorpotential_from_curlv!\n#md # vecfield_from_vectorpotential!\n#md # masked_curlv_from_curlv_masked!\n#md # curlv_masked_from_masked_curlv!\n#md # scalarpotential_from_divv!\n#md # masked_divv_from_divv_masked!\n#md # divv_masked_from_masked_divv!\n#md # vecfield_from_scalarpotential!\n#md # vecfield_helmholtz!\n#md # vectorpotential_uniformvecfield!\n#md # scalarpotential_uniformvecfield!\n#md # vecfield_uniformvecfield!\n#md # ```\n", "meta": {"hexsha": "67bd35797d15bd832e8447c5c25b96e28cac1c7e", "size": 3223, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test/literate/helmholtz.jl", "max_stars_repo_name": "mleprovost/ImmersedLayers.jl", "max_stars_repo_head_hexsha": "09d95ee2201ee01da943838edd4c28ac5aad5156", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test/literate/helmholtz.jl", "max_issues_repo_name": "mleprovost/ImmersedLayers.jl", "max_issues_repo_head_hexsha": "09d95ee2201ee01da943838edd4c28ac5aad5156", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test/literate/helmholtz.jl", "max_forks_repo_name": "mleprovost/ImmersedLayers.jl", "max_forks_repo_head_hexsha": "09d95ee2201ee01da943838edd4c28ac5aad5156", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.7974683544, "max_line_length": 130, "alphanum_fraction": 0.7610921502, "num_tokens": 905, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9353465098415279, "lm_q2_score": 0.8615382129861583, "lm_q1q2_score": 0.8058367606117101}}
{"text": "module CommonMath\n\nconst ∞ = Inf\nconst ∞⁻ = -Inf\nconst 𝐆 = MathConstants.golden\nconst 𝐆⁻¹ = 𝐆 - 1\nconst 𝐆1⁻¹ = 2 - 𝐆\nconst c½ = 0.5\n\nfunction mean_geometric(a::Float64, b::Float64; use_abs::Bool = false)\n prod = a * b\n return (use_abs ? sign(prod) : 1) * √(use_abs ? abs(prod) : prod)\nend\nfunction mean_geometric(V::Vector{Float64}; use_abs::Bool = false)\n @assert !isempty(V)\n prodV = prod(V)\n σ = use_abs ? sign(prodV) : 1\n return (use_abs ? abs(prodV) : prodV)^(1 / length(V))\nend\n\nfunction mean_harmonic(a::Float64, b::Float64)\n return 2 / (1 / a + 1 / b)\nend\nfunction mean_harmonic(V::Vector{Float64})\n @assert !isempty(V)\n return length(V) / sum(v -> 1 / v, V)\nend\n\nfunction scientific_notation_parts(a::Float64)\n o = floor(log10(abs(a)))\n υ = a / 10^o\n return (υ, o)\nend\n\nfunction mean_log10(a::Float64, b::Float64)\n υᵃ, oᵃ = scientific_notation_parts(a)\n υᵇ, oᵇ = scientific_notation_parts(b)\n return mean_geometric(υᵃ, υᵇ; use_abs = false) * 10^(mean_geometric(oᵃ, oᵇ; use_abs = true))\n return mean_geometric(υᵃ, υᵇ; use_abs = false) * 10^(mean_geometric(oᵃ, oᵇ; use_abs = true))\nend\nfunction mean_log10(V::Vector{Float64})\n @assert !isempty(V)\n vᵐⁱⁿ = minimum(V); vᵐᵃˣ = maximum(V)\n @assert vᵐⁱⁿ ≠ 0\n return mean_log10(vᵐⁱⁿ, vᵐᵃˣ)\nend\n\nfunction mean_log10_golden_lower(v¹::Float64, v²::Float64)\n a = min(v¹, v²); b = max(v¹, v²)\n υᵃ, oᵃ = scientific_notation_parts(a)\n υᵇ, oᵇ = scientific_notation_parts(b)\n return (𝐆⁻¹*υᵃ + 𝐆1⁻¹*υᵇ) * 10^(𝐆⁻¹*oᵃ + 𝐆1⁻¹*oᵇ)\nend\nfunction mean_log10_golden_lower(V::Vector{Float64})\n @assert !isempty(V)\n vᵐⁱⁿ = minimum(V); vᵐᵃˣ = maximum(V)\n @assert vᵐⁱⁿ ≠ 0\n return mean_log10_golden_lower(vᵐⁱⁿ, vᵐᵃˣ)\nend\n\nexport ∞, ∞⁻\nexport 𝐆, 𝐆⁻¹, 𝐆1⁻¹\nexport mean_geometric, mean_harmonic\nexport scientific_notation_parts\nexport mean_log10, mean_log10_golden_lower\n\nend # module CommonMath\n", "meta": {"hexsha": "ce57f76fda86942e9124804155f4e670f202d529", "size": 1907, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/commonmath/commonmath.jl", "max_stars_repo_name": "gitter-badger/NumericalDataManipulation.jl", "max_stars_repo_head_hexsha": "4f1bc43e8c2f94c3700c88619f51a9632f956306", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/commonmath/commonmath.jl", "max_issues_repo_name": "gitter-badger/NumericalDataManipulation.jl", "max_issues_repo_head_hexsha": "4f1bc43e8c2f94c3700c88619f51a9632f956306", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/commonmath/commonmath.jl", "max_forks_repo_name": "gitter-badger/NumericalDataManipulation.jl", "max_forks_repo_head_hexsha": "4f1bc43e8c2f94c3700c88619f51a9632f956306", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 28.0441176471, "max_line_length": 96, "alphanum_fraction": 0.6717357105, "num_tokens": 814, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9353465062370312, "lm_q2_score": 0.861538211208597, "lm_q1q2_score": 0.8058367558436627}}
{"text": "\"\"\"\n A, B, C = ss_to_tf(tf)\n\nthis formulation is based on the notes here: \nhttp://web.mit.edu/2.14/www/Handouts/StateSpace.pdf\n\n Y(s) bₙ sⁿ + ... + b₁s + b₀\n G(s) = ----- = -------------------------\n U(s) aₙ sⁿ + ... + a₁s + a₀\n\nthen we can formulate the system described by transfer\nfunction G(s) as a state space representation known\nas the controllable canonical form:\n\n dx\n---- = A * x(t) + B * u(t)\n dt\n\ny(t) = C * x(t) + bₙ/aₙ * u(t)\n\nwe compute the matrices A, B, C here.\n\"\"\"\nfunction tf_to_ss(tf::TransferFunction)\n @assert proper(tf) \"tf not proper!\"\n \n a_i(i::Int) = tf.denominator[i]\n b_i(i::Int) = tf.numerator[i]\n\n # get the highest degree in the num/den, which is that in the den if proper.\n n = degree(tf.denominator) # asserted proper above\n\n A = zeros(n, n)\n for i = 1:n-1\n A[i, i + 1] = 1.0\n end\n for i = 1:n\n A[n, i] = - a_i(i-1)\n end\n A[n, :] /= a_i(n)\n \n B = zeros(n, 1)\n B[n, 1] = 1 / a_i(n)\n \n C = zeros(1, n)\n for i = 1:n\n C[1, i] = b_i(i-1) - b_i(n) * a_i(i-1) / a_i(n)\n end\n return A, B, C\nend\n\n\n@doc raw\"\"\"\n t, y = simulate(Y, final_time, nb_time_points=100) # invert Y(s)\n\nSimulate the output $y(t)$ of an LTI system, given the Laplace transform of the output, $Y(s)$, `Y`.\n\nIn other words, `simulate` inverts an expression in the frequency domain into the time domain.\n\n# Arguments\n* `Y::TransferFunction`: the Laplace transform of the output $y(t)$. Usually formed by $g(s)U(s)$, where $U(s)$ is the Laplace transform of the input.\n* `final_time::Tuple{Float64, Float64}`: the duration over which to simulate the output of the LTI system, starting at time zero.\n* `nb_time_points::Int=100`: the number of time points at which to save the solution $y(t)$\n\nTwo time points preceding $t=0$ are included to illustrate that it is assumed $y(t)=0$ for $t<0$.\n\n# Returns\n* `t::Array{Float64, 1}`: array of times: $t_i$'s\n* `y::Array{Float64, 1}`: array of $y$ values at corresponding times in `t`: $y_i$'s, where $y_i=y(t_i)$.\n\n# Example\n\nOne can simulate the first order step response as, given the Laplace transform of the output, `Y`:\n\n```\njulia> g = 4 / (3 * s + 1) # first-order transfer function\njulia> U = 1 / s # unit step input\njulia> Y = g / s # output\njulia> t, y = simulate(Y, 12.0)\n```\n\"\"\"\nfunction simulate(Y::TransferFunction, final_time::Float64; nb_time_points::Int=100)\n if ! proper(Y)\n error(\"LTI system is not proper...\")\n end\n \n # coeffs on denominator, numerator polynomial\n a_i(i::Int) = Y.denominator[i]\n b_i(i::Int) = Y.numerator[i]\n n = degree(Y.denominator)\n\n # convert tf to state space form\n A, B, C = tf_to_ss(Y)\n\n x0 = deepcopy(B) # initial condition\n\n f(x, p, t) = A * x # RHS of ODE (ignore p for params)\n prob = ODEProblem(f, x0, (0.0, final_time))\n sol = solve(prob, d_discontinuities=[0.0])\n\n t = vcat([-0.05 * final_time, -1e-5], range(1e-5, final_time, length=nb_time_points - 2))\n y = [NaN for i = 1:nb_time_points]\n for (i, t_i) in enumerate(t)\n if t_i < Y.time_delay\n y[i] = 0.0\n else\n y[i] = (C * sol(t_i - Y.time_delay))[1] # put in shifted time\n end\n end\n return t, y\nend\n\n@doc raw\"\"\"\n y_at_t̃ = interpolate(t, y, t̃)\n\ngiven an array of times $t_i$ in `t` and corresponding values of $y(t)$, $y_i=y(t_i)$ in the array `y`, interpolate to approximate the function $y(t)$ at a new time `t̃`, i.e. $y(\\tilde{t})$.\n\nthe output of [`simulate`](@ref) is an array of times `t` and corresponding output values `y`. `interpolate` is useful for obtaining the solution at a particular time `t̃` that is not necessarily present in the array `t`.\n\nan error is thrown if `maximum(t) < t̃ < minimum(t)` (extrapolation) or if the `t` array is not sorted.\n\n# Arguments\n* `t::Array{Float64, 1}`: array of times $t_i$. these must be sorted.\n* `y::Array{Float64, 1}`: an array of corresponding $y$-values $y_i=y(t_i)$\n* `t̃::Float64`: the new time at which we wish to know $y$. i.e. we wish to know $y(\\tilde{t})$.\n\n# Returns\n* `y_at_t̃::Float64`: the value of $y$ when $t$ is `t̃`, $y(\\tilde{t})$, according to linear interpolation.\n\n# Example\n\nthe unit step response of a first-order process with time constant $\\tau$ is $\\approx63\\%$ of the final value when $t=\\tau$.\n\n```julia\njulia> τ = 3.45\njulia> g = 1 / (τ * s + 1) # FO system\njulia> t, y = simulate(g / s, 10.0) # unit step response\njulia> y_at_τ = interpolate(t, y, τ) # 0.63\n```\n\"\"\"\nfunction interpolate(t::Array{Float64, 1}, y::Array{Float64, 1}, t̃::Float64)\n if t̃ < minimum(t) || t̃ > maximum(t)\n error(@sprintf(\"t̃ = %f is outside the range of the t-array passed, %f to %f.\\n\", t̃, minimum(t), maximum(t)))\n end\n # from Interpolations.jl.\n # will throw error if t not sorted. \n return LinearInterpolation(t, y)(t̃)\nend\n", "meta": {"hexsha": "f1810d76befd1b26c8faeeb00c121b71ce3c7705", "size": 4874, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/sim.jl", "max_stars_repo_name": "dmolina/Controlz.jl", "max_stars_repo_head_hexsha": "8cf246a4a10fcaac2ab6b874f40a54c0392f52e7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/sim.jl", "max_issues_repo_name": "dmolina/Controlz.jl", "max_issues_repo_head_hexsha": "8cf246a4a10fcaac2ab6b874f40a54c0392f52e7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/sim.jl", "max_forks_repo_name": "dmolina/Controlz.jl", "max_forks_repo_head_hexsha": "8cf246a4a10fcaac2ab6b874f40a54c0392f52e7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.9324324324, "max_line_length": 221, "alphanum_fraction": 0.6153057037, "num_tokens": 1634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.921921841290738, "lm_q2_score": 0.8740772466456689, "lm_q1q2_score": 0.8058309046579135}}
{"text": "# -*- coding: utf-8 -*-\n# # Runge-Kutta methods\n\nusing Plots\ngr()\n\n# We will implement in Julia different numerical methods to solve\n#\n# $$\n# y'(t) = 1 - y(t)\n# $$\n#\n# $ t \\in [0,5] \\qquad \\mbox{ and } \\qquad y(0) = 0 $\n\n# ## Explicit Euler\n\n\"\"\"\n\n euler(f::Function, t::Float64, y::Float64, h::Float64)\n\nexplicit euler method function that returns\n\n```math\ny^{n+1} = y^n + h \\\\cdot f(t^n, y^n)\n```\n\n\n\"\"\"\nfunction euler(f::Function, t::Float64, y::Float64, h::Float64)\n t + h, y + h * f(t,y)\nend\n#----------------------------------------------------------------------------\n\n?euler\n\n# ## Runge-Kutta 2nd order\n\n\"\"\"\n\n rk2(f::Function, t::Float64, y::Float64, dt::Float64)\n\nRunge-Kutta second order method function\n\n```jldoctest\njulia> f(t,y) = 1 - y\njulia> rk2(f, 0.0, 0.0, 1.0)\n(1.0, 0.5)\n```\n\n\"\"\"\nfunction rk2(f::Function, t::Float64, y::Float64, h::Float64)\n ỹ = y + h/2 * f(t,y)\n t + h, y + h * f(t+h/2,ỹ)\nend\n#----------------------------------------------------------------------------\n\n?rk2\n\n# ## Runge-Kutta 4th order\n\n\"\"\"\n\n rk4(f::Function, t::Float64, y::Float64, dt::Float64)\n\nRunge-Kutta fourth order method function\n\n[Runge–Kutta methods on Wikipedia](https://en.wikipedia.org/wiki/Runge–Kutta_methods)\n\n\"\"\"\nfunction rk4(f::Function, t::Float64, y::Float64, dt::Float64)\n\n y₁ = dt * f(t,y)\n y₂ = dt * f(t+dt/2,y+y₁/2)\n y₃ = dt * f(t+dt/2,y+y₂/2)\n y₄ = dt * f(t+dt,y+y₃)\n\n t+dt, y+(y₁+2*y₂+2*y₃+y₄)/6\n\nend\n#----------------------------------------------------------------------------\n\n?rk4\n#----------------------------------------------------------------------------\n\n# ## Solver function\n\n\"\"\"\n\n solver(f::Function, Method::Function, t₀::Float64,\n y₀::Float64, dt::Float64, nsteps::Int64)\n\nSolve numerically the equation ``y' = f(t, y)``\n\nwith `y(t₀)= y₀` and `nsteps` steps `h`\n\n## Arguments\n- `f::Function`: the function `f` of equation ``y' = f(t,y)``.\n- `Method::Function`: numerical method from (tⁿ,yⁿ) returns ``(t^{n+1},y^{n+1})``\n\n\n\"\"\"\nfunction solver(f::Function,\n Method::Function,\n t₀::Float64,\n y₀::Float64, h::Float64, nsteps::Int64)\n\n t = zeros(Float64,nsteps)\n y = similar(t)\n\n t[1] = t₀\n y[1] = y₀\n\n for i in 2:nsteps\n t[i], y[i] = Method(f,t[i-1],y[i-1], h)\n end\n\n t, y\n\nend\n#----------------------------------------------------------------------------\n\n?solver\n#----------------------------------------------------------------------------\n\nnsteps, tfinal = 7, 5.0\nt₀, x₀ = 0., 0.\n\ndt = tfinal / (nsteps-1)\nf(t, x) = 1 - x\n#----------------------------------------------------------------------------\n\nplot( solver(f, euler, t₀, x₀, dt, nsteps); marker = :o, label=:euler)\nplot!(solver(f, rk2, t₀, x₀, dt, nsteps); marker = :d, label=:rk2)\nplot!(solver(f, rk4, t₀, x₀, dt, nsteps); marker = :p, label=:rk4)\nt = 0:0.1:5\nplot!(t, 1 .- exp.(-t); line = 3, label = :exact)\n#----------------------------------------------------------------------------\n\n# # Callable object\n#\n# We want to build a numerical method to solve the problem and then use it\n# as a function and still call the solver function.\n\n#\n#\n# Runge-Kutta scheme can be built using Butcher tableau :\n#\n# \\begin{array}{c|cccc}\n# c₁ & & & & \\\\\n# c₂ & a_{21} & & & \\\\\n# c₃ & a_{31} & a_{32} & & \\\\\n# c₄ & a_{41} & a_{42} & a_{43} & \\\\\n# \\hline\n# & b_1 & b_2 & b_3 & b_4 \\\\\n# \\end{array}\n#\n# $$\n# \\forall i = 1, \\dotsc, q, \\begin{cases}t_{n,i} &= t_n + c_i h_n, \\\\ \n# y_{n,i} &= y_n + h_n \\sum_{k = 1}^{i-1} a_{ik} p_{n,k}\\\\ \n# p_{n,i} &= f(t_{n,i}, y_{n,i}) \\end{cases}\n# $$\n#\n# $$\n# y_{n+1} = y_n + h_n \\sum_{k = 1}^q b_k p_{n,k}.\n# $$\n#\n\nmutable struct RungeKutta\n \n q :: Int64\n a :: Array{Float64, 2}\n b :: Array{Float64, 1}\n c :: Array{Float64, 1}\n \n tn :: Vector{Float64}\n yn :: Vector{Float64}\n pn :: Vector{Float64}\n \n function RungeKutta( a::Array{Float64,2}, b::Vector{Float64}, c::Vector{Float64})\n \n q = length(c)\n @assert ( length(c) == size(a,1))\n @assert ( length(b) == size(a,2))\n tn = zeros(Float64, q)\n yn = zeros(Float64, q)\n pn = zeros(Float64, q)\n new( q, a, b, c, tn, yn, pn)\n \n end\n\nend\n\nfunction (rk::RungeKutta)(f::Function, t::Float64, y::Float64, h::Float64)\n\n for i = 1:rk.q\n rk.tn[i] = t + rk.c[i] * h\n rk.yn[i] = y + h * sum([rk.a[i,k]*rk.pn[k] for k = 1:i-1])\n rk.pn[i] = f(rk.tn[i],rk.yn[i])\n end\n\n t + h, y + h * sum([rk.b[k]*rk.pn[k] for k in 1:rk.q ])\n\nend\n#----------------------------------------------------------------------------\n\nfunction solver(f::Function,\n Method::RungeKutta,\n t₀::Float64,\n y₀::Float64, h::Float64, nsteps::Int64)\n\n t = zeros(Float64,nsteps)\n y = similar(t)\n\n t[1] = t₀\n y[1] = y₀\n\n for i in 2:nsteps\n t[i], y[i] = Method(f,t[i-1],y[i-1], h)\n end\n\n t, y\n\nend\n#----------------------------------------------------------------------------\n\na = [ 0 0 0 0; \n 1/2 0 0 0; \n 0 1/2 0 0; \n 0 0 1 0]\n\nb = [1/6 ,1/3, 1/3, 1/6]\nc = [0 ,1/2, 1/2, 1 ]\n\nrk4_new = RungeKutta(a, b, c)\n#----------------------------------------------------------------------------\n\nt = 0:0.1:5\nplot(t, 1 .- exp.(-t),label = :exact)\nplot!(solver(f, rk4_new, t₀, x₀, dt, nsteps), marker = :o, label=:rk4_new)\n#----------------------------------------------------------------------------\n\na = [ 0 0 0 0; \n 1/3 0 0 0; \n -1/3 1 0 0; \n 1 -1 1 0]\n\nb = [1/8, 3/8, 3/8, 1/8]\nc = [0, 1/3, 2/3, 1]\nrk4_38 = RungeKutta(a, b, c)\nplot!(solver(f, rk4_38, t₀, x₀, dt, nsteps), marker = :r, label=:rk4_38)\n#----------------------------------------------------------------------------\n\n# [Reference](https://www.juliabloggers.com/runge-kutta-methods/):Ordinary Differential Equation Solvers: Runge-Kutta Methods by Christina Lee\n\n# # Creating expressions via interpolation\n#\n# Create a solver function with the method choosen at initialization.\n\nmacro add(x, y)\n return :($x + $y)\nend\n\n@add 2 3 ## or @add(2, 3) \n#----------------------------------------------------------------------------\n\nmacro abs(x)\n return :( $x > 0 ? $x : -$x)\nend\n@abs(-2), @abs(2)\n#----------------------------------------------------------------------------\n\nmacro make_method( meth)\n return quote\n function (f::Function, t₀::Float64,\n x₀::Float64, dt::Float64, nsteps::Int64)\n\n t = zeros(Float64,nsteps)\n x = zeros(Float64,nsteps)\n\n t[1] = t₀\n x[1] = x₀\n\n for i in 2:nsteps\n t[i], x[i] = $meth(f,t[i-1],x[i-1], dt)\n end\n\n return t, x\n end\n end\nend\n\nrk4_solver = @make_method rk4\n\nplot(rk4_solver(f, t₀, x₀, dt, nsteps))\n#----------------------------------------------------------------------------\n\n# [DifferentialEquations.jl](http://docs.juliadiffeq.org/latest/)\n#\n# [ODE Solvers](http://docs.juliadiffeq.org/latest/solvers/ode_solve.html#OrdinaryDiffEq.jl-1)\n\nusing OrdinaryDiffEq\nusing Plots\n\n\nf(y,p,t) = 1.0 - y\ny₀ = 0.0\nt = (0.0,5.0)\nprob = ODEProblem(f,y₀,t)\neuler = solve(prob,Euler(), dt=1.0)\nrk4 = solve(prob, RK4(), dt=1.0)\nplot(euler,label=\"Euler\")\nplot!(rk4,label=\"RK4\")\nplot!(1:0.1:5, t->1. - exp(-t),lw=3,ls=:dash,label=\"True Solution!\")\n#----------------------------------------------------------------------------\n\nusing DifferentialEquations\n\nfunction lorenz(du,u,p,t)\n du[1] = 10.0*(u[2]-u[1])\n du[2] = u[1]*(28.0-u[3]) - u[2]\n du[3] = u[1]*u[2] - (8/3)*u[3]\nend\n#----------------------------------------------------------------------------\n\nu0 = [1.0;0.0;0.0]\ntspan = (0.0,100.0)\nprob = ODEProblem(lorenz,u0,tspan)\nsol = solve(prob)\n#----------------------------------------------------------------------------\n\nplot(sol,vars=(1,2,3))\n\n# ----------------------------------------------------------------------------\n", "meta": {"hexsha": "95cf3640a96c1e93b816d8a99f582e64034812e1", "size": 8039, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "03.RungeKuttaMethods.jl", "max_stars_repo_name": "pnavaro/math-julia", "max_stars_repo_head_hexsha": "900db803b2ddd9e44eb2d4fabb481768a14953c0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2019-01-14T13:32:03.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T04:22:17.000Z", "max_issues_repo_path": "03.RungeKuttaMethods.jl", "max_issues_repo_name": "pnavaro/math-julia", "max_issues_repo_head_hexsha": "900db803b2ddd9e44eb2d4fabb481768a14953c0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "03.RungeKuttaMethods.jl", "max_forks_repo_name": "pnavaro/math-julia", "max_forks_repo_head_hexsha": "900db803b2ddd9e44eb2d4fabb481768a14953c0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-10-15T07:58:35.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-11T01:52:10.000Z", "avg_line_length": 24.0688622754, "max_line_length": 142, "alphanum_fraction": 0.429282249, "num_tokens": 2655, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218412907381, "lm_q2_score": 0.8740772351648677, "lm_q1q2_score": 0.8058308940735123}}
{"text": "### A Pluto.jl notebook ###\n# v0.14.4\n\nusing Markdown\nusing InteractiveUtils\n\n# ╔═╡ c7e41465-41ad-4f11-baec-bfaae9edbf59\nbegin\n\tusing PlutoUI\n\tPlutoUI.TableOfContents(aside=true)\nend\n\n# ╔═╡ d0233ece-1f69-11eb-3da1-43b8ddc9fbe1\nbegin\n\t# Necessary packages\n\tusing LinearAlgebra\n\tusing Random\n\tusing Statistics\n\tusing Clustering\n\tusing Plots\nend\n\n# ╔═╡ 00d72ad7-ea80-4954-8c75-590d4e9ccdc4\nmd\"\"\"\n# K-means Algorithm\n\n\nData clustering is one of the main mathematical applications variety of algorithms have been developed to tackle the problem. K-means is one of the basic algorithms for data clustering.\n\n__Prerequisites__\n\nThe reader should be familiar with basic linear algebra. \n \n__Competences__\n\nThe reader should be able to recognise applications where K-means algorithm can be efficiently used and use it.\n\n__Credits.__ The notebook was initially derived from M.Sc. Thesis of Ivančica Mirošević. \n\"\"\"\n\n# ╔═╡ 235e1901-eed3-4934-9214-ffdd82cf21d6\nmd\"\"\"\n## Definitions\n\n__Data clustering problem__ is the following: partition the given set of $m$ objects of the same type into $k$ subsets according to some criterion. Additional request may be to find the optimal $k$.\n\n__K-means clustering problem__ is the following: partition the set $X=\\{x_{1},x_{2},\\cdots ,x_{m}\\}$ , where $x_{i}\\in\\mathbb{R}^{n}$, into $k$ _clusters_ $\\pi=\\{C_{1},C_{2},...,C_{k}\\}$ such that\n\n$$\nJ(\\pi)=\\sum_{i=1}^{k}\\sum_{x\\in\nC_{i}}\\| x-c_{i}\\|_{2}^{2} \\to \\min$$\n\nover all possible partitions. Here $c_{i}=\\displaystyle\\frac{1}{|C_{i}|}\\sum_{x\\in C_{i}} x$ is the mean of points in $C_i$ and $|C_i|$ is the cardinality of $C_i$.\n\"\"\"\n\n# ╔═╡ 26640b90-1f69-11eb-029e-4dc3a825f639\nmd\"\n## K-means clustering algorithm\n\n1. __Initialization__: Choose initial set of $k$ means $\\{c_1,\\ldots,c_k\\}$ (for example, by choosing randomly $k$ points from $X$).\n2. __Assignment step__: Assign each point $x$ to one nearest mean $c_i$.\n3. __Update step__: Compute the new means.\n4. __Iteration__: Repeat Steps 2 and 3 until the assignment no longer changes.\n\"\n\n# ╔═╡ 48914840-1f69-11eb-30a9-3943767bc261\nmd\"\n## First Variation clustering algorithm\n\nA __first variation__ of a partition $\\pi=\\{C_1,\\ldots,C_k\\}$ is \na partition $\\pi^{\\prime}=\\{C_{1}^{\\prime},\\cdots ,C_{k}^{\\prime }\\}$ \nobtained by moving a single point $x$ from a cluster $C_{i}$ to a cluster $C_{j}$. Notice that $\\pi$ is a first variation of itself.\n\nA __next partition__ of the partition $\\pi$ is a partition \n$\\mathop{\\mathrm{next}}(\\pi)=\\mathop{\\mathrm{arg min}}\\limits_{\\pi^{\\prime}} J(\\pi^{\\prime})$.\n\nWe have the following algorithm:\n\n1. Choose initial partition $\\pi$.\n2. Compute $\\mathop{\\mathrm{next}}(\\pi)$\n3. If $J(\\mathop{\\mathrm{next}}(\\pi))\nFor data point $i$ and cluster $k$, this quantity is\n\n$$r_{ik} \\propto \\pi_k N(x_i | \\mu_k, \\Sigma_k)$$\n\nwhere $N(x_i | \\mu_k, \\Sigma_k)$ is the Gaussian distribution for cluster $k$ (with mean $\\mu_k$ and covariance $\\Sigma_k$).\n\nWe used $\\propto$ because the quantity $N(x_i | \\mu_k, \\Sigma_k)$ is not yet the responsibility we want.
To ensure that all responsibilities over each data point add up to 1, we add the normalization constant in the denominator:\n\n$$r_{ik} = \\frac{\\pi_k N(x_i | \\mu_k, \\Sigma_k)}{\\sum_{k=1}^{K} \\pi_k N(x_i | \\mu_k, \\Sigma_k)}$$\n\nComplete the following function that computes $r_{ik}$ for all data points $i$ and clusters $k$.\n\"\"\"\n\n# ╔═╡ 4d69bcec-830c-11eb-0c80-314c8cded6db\nfunction compute_responsibilities(data, weights, μ, σ)\n \"\"\"\n E-step fo EM Algorithm:\n - compute responsibilities, given the current parameters\n \"\"\"\n\t@assert length(data) > 0\n\n ndata, nclusters = length(data), length(μ)\n resp = zeros(eltype(data[1]), ndata, nclusters)\n\n ## Update resp. matrix so that resp[i, k] is the responsibility of\n\t## cluster k for data point i.\n for i ∈ 1:ndata\n for k ∈ 1:nclusters\n\t\t\td = MvNormal(μ[k], σ[k])\n\t\t\tresp[i, k] = weights[k] * pdf(d, data[i])\n\t\tend\n\tend\n ## Add up responsibilities over each data point and normalize\n row_sums = reshape(sum(resp, dims=2), :, 1)\n resp ./= row_sums\nend\n\n# ╔═╡ 31835154-830d-11eb-09fd-196046609868\nbegin\n # const ϵ = 1e-8\n resp₀ = compute_responsibilities([[1., 2.], [-1., -2.]],\n\t[0.3, 0.7], # weights\n [[0., 0.], [1., 1.]], # μ\n\t[[1.5 0.; 0. 2.5], [1. 1.; 1. 2.]] # σ\n )\n\n @test size(resp₀) == (2, 2)\n @test approx(resp₀, [0.10512733 0.89487267; 0.46468164 0.53531836])\nend\n\n# ╔═╡ 430e58a8-8310-11eb-2a53-ed2608f43122\nmd\"\"\"\n#### M-step: Update parameters, given current cluster responsibilities\n\nOnce the cluster responsibilities are computed, we update the parameters (*weights*, *means*, and *covariances*) associated with the clusters.\n\n**Computing soft counts**. Before updating the parameters, we first compute what is known as \"soft counts\".\n\nThe soft count of a cluster is the sum of all cluster responsibilities for that cluster:\n\n$$N^{\\text{soft}}_k = r_{1k} + r_{2k} + \\ldots + r_{Nk} = \\sum_{i=1}^{N} r_{ik}$$\n\nwhere we loop over data points. Note that, unlike k-means, we must loop over every single data point in the dataset. This is because all clusters are represented in all data points, to a varying degree.\n\n\"\"\"\n\n# ╔═╡ 42ef055c-8310-11eb-1ae5-d5ae8bf22c1f\n\"\"\"\n Compute the total responsibility assigned to each cluster, which will be useful when\n implementing M-steps below. In the lectures this is called N^soft\n\"\"\"\ncompute_soft_counts(resp) = sum(resp; dims=1)\n\n# ╔═╡ 93988ed8-8310-11eb-15f4-554b537eacd6\nmd\"\"\"\n**Updating weights.** The cluster weights show us how much each cluster is represented over all data points.\n\nThe weight of cluster $k$ is given by the ratio of the soft count\n$N^{\\text{soft}}_{k}$ to the total number of data points $N$:\n\n$$\\hat{\\pi}_k = \\frac{N^{\\text{soft}}_{k}}{N}$$\n\nNotice that $N$ is equal to the sum over the soft counts $N^{\\text{soft}}_{k}$ of all clusters.\n\nComplete the following function:\n\"\"\"\n\n# ╔═╡ 937da226-8310-11eb-1966-6bcc8ff3cb47\nfunction compute_weights(counts)\n nclusters, n = length(counts), sum(counts)\n weights = zeros(eltype(counts), nclusters)\n\n\tfor k ∈ 1:nclusters\n ## Update the weight for cluster k using the M-step update rule for\n\t\t## the cluster weight: π̂ₖ\n weights[k] = counts[k] / n\n\tend\n weights\nend\n\n# ╔═╡ 93653664-8310-11eb-23f2-d5236b8bc0ed\nbegin\n\tresp₁ = compute_responsibilities([[1., 2.], [-1.,-2.], [0., 0.]],\n\t\t[0.3, 0.7], # weights\n\t\t[[0., 0.], [1., 1.]], # μ\n\t\t[[1.5 0.; 0. 2.5], [1. 1.; 1. 2.]] # σ\n \t)\n\tcounts₁ = compute_soft_counts(resp₁)\n\tweights₁ = compute_weights(counts₁)\n\n\t@test approx(weights₁, [0.27904865942515705 0.720951340574843])\nend\n\n# ╔═╡ 93490854-8310-11eb-3de5-5b24918d8840\nwith_terminal() do\n println(\" - counts: $(counts₁)\")\n println(\" - weigths: $(weights₁)\")\n println(\"Checkpoint passed!\")\nend\n\n# ╔═╡ 42d1c078-8310-11eb-2262-bb5ae92a2b27\nmd\"\"\"\n**Updating means**. The mean of each cluster is set to the [weighted average](https://en.wikipedia.org/wiki/Weighted_arithmetic_mean) of all data points, weighted by the cluster responsibilities:\n\n$$\\hat{\\mu}_k = \\frac{1}{N_k^{\\text{soft}}} \\sum_{i=1}^N r_{ik}x_i$$\n\nComplete the following function:\n\"\"\"\n\n# ╔═╡ 1b27764e-8315-11eb-3854-f9e5b591744a\nfunction compute_means(data, resp, counts)\n nclusters, ndata = length(counts), length(data)\n means = [zeros(eltype(data[1]), length(data[1])) for _ ∈ 1:nclusters]\n\n for k in 1:nclusters\n ## Update means for cluster k using the M-step update rule\n\t\t## for the mean variables.\n ## This will assign the variable means[k] to be our estimate for μ̂ₖ.\n weighted_sum = sum(\n [resp[i, k] * data[i] for i ∈ 1:ndata]\n )\n means[k] = weighted_sum / counts[k]\n\tend\n means\nend\n\n# ╔═╡ 34b1bb5e-8317-11eb-0260-5f1d3132a7ba\nbegin\n\tdata_tmp = [[1., 2.], [-1.,-2.]]\n\tresp₂ = compute_responsibilities(data_tmp,\n\t\t[0.3, 0.7], # weights\n\t\t[[0., 0.], [1., 1.]], # μ\n\t\t[[1.5 0.; 0. 2.5], [1. 1.; 1. 2.]] # σ\n\t)\n\tcounts₂ = compute_soft_counts(resp₂)\n\tmeans₂ = compute_means(data_tmp, resp₂, counts₂)\n\texp_means₂ = [[-0.6310085, -1.262017], [0.25140299, 0.50280599]]\n\t@test approx(means₂[1], exp_means₂[1])\n\t@test approx(means₂[2], exp_means₂[2])\nend\n\n# ╔═╡ 1ad0cbe4-8315-11eb-341c-dbce64acb374\nwith_terminal() do\n println(\" - counts: $(counts₂)\")\n println(\" - means: $(means₂)\")\n println(\"Checkpoint passed!\")\nend\n\n# ╔═╡ 1a99af8a-8315-11eb-0f74-eb97741a40ca\nmd\"\"\"\n\n**Updating covariances**. The covariance of each cluster is set to the weighted average of all [outer products](https://people.duke.edu/~ccc14/sta-663/LinearAlgebraReview.html), weighted by the cluster responsibilities:\n\n$$\\hat{\\Sigma}_k = \\frac{1}{N^{\\text{soft}}_k}\\sum_{i=1}^N r_{ik} (x_i - \\hat{\\mu}_k)(x_i - \\hat{\\mu}_k)^T$$\n\nThe \"outer product\" in this context refers to the matrix product\n\n$$(x_i - \\hat{\\mu}_k)(x_i - \\hat{\\mu}_k)^T.$$\nLetting $(x_i - \\hat{\\mu}_k)$ to be $d \\times 1$ column vector, this product is a $d \\times d$ matrix. Taking the weighted average of all outer products gives us the covariance matrix, which is also $d \\times d$.\n\nReminder:\nThe inner product (of two column vectors $1 \\times n$) denoted as $
![\"Sketch](\"lyapunov.png\")
\n#\n#\n# * $\\lambda$ denotes the \"maximum Lyapunov exponent\".\n# * A $D$-dimensional system has $D$ exponents.\n# * In general, a trajectory is called \"chaotic\" if\n# 1. it follows nonlinear dynamics\n# 2. it is *bounded* (does not escape to infinity)\n# 2. it has a positive Lyapunov exponent\n#\n# *(please be aware that the above is an over-simplification! See the textbooks cited in our\n# documentation for more)*\n#\n# ---\n# ------------------------------------------------------------------------------------------\n\nusing DynamicalSystems, BenchmarkTools, PyPlot\n\n# ------------------------------------------------------------------------------------------\n# Before computing Lyapunov exponents, we'll demonstrate the concept of exponential\n# separation using a simple map: the *towel map*\n#\n# $$\n# \\begin{aligned}\n# x_{n+1} &= a x_n (1-x_n) -0.05 (y_n +0.35) (1-2z_n) \\\\\n# y_{n+1} &= 0.1 \\left( \\left( y_n +0.35 \\right)\\left( 1+2z_n\\right) -1 \\right)\n# \\left( 1 -1.9 x_n \\right) \\\\\n# z_{n+1} &= 3.78 z_n (1-z_n) + b y_n\n# \\end{aligned}\n# $$\n# ------------------------------------------------------------------------------------------\n\ntowel = Systems.towel()\n\n# ------------------------------------------------------------------------------------------\n# First we'll generate a trajectory for the towel map, `tr1`, from the default initial\n# condition,\n# ------------------------------------------------------------------------------------------\n\ntr1 = trajectory(towel, 100)\nsummary(tr1)\n\n# ------------------------------------------------------------------------------------------\n# and then we will generate a second trajectory, `tr2`, with a starting point slightly\n# shifted from the initial condition of `tr1`.\n# ------------------------------------------------------------------------------------------\n\nu2 = get_state(towel) + (1e-9 * rand(3))\ntr2 = trajectory(towel, 100, u2)\nsummary(tr2)\n\nfigure(figsize=(8,5))\n\n# Plot the x-coordinate of the two trajectories:\nax1 = subplot(2,1,1)\nplot(tr1[:, 1], alpha = 0.5)\nplot(tr2[:, 1], alpha = 0.5)\nylabel(\"x\")\n\n# Plot their distance in a semilog plot:\nax2 = subplot(2,1,2, sharex = ax1)\nd = [norm(tr1[i] - tr2[i]) for i in 1:length(tr2)]\nylabel(\"d\")\nxlabel(\"n\")\nsemilogy(d);\n\n# ------------------------------------------------------------------------------------------\n# # Maximum Lyapunov Exponent\n# `lyapunov` is a function that calculates the maximum Lyapunov exponent for a\n# `DynamicalSystem` (for a given starting point).\n#\n# Since `lyapunov` is not a trivial function, it is best to read the documentation string\n# first:\n# ------------------------------------------------------------------------------------------\n\n?lyapunov\n\n# ------------------------------------------------------------------------------------------\n# ---\n#\n# Let's apply this to the example of the previous section, the Shinriki oscillator!\n#\n# *Reminder:* we found something that \"could\" be chaotic behavior for the parameter `R1 =\n# 21.0`\n# \n# ------------------------------------------------------------------------------------------\n\nshi = Systems.shinriki(;R1 = 21.0)\n\nlyapunov(shi, 1000.0, Ttr = 10.0)\n\n# ------------------------------------------------------------------------------------------\n# Positive Lyapunov exponent!?!? That is definitely chaotic behavior, right?\n#\n# *Right?*\n# ------------------------------------------------------------------------------------------\n\n# ------------------------------------------------------------------------------------------\n# Let's increase accuracy of computation, as well as the transient time, to see whether an\n# orbit actually enters a limit cycle.\n# ------------------------------------------------------------------------------------------\n\nlyapunov(shi, 2000.0, Ttr = 1000.0)\n\n# ------------------------------------------------------------------------------------------\n# **surprise**\n#\n# Let's find out whats going on!\n#\n# To see why the lyapunov exponent was positive in one case and negative in another, we can\n# produce a more detailed orbit diagram, around the \"critical\" value of `R = 21.0`.\n# ------------------------------------------------------------------------------------------\n\npvalues = linspace(20.9,21.1,101)\ni = 1\nplane = (2, 0.0)\ntf = 1000.0\np_index = 1\n\n# use extremely long transient time:\noutput = produce_orbitdiagram(shi, plane, i, p_index, pvalues; tfinal = tf,\n Ttr = 2000.0, direction = -1, printparams = false);\nlength(output)\n\nusing PyPlot\nfigure(figsize=(6,4))\nfor (j, p) in enumerate(pvalues)\n plot(p .* ones(output[j]), output[j], lw = 0,\n marker = \"o\", ms = 0.2, color = \"black\")\nend\nplot([21, 21], [-2.1, 0.1], color = \"red\", alpha = 0.55)\nxlabel(\"\\$R_1\\$\"); ylabel(\"\\$V_1\\$\");\n\n# result of orbit diagram at R1 = 21.0\nvalues = output[51] \n\n# Amount of unique points\nun = unique(round.(output[51], 8))\nprintln(\"Total: $(length(output[51])), unique: $(length(un))\")\n\n# ------------------------------------------------------------------------------------------\n# ## `lyapunov` for discrete system\n#\n# * All functions that accept a `DynamicalSystem` work with *any* instance of\n# `DynamicalSystem`, regardless of whether it is continuous, discrete, in-place, out-of-\n# place or whatever.\n# \n# ------------------------------------------------------------------------------------------\n\n# Get the Henon map from the library of pre-defined systems:\nhen = Systems.henon()\nλ = lyapunov(hen, 10000)\n\n# ------------------------------------------------------------------------------------------\n# ---\n#\n# # Lyapunov Spectrum\n#\n# Besides the maximum Laypunov exponent, the function `lyapunovs` (with `s` at the end)\n# returns the entire Lyapunov spectrum (or as many exponents the user wants).\n#\n#\n#\n# Once again, because the function `lyapunovs` is not trivial, we will view the\n# documentation string first:\n# ------------------------------------------------------------------------------------------\n\n?lyapunovs\n\n# ------------------------------------------------------------------------------------------\n# ### Lyapunovs for discrete systems\n#\n# In our first example of calling `lyapunovs`, let's pass a discrete system.\n# ------------------------------------------------------------------------------------------\n\ntowel = Systems.towel()\nlyapunovs(towel, 2000; Ttr = 200)\n\n# ------------------------------------------------------------------------------------------\n# Here we're choosing to compute only the first two exponents.\n# ------------------------------------------------------------------------------------------\n\nlyapunovs(towel, 2000, 2; Ttr = 200)\n\n# ------------------------------------------------------------------------------------------\n# If you only want the first exponent (maximum), use the `lyapunov` function instead\n# ------------------------------------------------------------------------------------------\n\nlyapunov(towel, 2000; Ttr = 200)\n\n# ------------------------------------------------------------------------------------------\n# How much time does this take?\n# ------------------------------------------------------------------------------------------\n\nusing BenchmarkTools\n@btime lyapunovs($towel, 2000; Ttr = 200);\n\n@btime lyapunov($towel, 2000; Ttr = 200);\n\n# ------------------------------------------------------------------------------------------\n# ### Lyapunov exponents for continuous systems\n#\n# Next, let's initialize the Lorenz system with random initial condition\n# ------------------------------------------------------------------------------------------\n\nlor = Systems.lorenz()\n\n# ------------------------------------------------------------------------------------------\n# We'll compute the Lyapunov spectrum with specified initial parallepiped matrix `Q0`:\n# ------------------------------------------------------------------------------------------\n\nQ0 = eye(3)\nlyapunovs(lor, 2000, Q0; Ttr = 10.0)\n\n# ------------------------------------------------------------------------------------------\n# And we find that results \"converge\" already with 2000 iterations:\n# ------------------------------------------------------------------------------------------\n\nlyapunovs(lor, 3000, Q0; Ttr = 10.0)\n\n# ------------------------------------------------------------------------------------------\n# * Even the continuous systems are quite performant (note that compilation takes a **lot**\n# of time):\n# ------------------------------------------------------------------------------------------\n\n@btime lyapunovs($lor, 2000, $Q0; Ttr = 10.0);\n\n# ------------------------------------------------------------------------------------------\n# * The above integration is done with a 9th order solver and tolerances of `1e-9`. But you\n# can get away with lower tolerances.\n#\n# Let's load in `OrdinaryDiffEq` and specify keyword arguments for the integrators of\n# DifferentialEquations.jl.\n# ------------------------------------------------------------------------------------------\n\nusing OrdinaryDiffEq\n \ndek = Dict(:solver => Tsit5(), :abstol => 1e-6, :reltol => 1e-6)\n\n# ------------------------------------------------------------------------------------------\n# And now we can call `lyapunovs` with the keyword `diff_eq_kwargs`\n# ------------------------------------------------------------------------------------------\n\nlyapunovs(lor, 2000, Q0; Ttr = 10.0, diff_eq_kwargs = dek)\n\n@btime lyapunovs($lor, 2000.0, $Q0; Ttr = 10.0, diff_eq_kwargs = $dek);\n", "meta": {"hexsha": "aa0865e55edabc9ed6a222d9e79efb7ec5287c79", "size": 10292, "ext": "jl", "lang": "Julia", "max_stars_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/introduction-to-dynamicalsystems.jl/3. Quantifying Chaos (Lyapunov Exponents).jl", "max_stars_repo_name": "grenkoca/JuliaTutorials", "max_stars_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 535, "max_stars_repo_stars_event_min_datetime": "2020-07-15T14:56:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-25T12:50:32.000Z", "max_issues_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/introduction-to-dynamicalsystems.jl/3. Quantifying Chaos (Lyapunov Exponents).jl", "max_issues_repo_name": "grenkoca/JuliaTutorials", "max_issues_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 42, "max_issues_repo_issues_event_min_datetime": "2018-02-25T22:53:47.000Z", "max_issues_repo_issues_event_max_datetime": "2020-05-14T02:15:50.000Z", "max_forks_repo_path": ".nbexports/introductory-tutorials/broader-topics-and-ecosystem/introduction-to-dynamicalsystems.jl/3. Quantifying Chaos (Lyapunov Exponents).jl", "max_forks_repo_name": "grenkoca/JuliaTutorials", "max_forks_repo_head_hexsha": "3968e0430db77856112521522e10f7da0d7610a0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 394, "max_forks_repo_forks_event_min_datetime": "2020-07-14T23:22:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T20:12:57.000Z", "avg_line_length": 38.2602230483, "max_line_length": 92, "alphanum_fraction": 0.4318888457, "num_tokens": 2162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625069680097, "lm_q2_score": 0.8615382129861583, "lm_q1q2_score": 0.802490543716826}}
{"text": "# This file is part of the IntervalArithmetic.jl package; MIT licensed\n\n\n\"\"\"\n in(x, a)\n ∈(x, a)\n\nChecks if the number `x` is a member of the interval `a`, treated as a set.\nCorresponds to `isMember` in the ITF-1788 Standard.\n\"\"\"\nfunction in(x::T, a::Interval) where T<:Real\n isinf(x) && return false\n a.lo <= x <= a.hi\nend\n\n\n\n\"\"\"\n issubset(a,b)\n ⊆(a,b)\n\nChecks if all the points of the interval `a` are within the interval `b`.\n\"\"\"\nfunction ⊆(a::Interval, b::Interval)\n isempty(a) && return true\n b.lo ≤ a.lo && a.hi ≤ b.hi\nend\n\nfunction ⊂(a::Interval, b::Interval)\n a == b && return false\n a ⊆ b\nend\n\nfunction ⊇(a::Interval, b::Interval)\n b ⊆ a\nend\n\nfunction ⊃(a::Interval, b::Interval)\n b ⊂ a\nend\n\n# isinterior\nfunction isinterior(a::Interval, b::Interval)\n isempty(a) && return true\n islessprime(b.lo, a.lo) && islessprime(a.hi, b.hi)\nend\nconst ⪽ = isinterior # \\subsetdot\n\n# Disjoint:\nfunction isdisjoint(a::Interval, b::Interval)\n (isempty(a) || isempty(b)) && return true\n islessprime(b.hi, a.lo) || islessprime(a.hi, b.lo)\nend\n\n\n# Intersection\n\"\"\"\n intersect(a, b)\n ∩(a,b)\n\nReturns the intersection of the intervals `a` and `b`, considered as\n(extended) sets of real numbers. That is, the set that contains\nthe points common in `a` and `b`.\n\"\"\"\nfunction intersect(a::Interval{T}, b::Interval{T}) where T\n isdisjoint(a,b) && return emptyinterval(T)\n\n Interval(max(a.lo, b.lo), min(a.hi, b.hi))\nend\n# Specific promotion rule for intersect:\nintersect(a::Interval{T}, b::Interval{S}) where {T,S} =\n intersect(promote(a, b)...)\n\n\n## Hull\n\"\"\"\n hull(a, b)\n\nReturns the \"interval hull\" of the intervals `a` and `b`, considered as\n(extended) sets of real numbers, i.e. the smallest interval that contains\nall of `a` and `b`.\n\"\"\"\nhull(a::Interval, b::Interval) = Interval(min(a.lo, b.lo), max(a.hi, b.hi))\n#\n# hull{T,S}(a::Interval{T}, b::Interval{S}) = hull(promote(a, b)...)\n\n\"\"\"\n union(a, b)\n ∪(a,b)\n\nReturns the union (convex hull) of the intervals `a` and `b`; it is equivalent\nto `hull(a,b)`.\n\"\"\"\nunion(a::Interval, b::Interval) = hull(a, b)\n#\n# union(a::Interval, b::Interval) = union(promote(a, b)...)\n\n\n\"\"\"\n setdiff(x::Interval, y::Interval)\n\nCalculate the set difference `x ∖ y`, i.e. the set of values\nthat are inside the interval `x` but not inside `y`.\n\nReturns an array of intervals.\nThe array may:\n\n- be empty if `x ⊆ y`;\n- contain a single interval, if `y` overlaps `x`\n- contain two intervals, if `y` is strictly contained within `x`.\n\"\"\"\nfunction setdiff(x::Interval, y::Interval)\n intersection = x ∩ y\n\n isempty(intersection) && return [x]\n intersection == x && return typeof(x)[] # x is subset of y; setdiff is empty\n\n x.lo == intersection.lo && return [Interval(intersection.hi, x.hi)]\n x.hi == intersection.hi && return [Interval(x.lo, intersection.lo)]\n\n return [Interval(x.lo, y.lo),\n Interval(y.hi, x.hi)]\n\nend\n", "meta": {"hexsha": "11749217efda2447fa0a3a39aeb9c6890d829a33", "size": 2939, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/intervals/set_operations.jl", "max_stars_repo_name": "mewilhel/IntervalArithmetic.jl", "max_stars_repo_head_hexsha": "38a011b4f87eaa37bce97c0f2c38a526eb1e0841", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/intervals/set_operations.jl", "max_issues_repo_name": "mewilhel/IntervalArithmetic.jl", "max_issues_repo_head_hexsha": "38a011b4f87eaa37bce97c0f2c38a526eb1e0841", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/intervals/set_operations.jl", "max_forks_repo_name": "mewilhel/IntervalArithmetic.jl", "max_forks_repo_head_hexsha": "38a011b4f87eaa37bce97c0f2c38a526eb1e0841", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-06T06:15:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-06T06:15:24.000Z", "avg_line_length": 23.512, "max_line_length": 81, "alphanum_fraction": 0.636951344, "num_tokens": 900, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625069680098, "lm_q2_score": 0.8615382094310357, "lm_q1q2_score": 0.8024905404053627}}
{"text": "# This file is a part of AstroLib.jl. License is MIT \"Expat\".\n# Copyright (C) 2016 Mosè Giordano.\n\nfunction mphase(jd::AbstractFloat)\n ram, decm, dism = moonpos(jd, radians=true)\n ras, decs = sunpos(jd, radians=true)\n # phi: geocentric elongation of the Moon from the Sun\n # inc: selenocentric (Moon centered) elongation of the Earth from the Sun\n sin_decs, cos_decs = sincos(decs)\n sin_decm, cos_decm = sincos(decm)\n phi = acos(sin_decs * sin_decm + cos_decs * cos_decm * cos(ras - ram))\n # \"dism\" is in kilometers, AU in meters\n sin_phi, cos_phi = sincos(phi)\n inc = atan(AU * sin_phi, dism * 1000 - AU * cos_phi)\n return (1 + cos(inc))/2\nend\n\n\"\"\"\n mphase(jd) -> k\n\n### Purpose ###\n\nReturn the illuminated fraction of the Moon at given Julian date(s).\n\n### Arguments ###\n\n* `jd`: the Julian ephemeris date.\n\n### Output ###\n\nThe illuminated fraction \\$k\\$ of Moon's disk, with \\$0 \\\\leq k \\\\leq 1\\$. \\$k\n= 0\\$ indicates a new moon, while \\$k = 1\\$ stands for a full moon.\n\n### Method ###\n\nAlgorithm from Chapter 46 of \"Astronomical Algorithms\" by Jean Meeus\n(Willmann-Bell, Richmond) 1991. `sunpos` and `moonpos` are used to get\npositions of the Sun and the Moon, and the Moon distance. The selenocentric\nelongation of the Earth from the Sun (phase angle) is then computed, and used to\ndetermine the illuminated fraction.\n\n### Example ###\n\nPlot the illuminated fraction of the Moon for every day in January 2018 with a\nhourly sampling. Use [PyPlot.jl](https://github.com/JuliaPlots/Plots.jl/) for\nplotting\n\n```julia\nusing PyPlot\npoints = DateTime(2018,01,01):Dates.Hour(1):DateTime(2018,01,31,23,59,59);\nplot(points, mphase.(jdcnv.(points)))\n```\n\nNote that in this calendar month there are two full moons, this event is called\n[blue moon](https://en.wikipedia.org/wiki/Blue_moon).\n\n### Notes ###\n\nCode of this function is based on IDL Astronomy User's Library.\n\"\"\"\nmphase(jd::Real) = mphase(float(jd))\n", "meta": {"hexsha": "93ed59af60dc0d1cfe3283d925dda779fbf4fac8", "size": 1942, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/mphase.jl", "max_stars_repo_name": "UnofficialJuliaMirror/AstroLib.jl-c7932e45-9af1-51e7-9da9-f004cd3a462b", "max_stars_repo_head_hexsha": "fb2ef587a2ac68a1c864bf251e8d9c3601ec4719", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 62, "max_stars_repo_stars_event_min_datetime": "2016-09-11T14:59:20.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-23T20:45:36.000Z", "max_issues_repo_path": "src/mphase.jl", "max_issues_repo_name": "UnofficialJuliaMirror/AstroLib.jl-c7932e45-9af1-51e7-9da9-f004cd3a462b", "max_issues_repo_head_hexsha": "fb2ef587a2ac68a1c864bf251e8d9c3601ec4719", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 64, "max_issues_repo_issues_event_min_datetime": "2017-01-19T21:03:34.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-30T12:27:46.000Z", "max_forks_repo_path": "src/mphase.jl", "max_forks_repo_name": "UnofficialJuliaMirror/AstroLib.jl-c7932e45-9af1-51e7-9da9-f004cd3a462b", "max_forks_repo_head_hexsha": "fb2ef587a2ac68a1c864bf251e8d9c3601ec4719", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 26, "max_forks_repo_forks_event_min_datetime": "2016-07-12T02:11:58.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-30T11:55:21.000Z", "avg_line_length": 31.3225806452, "max_line_length": 80, "alphanum_fraction": 0.6941297631, "num_tokens": 589, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625069680097, "lm_q2_score": 0.8615382076534742, "lm_q1q2_score": 0.8024905387496308}}
{"text": "\n## coordinate.jl\n## Functions for coordinate transformation\n\n\n\"\"\"\n eq2ec(α, δ) -> λ, β\n\nCoordinate transform from the equatorial to ecliptic coordinate systems\n\n# Arguments\n- `α`: right ascension (RA)\n- `δ`: declination (Dec)\n# Returns\n- `λ`: ecliptic longitude\n- `β`: ecliptic latitude\n\"\"\"\nfunction eq2ec(α, δ)\n\n ε = deg2rad(23.439279444444445) # obliquity of the Earth's axis\n\n sin_ε = sin(ε)\n cos_ε = cos(ε)\n \n sin_α = sin(α)\n cos_α = cos(α)\n \n sin_δ = sin(δ)\n cos_δ = cos(δ)\n \n λ = atan(cos_ε*cos_δ*sin_α + sin_ε*sin_δ, cos_δ*cos_α)\n λ < 0 && (λ += 2π)\n\n β = asin(-sin_ε*cos_δ*sin_α + cos_ε*sin_δ)\n\n return λ, β\nend\n\n\nfunction ec2eq(λ, β)\n\n ε = deg2rad(23.439279444444445) # obliquity of the Earth's axis\n\n sin_ε = sin(ε)\n cos_ε = cos(ε)\n\n sin_λ = sin(λ)\n cos_λ = cos(λ)\n \n sin_β = sin(β)\n cos_β = cos(β)\n \n α = atan(cos_ε*cos_β*sin_λ - sin_ε*sin_β, cos_β*cos_λ)\n α < 0 && (α += 2π)\n\n δ = asin(sin_ε*cos_β*sin_λ + cos_ε*sin_β)\n\n return α, δ\nend\n\n\n# cpdef inline tuple eq2hz(double RA, double Dec, double obslon, double obslat):\n# \"\"\"\n# Coordinate transform from equatorial to horizontal coordinate system\n# for an observer in (obslon, obslat)\n# \"\"\"\n# cdef:\n# double hour_angle = obslon - RA # hour angle = GST + longitude - RA\n# double azimuth, elevation\n\n# azimuth = atan2(-cos(Dec)*sin(hour_angle), sin(Dec)*cos(obslat) - cos(Dec)*sin(obslat)*cos(hour_angle))\n\n# if azimuth < 0:\n# azimuth += 2*pi\n\n# elevation = asin(sin(Dec)*sin(obslat) + cos(Dec)*cos(obslat)*cos(hour_angle))\n\n# return azimuth, elevation\n\n\n################################################################\n# Rotation transformation (inpace)\n################################################################\n\n\"\"\"\nRotation transformation around X-axis\nParameters:\n v : vector to be transformed\n θ : rotation angle\n\"\"\"\nfunction rotateX!(v, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n v[1], v[2], v[3] = v[1], v[2]*cosθ + v[3]*sinθ, - v[2]*sinθ + v[3]*cosθ\nend\n\n\n\"\"\"\nRotation transformation around Y-axis\nParameters:\n v : vector to be transformed\n θ : rotation angleon\n\"\"\"\nfunction rotateY!(v, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n v[1], v[2], v[3] = v[1]*cosθ - v[3]*sinθ, v[2], v[1]*sinθ + v[3]*cosθ\nend\n\n\n\"\"\"\nRotation transformation around Z-axis\nParameters:\n v : vector to be transformed\n θ : rotation angleon\n\"\"\"\nfunction rotateZ!(v, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n v[1], v[2], v[3] = v[1]*cosθ + v[2]*sinθ, - v[1]*sinθ + v[2]*cosθ, v[3]\nend\n\n\n################################################################\n# Rotation transformation (non-inpalce)\n################################################################\n\n\nrotateX(v, θ) = rotateX!(copy(v), θ)\nrotateY(v, θ) = rotateY!(copy(v), θ)\nrotateZ(v, θ) = rotateZ!(copy(v), θ)\n\n\nfunction rotateX(v::SVector, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n x = v[1]\n y = v[2]*cosθ + v[3]*sinθ\n z = - v[2]*sinθ + v[3]*cosθ\n\n SA_F64[x, y, z]\nend\n\n\nfunction rotateY(v::SVector, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n x = v[1]*cosθ - v[3]*sinθ\n y = v[2]\n z = v[1]*sinθ + v[3]*cosθ\n\n SA_F64[x, y, z]\nend\n\n\nfunction rotateZ(v::SVector, θ)\n sinθ = sin(θ)\n cosθ = cos(θ)\n \n x = v[1]*cosθ + v[2]*sinθ\n y = - v[1]*sinθ + v[2]*cosθ\n z = v[3]\n\n SA_F64[x, y, z]\nend\n\n\n################################################################\n# Coordinate transformation\n################################################################\n\n\"\"\"\nTransform the inertial coordinate system to the orbital plane system\n\n# Parameters\n- v : vector to be transformed\n- ω : argument of periapsis\n- I : inclination\n- Ω : longitude of the ascneding node\n\"\"\"\nfunction inertia_to_orbit!(v, ω, I, Ω)\n rotateZ!(v, Ω)\n rotateX!(v, I)\n rotateZ!(v, ω)\nend\n\nfunction inertia_to_orbit(v, ω, I, Ω)\n v = rotateZ(v, Ω)\n v = rotateX(v, I)\n v = rotateZ(v, ω)\nend\n\ninertia_to_orbit!(v, orbit) = inertia_to_orbit!(v, orbit.ω, orbit.I, orbit.Ω)\ninertia_to_orbit(v, orbit) = inertia_to_orbit(v, orbit.ω, orbit.I, orbit.Ω)\n\n\n\"\"\"\nTransform the orbital plane system to the inertial coordinate system\n\n# Parameters\n- v : vector to be transformed\n- ω : argument of periapsis\n- I : inclination\n- Ω : longitude of the ascneding node\n\"\"\"\nfunction orbit_to_inertia!(v, ω, I, Ω)\n rotateZ!(v, -ω)\n rotateX!(v, -I)\n rotateZ!(v, -Ω)\nend\n\nfunction orbit_to_inertia(v, ω, I, Ω)\n v = rotateZ(v, -ω)\n v = rotateX(v, -I)\n v = rotateZ(v, -Ω)\nend\n\norbit_to_inertia!(v, orbit) = orbit_to_inertia!(v, orbit.ω, orbit.I, orbit.Ω)\norbit_to_inertia(v, orbit) = orbit_to_inertia(v, orbit.ω, orbit.I, orbit.Ω)\n\n\n\"\"\"\n orbit_to_body(v::SArray{Tuple{3},Float64,1,3}, γ, ε, ϕ) -> v\n\n# Parameters\n- `v` : vector in the orbital plane frame\n- `γ` : longitude of vernal equinox direction of the body\n- `ε` : obliquity of the spin pole\n- `ϕ` : spin phase of the body\n\n# Return\n- `v` : vector in the body-fixed frame\n\"\"\"\nfunction orbit_to_body(v, γ, ε, ϕ)\n v = rotateZ(v, γ) # body's ecliptic coordinate\n v = rotateX(v, ε) # body's equatorial coordinate\n v = rotateZ(v, ϕ) # body-fixed frame\nend\n\n\nfunction orbit_to_body!(v, γ, ε, ϕ)\n rotateZ!(v, γ) # body's ecliptic coordinate\n rotateX!(v, ε) # body's equatorial coordinate\n rotateZ!(v, ϕ) # body-fixed frame\nend\n\n\n\"\"\"\n body_to_orbit(v::SArray{Tuple{3},Float64,1,3}, γ, ε, ϕ) -> v\n\n# Parameters\n- `v` : vector in the body-fixed frame\n- `γ` : longitude of vernal equinox direction of the body\n- `ε` : obliquity of the spin pole\n- `ϕ` : spin phase of the body\n\n# Return\n- `v` : vector in the orbital plane frame\n\"\"\"\nfunction body_to_orbit(v, γ, ε, ϕ)\n v = rotateZ(v, -ϕ) # body's equatorial coordinate\n v = rotateX(v, -ε) # body's ecliptic coordinate\n v = rotateZ(v, -γ) # orbital plane frame \nend\n\n\nfunction body_to_orbit!(v, γ, ε, ϕ)\n rotateZ!(v, -ϕ) # body's equatorial coordinate\n rotateX!(v, -ε) # body's ecliptic coordinate\n rotateZ!(v, -γ) # orbital plane frame \nend\n\n", "meta": {"hexsha": "5d4fc216dac6e6640d0bc0b41e5690faf76952d5", "size": 6166, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/.ipynb_checkpoints/coordinates-checkpoint.jl", "max_stars_repo_name": "MasanoriKanamaru/Astroshaper", "max_stars_repo_head_hexsha": "0040c4c9260252859d4ce3225b3e8616258d0a36", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 23, "max_stars_repo_stars_event_min_datetime": "2021-05-25T07:22:50.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-22T15:42:50.000Z", "max_issues_repo_path": "src/coordinates.jl", "max_issues_repo_name": "MasanoriKanamaru/Astroshaper", "max_issues_repo_head_hexsha": "0040c4c9260252859d4ce3225b3e8616258d0a36", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/coordinates.jl", "max_forks_repo_name": "MasanoriKanamaru/Astroshaper", "max_forks_repo_head_hexsha": "0040c4c9260252859d4ce3225b3e8616258d0a36", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.1798561151, "max_line_length": 109, "alphanum_fraction": 0.5647096983, "num_tokens": 2101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9706877684006774, "lm_q2_score": 0.8267118026095992, "lm_q1q2_score": 0.8024790347856131}}
{"text": "function linearTrussStiffnessMatrix( E, A, nodalCoords )\n\n diff = nodalCoords[ 2, : ] - nodalCoords[ 1, : ] ;\n\n length = sqrt( diff' * diff ) ;\n c = diff[1] / length ;\n s = diff[3] / length ;\n\n Qloc2glo = [ c -s 0 0 ;\n s c 0 0 ;\n 0 0 c -s ;\n 0 0 s c ] ;\n\n Kloc = E*A/length * [ 1 0 -1 0 ;\n 0 0 0 0 ;\n -1 0 1 0 ;\n 0 0 0 0 ] ;\n Kglo = Qloc2glo * Kloc * transpose(Qloc2glo) ;\n\n return Kglo\nend\n#\n", "meta": {"hexsha": "2a90fdea0f671ddd14d1d6eccdc781bb7c586d1a", "size": 565, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/elements/linearTrussStiffnessMatrix.jl", "max_stars_repo_name": "jorgepz/FEMAssembler.jl", "max_stars_repo_head_hexsha": "2d4386b532c1ba41feaf82131454fa07d61f0c96", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/elements/linearTrussStiffnessMatrix.jl", "max_issues_repo_name": "jorgepz/FEMAssembler.jl", "max_issues_repo_head_hexsha": "2d4386b532c1ba41feaf82131454fa07d61f0c96", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2022-03-07T16:18:07.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-13T10:50:46.000Z", "max_forks_repo_path": "src/elements/linearTrussStiffnessMatrix.jl", "max_forks_repo_name": "jorgepz/FEMAssembler.jl", "max_forks_repo_head_hexsha": "2d4386b532c1ba41feaf82131454fa07d61f0c96", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-25T15:21:19.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-25T15:21:19.000Z", "avg_line_length": 24.5652173913, "max_line_length": 57, "alphanum_fraction": 0.3805309735, "num_tokens": 211, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9706877726405082, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.8024790300021005}}
{"text": "\n#Import Turing, Distributions and DataFrames\nusing Turing, Distributions, DataFrames, Distributed\n\n# Import MCMCChain, Plots, and StatsPlots for visualizations and diagnostics.\nusing MCMCChains, Plots, StatsPlots\n\n# Set a seed for reproducibility.\nusing Random\nRandom.seed!(12);\n\n# Turn off progress monitor.\nTuring.setprogress!(false)\n\n\ntheta_noalcohol_meds = 1 # no alcohol, took medicine\ntheta_alcohol_meds = 3 # alcohol, took medicine\ntheta_noalcohol_nomeds = 6 # no alcohol, no medicine\ntheta_alcohol_nomeds = 36 # alcohol, no medicine\n\n# no of samples for each of the above cases\nq = 100\n\n#Generate data from different Poisson distributions\nnoalcohol_meds = Poisson(theta_noalcohol_meds)\nalcohol_meds = Poisson(theta_alcohol_meds)\nnoalcohol_nomeds = Poisson(theta_noalcohol_nomeds)\nalcohol_nomeds = Poisson(theta_alcohol_nomeds)\n\nnsneeze_data = vcat(\n rand(noalcohol_meds, q),\n rand(alcohol_meds, q),\n rand(noalcohol_nomeds, q),\n rand(alcohol_nomeds, q),\n)\nalcohol_data = vcat(zeros(q), ones(q), zeros(q), ones(q))\nmeds_data = vcat(zeros(q), zeros(q), ones(q), ones(q))\n\ndf = DataFrame(;\n nsneeze=nsneeze_data,\n alcohol_taken=alcohol_data,\n nomeds_taken=meds_data,\n product_alcohol_meds=meds_data .* alcohol_data,\n)\ndf[sample(1:nrow(df), 5; replace=false), :]\n\n\n#Data Plotting\n\np1 = Plots.histogram(\n df[(df[:, :alcohol_taken] .== 0) .& (df[:, :nomeds_taken] .== 0), 1];\n title=\"no_alcohol+meds\",\n)\np2 = Plots.histogram(\n (df[(df[:, :alcohol_taken] .== 1) .& (df[:, :nomeds_taken] .== 0), 1]);\n title=\"alcohol+meds\",\n)\np3 = Plots.histogram(\n (df[(df[:, :alcohol_taken] .== 0) .& (df[:, :nomeds_taken] .== 1), 1]);\n title=\"no_alcohol+no_meds\",\n)\np4 = Plots.histogram(\n (df[(df[:, :alcohol_taken] .== 1) .& (df[:, :nomeds_taken] .== 1), 1]);\n title=\"alcohol+no_meds\",\n)\nplot(p1, p2, p3, p4; layout=(2, 2), legend=false)\n\n\n# Convert the DataFrame object to matrices.\ndata = Matrix(df[:, [:alcohol_taken, :nomeds_taken, :product_alcohol_meds]])\ndata_labels = df[:, :nsneeze]\ndata\n\n\n# # Rescale our matrices.\ndata = (data .- mean(data; dims=1)) ./ std(data; dims=1)\n\n\n# Bayesian poisson regression (LR)\n@model function poisson_regression(x, y, n, σ²)\n b0 ~ Normal(0, σ²)\n b1 ~ Normal(0, σ²)\n b2 ~ Normal(0, σ²)\n b3 ~ Normal(0, σ²)\n for i in 1:n\n theta = b0 + b1 * x[i, 1] + b2 * x[i, 2] + b3 * x[i, 3]\n y[i] ~ Poisson(exp(theta))\n end\nend;\n\n\n# Retrieve the number of observations.\nn, _ = size(data)\n\n# Sample using NUTS.\n\nnum_chains = 4\nm = poisson_regression(data, data_labels, n, 10)\nchain = sample(m, NUTS(200, 0.65), MCMCThreads(), 2_500, num_chains; discard_adapt=false)\n\n\ngelmandiag(chain)\n\n\n# Taking the first chain\nc1 = chain[:, :, 1]\n\n# Calculating the exponentiated means\nb0_exp = exp(mean(c1[:b0]))\nb1_exp = exp(mean(c1[:b1]))\nb2_exp = exp(mean(c1[:b2]))\nb3_exp = exp(mean(c1[:b3]))\n\nprint(\"The exponent of the meaned values of the weights (or coefficients are): \\n\")\nprintln(\"b0: \", b0_exp)\nprintln(\"b1: \", b1_exp)\nprintln(\"b2: \", b2_exp)\nprintln(\"b3: \", b3_exp)\nprint(\"The posterior distributions obtained after sampling can be visualised as :\\n\")\n\n\nplot(chain)\n\n\n# Note the standard deviation before removing the warmup samples\ndescribe(chain)\n\n\n# Removing the first 200 values of the chains.\nchains_new = chain[201:2500, :, :]\ndescribe(chains_new)\n\n\nplot(chains_new)\n\n\nif isdefined(Main, :TuringTutorials)\n Main.TuringTutorials.tutorial_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file])\nend\n\n", "meta": {"hexsha": "7ce8fdeeb722292ed012ff0649c71da782fc0aa2", "size": 3500, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "script/07-poisson-regression/07_poisson-regression.jl", "max_stars_repo_name": "TuringLang/TuringTutorialsOutput", "max_stars_repo_head_hexsha": "0928cc205032f4b117b5d1dd1512b1621fb5674a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "script/07-poisson-regression/07_poisson-regression.jl", "max_issues_repo_name": "TuringLang/TuringTutorialsOutput", "max_issues_repo_head_hexsha": "0928cc205032f4b117b5d1dd1512b1621fb5674a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "script/07-poisson-regression/07_poisson-regression.jl", "max_forks_repo_name": "TuringLang/TuringTutorialsOutput", "max_forks_repo_head_hexsha": "0928cc205032f4b117b5d1dd1512b1621fb5674a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.8226950355, "max_line_length": 89, "alphanum_fraction": 0.6857142857, "num_tokens": 1127, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9334308091776495, "lm_q2_score": 0.8596637451167997, "lm_q1q2_score": 0.802436625225063}}
{"text": "\"\"\"\n\nLicense: MIT\nJulia 1.02\nDynamic programming formulation to count number of matchings of maximum cardinality\n\"\"\"\n\nusing Printf\n# INPUT\n# G - graph generated using Graph_generation\n# Recursive_table - dictionary computed in CountingMatchings_rosso.jl with recursive approach\n# maxCard - size of the maximum cardinality matching\nusing Distributions\nfunction UniformMaxMatching_julia(G,E,Recursive_table,maxCard)\n set_V = sort(collect(keys(G)))\n s_aux = [1 for v in set_V]\n r = length(E)\n sol_M = [] # it will contain the edges in the generated matching\n for e in reverse(E)\n # e is in set_E with prob:\n aux = rand()\n if r-1== 0 # if r-1 = 0 then x_r must be added, otherwise sol_M alread had maxCard nodes and the algorithm had stopped\n prob_no_e = 0\n else\n # probability of x_e = 0\n prob_no_e = Recursive_table[r-1,maxCard,s_aux]/(Recursive_table[r,maxCard,s_aux])\n end\n if aux > prob_no_e\n # then x_e = 1\n append!(sol_M,[Set(e)])\n #s_aux = [1 if v not in e else 0 for v in set_V]\n s_aux = [if !(v in e) s_aux[j] else s_aux[j]-1 end for (j,v) in enumerate(set_V)]\n maxCard = maxCard - 1\n end\n r = r-1\n if maxCard == 0\n return sol_M\n end\n end\n return sol_M\nend\n\n# ask the number of matching to generate\nfunction Uniform_Number(G,E,gen_num,info)\n # maximum cardinality\n maxCard = OptimalMatching(G,E)\n ## save maxCard\n if info==1\n maxCardinality = @sprintf \"%d\" maxCard\n fres = open(\"results.txt\",\"a\")\n write(fres,\"maxCard= \",maxCardinality, \" \\n\")\n close(fres)\n end\n ## end save\n r = length(E)\n s = Dict(v=>1 for v in keys(G))\n F_table,Numb_matchings= Count_Mat_Paper(G, maxCard,E,r,s,1)\n ## save number of maximum matchings\n Numb_OPT_matchings = @sprintf \"%d\" Numb_matchings\n if info==1\n fres = open(\"results.txt\",\"a\")\n write(fres,\"Numb_matchings= \",Numb_OPT_matchings, \" \\n\")\n close(fres)\n end\n ## end save\n fgen = open(\"tmp.txt\",\"w\")\n for _ in 1:gen_num\n gen_M =UniformMaxMatching_julia(G,E,F_table,maxCard)\n aux = 0\n for e in gen_M\n e = [v for v in e]\n if aux ==0\n write(fgen,\"[{\")\n else\n write(fgen,\",{\")\n end\n write(fgen,string(e[1])*\",\"*string(e[2])*\"}\")\n aux = aux+1\n end\n if length(gen_M)==0\n write(fgen,\"[{}]\\n\")\n else\n write(fgen,\"] \\n\")\n end\n end\n close(fgen)\nend\n", "meta": {"hexsha": "0c05acb8c7adb6a1b0c51eeb51f6ef737ed01441", "size": 2638, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "UniformGeneration_julia.jl", "max_stars_repo_name": "mxmmargarida/KEG", "max_stars_repo_head_hexsha": "0e8e1920b2bfe13d0d3c60863f38ca0a5f5793a1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "UniformGeneration_julia.jl", "max_issues_repo_name": "mxmmargarida/KEG", "max_issues_repo_head_hexsha": "0e8e1920b2bfe13d0d3c60863f38ca0a5f5793a1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "UniformGeneration_julia.jl", "max_forks_repo_name": "mxmmargarida/KEG", "max_forks_repo_head_hexsha": "0e8e1920b2bfe13d0d3c60863f38ca0a5f5793a1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.9772727273, "max_line_length": 126, "alphanum_fraction": 0.5750568613, "num_tokens": 744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632916317103, "lm_q2_score": 0.865224084314688, "lm_q1q2_score": 0.8023770548291015}}
{"text": "\n\"\"\"\n transformLog(data::T, base::Real = ℯ) where T <: AbstractArray{<: Number}\n\nLog transform `data` using log-base, `base`.\n\n# Examples:\n\n```julia-repl\njulia> data = [0:5;];\n\njulia> transformLog(data)\n6-element Array{Float64,1}:\n -Inf\n 0.0\n 0.6931471805599453\n 1.0986122886681098\n 1.3862943611198906\n 1.6094379124341003\n\njulia> transformLog(data,2)\n6-element Array{Float64,1}:\n -Inf\n 0.0\n 1.0\n 1.5849625007211563\n 2.0\n 2.321928094887362\n\n```\n\nSee also: [`transformRoot`](@ref), [`transformBoxCox`](@ref)\n\"\"\"\nfunction transformLog(data::T, base::Real = ℯ) where T <: AbstractArray{<: Number}\n log.(base,data)\nend\n\n\"\"\"\n transformRoot(data::T, index::Real = 10) where T <: AbstractArray{<: Number}\n\nRoot transform `data` using root index, `index`.\n\n# Examples:\n\n```julia-repl\njulia> data = [0:5;];\n\njulia> transformRoot(data)\n6-element Array{Float64,1}:\n 0.0\n 1.0\n 1.0717734625362931\n 1.1161231740339044\n 1.148698354997035\n 1.174618943088019\n\njulia> transformRoot(data,2)\n6-element Array{Float64,1}:\n 0.0\n 1.0\n 1.4142135623730951\n 1.7320508075688772\n 2.0\n 2.23606797749979\n```\n\nSee also: [`transformLog`](@ref), [`transformBoxCox`](@ref)\n\"\"\"\nfunction transformRoot(data::T, index::Real = 10) where T <: AbstractArray{<: Number}\n data .^ (1/index)\nend\n\n\"\"\"\n transformBoxCox(data::T, λ::Real = 0.0) where T <: AbstractArray{<: Number}\n\n[Box-Cox power transformation](https://en.wikipedia.org/wiki/Power_transform#Box%E2%80%93Cox_transformation)\nfollowing the following function:\n\n```math\ny_i^{(\\\\lambda)} = \\\\left\\\\{\\\\begin{matrix}\n\\\\frac{y_i^\\\\lambda - 1}{\\\\lambda} & \\\\mathrm{if} \\\\lambda \\\\neq 0, \\\\\\\\ \n\\\\mathrm{ln} y_i & \\\\mathrm{if} \\\\lambda = 0, \n\\\\end{matrix}\\\\right.\n```\n\n# Examples:\n\n```julia-repl\njulia> data = [0:5;];\n\njulia> transformBoxCox(data)\n6-element Array{Float64,1}:\n -Inf\n 0.0\n 0.6931471805599453\n 1.0986122886681098\n 1.3862943611198906\n 1.6094379124341003\n\njulia> transformBoxCox(data,.1)\n6-element Array{Float64,1}:\n -10.0\n 0.0\n 0.7177346253629313\n 1.1612317403390437\n 1.486983549970351\n 1.7461894308801895\n\njulia> transformBoxCox(data,1)\n6-element Array{Float64,1}:\n -1.0\n 0.0\n 1.0\n 2.0\n 3.0\n 4.0\n```\n\nSee also: [`transformLog`](@ref), [`transformRoot`](@ref)\n\"\"\"\nfunction transformBoxCox(data::T, λ::Real = 0.0) where T <: AbstractArray{<: Number}\n λ == 0 ? log.(data) : (data .^ λ .- 1) ./ λ \nend\n\n\n", "meta": {"hexsha": "51e8e851076576de61a74ecd48a4339bffc8b011", "size": 2386, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/numeric/transforming.jl", "max_stars_repo_name": "a-poor/FeatureEng.jl", "max_stars_repo_head_hexsha": "d8aa98b29dd707c66812582dc0dd68506b46f015", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-05T14:39:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-26T18:50:49.000Z", "max_issues_repo_path": "src/numeric/transforming.jl", "max_issues_repo_name": "a-poor/FeatureEng.jl", "max_issues_repo_head_hexsha": "d8aa98b29dd707c66812582dc0dd68506b46f015", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2021-01-29T00:40:58.000Z", "max_issues_repo_issues_event_max_datetime": "2021-02-03T16:33:04.000Z", "max_forks_repo_path": "src/numeric/transforming.jl", "max_forks_repo_name": "a-poor/FeatureEng.jl", "max_forks_repo_head_hexsha": "d8aa98b29dd707c66812582dc0dd68506b46f015", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.9365079365, "max_line_length": 108, "alphanum_fraction": 0.6647108131, "num_tokens": 926, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632976542185, "lm_q2_score": 0.865224073888819, "lm_q1q2_score": 0.8023770503713524}}
{"text": "module SimpleQuaternions\n\nusing LinearAlgebra\n\nexport SimpleQuaternion, jm, km, get_parts\n\nimport Base: (+), (-), (*), (/), (//), (==), inv, string, show, adjoint, conj\nimport Base: real, isreal, zero, one\nimport LinearAlgebra: norm\nexport norm\n\n\nstruct SimpleQuaternion{T} <: Number\n a::T \n b::T\n c::T \n d::T \n function SimpleQuaternion(x::Real=0, y::Real=0, z::Real=0, w::Real=0)\n (aa,bb,cc,dd) = promote(x,y,z,w)\n T = typeof(aa)\n new{T}(aa,bb,cc,dd)\n end \nend\n\n\"\"\"\n`get_parts(x::SimpleQuaternion)` returns the tuple `(a,b,c,d)` that are \nreal and various imaginary get_parts of `x`. Specifically, \n`x == a + b*im + c*jm + d*km`.\n\"\"\"\nget_parts(x::SimpleQuaternion) = (x.a,x.b,x.c,x.d)\n\n\"\"\"\n`SimpleQuaternion` creates a quaternion number. The basic constructor \nis `SimpleQuaternion(a,b,c,d)` which is equivalent to \n`a + b*im + c*jm + d*km`.\n\n`SimpleQuaternion(x::Real)` yields `x + 0*im + 0*jm + 0*km`.\n\nIf `z` is the `Complex` number `a+b*im` then `SimpleQuaternion(z)`\nyields `a + b*im + 0*jm + 0*km`.\n\"\"\"\nSimpleQuaternion(z::Complex) = SimpleQuaternion(real(z),imag(z),0,0)\n\n\nfunction SimpleQuaternion{T}(a::Real=0,b::Real=0,c::Real=0,d::Real=0) where T<:Real\n (x,aa,bb,cc,dd) = promote(one(T),a,b,c,d)\n SimpleQuaternion(aa,bb,cc,dd)\nend \n\n\"\"\"\n`jm` is the quaternion unit *j*.\n\"\"\"\njm = SimpleQuaternion(false,false,true,false)\n\n\"\"\"\n`km` is the quaternion unit *k*.\n\"\"\"\nkm = SimpleQuaternion(false,false,false,true)\n\nfunction (==)(x::SimpleQuaternion,y::SimpleQuaternion)\n get_parts(x) == get_parts(y)\nend\n\n\"\"\"\n`real(x::SimpleQuaternion)` returns the real part of `x`.\n\"\"\"\nreal(x::SimpleQuaternion) = x.a\n\n\"\"\"\n`isreal(x::SimpleQuaternion)` returns `true` exactly when `x` is a real \nnumber (all complex get_parts are zero).\n\"\"\"\nisreal(x::SimpleQuaternion) = x == x.a \n\n(==)(x::SimpleQuaternion, y::Number) = x == SimpleQuaternion(y)\n(==)(x::Number, y::SimpleQuaternion) = SimpleQuaternion(x) == y\n\none(a::SimpleQuaternion{T}) where T = SimpleQuaternion(one(T))\nzero(a::SimpleQuaternion{T}) where T = SimpleQuaternion(zero(T))\n\none(::Type{SimpleQuaternion{T}}) where T = SimpleQuaternion(one(T))\nzero(::Type{SimpleQuaternion{T}}) where T = SimpleQuaternion(zero(T))\n\ninclude(\"arithmetic.jl\")\ninclude(\"show.jl\")\ninclude(\"matrices.jl\")\ninclude(\"random.jl\")\n\nend # module\n", "meta": {"hexsha": "5da7377922d79c8e21e26af75806eb71e43d4986", "size": 2321, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/SimpleQuaternions.jl", "max_stars_repo_name": "scheinerman/SimpleQuaternions.jl", "max_stars_repo_head_hexsha": "ad494b7d866fc74c359177155bc58fa2c6b66f07", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/SimpleQuaternions.jl", "max_issues_repo_name": "scheinerman/SimpleQuaternions.jl", "max_issues_repo_head_hexsha": "ad494b7d866fc74c359177155bc58fa2c6b66f07", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/SimpleQuaternions.jl", "max_forks_repo_name": "scheinerman/SimpleQuaternions.jl", "max_forks_repo_head_hexsha": "ad494b7d866fc74c359177155bc58fa2c6b66f07", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-04-13T11:56:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-04-13T11:56:34.000Z", "avg_line_length": 25.7888888889, "max_line_length": 83, "alphanum_fraction": 0.6591986213, "num_tokens": 711, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.927363293639213, "lm_q2_score": 0.865224072151174, "lm_q1q2_score": 0.8023770452860448}}
{"text": "#solve eq x=cos(x)\n#Reference: http://nbviewer.jupyter.org/gist/genkuroki/ce7f62d10365b9a25c82aad499c886c2\n\nfunction bisection(f, x₀::T, x₁::T; ε=eps(T), maxiter=10^8) where T<:AbstractFloat\n y₀, y₁ = f(x₀), f(x₁)\n iter = 0\n y₀ == 0 && return x₀, iter\n y₁ == 0 && return x₁, iter\n y₀ * y₁ > 0 && return T(NaN), -1\n for iter in 1:maxiter\n x₂ = (x₀ + x₁)/2\n y₂ = f(x₂)\n y₂ == 0 && return x₂, iter\n if y₀ * y₂ > 0\n x₀, y₀ = x₂, y₂\n else\n x₁, y₁ = x₂, y₂\n end\n abs(x₁ - x₀) < ε && return x₂, iter\n end\n return T(NaN), iter\nend\n\nfunction main()\n f(x)=x-cos(x)\n x₀, x₁ = 0.0, 3.14\n @show α, iter = bisection(f, x₀, x₁)\n @show f(α)\nend\n\nmain()", "meta": {"hexsha": "681a33de958ba7629ee93efe60ce04f224675f99", "size": 743, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "solveNonLinear/bisection/bisection_gen_kuroki.jl", "max_stars_repo_name": "terasakisatoshi/juliaExer", "max_stars_repo_head_hexsha": "e3c2195f39de858915a3dcd47684eccbb7ecb552", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-05-02T01:24:20.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-04T12:03:25.000Z", "max_issues_repo_path": "solveNonLinear/bisection/bisection_gen_kuroki.jl", "max_issues_repo_name": "terasakisatoshi/juliaExer", "max_issues_repo_head_hexsha": "e3c2195f39de858915a3dcd47684eccbb7ecb552", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "solveNonLinear/bisection/bisection_gen_kuroki.jl", "max_forks_repo_name": "terasakisatoshi/juliaExer", "max_forks_repo_head_hexsha": "e3c2195f39de858915a3dcd47684eccbb7ecb552", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.9677419355, "max_line_length": 87, "alphanum_fraction": 0.5181695828, "num_tokens": 319, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9653811591688146, "lm_q2_score": 0.8311430415844385, "lm_q1q2_score": 0.8023698329198795}}
{"text": "function energy(planets)\n eng = 0.0\n # kinetic energy\n for p in planets\n eng += 1/2*p.m*norm2(p.v)\n end\n # potential energy\n for j in 1:nplanets\n pj = planets[j]\n for k in j+1:nplanets\n pk = planets[k]\n eng -= G*pj.m*pk.m/sqdist(pj.r, pk.r)\n end\n end\n eng\nend\n \nfunction barycenter(m,mTot,x,y,z) # Find Barycenter\n # m : mass of planet\n # mTot : total mass of system\n # x,y,z: position coordinates of planet\n #\n baryX = (m*x)/mTot\n baryY = (m*y)/mTot\n baryZ = (m*z)/mTot\n return baryX,baryY,baryZ;\nend\n\nfunction momentum(m,x,y,z,vx,vy,vz)\n r = [x,y,z];\n vel = [vx,vy,vz];\n H = m*cross(r,vel);\n return H;\nend\n\n", "meta": {"hexsha": "86680af3f6f78b61d67052dd6d3ef1560a64a154", "size": 725, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/utils.jl", "max_stars_repo_name": "JuliaReverse/NBodyLeapFrog.jl", "max_stars_repo_head_hexsha": "6ac5e78760ad0078bcc71c7afb525630326bd5e3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/utils.jl", "max_issues_repo_name": "JuliaReverse/NBodyLeapFrog.jl", "max_issues_repo_head_hexsha": "6ac5e78760ad0078bcc71c7afb525630326bd5e3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/utils.jl", "max_forks_repo_name": "JuliaReverse/NBodyLeapFrog.jl", "max_forks_repo_head_hexsha": "6ac5e78760ad0078bcc71c7afb525630326bd5e3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.1388888889, "max_line_length": 51, "alphanum_fraction": 0.5379310345, "num_tokens": 257, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122720843811, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.802366442219827}}
{"text": "using Distributions\nusing Gadfly\nusing PMTK4Stats\n\n\"\"\"\nPlot CI and HPD for beta posterior\n\"\"\"\nfunction drawBetaHPD()\n a = 3\n b = 9\n α = 0.05\n β = Beta(a, b) \n\n CI = quantile(β, [α/2, 1-α/2])\n \n xs = linspace(0.001, 0.999, 40)\n ps = pdf(β, xs)\n \n icdf(p::Real) = quantile(β, p)\n HPD = hdiFromIcdf(icdf)\n \n ints = hcat(CI, HPD)\n names = [\"CI\"; \"HPD\"]\n \n function plot_ci(int, name)\n l, u = int\n pl = pdf(β, l)\n pu = pdf(β, u)\n vert_up_l = layer(x=[l, l], y=[0, pl], Geom.line, Theme(default_color=colorant\"blue\", line_width=3px))\n hor_l = layer(x=[l, u], y=[pl, pu], Geom.line, Theme(default_color=colorant\"blue\", line_width=3px))\n vert_dn_l = layer(x=[u, u], y=[pu, 0], Geom.line, Theme(default_color=colorant\"blue\", line_width=3px))\n pdf_l = layer(x=xs, y=ps, Geom.line, Theme(default_color=colorant\"black\", line_width=3px))\n \n pp = plot(vert_up_l, hor_l, vert_dn_l, pdf_l)\n end\n \n [plot_ci(slicedim(ints, 2, i), names[i]) for i=1:length(names)]\nend\n\n", "meta": {"hexsha": "ff840d2ec5fe7abd755eaa0b6868a847ea70811a", "size": 1077, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/BayesianStatistics/betaHPD.jl", "max_stars_repo_name": "aoboturov/PMTK4BookDemos.jl", "max_stars_repo_head_hexsha": "f44853081139b358e0e9a2bb36a3d35ac35dcb39", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-06-25T15:50:42.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-25T15:50:42.000Z", "max_issues_repo_path": "src/BayesianStatistics/betaHPD.jl", "max_issues_repo_name": "aoboturov/PMTK4BookDemos.jl", "max_issues_repo_head_hexsha": "f44853081139b358e0e9a2bb36a3d35ac35dcb39", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/BayesianStatistics/betaHPD.jl", "max_forks_repo_name": "aoboturov/PMTK4BookDemos.jl", "max_forks_repo_head_hexsha": "f44853081139b358e0e9a2bb36a3d35ac35dcb39", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.925, "max_line_length": 111, "alphanum_fraction": 0.5710306407, "num_tokens": 382, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122720843811, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.802366442219827}}
{"text": "function lucrout(A)\r\n A = convert(Array{Float64,2},A);\r\n numRow, numCol = size(A);\r\n if numRow != numCol\r\n return println(\"Non-square matrix\")\r\n end\r\n\r\n L, U = zeros(numRow,numCol), zeros(numRow,numCol); # initialize\r\n\r\n # We first put the first column of L and the ones in U\r\n for j in 1:numRow\r\n L[j,1] = A[j,1];\r\n U[j,j] = 1.0;\r\n end\r\n\r\n # We then obtain the fist row of U by dividing\r\n for j = 2:numRow\r\n U[1,j] = A[1,j]/L[1,1];\r\n end\r\n\r\n # Now we try to calculate the rest\r\n\r\n for i = 2:numCol\r\n # We calculate all the i-th column of L\r\n for j in i:numRow\r\n S = 0; #initialize\r\n for k in 1:i-1\r\n S += L[j,k]*U[k,i];\r\n end\r\n L[j,i] = A[j,i] - S;\r\n end\r\n \r\n #We know calculate the U-th row of U \r\n for j = (i+1):numCol\r\n S = 0; # Initialize\r\n for k= 1:(i-1)\r\n S += L[i,k]*U[k,j];\r\n end\r\n U[i,j] = 1/L[i,i]*(A[i,j]-S);\r\n end\r\n\r\n end\r\n return L,U\r\nend\r\n\r\n", "meta": {"hexsha": "3150049df9f88f20e7b4ded08c24a322867d02ad", "size": 1100, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/lucrout.jl", "max_stars_repo_name": "leogabac/FiscomTools.jl", "max_stars_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/lucrout.jl", "max_issues_repo_name": "leogabac/FiscomTools.jl", "max_issues_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/lucrout.jl", "max_forks_repo_name": "leogabac/FiscomTools.jl", "max_forks_repo_head_hexsha": "6f47512c3ccc1a866088141fc367554cdf88dc84", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.9130434783, "max_line_length": 68, "alphanum_fraction": 0.4427272727, "num_tokens": 338, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122672782974, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.8023664401356022}}
{"text": "\"\"\"\n determinant(mat)\n\nGiven a non singluar matrix, calculate its determinant using LU decomposition.\n\nL and U are lower triangular and upper triangular matrices respectively such that\n\nA = L*U\n\nIf we want to find the determinant, then\n\ndet(A) = det(LU) = det(L)*det(U)\n\nDeterminant of triangualar matrices is the product of their diagonal entries. Hence, makes finding the determinant easy.\n\"\"\"\nfunction determinant(mat)\n n, m = size(mat)\n if n != m\n DomainError(mat, \"The matrix should be a square matrix.\")\n end\n L, U = lu_decompose(mat)\n l_prod = prod([L[i, i] for i in 1:n])\n u_prod = prod([U[i, i] for i in 1:n])\n\n return l_prod * u_prod\nend\n", "meta": {"hexsha": "b4bf2faf7ec2b6bd82d7ca68e5cce8a0622bb43c", "size": 679, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/matrix/determinant.jl", "max_stars_repo_name": "Whiteshark-314/Julia", "max_stars_repo_head_hexsha": "3285d8d6b7585cc1075831c2c210b891151da0c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-10-14T21:48:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-14T21:48:50.000Z", "max_issues_repo_path": "src/matrix/determinant.jl", "max_issues_repo_name": "AugustoCL/Julia", "max_issues_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/matrix/determinant.jl", "max_forks_repo_name": "AugustoCL/Julia", "max_forks_repo_head_hexsha": "1bf4e4a7829fafc64290d903bcbfdd48eab839e7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.1481481481, "max_line_length": 120, "alphanum_fraction": 0.6833578792, "num_tokens": 180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9579122720843812, "lm_q2_score": 0.8376199552262967, "lm_q1q2_score": 0.8023664344540395}}
{"text": "\"\"\"\n\n```\nsphericalshellvolume(i,step)\n```\n\nComputes the volume of the spherical shell defined within [(i-1)*step,i*step].\n\n\"\"\"\nfunction sphericalshellvolume(i,step)\n rmin = (i-1)*step\n return (4*pi/3)*( (rmin+step)^3 - rmin^3 )\nend\n\n\n", "meta": {"hexsha": "3967f9d6eccebe921bb2a09451692c98e4a29281", "size": 236, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/sphericalshellvolume.jl", "max_stars_repo_name": "m3g/ComplexMixtures.jl", "max_stars_repo_head_hexsha": "0ba3d05f31742752dae37bd4192122d95c984c7f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2021-05-14T02:35:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-22T15:56:19.000Z", "max_issues_repo_path": "src/sphericalshellvolume.jl", "max_issues_repo_name": "m3g/ComplexMixtures.jl", "max_issues_repo_head_hexsha": "0ba3d05f31742752dae37bd4192122d95c984c7f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2021-03-18T19:41:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-26T13:56:17.000Z", "max_forks_repo_path": "src/sphericalshellvolume.jl", "max_forks_repo_name": "m3g/ComplexMixtures.jl", "max_forks_repo_head_hexsha": "0ba3d05f31742752dae37bd4192122d95c984c7f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 14.75, "max_line_length": 78, "alphanum_fraction": 0.656779661, "num_tokens": 76, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9579122672782974, "lm_q2_score": 0.8376199572530448, "lm_q1q2_score": 0.8023664323698148}}
{"text": "using Plots\n\n\nfunction rosenbrock(x; a=1, b=100)\n (a-x[1])^2 + b * (x[2] - x[1]^2)^2;\nend\n\n# plot\n\n# contour(-2:0.1:2, -1:0.1:3, (x, y)->rosenbrock([x, y]))\nheatmap(-2:0.01:2, -1:0.01:3, (x, y)->rosenbrock([x, y]))\n", "meta": {"hexsha": "934f9e8ee9a30c5f0a3d2107b9421bceff4cee79", "size": 218, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "test_function/rosenbrocks_banana_function.jl", "max_stars_repo_name": "tor4z/convex_optimization", "max_stars_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "test_function/rosenbrocks_banana_function.jl", "max_issues_repo_name": "tor4z/convex_optimization", "max_issues_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "test_function/rosenbrocks_banana_function.jl", "max_forks_repo_name": "tor4z/convex_optimization", "max_forks_repo_head_hexsha": "15fd3aa09fbc3306ff68cc301bbddac3d2006f3f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.1666666667, "max_line_length": 57, "alphanum_fraction": 0.5183486239, "num_tokens": 117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9603611631680358, "lm_q2_score": 0.8354835350552603, "lm_q1q2_score": 0.8023659395334122}}
{"text": "using LinearAlgebra\n\n\"\"\"Optimize some function f\n# Args:\n ∇f: Gradient of f\n x_0: Initial guess\n η: learning rate\n ε: cancellation tolerance\n max_iters: maximum number of iterations\n\"\"\"\nfunction gradient_descent(∇f::Function,\n x_0,\n η::Number,\n ε::Number,\n max_iters::Integer)\n x_k = x_0\n count = 0\n for _ = 1:max_iters\n if any(isnan.(x_k))\n error(\"Encoutered NaN\")\n end\n if any(isinf.(x_k))\n error(\"Encoutered Inf\")\n end\n count += 1\n p_k = -∇f(x_k)\n if norm(∇f(p_k)) < ε\n break\n end\n x_k += η * p_k\n end\n x_k\nend\n\n#numeric_differentiation(f, h) = x -> (f(x + h) - f(x)) / h\nnumeric_differentiation(f, h) = x -> (f(x + h) - f(x - h)) / (2 * h)\n\n\"\"\"BFGS Optimization Algorithm\nBroyden–Fletcher–Goldfarb–Shanno algorithm\nfor details see:\n https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm\n\n# Args:\n ∇f: Gradient of function to optimize\n x_0: Initial guess for optimal value\n iters: Maximum number of iterations before cancellation\n line_search: Line search function that's used\n ε: Optimization stops once the norm of the gradient is below this value\n ls_x_0: Initial value for the line search\n ls_η: Learning rate for the line search\n ε_ls: same as ε but for the line search\n\"\"\"\nfunction BFGS(f::Function,\n ∇f::Function, \n x_0::Vector{<:Number}, \n iters::Integer,\n line_search::Function,\n ε = 10e-12::Real,\n )\n n = size(x_0)[1]\n x_k = x_0\n B_k = Matrix{typeof(x_0[1])}(I, n, n)\n did_iters = 0\n for i = 1:iters\n did_iters += 1\n if norm(∇f(x_k)) < ε\n break\n end\n # Step 1: obtain direction p_k by solving B_k ∙ p_k = - (gradient of f at x_k)\n p_k = B_k \\ -∇f(x_k)\n\n # Step 2.: Find stepsize α_k such that α_k = arg min ∂f(x_k + α_k * p_k)/∂α\n α_k = line_search(numeric_differentiation(α->(f(x_k + α * p_k)), 10e-10))\n if isnan(α_k)\n break\n end\n \n # Step 3.\n s_k = α_k * p_k\n x_k_prime = x_k + s_k\n \n # Step 4.\n y_k = ∇f(x_k_prime) - ∇f(x_k)\n\n # Step 5.\n B_k += (y_k * y_k') / (y_k' * s_k) - (B_k * s_k * s_k' * B_k) / (s_k' * B_k * s_k)\n x_k = x_k_prime\n end\n println(did_iters)\n x_k\nend\n\n# f(x) = -5 * x[1] + 10 * x[1]^2\n# ∇f(x) = [-5 + 20 * x[1]]\nf(x) = x[1]^5 / 5000 + 21 * x[1]^4 / 4000 + 17 * x[1]^3 / 375 + 293 * x[1]^2 / 1000 + 521 * x[1] / 1000\n∇f(x) = [x[1]^4 / 1000 + 21 * x[1]^3 / 1000 + 17 * x[1]^2 / 125 + 293 * x[1] / 500 + 521 / 1000]\na = 1\nb = 100\nr(X) = begin\n x = X[1]\n y = X[2]\n (a - x)^2 + b*(y - x^2)^2\nend\n∇r(X) = begin\n x = X[1]\n y = X[2]\n [-2*a+4*b*x^3-4*b*x*y+2*x, 2*b*(y-x^2)]\nend\n#f(X) = begin\n# x = X[1]\n# y = X[2]\n# x^2 + y^2\n#end\n#∇f(X) = begin\n# x = X[1]\n# y = X[2]\n# [2*x, 2*y]\n#end\nprintln(\"optimal x = \", BFGS(\n f,\n ∇f,\n [0],\n 500,\n ∇f -> gradient_descent(∇f, 10e-10, 1, 1e-10, 1000),\n 10e-5))\n\n\n# f(w, x) = w[1] * x[1] + w[2] * x[1]^2\n\n# ∇f(w, x) = [w[1] + 2 * x[1] * w[2]] # [x; x^2]\n\n# # initialize parameters\n# w_0 = [-5, 10]\n# x_0 = [-5]\n# B_0 = Matrix{Float64}(I, 1, 1) # initialize approximate hessian with identity matrix\n# ε = 1e-10 # cancellation parameter\n\n# w_k = w_0\n# x_k = x_0\n# B_k = B_0\n\n# while begin\n# x = norm(∇f(w_k, x_k))\n# println(\"∇f = \", x)\n# x > ε\n# end\n# global B_k\n# global x_k\n# # Step 1.: obtain direction p_k by solving B_k*p_k = - (gradient of f at x_k)\n\n# p_k = B_k \\ -∇f(w_k, x_k)\n\n# # Step 2.: Find stepsize α_k such that α_k minimizes f(x_k + α_k * p_k)\n\n# g_k(α) = w_k[1] + 2 * (x_k[1] + α * p_k[1]) * w_k[2]\n# α_k = line_search_gradient_descent(g_k, 0, 0.000001, 1e-10)\n# println(\"α_k = \", α_k)\n\n# # Step 3.\n# s_k = α_k * p_k\n# x_k_prime = x_k + s_k\n\n# # Step 4.\n# y_k = ∇f(w_k, x_k_prime) - ∇f(w_k, x_k)\n\n# # Step 5.\n# B_k += (y_k * y_k') / (y_k' * s_k) - (B_k * s_k * s_k' * B_k) / (s_k' * B_k * s_k)\n# x_k = x_k_prime\n# end\n\n# println(x_k)\n\n\"\"\"\nbegin\n inv_B = inv(B_k)\n (s_k' * y_k + y_k' * inv_B * y_k) * (s_k * s_k') / (s_k' * y_k)^2 - (inv_B * y_k * s_k' + s_k * y_k' * inv_B) / (s_k' * y_k)\nend\n\"\"\"\n", "meta": {"hexsha": "eea1942c4a00bf0271e1a056c3b42e3238593e20", "size": 4308, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "BFGS.jl", "max_stars_repo_name": "SV-97/LinearRegression", "max_stars_repo_head_hexsha": "32c5efa06bf3bd5e541fd9b084a0827c642cdf10", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "BFGS.jl", "max_issues_repo_name": "SV-97/LinearRegression", "max_issues_repo_head_hexsha": "32c5efa06bf3bd5e541fd9b084a0827c642cdf10", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "BFGS.jl", "max_forks_repo_name": "SV-97/LinearRegression", "max_forks_repo_head_hexsha": "32c5efa06bf3bd5e541fd9b084a0827c642cdf10", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.0670391061, "max_line_length": 128, "alphanum_fraction": 0.519266481, "num_tokens": 1774, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096204605946, "lm_q2_score": 0.8757869965109764, "lm_q1q2_score": 0.8023168929779947}}
{"text": "using Printf;\nusing Plots;\n\nfunction init()\n\t## 横の長さ\n\tglobal lx = 1.0;\n\t## 縦の長さ\n\tglobal ly = 1.0;\n\t## 横の分割数\n\tglobal nx = 11;\n\t## 縦の分割数\n\tglobal ny = 11;\n\tglobal dx = lx / (nx - 1);\n\tglobal dy = ly / (ny - 1);\n\t## 係数\n\tbeta_x = 1.0 / dx^2;\n\tbeta_y = 1.0 / dy^2;\n\tglobal x = range(0.0, stop=lx, length=nx);\n\tglobal y = range(0.0, stop=ly, length=ny);\n\tglobal u = zeros(Float64, (ny, nx));\n\t## 境界条件の適用\n\tu[:, begin] .= 0.0;\n\tu[:, end] = 100y;\n\tu[begin, :] .= 0.0;\n\tu[end, :] = 100x;\n\tglobal u_t = zeros(Float64, (ny, nx));\n\tfor j = 1:ny\n\t\tfor i = 1:nx\n\t\t\tu_t[j, i] = 100*x[i]*y[j];\n\t\tend\n\tend\n\tglobal pre_u = copy(u);\n\t## SOR法のωの値を決定\n\tmu = cos(pi/nx) + cos(pi/ny);\n\tglobal omega = 2.0/(1.0 + sqrt(1-(mu/2)^2));\n\tglobal epsilon = 2e-4;\n\tglobal a_ir = beta_x;\n\tglobal a_il = beta_x;\n\tglobal a_ia = beta_y;\n\tglobal a_ib = beta_y;\n\tglobal a_0 = -2.0(beta_x + beta_y);\n\tglobal rhs = 0.0;\n\tglobal iter_max = 100;\n\tglobal epsilon = 2e-4;\nend\n\nfunction main()\n\tinit();\n\tfor iter=1:iter_max\n\t\terr = 0.0;\n\t\tfor j = 1:ny-2\n\t\t\tfor i = 1:nx-2\n\t\t\t\ttmp = (rhs - a_ir*u[j+1, i+2] - a_il*u[j+1, i] - a_ia*u[j+2, i+1] - a_ib*u[j, i+1]) / a_0;\n\t\t\t\tu[j+1, i+1] = u[j+1, i+1] + (tmp - u[j+1, i+1])*omega;;\n\t\t\t\terr += sqrt((u_t[j+1, i+1] - u[j+1, i+1])^2)/(nx*ny);\n\t\t\tend\n\t\tend\n\t\terr = sqrt(sum((u - u_t).^2)/(nx*ny))\n\t\t#err = sqrt(sum((u - pre_u).^2)/(nx*ny)) global pre_u = copy(u);\n\t\t@printf(\"iter: %d, err: %5e\\n\", iter, err);\n\t\tif err < epsilon\n\t\t\tbreak;\n\t\tend\n\tend\n\tcontour(u);\n\tsavefig(\"Laplace.png\");\nend\n\nmain()\n", "meta": {"hexsha": "9e0407c541f8dbb07a8887e3b4d26f090add8622", "size": 1493, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "laplace.jl", "max_stars_repo_name": "Makoto523sys/laplace-equation", "max_stars_repo_head_hexsha": "a1abb57c3cdbf8691142e3df7f8dfa2a8feccf68", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "laplace.jl", "max_issues_repo_name": "Makoto523sys/laplace-equation", "max_issues_repo_head_hexsha": "a1abb57c3cdbf8691142e3df7f8dfa2a8feccf68", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "laplace.jl", "max_forks_repo_name": "Makoto523sys/laplace-equation", "max_forks_repo_head_hexsha": "a1abb57c3cdbf8691142e3df7f8dfa2a8feccf68", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.3285714286, "max_line_length": 94, "alphanum_fraction": 0.5472203617, "num_tokens": 684, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425333801889, "lm_q2_score": 0.8723473779969194, "lm_q1q2_score": 0.8022477526886522}}
{"text": "\n\"\"\"\n unitvector(v)\n\nNormalize input vector `v`.\n\"\"\"\nfunction unitvector(v)\n n = norm(v)\n if n > 0\n return v / n\n else\n # zero vector\n return v\n end\nend\n\n\n\"\"\"\n random_in_unit_disk()\n\nGenerate a uniformly random point within the unit disk.\n\"\"\"\nfunction random_in_unit_disk()\n while true\n p = 2*rand(2) .- 1\n if dot(p, p) < 1\n return p\n end\n end\nend\n\n\n\"\"\"\n random_in_unit_sphere()\n\nGenerate a uniformly random point within the unit sphere.\n\"\"\"\nfunction random_in_unit_sphere()\n while true\n p = 2*rand(3) .- 1\n if dot(p, p) < 1\n return p\n end\n end\nend\n", "meta": {"hexsha": "9254d3c4a9cf1ff8500874545b6dbc5dfa1c69c3", "size": 667, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/util.jl", "max_stars_repo_name": "cmendl/Raytracing.jl", "max_stars_repo_head_hexsha": "896bd19020940bb246804bc6b9546aba45180ec6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/util.jl", "max_issues_repo_name": "cmendl/Raytracing.jl", "max_issues_repo_head_hexsha": "896bd19020940bb246804bc6b9546aba45180ec6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/util.jl", "max_forks_repo_name": "cmendl/Raytracing.jl", "max_forks_repo_head_hexsha": "896bd19020940bb246804bc6b9546aba45180ec6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 14.5, "max_line_length": 57, "alphanum_fraction": 0.5517241379, "num_tokens": 187, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425267730008, "lm_q2_score": 0.8723473829749844, "lm_q1q2_score": 0.8022477515029293}}
{"text": "#=\nTakes one step in the Markov chain according to the given state's transition\nvector.\nInputs:\n- transition_vector: An array of length p where the ith element denotes the\n probability of transitioning into state i from the current\n state (which is implicitly given).\nOutputs:\n- An integer in {1,2,..., p} denoting the index of the next state which occurs\nwith the probability specified in transition_vector.\n=#\nfunction one_step(transition_vector)\n cumulative_vector = cumsum(transition_vector)\n state_index = 1\n U = rand()\n while(U > cumulative_vector[state_index])\n state_index += 1\n end\n\n return state_index\nend\n\n#=\nRuns a Markov chain a fixed number of times and returns the final state.\nInputs:\n- states: An array of length p where each element represents a state of the\n chain. Assumes the ordering is such that the probability of\n transitioning from state i to state j is given by P[i,j].\n Examples:\n [0,1,2,3,4,5]\n [\"red\", \"green\", \"blue\"]\n [0,\"blue\",3,2]\n- P: A (p x p) matrix denoting the transition matrix of the Markov chain.\n Assumes the rows of P sum to 1 and P[i,j] >= 0 for all i,j.\n- S0: The initial state of the Markov chain. Assumed to be an element of the\n state variable.\n- steps: A nonnegative integer specifying the number of steps for the Markov\n chain to go through.\nOutputs:\n- An element of the states variable denoting the state the chain is in after\n the specified number of steps.\n=#\nfunction run_fixed_simulation(states, P, S0, steps)\n\n # Define a dictionary mapping state to their index in P\n state_dict = Dict(states[i] => i for i = 1:length(states))\n\n # Define the current state of the chain\n Sn = S0\n\n # Run through the specified number of steps\n for i = 1:steps\n state_index = get(state_dict, Sn, -1)\n next_state_index = one_step(P[state_index,:])\n Sn = states[next_state_index]\n end\n\n return Sn\nend\n\n#=\nRuns a Markov chain until a state is reached.\nInputs:\n- states: Same as defined in ``run_fixed_simulation`` function\n- P: Same as defined in ``run_fixed_simulation`` function\n- S0: Same as defined in ``run_fixed_simulation`` function\n- Sf: An array of states defining a possible final state. Assumed to be a\n subset of the states variable.\nOutputs:\n- An integer representing the number of steps it took to get to get to state\n contained in the Sf array.\n=#\nfunction run_simulation(states, P, S0, Sf)\n\n # Define a dictionary mapping state to their index in P\n state_dict = Dict(states[i] => i for i = 1:length(states))\n\n # Define the current state of the chain\n Sn = S0\n\n # Keep tracks of number of steps taken\n steps = 0\n\n # Run through the Markov changing\n while !(Sn in Sf) && steps < 1000\n state_index = get(state_dict, Sn, -1)\n next_state_index = one_step(P[state_index,:])\n Sn = states[next_state_index]\n steps += 1\n end\n\n return steps\nend\n\n\n# Define gambler's ruin Markov chain with N = 5 and p = .3\n\n# gambler's ruin state space (χ)\nstates = [0,1,2,3,4,5]\n\n# Transition matrix for gambler's ruin\nP = [1 0 0 0 0 0;\n .7 0 .3 0 0 0;\n 0 .7 0 .3 0 0;\n 0 0 .7 0 .3 0;\n 0 0 0 .7 0 .3;\n 0 0 0 0 0 1]\n\n# Estimate probability of having $5 after 6 turns when starting with $2\nnum_simulations = 100000\nmc_est = mean([run_fixed_simulation(states, P, 2, 6) == 5\n for i = 1:num_simulations])\n\n# Define masters program Markov chain\nstates = [1,2,\"G\",\"D\"]\n\n# Transition matrix for masters Markov chain\nP = [.4 .5 0 .1;\n 0 .3 .6 .1;\n 0 0 1 0;\n 0 0 0 1]\n\nmc_est = mean([run_simulation(states, P, 1, [\"G\", \"D\"]) for i = 1:10000])\n", "meta": {"hexsha": "102c661345aab422f9f282cbefc24501dff1c27f", "size": 3730, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "content/stochastic-approximations/code/markov_chain_simulation.jl", "max_stars_repo_name": "seanrattana/courses", "max_stars_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2019-08-21T07:33:09.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-24T15:46:58.000Z", "max_issues_repo_path": "content/stochastic-approximations/code/markov_chain_simulation.jl", "max_issues_repo_name": "seanrattana/courses", "max_issues_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2019-08-23T06:04:36.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-26T12:47:12.000Z", "max_forks_repo_path": "content/stochastic-approximations/code/markov_chain_simulation.jl", "max_forks_repo_name": "seanrattana/courses", "max_forks_repo_head_hexsha": "9453abc38af26ea49cb00fe1744066fc0de237b3", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2019-08-18T21:23:16.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T19:14:33.000Z", "avg_line_length": 30.0806451613, "max_line_length": 79, "alphanum_fraction": 0.6648793566, "num_tokens": 1047, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425289753969, "lm_q2_score": 0.8723473697001441, "lm_q1q2_score": 0.8022477412160761}}
{"text": "################################################################################\n# loggingOptimisation Problem\n#\n# You are the decision-maker of a production unit (here a logging company) and have a set of possible activities to undergo.\n# How do you choose the activities that lead to a maximisation of the unit's profit? \n# The objective of this problem is hence to find the optimal level of activities to maximise the company profit, given (a) the profitability (gross margin) of each activity, (b) the resources available to the company and (c) the matrix of technical coefficients that link each activity to the resources required (positive coefficient) or provided (negative coefficient) by that specific activity.\n\n# The problem is the same as those in the SpreadSheet file \"Optimal production mix.ods\" (available in the `data` folder) and can be solved in LibreOffice by using the \"Solver tool\" as indicated in that file. You can use it to check your results.\n#\n#\n\n# ### Environment set-up and data loading\n\n# 1) Start by setting the working directory to the directory of this file and activate it. If you have the `Manifest.toml` file in the directory, just run `Pkg.instantiate()`, otherwise manually add the packages JuMP, GLPK, DataFrames, CSV, Pipe, HTTP.\n\n# 2) Load the packages DelimitedFiles, JuMP, GLPK, DataFrames, CSV, Pipe, HTTP\n\n# 3) Load from internet or from local files the following data:\n\nurlActivities = \"https://raw.githubusercontent.com/sylvaticus/IntroSPMLJuliaCourse/main/lessonsMaterial/02_JULIA2/loggingOptimisation/data/activities.csv\"\nurlResources = \"https://raw.githubusercontent.com/sylvaticus/IntroSPMLJuliaCourse/main/lessonsMaterial/02_JULIA2/loggingOptimisation/data/resources.csv\"\nurlCoefficients = \"https://raw.githubusercontent.com/sylvaticus/IntroSPMLJuliaCourse/main/lessonsMaterial/02_JULIA2/loggingOptimisation/data/coefficients.csv\"\n\n# The result must be:\n# - `activities`, a dataframe with the columns `label`, `gm` and `integer`\n# - `resources`, a dataframe with the columns `label`, `initial` and `initial2`\n# - `coeff`, a 10x12 Matrix of Float64\n\n# For example to download from internet and import the matrix you can use something like:\ncoef = @pipe HTTP.get(urlCoefficients).body |> readdlm(_,';')\n\n# 4) Determine `nA` and `nR` as the number of activities and the number of resources (use the `size` function)\n\n\n# ### Optimisation model definition\n\n# 5) Define `profitModel` as a model to be optimised using the `GLPK.Optimizer`\n# 6) [OPTIONAL] set `GLPK.GLP_MSG_ALL` as the `msg_lev` of GLPK\n\n# ### Model's endogenous variables definition\n\n# 7) Define the non-negative model `x` variable, indexed by the positions between 1 and `nA` (i.e. `x[1:nA] >= 0`) (you could use the @variables macro). Don't set the `x` variable to be integer at this step, as some variables are continuous, just set them to be non-negative.\n\n# 8) Set the variables for which the corresponding `integer` column in the `activity` dataframe is equal to 1 as a integer variable.\n# To set the specific vaciable `x[i]` as integer use `set_integer(x[i])`\n\n# ### Model's constraint definition\n\n# 9) Define the `resLimit[r in 1:nR]` family of contraints, such that when you sum `coef[r,a]*x[a]` for all the `1:nA` activities you must have a value not greater than `resources.initial[r]`\n\n# ### Objective definition\n\n# 10) Define the objective as the maximisation of the profit given by summing for each of the `1:nA` activities `activities.gm[a] * x[a]`\n\n# ### Model resolution\n# 11) [OPTIONAL] Print the model to check it\n# 12) Optimize the model\n# 13) Check with the function `status = termination_status(profitModel)` that the status is `OPTIMAL` (it should be!)\n\n# ### Print optimal level of activities\n\n# 14) Run the following code to print the results (optimal level of activities)\n\nif (status == MOI.OPTIMAL || status == MOI.LOCALLY_SOLVED || status == MOI.TIME_LIMIT) && has_values(profitModel)\n println(\"#################################################################\")\n if (status == MOI.OPTIMAL)\n println(\"** Problem solved correctly **\")\n else\n println(\"** Problem returned a (possibly suboptimal) solution **\")\n end\n println(\"- Objective value (total costs): \", objective_value(profitModel))\n println(\"- Optimal Activities:\\n\")\n optValues = value.(x)\n for a in 1:nA\n println(\"* $(activities.label[a]):\\t $(optValues[a])\")\n end\n if JuMP.has_duals(profitModel)\n println(\"\\n\\n- Shadow prices of the resources:\\n\")\n for r in 1:nR\n println(\"* $(resources.label[r]):\\t $(dual(resLimit[r]))\")\n end\n end\nelse\n println(\"The model was not solved correctly.\")\n println(status)\nend\n\n\n# 15) [OPTIONAL] Observe the emergence of scale effects\n# Optionally re-run the model with `initial2` initial resources (instead of `initial`) and notice how this larger company can afford to perform different types of activities (logging high forest instead of coppices in this example) and obtain a better profitability per unit of resource employed.\n# Can you guess which are the aspects of the optimisation model that allow for the emergence of these scale effects?\n", "meta": {"hexsha": "ef20fc4fa94376a39bb10250404e02434a98285d", "size": 5178, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "lessonsMaterial/02_JULIA2/loggingOptimisation/loggingOptimisation.jl", "max_stars_repo_name": "sylvaticus/IntroSPMLJuliaCourse", "max_stars_repo_head_hexsha": "84ba4432548dcfe826353c03616c03e86f46e9eb", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2022-02-26T15:44:12.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-02T13:05:52.000Z", "max_issues_repo_path": "lessonsMaterial/02_JULIA2/loggingOptimisation/loggingOptimisation.jl", "max_issues_repo_name": "sylvaticus/IntroSPMLJuliaCourse", "max_issues_repo_head_hexsha": "84ba4432548dcfe826353c03616c03e86f46e9eb", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lessonsMaterial/02_JULIA2/loggingOptimisation/loggingOptimisation.jl", "max_forks_repo_name": "sylvaticus/IntroSPMLJuliaCourse", "max_forks_repo_head_hexsha": "84ba4432548dcfe826353c03616c03e86f46e9eb", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-26T08:06:40.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-26T08:06:40.000Z", "avg_line_length": 56.2826086957, "max_line_length": 396, "alphanum_fraction": 0.7168791039, "num_tokens": 1268, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541577509315, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.8022449034904532}}
{"text": "#\n#Simple example of a Poisson problem with a variable coefficient:\n#\n# y\n# |\n# | u_y = 0\n# Ly ----------------------------------\n# | |\n# | |\n# | |\n# u = 0 | -(au_x)_x - (au_y)_y = f | u = 0\n# | |\n# | |\n# | |\n# -----------------------------------|----- x\n# u_y = 0 Lx\n#\n#Here, \n#\n# u = sin(ωx x) * cos(ωy y) \n# a = exp(x-y)\n# f = ω exp(x-y) ( 2ω sin(ω x) cos(ω y) - cos ω(x-y) )\n# ωx = mπ / Lx\n# ωy = nπ / Ly\n#\n\nusing FinElt\nusing FinElt.PlanarPoisson\n\ninclude(\"params.jl\")\nconst ωx = pi / Lx\nconst ωy = pi / Ly\n\nfunction exact_u(x)\n return sin(ωx*x[1]) * cos(ωy*x[2])\nend\n\nfunction a(x)\n return exp(x[1]-x[2])\nend\n\nfunction f(x)\n return exp(x[1]-x[2]) * ( \n ωx * ( ωx * sin(ωx*x[1]) - cos(ωx*x[1]) ) * cos(ωy*x[2])\n + ωy * ( ωy * cos(ωy*x[2]) - sin(ωy*x[2]) ) * sin(ωx*x[1])\n )\nend\n\nell_f(v, z) = P1SourceTimesFunc!(v, z, f)\nlin_functionals = [ (\"Omega\", ell_f) ]\n\nmaxerr = zeros(refinements+1)\n@printf(\"%10s %12s %8s %8s\\n\\n\", \n \"N\", \"max error\", \"rate\", \"seconds\")\nfor k = 0:refinements\n start = time()\n mesh = read_msh_file(\"rect$k.msh\") \n vp = VariationalProblem(mesh, [\"Left\", \"Right\"])\n add_bilin_form!(vp, \"Omega\", grad_dot_grad!, a)\n add_lin_functnl!(vp, \"Omega\", source_times_func!, f)\n # Implicit zero boundary conditions.\n A, b = assembled_linear_system(vp)\n ufree = A \\ b\n uh = complete_soln(ufree, vp)\n u = get_nodal_vals(exact_u, mesh)\n finish = time()\n maxerr[k+1] = maximum(abs.(uh-u))\n N = length(ufree)\n if k == 0\n @printf(\"%10d %12.4e\\n\", N, maxerr[k+1])\n else\n rate = log2(maxerr[k]/maxerr[k+1])\n elapsed = finish - start\n @printf(\"%10d %12.4e %8.4f %8.4f\\n\", \n N, maxerr[k+1], rate, elapsed)\n end\nend\n\n", "meta": {"hexsha": "942c1c0c8c21cee6a576b0b7ba96d5ac9ef3c90d", "size": 2170, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "examples/rectangle/var_coef.jl", "max_stars_repo_name": "billmclean/FinElt.jl", "max_stars_repo_head_hexsha": "5153f1624fe1c7dcadd646d60c716e6153fedb2a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 10, "max_stars_repo_stars_event_min_datetime": "2015-07-18T20:04:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T21:29:09.000Z", "max_issues_repo_path": "examples/rectangle/var_coef.jl", "max_issues_repo_name": "billmclean/FinElt.jl", "max_issues_repo_head_hexsha": "5153f1624fe1c7dcadd646d60c716e6153fedb2a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/rectangle/var_coef.jl", "max_forks_repo_name": "billmclean/FinElt.jl", "max_forks_repo_head_hexsha": "5153f1624fe1c7dcadd646d60c716e6153fedb2a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2017-06-29T15:15:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-04T18:01:08.000Z", "avg_line_length": 27.4683544304, "max_line_length": 69, "alphanum_fraction": 0.4156682028, "num_tokens": 715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9416541561135441, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.8022449020954764}}
{"text": "using Random\nusing LinearAlgebra\n\n\n\"\"\"\nGenerate an N×N matrix with uniformly random entries in [0,1].\n\"\"\"\nfunction uniform_matrix(N)\n\treturn rand(N, N)\nend\n\n\"\"\"\nGenerate a random N×N matrix with entries satisfying the triangle inequality.\n\"\"\"\nfunction metric_matrix(N; dim=5, p=2)\n\tpoints = rand(N, dim)\n\tM = [norm(points[i,:] - points[j,:], p) for i in 1:N, j in 1:N]\n\treturn M\nend\n\n\"\"\"\nGenerate a random 0-1 N×N matrix with edge probability p.\n\"\"\"\nfunction zeroone_matrix(N; p=0.25, selfedges=true, symmetric=false)\n\tM = rand(N, N) .≤ p\n\tif symmetric\n\t\tM = triu(M) .| triu(M)'\n\tend\n\tif !selfedges\n\t\tfor i in 1:N\n\t\t\tM[i,i] = 0\n\t\tend\n\tend\n\treturn M\nend\n\n\"\"\"\nGenerate a pair of N×N matricies with known optimal permutation using\nAlgorithm 4 from \"Generating Quadratic Assignment Test Problems with\nKnown Optimal Permutations\" by Li and Pardalos.\n\"\"\"\nfunction generate_qap(N)\n\tD = Matrix(I, N, N)\n\tO = .~D\n\n\tΔA = rand(1:100)\n\tΔB = rand(1:100)\n\n\tA = fill(ΔA, N, N)\n\tA[D] .= 0\n\n\tB = Array{Int,2}(undef, N, N)\n\tfor i in 1:N, j in 1:N\n\t\tif i == j\n\t\t\tB[i,j] = 0\n\t\telse\n\t\t\tB[i,j] = rand(0:ΔB)\n\t\tend\n\tend\n\n\tR = findall(O)\n\tsort!(R, by=(ij -> B[ij]))\n\n\tX = rand(1:ΔA, N*(N-1))\n\tsort!(X, rev=true)\n\n\tfor (k, ij) in enumerate(R)\n\t\tA[ij] = A[ij] - X[k]\n\tend\n\n\tσ = randperm(N)\n\tσinv = invperm(σ)\n\tB = B[σinv,σinv]\n\n\treturn A, B, σ\nend\n", "meta": {"hexsha": "c63da5b50eea715abcad68df771727e678530396", "size": 1321, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/util/generators.jl", "max_stars_repo_name": "flixpar/QuadraticAssignmentProblem.jl-", "max_stars_repo_head_hexsha": "0d769a70bba87e00c41d0d0bb1cb2eeebb06c57c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-05-11T20:05:37.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-29T09:25:06.000Z", "max_issues_repo_path": "src/util/generators.jl", "max_issues_repo_name": "flixpar/QuadraticAssignmentProblem.jl-", "max_issues_repo_head_hexsha": "0d769a70bba87e00c41d0d0bb1cb2eeebb06c57c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-16T03:37:34.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-20T14:06:30.000Z", "max_forks_repo_path": "src/util/generators.jl", "max_forks_repo_name": "flixpar/QuadraticAssignmentProblem.jl", "max_forks_repo_head_hexsha": "0d769a70bba87e00c41d0d0bb1cb2eeebb06c57c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.1558441558, "max_line_length": 77, "alphanum_fraction": 0.6298258895, "num_tokens": 479, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9553191309994467, "lm_q2_score": 0.8397339756938818, "lm_q1q2_score": 0.8022139319305895}}
{"text": "using PyPlot\n\npospart(x) = ( x > 0 ? x : zero(x) )\n\nfunction K2(x, y, a, b)\n pt = [a, (a+b)/2, b]\n ℓ2 = (x-pt[1])*(x-pt[3])/((pt[2]-pt[1])*(pt[2]-pt[3]))\n ℓ3 = (x-pt[1])*(x-pt[2])/((pt[3]-pt[1])*(pt[3]-pt[2]))\n Qπ = pospart(pt[2]-y)^2 * ℓ2 / 2 + (b-y)^2 * ℓ3 / 2\n return Qπ - pospart(x-y)^2/2\nend\n\na = -1.0\nb = 1.0\nN = 200\nx = LinRange(a, b, N+1)\ny = LinRange(a, b, N+1)\nz = Float64[ K2(x[k], y[j], a, b) for j=1:N+1, k=1:N+1 ]\nfig = figure(1)\nsurf(x, y, z, cstride=4, rstride=4, cmap=\"jet\", linewidth=0.25)\nxlabel(L\"x\")\nylabel(L\"y\")\nax = gca()\nax[:view_init](elev=40,azim=-100)\nsavefig(\"Quadratic_PeanoK_3d.pdf\")\n\nfigure(2)\ncontour(x, y, z, 12)\ncolorbar()\nxlabel(L\"x\")\nylabel(L\"y\")\ngrid(true)\nsavefig(\"Quadratic_PeanoK_contour.pdf\")\n", "meta": {"hexsha": "825b68cb9b0f51ea64c89b87429258869aae6013", "size": 749, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/chap2/PeanoK_quadratic.jl", "max_stars_repo_name": "billmclean/ComputationalMathsNotes", "max_stars_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-30T21:30:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-30T21:30:20.000Z", "max_issues_repo_path": "src/chap2/PeanoK_quadratic.jl", "max_issues_repo_name": "billmclean/ComputationalMathsNotes", "max_issues_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/chap2/PeanoK_quadratic.jl", "max_forks_repo_name": "billmclean/ComputationalMathsNotes", "max_forks_repo_head_hexsha": "9d521fdf7ec407cca287997885d81c3150973415", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 22.0294117647, "max_line_length": 63, "alphanum_fraction": 0.5487316422, "num_tokens": 360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9553191322715435, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.8022139234197769}}
{"text": "## sigmoidal functions\n\nexport logistic, logistic_prime, weibull, weibull_prime\n\nlogistic(x) = 1/(1 + exp(-x))\nlogistic_prime(x) = exp(-x)/(1 + exp(-x))^2\nweibull(x, coef, scale, shape) = (1 - exp(-((x/scale)^shape)))*coef\nweibull_prime(x, coef, scale, shape) = coef*(shape/scale)*((x/scale)^(shape-1))*exp(-((x/scale)^shape))\n", "meta": {"hexsha": "91c5b69e0f2bf3fb8f2228c2dfb81172d413f031", "size": 327, "ext": "jl", "lang": "Julia", "max_stars_repo_path": "src/functions.jl", "max_stars_repo_name": "marketingattribution/SigmoidalProgramming.jl", "max_stars_repo_head_hexsha": "f8ad8908c841aa39522a5bfbeeae1b5ee4122b29", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2015-03-11T19:34:07.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-09T04:00:55.000Z", "max_issues_repo_path": "src/functions.jl", "max_issues_repo_name": "madeleineudell/SigmoidalProgramming.jl", "max_issues_repo_head_hexsha": "78359012088f1e14ee13dec3f06f5023d9a5c9ba", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2015-03-11T20:35:13.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-10T04:50:28.000Z", "max_forks_repo_path": "src/functions.jl", "max_forks_repo_name": "marketingattribution/SigmoidalProgramming.jl", "max_forks_repo_head_hexsha": "f8ad8908c841aa39522a5bfbeeae1b5ee4122b29", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2015-06-23T18:17:35.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-24T14:08:51.000Z", "avg_line_length": 36.3333333333, "max_line_length": 103, "alphanum_fraction": 0.6574923547, "num_tokens": 117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9811668657039606, "lm_q2_score": 0.8175744806385542, "lm_q1q2_score": 0.8021769906476737}}
{"text": "# ---\n# title: 633. Sum of Square Numbers\n# id: problem633\n# author: Indigo\n# date: 2021-06-27\n# difficulty: Medium\n# categories: Math\n# link: