idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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41,001 | How to prove that the probability of spurious correlation increases with random walk length? | This isn't a direct answer to your question but provides some pointers that relate to the asymptotic distribution of the Pearson correlation in a Gaussian random walk. (Neither replacing Spearman with Pearson nor a Bernoulli random walk with a Gaussian is going to be of much consequence asymptotically; the Spearman may... | How to prove that the probability of spurious correlation increases with random walk length? | This isn't a direct answer to your question but provides some pointers that relate to the asymptotic distribution of the Pearson correlation in a Gaussian random walk. (Neither replacing Spearman with | How to prove that the probability of spurious correlation increases with random walk length?
This isn't a direct answer to your question but provides some pointers that relate to the asymptotic distribution of the Pearson correlation in a Gaussian random walk. (Neither replacing Spearman with Pearson nor a Bernoulli ra... | How to prove that the probability of spurious correlation increases with random walk length?
This isn't a direct answer to your question but provides some pointers that relate to the asymptotic distribution of the Pearson correlation in a Gaussian random walk. (Neither replacing Spearman with |
41,002 | How to prove that the probability of spurious correlation increases with random walk length? | Ok, I am gonna try and take whuber's advice to attend to the distribution of $\rho_{\text{s}}$. I do not have a mathematical statistics background, so gentle prodding me to make my thinking more rigorous is welcome.
Here goes:
The random walk length $T$ is a natural number.
There are $2^{T}$ possible random walks of t... | How to prove that the probability of spurious correlation increases with random walk length? | Ok, I am gonna try and take whuber's advice to attend to the distribution of $\rho_{\text{s}}$. I do not have a mathematical statistics background, so gentle prodding me to make my thinking more rigor | How to prove that the probability of spurious correlation increases with random walk length?
Ok, I am gonna try and take whuber's advice to attend to the distribution of $\rho_{\text{s}}$. I do not have a mathematical statistics background, so gentle prodding me to make my thinking more rigorous is welcome.
Here goes:
... | How to prove that the probability of spurious correlation increases with random walk length?
Ok, I am gonna try and take whuber's advice to attend to the distribution of $\rho_{\text{s}}$. I do not have a mathematical statistics background, so gentle prodding me to make my thinking more rigor |
41,003 | 2 hidden layers are more powerful than 1 | I found the answer to my question in the paper Feedback stabilization using two-hidden-layer nets by E.D. Sontag. From the introduction:
It is by now well-known that functions computable by nets with a single hidden layer can approximate continuous functions, uniformly on compacts, under only weak assumptions on $\the... | 2 hidden layers are more powerful than 1 | I found the answer to my question in the paper Feedback stabilization using two-hidden-layer nets by E.D. Sontag. From the introduction:
It is by now well-known that functions computable by nets with | 2 hidden layers are more powerful than 1
I found the answer to my question in the paper Feedback stabilization using two-hidden-layer nets by E.D. Sontag. From the introduction:
It is by now well-known that functions computable by nets with a single hidden layer can approximate continuous functions, uniformly on compa... | 2 hidden layers are more powerful than 1
I found the answer to my question in the paper Feedback stabilization using two-hidden-layer nets by E.D. Sontag. From the introduction:
It is by now well-known that functions computable by nets with |
41,004 | 2 hidden layers are more powerful than 1 | Another paper about a qualitative difference between 1- and 2-hidden layer networks:
Neural networks for localized approximation (1994)
We prove that feedforward artificial neural networks with a single hidden layer and an ideal sigmoidal response function cannot provide localized approximation in a Euclidean space of... | 2 hidden layers are more powerful than 1 | Another paper about a qualitative difference between 1- and 2-hidden layer networks:
Neural networks for localized approximation (1994)
We prove that feedforward artificial neural networks with a sin | 2 hidden layers are more powerful than 1
Another paper about a qualitative difference between 1- and 2-hidden layer networks:
Neural networks for localized approximation (1994)
We prove that feedforward artificial neural networks with a single hidden layer and an ideal sigmoidal response function cannot provide locali... | 2 hidden layers are more powerful than 1
Another paper about a qualitative difference between 1- and 2-hidden layer networks:
Neural networks for localized approximation (1994)
We prove that feedforward artificial neural networks with a sin |
41,005 | Understanding batch normalization | Their notation is confusing. For instance, they use the hat symbol to denote the normalized variable $\hat x=x-E[x]$, then even worse they use the expectation symbol to denote what is clearly a sample average estimator: $E[x]=\frac 1 N\sum_{i=1}^Nx_i$
I assume that when you wrote $E[b]=b$, you mean the expectation of b... | Understanding batch normalization | Their notation is confusing. For instance, they use the hat symbol to denote the normalized variable $\hat x=x-E[x]$, then even worse they use the expectation symbol to denote what is clearly a sample | Understanding batch normalization
Their notation is confusing. For instance, they use the hat symbol to denote the normalized variable $\hat x=x-E[x]$, then even worse they use the expectation symbol to denote what is clearly a sample average estimator: $E[x]=\frac 1 N\sum_{i=1}^Nx_i$
I assume that when you wrote $E[b]... | Understanding batch normalization
Their notation is confusing. For instance, they use the hat symbol to denote the normalized variable $\hat x=x-E[x]$, then even worse they use the expectation symbol to denote what is clearly a sample |
41,006 | Feature scaling in svm: Does it depend on the Kernel? | Yes feature scaling depends on the kernel and in general it's a good idea. The kernel is effectively a distance and if different features vary on different scales then this can matter. For the RBF kernel, for instance, we have
$$
K(x, x') = \exp\left(-\gamma ||x-x'||^2\right)
$$
so if one dimension takes much larger va... | Feature scaling in svm: Does it depend on the Kernel? | Yes feature scaling depends on the kernel and in general it's a good idea. The kernel is effectively a distance and if different features vary on different scales then this can matter. For the RBF ker | Feature scaling in svm: Does it depend on the Kernel?
Yes feature scaling depends on the kernel and in general it's a good idea. The kernel is effectively a distance and if different features vary on different scales then this can matter. For the RBF kernel, for instance, we have
$$
K(x, x') = \exp\left(-\gamma ||x-x'|... | Feature scaling in svm: Does it depend on the Kernel?
Yes feature scaling depends on the kernel and in general it's a good idea. The kernel is effectively a distance and if different features vary on different scales then this can matter. For the RBF ker |
41,007 | How to know when to use linear dimensionality reduction vs non-linear dimensionality reduction? | One approach is to learn more about the structure of the data. Dimensionality reduction supposes that the data are distributed near a low dimensional manifold. If this is the case, one might choose PCA if the manifold is (approximately) linear, and nonlinear dimensionality reduction (NLDR) if the manifold is nonlinear.... | How to know when to use linear dimensionality reduction vs non-linear dimensionality reduction? | One approach is to learn more about the structure of the data. Dimensionality reduction supposes that the data are distributed near a low dimensional manifold. If this is the case, one might choose PC | How to know when to use linear dimensionality reduction vs non-linear dimensionality reduction?
One approach is to learn more about the structure of the data. Dimensionality reduction supposes that the data are distributed near a low dimensional manifold. If this is the case, one might choose PCA if the manifold is (ap... | How to know when to use linear dimensionality reduction vs non-linear dimensionality reduction?
One approach is to learn more about the structure of the data. Dimensionality reduction supposes that the data are distributed near a low dimensional manifold. If this is the case, one might choose PC |
41,008 | What does this E symbol mean? [duplicate] | In mathematics and statistics many symbols are used to denote different things, e.g. $\pi$ does not always have to be the $3.14.15...$ constant. However the uppercase "E" (written in many forms, e.g. $\mathbb{E}, E, \mathrm{E}$) is almost always used in statistics to denote the expected value. In the case of your equat... | What does this E symbol mean? [duplicate] | In mathematics and statistics many symbols are used to denote different things, e.g. $\pi$ does not always have to be the $3.14.15...$ constant. However the uppercase "E" (written in many forms, e.g. | What does this E symbol mean? [duplicate]
In mathematics and statistics many symbols are used to denote different things, e.g. $\pi$ does not always have to be the $3.14.15...$ constant. However the uppercase "E" (written in many forms, e.g. $\mathbb{E}, E, \mathrm{E}$) is almost always used in statistics to denote the... | What does this E symbol mean? [duplicate]
In mathematics and statistics many symbols are used to denote different things, e.g. $\pi$ does not always have to be the $3.14.15...$ constant. However the uppercase "E" (written in many forms, e.g. |
41,009 | The fundamental theorem of simulation | Your question is unclear: when simulating $X\sim f(x)$, one can instead simulate$$(X,U)\sim\mathcal{U}(\{(x,u);\ 0<u<f(x)\})$$which has the joint density$$\mathbb{I}_{(0,f(x)}(u)\mathbb{I}_\mathcal{X}(x)$$and then use only the $X$ component in the simulation. For instance, here is a figure from our book depicting many ... | The fundamental theorem of simulation | Your question is unclear: when simulating $X\sim f(x)$, one can instead simulate$$(X,U)\sim\mathcal{U}(\{(x,u);\ 0<u<f(x)\})$$which has the joint density$$\mathbb{I}_{(0,f(x)}(u)\mathbb{I}_\mathcal{X} | The fundamental theorem of simulation
Your question is unclear: when simulating $X\sim f(x)$, one can instead simulate$$(X,U)\sim\mathcal{U}(\{(x,u);\ 0<u<f(x)\})$$which has the joint density$$\mathbb{I}_{(0,f(x)}(u)\mathbb{I}_\mathcal{X}(x)$$and then use only the $X$ component in the simulation. For instance, here is ... | The fundamental theorem of simulation
Your question is unclear: when simulating $X\sim f(x)$, one can instead simulate$$(X,U)\sim\mathcal{U}(\{(x,u);\ 0<u<f(x)\})$$which has the joint density$$\mathbb{I}_{(0,f(x)}(u)\mathbb{I}_\mathcal{X} |
41,010 | Optimum number of epochs and neurons for an LSTM network | The optimum parametrization depends on the problem (there is not a rule that can do what you want), but there are some techniques that can help you, see the link below.
http://machinelearningmastery.com/tune-lstm-hyperparameters-keras-time-series-forecasting/
Basically what the upper link does is:
First calculate the... | Optimum number of epochs and neurons for an LSTM network | The optimum parametrization depends on the problem (there is not a rule that can do what you want), but there are some techniques that can help you, see the link below.
http://machinelearningmastery. | Optimum number of epochs and neurons for an LSTM network
The optimum parametrization depends on the problem (there is not a rule that can do what you want), but there are some techniques that can help you, see the link below.
http://machinelearningmastery.com/tune-lstm-hyperparameters-keras-time-series-forecasting/
Ba... | Optimum number of epochs and neurons for an LSTM network
The optimum parametrization depends on the problem (there is not a rule that can do what you want), but there are some techniques that can help you, see the link below.
http://machinelearningmastery. |
41,011 | Logistic regression for a continuous dependent variable | Your first option could work. It assumes that the residuals from the model on the transformed data are normally distributed. You need to check this. If it's true, you will be OK.
Option 2 depends on how you set up the GLM. Simply using a logit link function does not necessitate that you use any particular respons... | Logistic regression for a continuous dependent variable | Your first option could work. It assumes that the residuals from the model on the transformed data are normally distributed. You need to check this. If it's true, you will be OK.
Option 2 depends | Logistic regression for a continuous dependent variable
Your first option could work. It assumes that the residuals from the model on the transformed data are normally distributed. You need to check this. If it's true, you will be OK.
Option 2 depends on how you set up the GLM. Simply using a logit link function ... | Logistic regression for a continuous dependent variable
Your first option could work. It assumes that the residuals from the model on the transformed data are normally distributed. You need to check this. If it's true, you will be OK.
Option 2 depends |
41,012 | Logistic regression for a continuous dependent variable | Your idea is to use the logit (I call it $f$ to make clear that what $f$ is does not matter in what I will explain) in order to fall between 0 and 1. This idea of yours is not especially related to logistic regression. It is similar to using a log link function in order to deal with a positive $Y$.
So the difference be... | Logistic regression for a continuous dependent variable | Your idea is to use the logit (I call it $f$ to make clear that what $f$ is does not matter in what I will explain) in order to fall between 0 and 1. This idea of yours is not especially related to lo | Logistic regression for a continuous dependent variable
Your idea is to use the logit (I call it $f$ to make clear that what $f$ is does not matter in what I will explain) in order to fall between 0 and 1. This idea of yours is not especially related to logistic regression. It is similar to using a log link function in... | Logistic regression for a continuous dependent variable
Your idea is to use the logit (I call it $f$ to make clear that what $f$ is does not matter in what I will explain) in order to fall between 0 and 1. This idea of yours is not especially related to lo |
41,013 | Logistic regression for a continuous dependent variable | The difference between the two models you've described is that the first supposes that the DV is a continuous variable that varies between 0 and 1, whereas the second (usually called "logistic regression") supposes that the DV is a discrete variable that can take only the values 0 and 1. So the second one is inappropri... | Logistic regression for a continuous dependent variable | The difference between the two models you've described is that the first supposes that the DV is a continuous variable that varies between 0 and 1, whereas the second (usually called "logistic regress | Logistic regression for a continuous dependent variable
The difference between the two models you've described is that the first supposes that the DV is a continuous variable that varies between 0 and 1, whereas the second (usually called "logistic regression") supposes that the DV is a discrete variable that can take ... | Logistic regression for a continuous dependent variable
The difference between the two models you've described is that the first supposes that the DV is a continuous variable that varies between 0 and 1, whereas the second (usually called "logistic regress |
41,014 | What is the intuition behind convolutional neural network? | Fourier basis functions are "global", extending over the entire signal/image domain. More typically the convolution filters used in image processing/computer vision will be local. For example moving average or derivative style filters.
My understanding of ConvNets is that the filters are typically local. But rather tha... | What is the intuition behind convolutional neural network? | Fourier basis functions are "global", extending over the entire signal/image domain. More typically the convolution filters used in image processing/computer vision will be local. For example moving a | What is the intuition behind convolutional neural network?
Fourier basis functions are "global", extending over the entire signal/image domain. More typically the convolution filters used in image processing/computer vision will be local. For example moving average or derivative style filters.
My understanding of ConvN... | What is the intuition behind convolutional neural network?
Fourier basis functions are "global", extending over the entire signal/image domain. More typically the convolution filters used in image processing/computer vision will be local. For example moving a |
41,015 | What is the intuition behind convolutional neural network? | In this blog post (Deep Learning in a Nutshell: Core Concepts), the author had a very good explanation of the intuition behind convolution.
Convolution is important in physics and mathematics as it defines a bridge between the spatial and time domains (pixel with intensity 147 at position (0,30)) and the frequency dom... | What is the intuition behind convolutional neural network? | In this blog post (Deep Learning in a Nutshell: Core Concepts), the author had a very good explanation of the intuition behind convolution.
Convolution is important in physics and mathematics as it d | What is the intuition behind convolutional neural network?
In this blog post (Deep Learning in a Nutshell: Core Concepts), the author had a very good explanation of the intuition behind convolution.
Convolution is important in physics and mathematics as it defines a bridge between the spatial and time domains (pixel w... | What is the intuition behind convolutional neural network?
In this blog post (Deep Learning in a Nutshell: Core Concepts), the author had a very good explanation of the intuition behind convolution.
Convolution is important in physics and mathematics as it d |
41,016 | Some general questions on Generative Adversarial Networks | after nothing came from anyone, I like to summarize my own research on the topic.
Question number 1: Is it ok to use a Generated sample for extracting data from an actual sample?
There will be some "changed" values in the generated sample but a practical view can help us understand why we might use it. in many applicat... | Some general questions on Generative Adversarial Networks | after nothing came from anyone, I like to summarize my own research on the topic.
Question number 1: Is it ok to use a Generated sample for extracting data from an actual sample?
There will be some "c | Some general questions on Generative Adversarial Networks
after nothing came from anyone, I like to summarize my own research on the topic.
Question number 1: Is it ok to use a Generated sample for extracting data from an actual sample?
There will be some "changed" values in the generated sample but a practical view ca... | Some general questions on Generative Adversarial Networks
after nothing came from anyone, I like to summarize my own research on the topic.
Question number 1: Is it ok to use a Generated sample for extracting data from an actual sample?
There will be some "c |
41,017 | Some general questions on Generative Adversarial Networks | Not a complete answer from my side, but adding my 2 cents to Question 3: What happens if you change Noise to another type of distribution?
Essentially you get a Conditional Generative Adversarial Network (CGAN) (Mirza & Osindero, 2014). CGAN's have been used in many applications including image super-resolution (which... | Some general questions on Generative Adversarial Networks | Not a complete answer from my side, but adding my 2 cents to Question 3: What happens if you change Noise to another type of distribution?
Essentially you get a Conditional Generative Adversarial Net | Some general questions on Generative Adversarial Networks
Not a complete answer from my side, but adding my 2 cents to Question 3: What happens if you change Noise to another type of distribution?
Essentially you get a Conditional Generative Adversarial Network (CGAN) (Mirza & Osindero, 2014). CGAN's have been used in... | Some general questions on Generative Adversarial Networks
Not a complete answer from my side, but adding my 2 cents to Question 3: What happens if you change Noise to another type of distribution?
Essentially you get a Conditional Generative Adversarial Net |
41,018 | How does GBM work with a Poisson loss fonction | I am going to define the GBM algorithm first to clarify the question:
Algorithm 10.3: Gradient Tree Boosting Algorithm from Hastie's The Elements of Statistical Learning states the following:
Where $N$ is the number of samples, $M$ is the number of iterations and $J_m$ is the number of terminal regions or size of the ... | How does GBM work with a Poisson loss fonction | I am going to define the GBM algorithm first to clarify the question:
Algorithm 10.3: Gradient Tree Boosting Algorithm from Hastie's The Elements of Statistical Learning states the following:
Where $ | How does GBM work with a Poisson loss fonction
I am going to define the GBM algorithm first to clarify the question:
Algorithm 10.3: Gradient Tree Boosting Algorithm from Hastie's The Elements of Statistical Learning states the following:
Where $N$ is the number of samples, $M$ is the number of iterations and $J_m$ is... | How does GBM work with a Poisson loss fonction
I am going to define the GBM algorithm first to clarify the question:
Algorithm 10.3: Gradient Tree Boosting Algorithm from Hastie's The Elements of Statistical Learning states the following:
Where $ |
41,019 | "Random variable" vs. "random value" (when translating from Russian into English) | I side with your editor. You plot the obtained values on the plot, then draw a mean (average) line, then calculate three standard deviations above and below the line to show the boundaries. There's really no modeling involved here, hence, there's no random variable defined explicitly. You could argue that implicitly th... | "Random variable" vs. "random value" (when translating from Russian into English) | I side with your editor. You plot the obtained values on the plot, then draw a mean (average) line, then calculate three standard deviations above and below the line to show the boundaries. There's re | "Random variable" vs. "random value" (when translating from Russian into English)
I side with your editor. You plot the obtained values on the plot, then draw a mean (average) line, then calculate three standard deviations above and below the line to show the boundaries. There's really no modeling involved here, hence,... | "Random variable" vs. "random value" (when translating from Russian into English)
I side with your editor. You plot the obtained values on the plot, then draw a mean (average) line, then calculate three standard deviations above and below the line to show the boundaries. There's re |
41,020 | "Random variable" vs. "random value" (when translating from Russian into English) | The correct phrase in order to make statistical sense, is "random variable". And this is because it refers to the "mean line" - and random values do not have a mean, only random variables do.
If the Russian word is "value", and even if the Russian author meant "value", (s)he was simply mistaken.
Alternatively, one co... | "Random variable" vs. "random value" (when translating from Russian into English) | The correct phrase in order to make statistical sense, is "random variable". And this is because it refers to the "mean line" - and random values do not have a mean, only random variables do.
If the | "Random variable" vs. "random value" (when translating from Russian into English)
The correct phrase in order to make statistical sense, is "random variable". And this is because it refers to the "mean line" - and random values do not have a mean, only random variables do.
If the Russian word is "value", and even if ... | "Random variable" vs. "random value" (when translating from Russian into English)
The correct phrase in order to make statistical sense, is "random variable". And this is because it refers to the "mean line" - and random values do not have a mean, only random variables do.
If the |
41,021 | Solving constrained optimization problem: projected gradient vs. dual? | Essentially yes, projected gradient descent is another method for solving constrained optimization problems. It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
Besides directly solving constrained problems with simple constraint sets, it ... | Solving constrained optimization problem: projected gradient vs. dual? | Essentially yes, projected gradient descent is another method for solving constrained optimization problems. It's only useful when the projection operation is easy or has a closed form, for example, b | Solving constrained optimization problem: projected gradient vs. dual?
Essentially yes, projected gradient descent is another method for solving constrained optimization problems. It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
Besides... | Solving constrained optimization problem: projected gradient vs. dual?
Essentially yes, projected gradient descent is another method for solving constrained optimization problems. It's only useful when the projection operation is easy or has a closed form, for example, b |
41,022 | Solving constrained optimization problem: projected gradient vs. dual? | AaronDefazio had a very good answer:
It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
I am just adding some details and an example of this statement.
Note that, in Projected Gradient Decent, the projection step is another optimizatio... | Solving constrained optimization problem: projected gradient vs. dual? | AaronDefazio had a very good answer:
It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
I am just adding some detail | Solving constrained optimization problem: projected gradient vs. dual?
AaronDefazio had a very good answer:
It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
I am just adding some details and an example of this statement.
Note that, i... | Solving constrained optimization problem: projected gradient vs. dual?
AaronDefazio had a very good answer:
It's only useful when the projection operation is easy or has a closed form, for example, box constraints or linear constraint sets.
I am just adding some detail |
41,023 | MCMC - Metropolis Hasting: formal derivation of detailed balance | That follows by an easy case distinction: If $P(x')g(x|x')>P(x)g(x'|x)$, then $A(x'|x)=1$ and, by symmetry, $A(x|x')=\frac{P(x)g(x'|x)}{P(x')g(x|x')}$ and the claim holds. The case $P(x')g(x|x')\le P(x)g(x'|x)$ works similarly.
Maybe, the equality
$$A(x'|x)\cdot P(x)g(x'|x)= A(x|x')\cdot P(x')g(x|x')$$
is easier to see... | MCMC - Metropolis Hasting: formal derivation of detailed balance | That follows by an easy case distinction: If $P(x')g(x|x')>P(x)g(x'|x)$, then $A(x'|x)=1$ and, by symmetry, $A(x|x')=\frac{P(x)g(x'|x)}{P(x')g(x|x')}$ and the claim holds. The case $P(x')g(x|x')\le P( | MCMC - Metropolis Hasting: formal derivation of detailed balance
That follows by an easy case distinction: If $P(x')g(x|x')>P(x)g(x'|x)$, then $A(x'|x)=1$ and, by symmetry, $A(x|x')=\frac{P(x)g(x'|x)}{P(x')g(x|x')}$ and the claim holds. The case $P(x')g(x|x')\le P(x)g(x'|x)$ works similarly.
Maybe, the equality
$$A(x'|... | MCMC - Metropolis Hasting: formal derivation of detailed balance
That follows by an easy case distinction: If $P(x')g(x|x')>P(x)g(x'|x)$, then $A(x'|x)=1$ and, by symmetry, $A(x|x')=\frac{P(x)g(x'|x)}{P(x')g(x|x')}$ and the claim holds. The case $P(x')g(x|x')\le P( |
41,024 | How to prove a Hamming distance as a Kernel? | Step 1. For an arbitrary set of strings $\{x_i\}$, first sort them by their length. Then the kernel matrix is block-diagonal, since the kernel value between any two strings of different lengths is zero.
This lets us see that we only need to consider strings of the same length; how?
Step 2. So, we only need to consider ... | How to prove a Hamming distance as a Kernel? | Step 1. For an arbitrary set of strings $\{x_i\}$, first sort them by their length. Then the kernel matrix is block-diagonal, since the kernel value between any two strings of different lengths is zer | How to prove a Hamming distance as a Kernel?
Step 1. For an arbitrary set of strings $\{x_i\}$, first sort them by their length. Then the kernel matrix is block-diagonal, since the kernel value between any two strings of different lengths is zero.
This lets us see that we only need to consider strings of the same lengt... | How to prove a Hamming distance as a Kernel?
Step 1. For an arbitrary set of strings $\{x_i\}$, first sort them by their length. Then the kernel matrix is block-diagonal, since the kernel value between any two strings of different lengths is zer |
41,025 | How to calculate $R^2$ for LASSO (glmnet) | I'm using
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.min)]
or if you have chosen the lambda.1se
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.1se)]
If you do a cross check with the traditional regression lm() and summary()$r.squared it will match... | How to calculate $R^2$ for LASSO (glmnet) | I'm using
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.min)]
or if you have chosen the lambda.1se
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == | How to calculate $R^2$ for LASSO (glmnet)
I'm using
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.min)]
or if you have chosen the lambda.1se
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.1se)]
If you do a cross check with the traditional regression ... | How to calculate $R^2$ for LASSO (glmnet)
I'm using
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == fitnet$lambda.min)]
or if you have chosen the lambda.1se
r2 <- fit$glmnet.fit$dev.ratio[which(fitnet$glmnet.fit$lambda == |
41,026 | How to calculate $R^2$ for LASSO (glmnet) | I think I know why the two calculations produce different answers. The cvm variable from the cvm.glmnet object is a cross-validated error. It's calculated from the residuals in the validation folds. The predict() function on the other hand, is not cross-validated. It's calculated from the residuals of predictions o... | How to calculate $R^2$ for LASSO (glmnet) | I think I know why the two calculations produce different answers. The cvm variable from the cvm.glmnet object is a cross-validated error. It's calculated from the residuals in the validation folds. | How to calculate $R^2$ for LASSO (glmnet)
I think I know why the two calculations produce different answers. The cvm variable from the cvm.glmnet object is a cross-validated error. It's calculated from the residuals in the validation folds. The predict() function on the other hand, is not cross-validated. It's calc... | How to calculate $R^2$ for LASSO (glmnet)
I think I know why the two calculations produce different answers. The cvm variable from the cvm.glmnet object is a cross-validated error. It's calculated from the residuals in the validation folds. |
41,027 | Over-represented values in FDR-adjusted p-values | There is nothing wrong with these q-values. Correcting p-values (corrected p-values are often referred to as q-values) is a concept that's newer than the Benjamini-Hochberg (BH) procedure, which in its purest form only outputs significant: yes or no. To understand why your q-values look the way they look we need to loo... | Over-represented values in FDR-adjusted p-values | There is nothing wrong with these q-values. Correcting p-values (corrected p-values are often referred to as q-values) is a concept that's newer than the Benjamini-Hochberg (BH) procedure, which in it | Over-represented values in FDR-adjusted p-values
There is nothing wrong with these q-values. Correcting p-values (corrected p-values are often referred to as q-values) is a concept that's newer than the Benjamini-Hochberg (BH) procedure, which in its purest form only outputs significant: yes or no. To understand why yo... | Over-represented values in FDR-adjusted p-values
There is nothing wrong with these q-values. Correcting p-values (corrected p-values are often referred to as q-values) is a concept that's newer than the Benjamini-Hochberg (BH) procedure, which in it |
41,028 | In online convex optimization, what is a leader in FTL algorithm? | The Follow-The-Leader (FTL) algorithm is a simple algorithm for solving online prediction problems. Imagine that you have a committee of experts, each of which suggests a strategy. At each time point, you pick an expert and follow his or her advice, which yields some associated cost (or reward) at the next time step. Y... | In online convex optimization, what is a leader in FTL algorithm? | The Follow-The-Leader (FTL) algorithm is a simple algorithm for solving online prediction problems. Imagine that you have a committee of experts, each of which suggests a strategy. At each time point, | In online convex optimization, what is a leader in FTL algorithm?
The Follow-The-Leader (FTL) algorithm is a simple algorithm for solving online prediction problems. Imagine that you have a committee of experts, each of which suggests a strategy. At each time point, you pick an expert and follow his or her advice, whic... | In online convex optimization, what is a leader in FTL algorithm?
The Follow-The-Leader (FTL) algorithm is a simple algorithm for solving online prediction problems. Imagine that you have a committee of experts, each of which suggests a strategy. At each time point, |
41,029 | In online convex optimization, what is a leader in FTL algorithm? | In the particular case of on-line convex optimization a leader corresponds to a point in the convex domain of optimization. So in principle you have an infinite number of experts. | In online convex optimization, what is a leader in FTL algorithm? | In the particular case of on-line convex optimization a leader corresponds to a point in the convex domain of optimization. So in principle you have an infinite number of experts. | In online convex optimization, what is a leader in FTL algorithm?
In the particular case of on-line convex optimization a leader corresponds to a point in the convex domain of optimization. So in principle you have an infinite number of experts. | In online convex optimization, what is a leader in FTL algorithm?
In the particular case of on-line convex optimization a leader corresponds to a point in the convex domain of optimization. So in principle you have an infinite number of experts. |
41,030 | The discriminant function in linear discriminant analysis | it can be shown that this is equivalent to assigning the observation to the class for which the below equation is largest
Deriving the discriminant function for LDA
For LDA we assume that the random variable $X$ is a vector $\mathbf{X} = (X_1,X_2,...,X_p)$ which is drawn from a multivariate Gaussian with class-specifi... | The discriminant function in linear discriminant analysis | it can be shown that this is equivalent to assigning the observation to the class for which the below equation is largest
Deriving the discriminant function for LDA
For LDA we assume that the random | The discriminant function in linear discriminant analysis
it can be shown that this is equivalent to assigning the observation to the class for which the below equation is largest
Deriving the discriminant function for LDA
For LDA we assume that the random variable $X$ is a vector $\mathbf{X} = (X_1,X_2,...,X_p)$ whic... | The discriminant function in linear discriminant analysis
it can be shown that this is equivalent to assigning the observation to the class for which the below equation is largest
Deriving the discriminant function for LDA
For LDA we assume that the random |
41,031 | The discriminant function in linear discriminant analysis | Given $x$, the summation term is the same for each $p_k(x)$, where $k=1,...K$.
Therefore when comparing the probabilities, you can delete it. | The discriminant function in linear discriminant analysis | Given $x$, the summation term is the same for each $p_k(x)$, where $k=1,...K$.
Therefore when comparing the probabilities, you can delete it. | The discriminant function in linear discriminant analysis
Given $x$, the summation term is the same for each $p_k(x)$, where $k=1,...K$.
Therefore when comparing the probabilities, you can delete it. | The discriminant function in linear discriminant analysis
Given $x$, the summation term is the same for each $p_k(x)$, where $k=1,...K$.
Therefore when comparing the probabilities, you can delete it. |
41,032 | The advantages and disadvantages of using Kaiser Rule to select the number of principal components | The advantage of the rule is that it is easy to calculate, especially if you live in the 1950s, and don't have access to a fast computer.
The disadvantages ... well, I'm going to quote Preacher and MacCallum, in their paper "Repairing Tom Swift’s Electric Factor Analysis Machine". It's worth reading the whole paper, av... | The advantages and disadvantages of using Kaiser Rule to select the number of principal components | The advantage of the rule is that it is easy to calculate, especially if you live in the 1950s, and don't have access to a fast computer.
The disadvantages ... well, I'm going to quote Preacher and Ma | The advantages and disadvantages of using Kaiser Rule to select the number of principal components
The advantage of the rule is that it is easy to calculate, especially if you live in the 1950s, and don't have access to a fast computer.
The disadvantages ... well, I'm going to quote Preacher and MacCallum, in their pap... | The advantages and disadvantages of using Kaiser Rule to select the number of principal components
The advantage of the rule is that it is easy to calculate, especially if you live in the 1950s, and don't have access to a fast computer.
The disadvantages ... well, I'm going to quote Preacher and Ma |
41,033 | Why use the normal approximation to the binomial? | I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what you get when you use the binomial RNGs from R, SciPy, etc. The only good reason I can think of to discuss the method in... | Why use the normal approximation to the binomial? | I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what | Why use the normal approximation to the binomial?
I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what you get when you use the binomial RNGs from R, SciPy, etc. The only g... | Why use the normal approximation to the binomial?
I know of no reason to use the normal approximation to the binomial distribution in practice. There are a variety of exact algorithms that are more than good enough for general use, and these are what |
41,034 | Why use the normal approximation to the binomial? | The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes. Sufficiently large depends on the success parameter $p$. When $p=0.5$, the binomial is symmetric and so the sample size does not need to be as much as if $p=0.95$, in which case the binomial c... | Why use the normal approximation to the binomial? | The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes. Sufficiently large depends on the success parameter $p$. When $p=0.5$, t | Why use the normal approximation to the binomial?
The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes. Sufficiently large depends on the success parameter $p$. When $p=0.5$, the binomial is symmetric and so the sample size does not need to be as... | Why use the normal approximation to the binomial?
The central limit theorem provides the reason why the normal can approximate the binomial in sufficiently large sample sizes. Sufficiently large depends on the success parameter $p$. When $p=0.5$, t |
41,035 | Test for the significance of the effect of an intervention in a time series | What you are referring to is called a test for structural change/break or a changepoint model.
As you have a known change date, you can simply add an interaction in the model, and use a standard t-test for this coefficient. If you test for more coefficients, use the Chow test formulation (see this post for example). T... | Test for the significance of the effect of an intervention in a time series | What you are referring to is called a test for structural change/break or a changepoint model.
As you have a known change date, you can simply add an interaction in the model, and use a standard t-te | Test for the significance of the effect of an intervention in a time series
What you are referring to is called a test for structural change/break or a changepoint model.
As you have a known change date, you can simply add an interaction in the model, and use a standard t-test for this coefficient. If you test for mor... | Test for the significance of the effect of an intervention in a time series
What you are referring to is called a test for structural change/break or a changepoint model.
As you have a known change date, you can simply add an interaction in the model, and use a standard t-te |
41,036 | Why is Binary Classification not a Hypothesis Test? | It seems perfectly reasonable to me that the binary classification of a given pattern could be analogous to a hypothesis test, but it isn't necessarily. It also depends on exactly what situation you have in mind here. Let us assume that you have a classifier (hopefully a good one), and want to use it determine which ... | Why is Binary Classification not a Hypothesis Test? | It seems perfectly reasonable to me that the binary classification of a given pattern could be analogous to a hypothesis test, but it isn't necessarily. It also depends on exactly what situation you | Why is Binary Classification not a Hypothesis Test?
It seems perfectly reasonable to me that the binary classification of a given pattern could be analogous to a hypothesis test, but it isn't necessarily. It also depends on exactly what situation you have in mind here. Let us assume that you have a classifier (hopefu... | Why is Binary Classification not a Hypothesis Test?
It seems perfectly reasonable to me that the binary classification of a given pattern could be analogous to a hypothesis test, but it isn't necessarily. It also depends on exactly what situation you |
41,037 | Why is Binary Classification not a Hypothesis Test? | The two are not necessarily non-overlapping in practice. I would also note that hypothesis-testing and statistical-significance seem slightly different to me (at least going by the tag descriptions on this site): Not all hypothesis testing has to be comparing to a "null" (or "random/by chance/no effect") alternative.
T... | Why is Binary Classification not a Hypothesis Test? | The two are not necessarily non-overlapping in practice. I would also note that hypothesis-testing and statistical-significance seem slightly different to me (at least going by the tag descriptions on | Why is Binary Classification not a Hypothesis Test?
The two are not necessarily non-overlapping in practice. I would also note that hypothesis-testing and statistical-significance seem slightly different to me (at least going by the tag descriptions on this site): Not all hypothesis testing has to be comparing to a "nu... | Why is Binary Classification not a Hypothesis Test?
The two are not necessarily non-overlapping in practice. I would also note that hypothesis-testing and statistical-significance seem slightly different to me (at least going by the tag descriptions on |
41,038 | Generalized Gamma GLM | There are several reasonably clear options.
You can use the survival model. Treat the response values as all-uncensored survival times. I've used this strategy to fit Weibull models for example; it often works quite well.
There's an example here that shows doing it with a Weibull model for non-survival data. [There'... | Generalized Gamma GLM | There are several reasonably clear options.
You can use the survival model. Treat the response values as all-uncensored survival times. I've used this strategy to fit Weibull models for example; it | Generalized Gamma GLM
There are several reasonably clear options.
You can use the survival model. Treat the response values as all-uncensored survival times. I've used this strategy to fit Weibull models for example; it often works quite well.
There's an example here that shows doing it with a Weibull model for non-... | Generalized Gamma GLM
There are several reasonably clear options.
You can use the survival model. Treat the response values as all-uncensored survival times. I've used this strategy to fit Weibull models for example; it |
41,039 | Similarity metrics for more than two vectors? | The cosine similarity between two column vectors $x_1$ and $x_2$ is simply the dot product between their unit vectors
$$\mathrm{CosSim}[x_1,x_2]=\frac{x_1}{\|x_1\|}\bullet\frac{x_2}{\|x_2\|}$$
and varies from -1 to +1, similar to a correlation coefficient $R$ (to which it is closely related).
There are many ways to gen... | Similarity metrics for more than two vectors? | The cosine similarity between two column vectors $x_1$ and $x_2$ is simply the dot product between their unit vectors
$$\mathrm{CosSim}[x_1,x_2]=\frac{x_1}{\|x_1\|}\bullet\frac{x_2}{\|x_2\|}$$
and var | Similarity metrics for more than two vectors?
The cosine similarity between two column vectors $x_1$ and $x_2$ is simply the dot product between their unit vectors
$$\mathrm{CosSim}[x_1,x_2]=\frac{x_1}{\|x_1\|}\bullet\frac{x_2}{\|x_2\|}$$
and varies from -1 to +1, similar to a correlation coefficient $R$ (to which it i... | Similarity metrics for more than two vectors?
The cosine similarity between two column vectors $x_1$ and $x_2$ is simply the dot product between their unit vectors
$$\mathrm{CosSim}[x_1,x_2]=\frac{x_1}{\|x_1\|}\bullet\frac{x_2}{\|x_2\|}$$
and var |
41,040 | Model with complications | Sandwich based robust error estimation handles both heteroscedasticity and non-normal error distribution asymptotically. That also happens to mean that you get approximately valid inference in relatively samples.
One criticism might be that a method which is so robust must be of low power. Generally, not as true as one... | Model with complications | Sandwich based robust error estimation handles both heteroscedasticity and non-normal error distribution asymptotically. That also happens to mean that you get approximately valid inference in relativ | Model with complications
Sandwich based robust error estimation handles both heteroscedasticity and non-normal error distribution asymptotically. That also happens to mean that you get approximately valid inference in relatively samples.
One criticism might be that a method which is so robust must be of low power. Gene... | Model with complications
Sandwich based robust error estimation handles both heteroscedasticity and non-normal error distribution asymptotically. That also happens to mean that you get approximately valid inference in relativ |
41,041 | Model with complications | Both heteroscedasticity and heavy-tailedness can be considered violations of the distributional assumptions of a standard linear model. If the distribution is nonetheless symmetrical, and the relationship is between $x$ and $y$ is rectilinear, your model should not be biased. Instead, interval estimates and inference... | Model with complications | Both heteroscedasticity and heavy-tailedness can be considered violations of the distributional assumptions of a standard linear model. If the distribution is nonetheless symmetrical, and the relatio | Model with complications
Both heteroscedasticity and heavy-tailedness can be considered violations of the distributional assumptions of a standard linear model. If the distribution is nonetheless symmetrical, and the relationship is between $x$ and $y$ is rectilinear, your model should not be biased. Instead, interva... | Model with complications
Both heteroscedasticity and heavy-tailedness can be considered violations of the distributional assumptions of a standard linear model. If the distribution is nonetheless symmetrical, and the relatio |
41,042 | Violin plot or bean plot? What are the advantages of each? | I don't know about bean plots but for small sample sizes violin plots may be unstable and I would prefer to just show the raw data with a rug plot or spike histogram. Sometimes I superimpose a violin plot with an extended box plot and the raw data. An extended box plot shows many more quantiles than a regular box plo... | Violin plot or bean plot? What are the advantages of each? | I don't know about bean plots but for small sample sizes violin plots may be unstable and I would prefer to just show the raw data with a rug plot or spike histogram. Sometimes I superimpose a violin | Violin plot or bean plot? What are the advantages of each?
I don't know about bean plots but for small sample sizes violin plots may be unstable and I would prefer to just show the raw data with a rug plot or spike histogram. Sometimes I superimpose a violin plot with an extended box plot and the raw data. An extende... | Violin plot or bean plot? What are the advantages of each?
I don't know about bean plots but for small sample sizes violin plots may be unstable and I would prefer to just show the raw data with a rug plot or spike histogram. Sometimes I superimpose a violin |
41,043 | Confused by a simple setting in Bayesian inference | For continuous models, the Bayes factor is defined as the ratio of marginal likelihoods (marginal density functions of the data):
$$P(M_i\vert x) = \int p(x\vert\theta_i, M_i)\pi(\theta_i)d\theta_i \neq 0,$$
where $p$ denotes the likelihood function (joint density of the data given the parameters of model $M_i$).
See:
... | Confused by a simple setting in Bayesian inference | For continuous models, the Bayes factor is defined as the ratio of marginal likelihoods (marginal density functions of the data):
$$P(M_i\vert x) = \int p(x\vert\theta_i, M_i)\pi(\theta_i)d\theta_i \n | Confused by a simple setting in Bayesian inference
For continuous models, the Bayes factor is defined as the ratio of marginal likelihoods (marginal density functions of the data):
$$P(M_i\vert x) = \int p(x\vert\theta_i, M_i)\pi(\theta_i)d\theta_i \neq 0,$$
where $p$ denotes the likelihood function (joint density of t... | Confused by a simple setting in Bayesian inference
For continuous models, the Bayes factor is defined as the ratio of marginal likelihoods (marginal density functions of the data):
$$P(M_i\vert x) = \int p(x\vert\theta_i, M_i)\pi(\theta_i)d\theta_i \n |
41,044 | Confused by a simple setting in Bayesian inference | In model $M_1$, there is no parameter, hence no need for a prior distribution. The marginal density of $X$ under model $M_1$ is thus the normal $N(0,1)$ in this case. The marginal density under model $M_2$ is indeed the integral
$$ \int f_{X}(x|\mu)f_\mu(\mu)\text{d}\mu = \int_1^2 \frac{1}{\sqrt{2\pi}}e^{-\frac{(x-\mu... | Confused by a simple setting in Bayesian inference | In model $M_1$, there is no parameter, hence no need for a prior distribution. The marginal density of $X$ under model $M_1$ is thus the normal $N(0,1)$ in this case. The marginal density under model | Confused by a simple setting in Bayesian inference
In model $M_1$, there is no parameter, hence no need for a prior distribution. The marginal density of $X$ under model $M_1$ is thus the normal $N(0,1)$ in this case. The marginal density under model $M_2$ is indeed the integral
$$ \int f_{X}(x|\mu)f_\mu(\mu)\text{d}\m... | Confused by a simple setting in Bayesian inference
In model $M_1$, there is no parameter, hence no need for a prior distribution. The marginal density of $X$ under model $M_1$ is thus the normal $N(0,1)$ in this case. The marginal density under model |
41,045 | An Information Criterion that considers how many variables we can choose from | I think simple cross validation is the best fit.
Both AIC and BIC consider the balance between model complexity and the amount of information available. With more data, more complex models can be learned. However, this balance is fixed and not based on the data.
Cross validation is based on the data. It also balances... | An Information Criterion that considers how many variables we can choose from | I think simple cross validation is the best fit.
Both AIC and BIC consider the balance between model complexity and the amount of information available. With more data, more complex models can be lea | An Information Criterion that considers how many variables we can choose from
I think simple cross validation is the best fit.
Both AIC and BIC consider the balance between model complexity and the amount of information available. With more data, more complex models can be learned. However, this balance is fixed and n... | An Information Criterion that considers how many variables we can choose from
I think simple cross validation is the best fit.
Both AIC and BIC consider the balance between model complexity and the amount of information available. With more data, more complex models can be lea |
41,046 | What is an orthogonal design? | Perhaps you haven't fully grasped the definition yet. The requirements for orthogonal designs are that the blocking is orthogonal and the treatment is orthogonal. This simply means that crossproducts total to zero, whether the blocking or treatment is continuous, pseudo-continuous, polytomous, or binary. As @whuber cor... | What is an orthogonal design? | Perhaps you haven't fully grasped the definition yet. The requirements for orthogonal designs are that the blocking is orthogonal and the treatment is orthogonal. This simply means that crossproducts | What is an orthogonal design?
Perhaps you haven't fully grasped the definition yet. The requirements for orthogonal designs are that the blocking is orthogonal and the treatment is orthogonal. This simply means that crossproducts total to zero, whether the blocking or treatment is continuous, pseudo-continuous, polytom... | What is an orthogonal design?
Perhaps you haven't fully grasped the definition yet. The requirements for orthogonal designs are that the blocking is orthogonal and the treatment is orthogonal. This simply means that crossproducts |
41,047 | PDF/CDF of max-min type random variable | I will illustrate with the example in the question, because a general answer is too complicated to write down.
Let $F$ be the common distribution function. We will need the distributions of the order statistics $x_{[1]} \le x_{[2]} \le \cdots \le x_{[n]}$. Their distribution functions $f_{[k]}$ are easy to express in... | PDF/CDF of max-min type random variable | I will illustrate with the example in the question, because a general answer is too complicated to write down.
Let $F$ be the common distribution function. We will need the distributions of the order | PDF/CDF of max-min type random variable
I will illustrate with the example in the question, because a general answer is too complicated to write down.
Let $F$ be the common distribution function. We will need the distributions of the order statistics $x_{[1]} \le x_{[2]} \le \cdots \le x_{[n]}$. Their distribution fu... | PDF/CDF of max-min type random variable
I will illustrate with the example in the question, because a general answer is too complicated to write down.
Let $F$ be the common distribution function. We will need the distributions of the order |
41,048 | How to Understand the Relationships Among Random Variables, Samples, and Populations? | Between the question and the existing answers there is some contradiction and confusion. Let's start by clearing up the definitions. Then I will provide an intuitive analog for thinking about them. I will finish by sketching out how a sample really doesn't have to be thought of any differently than a single element ... | How to Understand the Relationships Among Random Variables, Samples, and Populations? | Between the question and the existing answers there is some contradiction and confusion. Let's start by clearing up the definitions. Then I will provide an intuitive analog for thinking about them. | How to Understand the Relationships Among Random Variables, Samples, and Populations?
Between the question and the existing answers there is some contradiction and confusion. Let's start by clearing up the definitions. Then I will provide an intuitive analog for thinking about them. I will finish by sketching out ho... | How to Understand the Relationships Among Random Variables, Samples, and Populations?
Between the question and the existing answers there is some contradiction and confusion. Let's start by clearing up the definitions. Then I will provide an intuitive analog for thinking about them. |
41,049 | How to Understand the Relationships Among Random Variables, Samples, and Populations? | Neither a sample nor a population can be considered variables. I think measurement is the missing piece of the puzzle in your question.
Strictly speaking, a sample is considered random when every element in the population has an equal chance of being sampled (simple random sampling) or at least a known probability (ran... | How to Understand the Relationships Among Random Variables, Samples, and Populations? | Neither a sample nor a population can be considered variables. I think measurement is the missing piece of the puzzle in your question.
Strictly speaking, a sample is considered random when every elem | How to Understand the Relationships Among Random Variables, Samples, and Populations?
Neither a sample nor a population can be considered variables. I think measurement is the missing piece of the puzzle in your question.
Strictly speaking, a sample is considered random when every element in the population has an equal... | How to Understand the Relationships Among Random Variables, Samples, and Populations?
Neither a sample nor a population can be considered variables. I think measurement is the missing piece of the puzzle in your question.
Strictly speaking, a sample is considered random when every elem |
41,050 | How to Understand the Relationships Among Random Variables, Samples, and Populations? | A 'random variable' is a mathematical object, more precisely a measurable function.
A 'random' sample is not a mathematical object. It is only a way to pick individuals at random in a population. A way to do it is to simply consider that the population is uniformly distributed, so every individuals have the same chanc... | How to Understand the Relationships Among Random Variables, Samples, and Populations? | A 'random variable' is a mathematical object, more precisely a measurable function.
A 'random' sample is not a mathematical object. It is only a way to pick individuals at random in a population. A w | How to Understand the Relationships Among Random Variables, Samples, and Populations?
A 'random variable' is a mathematical object, more precisely a measurable function.
A 'random' sample is not a mathematical object. It is only a way to pick individuals at random in a population. A way to do it is to simply consider ... | How to Understand the Relationships Among Random Variables, Samples, and Populations?
A 'random variable' is a mathematical object, more precisely a measurable function.
A 'random' sample is not a mathematical object. It is only a way to pick individuals at random in a population. A w |
41,051 | How does R function summary.glm calculate the covariance matrix for glm model? | I was confused about this several months ago. After reading some material, I have a clue about the problem. But I am not sure that the following is correct.
Suppose the distribution of response $Y$ is an exponential family, $X^TX>0$. $p(Y|\eta)\sim \exp(\phi^{-1}\eta^T T(Y)-A(\eta)+B(Y))$, $\eta=g(E(Y|X))=X\beta$, $g$ ... | How does R function summary.glm calculate the covariance matrix for glm model? | I was confused about this several months ago. After reading some material, I have a clue about the problem. But I am not sure that the following is correct.
Suppose the distribution of response $Y$ is | How does R function summary.glm calculate the covariance matrix for glm model?
I was confused about this several months ago. After reading some material, I have a clue about the problem. But I am not sure that the following is correct.
Suppose the distribution of response $Y$ is an exponential family, $X^TX>0$. $p(Y|\e... | How does R function summary.glm calculate the covariance matrix for glm model?
I was confused about this several months ago. After reading some material, I have a clue about the problem. But I am not sure that the following is correct.
Suppose the distribution of response $Y$ is |
41,052 | How does R function summary.glm calculate the covariance matrix for glm model? | ?chol2inv says:
Invert a symmetric, positive definite square matrix from its
Choleski decomposition. Equivalently, compute (X'X)^(-1) from the
(R part) of the QR decomposition of X.
and if you want to dig deeper into the numerics:
This is an interface to the LAPACK routine ‘DPOTRI’
and the documentat... | How does R function summary.glm calculate the covariance matrix for glm model? | ?chol2inv says:
Invert a symmetric, positive definite square matrix from its
Choleski decomposition. Equivalently, compute (X'X)^(-1) from the
(R part) of the QR decomposition of X.
a | How does R function summary.glm calculate the covariance matrix for glm model?
?chol2inv says:
Invert a symmetric, positive definite square matrix from its
Choleski decomposition. Equivalently, compute (X'X)^(-1) from the
(R part) of the QR decomposition of X.
and if you want to dig deeper into the num... | How does R function summary.glm calculate the covariance matrix for glm model?
?chol2inv says:
Invert a symmetric, positive definite square matrix from its
Choleski decomposition. Equivalently, compute (X'X)^(-1) from the
(R part) of the QR decomposition of X.
a |
41,053 | How to determine a prior, if there is no relevant prior empirical information? | If you have no strong prior beliefs or assumptions, you can use non-informative priors, which don't impose strong structure or constraints. Rather, they try to express ignorance about the parameters in a principled way. The Jeffreys prior and reference priors are prominent examples. Very general constraints may be impo... | How to determine a prior, if there is no relevant prior empirical information? | If you have no strong prior beliefs or assumptions, you can use non-informative priors, which don't impose strong structure or constraints. Rather, they try to express ignorance about the parameters i | How to determine a prior, if there is no relevant prior empirical information?
If you have no strong prior beliefs or assumptions, you can use non-informative priors, which don't impose strong structure or constraints. Rather, they try to express ignorance about the parameters in a principled way. The Jeffreys prior an... | How to determine a prior, if there is no relevant prior empirical information?
If you have no strong prior beliefs or assumptions, you can use non-informative priors, which don't impose strong structure or constraints. Rather, they try to express ignorance about the parameters i |
41,054 | How to determine a prior, if there is no relevant prior empirical information? | In my case, I have an empirical prior probability to use in calculating the posterior probability. But this isn’t the subjective way of doing it, I believe.
On the contrary. The 'subjective way of doing it' is to ask what information you want to assume and then construct a prior accordingly. If you're happy to make e... | How to determine a prior, if there is no relevant prior empirical information? | In my case, I have an empirical prior probability to use in calculating the posterior probability. But this isn’t the subjective way of doing it, I believe.
On the contrary. The 'subjective way of d | How to determine a prior, if there is no relevant prior empirical information?
In my case, I have an empirical prior probability to use in calculating the posterior probability. But this isn’t the subjective way of doing it, I believe.
On the contrary. The 'subjective way of doing it' is to ask what information you w... | How to determine a prior, if there is no relevant prior empirical information?
In my case, I have an empirical prior probability to use in calculating the posterior probability. But this isn’t the subjective way of doing it, I believe.
On the contrary. The 'subjective way of d |
41,055 | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom? | The pdf of a $\chi^2$ distribution is
$f(x;k)=\frac{1}{2^{\frac{k}{2}}\Gamma(k/2)}x^{k/2-1}\exp(-x/2).$
So we just need to evaluate the expression for $f(0;k)$.
$$
f(0;1)=\infty
$$
$$
f(0;2)=0.5
$$
$$
f(0;3)=0
$$
And so on. The R code for this is dchisq(0,k) for some positive k. It's really only interesting for $k= 2$... | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom? | The pdf of a $\chi^2$ distribution is
$f(x;k)=\frac{1}{2^{\frac{k}{2}}\Gamma(k/2)}x^{k/2-1}\exp(-x/2).$
So we just need to evaluate the expression for $f(0;k)$.
$$
f(0;1)=\infty
$$
$$
f(0;2)=0.5
$$
$ | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom?
The pdf of a $\chi^2$ distribution is
$f(x;k)=\frac{1}{2^{\frac{k}{2}}\Gamma(k/2)}x^{k/2-1}\exp(-x/2).$
So we just need to evaluate the expression for $f(0;k)$.
$$
f(0;1)=\infty
$$
$$
f(0;2)=0.5
$$
$$
f(0;3)=0
$$
And so on. The R code for ... | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom?
The pdf of a $\chi^2$ distribution is
$f(x;k)=\frac{1}{2^{\frac{k}{2}}\Gamma(k/2)}x^{k/2-1}\exp(-x/2).$
So we just need to evaluate the expression for $f(0;k)$.
$$
f(0;1)=\infty
$$
$$
f(0;2)=0.5
$$
$ |
41,056 | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom? | Let's try to go back to the definition of this distribution and see what happens around 0.
By definition, the $\chi^2$ distribution is that of the sum of the squares of independent normal random variables:
$$
Y = \sum_{i=1}^k Z_i^2 ,
$$
(see the wikipedia page). We see easily that the value of the density is $0$ f... | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom? | Let's try to go back to the definition of this distribution and see what happens around 0.
By definition, the $\chi^2$ distribution is that of the sum of the squares of independent normal random vari | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom?
Let's try to go back to the definition of this distribution and see what happens around 0.
By definition, the $\chi^2$ distribution is that of the sum of the squares of independent normal random variables:
$$
Y = \sum_{i=1}^k Z_i^2 ,
$... | How is the $ \chi^2 $ distribution defined at 0 for various degrees of freedom?
Let's try to go back to the definition of this distribution and see what happens around 0.
By definition, the $\chi^2$ distribution is that of the sum of the squares of independent normal random vari |
41,057 | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1 | This is a categorical distribution, also known as a multinomial distribution with number of trials equal to $1$.
If the binomial probabilities are $q_k, \;k=1,\dots n$ then the multinomial probability is
$$p_k=\frac {q_k\prod_{j\neq k} (1-q_j)}{\sum_r q_r\prod_{s\neq r}(1-q_s)}$$
To derive this you simply use the condi... | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1 | This is a categorical distribution, also known as a multinomial distribution with number of trials equal to $1$.
If the binomial probabilities are $q_k, \;k=1,\dots n$ then the multinomial probability | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1
This is a categorical distribution, also known as a multinomial distribution with number of trials equal to $1$.
If the binomial probabilities are $q_k, \;k=1,\dots n$ then the multinomial probability is
$$p_k=\frac {q_k... | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1
This is a categorical distribution, also known as a multinomial distribution with number of trials equal to $1$.
If the binomial probabilities are $q_k, \;k=1,\dots n$ then the multinomial probability |
41,058 | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1 | There are only $n$ ways the variables can sum to $1$: one of them will equal $1$ and the other $n-1$ will equal zero. The very phrasing of the question indicates the variables are exchangeable: thus, the joint distribution will not change when the variables are permuted. Since permutations of the variables merely cha... | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1 | There are only $n$ ways the variables can sum to $1$: one of them will equal $1$ and the other $n-1$ will equal zero. The very phrasing of the question indicates the variables are exchangeable: thus, | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1
There are only $n$ ways the variables can sum to $1$: one of them will equal $1$ and the other $n-1$ will equal zero. The very phrasing of the question indicates the variables are exchangeable: thus, the joint distribut... | Question about joint distribution of Bernoulli random variables under constraint that sum must be 1
There are only $n$ ways the variables can sum to $1$: one of them will equal $1$ and the other $n-1$ will equal zero. The very phrasing of the question indicates the variables are exchangeable: thus, |
41,059 | A few clarifications regarding convolutional neural networks | Fully connected layers can only deal with input of a fixed size, because it requires a certain amount of parameters to "fully connect" the input and output. While convolutional layers just "slide" the same filters across the input, so it can basically deal with input of an arbitrary spatial size.
In the example network... | A few clarifications regarding convolutional neural networks | Fully connected layers can only deal with input of a fixed size, because it requires a certain amount of parameters to "fully connect" the input and output. While convolutional layers just "slide" the | A few clarifications regarding convolutional neural networks
Fully connected layers can only deal with input of a fixed size, because it requires a certain amount of parameters to "fully connect" the input and output. While convolutional layers just "slide" the same filters across the input, so it can basically deal wi... | A few clarifications regarding convolutional neural networks
Fully connected layers can only deal with input of a fixed size, because it requires a certain amount of parameters to "fully connect" the input and output. While convolutional layers just "slide" the |
41,060 | What is cross-lagged panel design? | The cross-lagged panel model (CLPM) is a type of structural equation model (specifically a path analysis model) that is used where two or more variables are measured at two or more occasions and interest is centered on the associations (often causal theories) with each other over time. For example, there may be interes... | What is cross-lagged panel design? | The cross-lagged panel model (CLPM) is a type of structural equation model (specifically a path analysis model) that is used where two or more variables are measured at two or more occasions and inter | What is cross-lagged panel design?
The cross-lagged panel model (CLPM) is a type of structural equation model (specifically a path analysis model) that is used where two or more variables are measured at two or more occasions and interest is centered on the associations (often causal theories) with each other over time... | What is cross-lagged panel design?
The cross-lagged panel model (CLPM) is a type of structural equation model (specifically a path analysis model) that is used where two or more variables are measured at two or more occasions and inter |
41,061 | Scoring quantile regressor | The first thing we expect from a quantile forecast is that it respects the prespecified quantile, i.e., that it provides quantile predictions that are larger than a proportion $\tau$ of your realizations. You can check this by looking at your $n$ quantile predictions $\hat{y}_i$ and assessing whether
$$\hat{\tau} := \f... | Scoring quantile regressor | The first thing we expect from a quantile forecast is that it respects the prespecified quantile, i.e., that it provides quantile predictions that are larger than a proportion $\tau$ of your realizati | Scoring quantile regressor
The first thing we expect from a quantile forecast is that it respects the prespecified quantile, i.e., that it provides quantile predictions that are larger than a proportion $\tau$ of your realizations. You can check this by looking at your $n$ quantile predictions $\hat{y}_i$ and assessing... | Scoring quantile regressor
The first thing we expect from a quantile forecast is that it respects the prespecified quantile, i.e., that it provides quantile predictions that are larger than a proportion $\tau$ of your realizati |
41,062 | Testing linear restriction in R | I think your current parameterization is sufficient to test either coefficient restriction (via the t-tests for each coefficient), but not both restrictions at the same time. To do that one can conduct an F-test between the unrestricted and the restricted model (since it is nested).
Example below in R using the car pa... | Testing linear restriction in R | I think your current parameterization is sufficient to test either coefficient restriction (via the t-tests for each coefficient), but not both restrictions at the same time. To do that one can conduc | Testing linear restriction in R
I think your current parameterization is sufficient to test either coefficient restriction (via the t-tests for each coefficient), but not both restrictions at the same time. To do that one can conduct an F-test between the unrestricted and the restricted model (since it is nested).
Exa... | Testing linear restriction in R
I think your current parameterization is sufficient to test either coefficient restriction (via the t-tests for each coefficient), but not both restrictions at the same time. To do that one can conduc |
41,063 | Testing linear restriction in R | You can test the hypothesized value of $\beta_k$ much in the same way that it would be tested for $\beta_k = 0$. You can get a t-value on n-2 df with the formula:
$$T = \frac{\beta_i- hypothesezed~~ \beta_i}{Standard~~ Error~~ \beta_i}$$
In many cases this simplifies to beta over standard error because making the hypot... | Testing linear restriction in R | You can test the hypothesized value of $\beta_k$ much in the same way that it would be tested for $\beta_k = 0$. You can get a t-value on n-2 df with the formula:
$$T = \frac{\beta_i- hypothesezed~~ \ | Testing linear restriction in R
You can test the hypothesized value of $\beta_k$ much in the same way that it would be tested for $\beta_k = 0$. You can get a t-value on n-2 df with the formula:
$$T = \frac{\beta_i- hypothesezed~~ \beta_i}{Standard~~ Error~~ \beta_i}$$
In many cases this simplifies to beta over standar... | Testing linear restriction in R
You can test the hypothesized value of $\beta_k$ much in the same way that it would be tested for $\beta_k = 0$. You can get a t-value on n-2 df with the formula:
$$T = \frac{\beta_i- hypothesezed~~ \ |
41,064 | Testing linear restriction in R | Let's say you want to test the hypothesis of the form $R\beta = r$. In your case:
$$\underbrace{\left[ \begin{array}{cccc} \ldots & 1& 0 & \ldots \\ \ldots & 0& 1 & \ldots \end{array}\right]}_R \underbrace{\left[ \begin{array}{c}\ldots \\ \beta_j \\ \beta_k \\ \ldots \end{array} \right]}_\beta = \underbrace{\left[ \be... | Testing linear restriction in R | Let's say you want to test the hypothesis of the form $R\beta = r$. In your case:
$$\underbrace{\left[ \begin{array}{cccc} \ldots & 1& 0 & \ldots \\ \ldots & 0& 1 & \ldots \end{array}\right]}_R \unde | Testing linear restriction in R
Let's say you want to test the hypothesis of the form $R\beta = r$. In your case:
$$\underbrace{\left[ \begin{array}{cccc} \ldots & 1& 0 & \ldots \\ \ldots & 0& 1 & \ldots \end{array}\right]}_R \underbrace{\left[ \begin{array}{c}\ldots \\ \beta_j \\ \beta_k \\ \ldots \end{array} \right]... | Testing linear restriction in R
Let's say you want to test the hypothesis of the form $R\beta = r$. In your case:
$$\underbrace{\left[ \begin{array}{cccc} \ldots & 1& 0 & \ldots \\ \ldots & 0& 1 & \ldots \end{array}\right]}_R \unde |
41,065 | Rotating a new matrix into the same basis as another using SVD | What you are doing is essentially PCA, even though you did not use this term. It is exactly PCA if your $X$ is centered; if not, then it is a sort of "uncentered PCA".
In any case, to get from $A$ to $UD$ you need to right-multiply it with $V$: $$AV=UDV^\top V = UD.$$
So to apply the same transformation to $B$, you sim... | Rotating a new matrix into the same basis as another using SVD | What you are doing is essentially PCA, even though you did not use this term. It is exactly PCA if your $X$ is centered; if not, then it is a sort of "uncentered PCA".
In any case, to get from $A$ to | Rotating a new matrix into the same basis as another using SVD
What you are doing is essentially PCA, even though you did not use this term. It is exactly PCA if your $X$ is centered; if not, then it is a sort of "uncentered PCA".
In any case, to get from $A$ to $UD$ you need to right-multiply it with $V$: $$AV=UDV^\to... | Rotating a new matrix into the same basis as another using SVD
What you are doing is essentially PCA, even though you did not use this term. It is exactly PCA if your $X$ is centered; if not, then it is a sort of "uncentered PCA".
In any case, to get from $A$ to |
41,066 | Recurrent event analysis | The main decision to be made is about the time scale that you plan on using. Is it time since the origin of the recurrent event process (like some diagnosis or some intervention, or birth) or is it time between two visits? These two approaches are called calendar time and gap time. As a general rule, what answers one o... | Recurrent event analysis | The main decision to be made is about the time scale that you plan on using. Is it time since the origin of the recurrent event process (like some diagnosis or some intervention, or birth) or is it ti | Recurrent event analysis
The main decision to be made is about the time scale that you plan on using. Is it time since the origin of the recurrent event process (like some diagnosis or some intervention, or birth) or is it time between two visits? These two approaches are called calendar time and gap time. As a general... | Recurrent event analysis
The main decision to be made is about the time scale that you plan on using. Is it time since the origin of the recurrent event process (like some diagnosis or some intervention, or birth) or is it ti |
41,067 | The Frog Riddle - Conditional Probabilities | Let's look at the pair of frogs. Male frogs are identified by croaking in the video.
As explained in the video, before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Frog 2 is Female
Frog 1 is Female, Frog 2 is F... | The Frog Riddle - Conditional Probabilities | Let's look at the pair of frogs. Male frogs are identified by croaking in the video.
As explained in the video, before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 | The Frog Riddle - Conditional Probabilities
Let's look at the pair of frogs. Male frogs are identified by croaking in the video.
As explained in the video, before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Fr... | The Frog Riddle - Conditional Probabilities
Let's look at the pair of frogs. Male frogs are identified by croaking in the video.
As explained in the video, before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 |
41,068 | The Frog Riddle - Conditional Probabilities | Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male. The former case carries less information and this effectively increases our chances over the latter situation.
Call the frogs left... | The Frog Riddle - Conditional Probabilities | Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male. The former | The Frog Riddle - Conditional Probabilities
Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male. The former case carries less information and this effectively increases our chances ov... | The Frog Riddle - Conditional Probabilities
Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male. The former |
41,069 | The Frog Riddle - Conditional Probabilities | Your intuition is correct in this case. As the problem is stated your odds of survival are 50%. The video incorrectly states the problem space based on the information we have and therefore comes to an incorrect conclusion. The correct problem space contains 8 conditions and is as follows.
We have two frogs on a log, ... | The Frog Riddle - Conditional Probabilities | Your intuition is correct in this case. As the problem is stated your odds of survival are 50%. The video incorrectly states the problem space based on the information we have and therefore comes to a | The Frog Riddle - Conditional Probabilities
Your intuition is correct in this case. As the problem is stated your odds of survival are 50%. The video incorrectly states the problem space based on the information we have and therefore comes to an incorrect conclusion. The correct problem space contains 8 conditions and ... | The Frog Riddle - Conditional Probabilities
Your intuition is correct in this case. As the problem is stated your odds of survival are 50%. The video incorrectly states the problem space based on the information we have and therefore comes to a |
41,070 | The Frog Riddle - Conditional Probabilities | A clearer answer to this, since the previous was too long and not easy to understand.
The possible outcomes are different, although I used same letters. To make clear the sample space, I will describe the possible outcomes
M M --> "The male is on the left" - "A random male on the right"
M F --> "The male is on the le... | The Frog Riddle - Conditional Probabilities | A clearer answer to this, since the previous was too long and not easy to understand.
The possible outcomes are different, although I used same letters. To make clear the sample space, I will describ | The Frog Riddle - Conditional Probabilities
A clearer answer to this, since the previous was too long and not easy to understand.
The possible outcomes are different, although I used same letters. To make clear the sample space, I will describe the possible outcomes
M M --> "The male is on the left" - "A random male ... | The Frog Riddle - Conditional Probabilities
A clearer answer to this, since the previous was too long and not easy to understand.
The possible outcomes are different, although I used same letters. To make clear the sample space, I will describ |
41,071 | The Frog Riddle - Conditional Probabilities | The problem I have with this problem, is that the solution seems to be using different rules for what it considers a possible result for the two frogs being male and female, and male and male.
The F/M pair, and the M/F pair, are different because we don't know whether the first frog or the second frog is male, so F/M a... | The Frog Riddle - Conditional Probabilities | The problem I have with this problem, is that the solution seems to be using different rules for what it considers a possible result for the two frogs being male and female, and male and male.
The F/M | The Frog Riddle - Conditional Probabilities
The problem I have with this problem, is that the solution seems to be using different rules for what it considers a possible result for the two frogs being male and female, and male and male.
The F/M pair, and the M/F pair, are different because we don't know whether the fir... | The Frog Riddle - Conditional Probabilities
The problem I have with this problem, is that the solution seems to be using different rules for what it considers a possible result for the two frogs being male and female, and male and male.
The F/M |
41,072 | The Frog Riddle - Conditional Probabilities | Not knowing anything: $\{(M,M), (M,F), (F,M), (F,F)\}$.
Three pairs with at least one female out of four possible combinations: $3/4$ or $75\%$
Knowing the first one is male: $\{(M,M), (M,F)\}$. One pair with at least one female out of two possible combinations: $1/2$ or $50\%$
Knowing that there is at least one male: ... | The Frog Riddle - Conditional Probabilities | Not knowing anything: $\{(M,M), (M,F), (F,M), (F,F)\}$.
Three pairs with at least one female out of four possible combinations: $3/4$ or $75\%$
Knowing the first one is male: $\{(M,M), (M,F)\}$. One p | The Frog Riddle - Conditional Probabilities
Not knowing anything: $\{(M,M), (M,F), (F,M), (F,F)\}$.
Three pairs with at least one female out of four possible combinations: $3/4$ or $75\%$
Knowing the first one is male: $\{(M,M), (M,F)\}$. One pair with at least one female out of two possible combinations: $1/2$ or $50\... | The Frog Riddle - Conditional Probabilities
Not knowing anything: $\{(M,M), (M,F), (F,M), (F,F)\}$.
Three pairs with at least one female out of four possible combinations: $3/4$ or $75\%$
Knowing the first one is male: $\{(M,M), (M,F)\}$. One p |
41,073 | The Frog Riddle - Conditional Probabilities | The correct answer is given above by tomciopp and the video is incorrect. I want to elaborate on their answer with a little more to diagnose the video's mistake and give an intuitive illustration or two.
The Video's Mistake
In the video, they conditionalize on the statement "at least one frog is male." However, the inf... | The Frog Riddle - Conditional Probabilities | The correct answer is given above by tomciopp and the video is incorrect. I want to elaborate on their answer with a little more to diagnose the video's mistake and give an intuitive illustration or t | The Frog Riddle - Conditional Probabilities
The correct answer is given above by tomciopp and the video is incorrect. I want to elaborate on their answer with a little more to diagnose the video's mistake and give an intuitive illustration or two.
The Video's Mistake
In the video, they conditionalize on the statement "... | The Frog Riddle - Conditional Probabilities
The correct answer is given above by tomciopp and the video is incorrect. I want to elaborate on their answer with a little more to diagnose the video's mistake and give an intuitive illustration or t |
41,074 | The Frog Riddle - Conditional Probabilities | The issue with considering Mm and mM separately is that there's a 50% chance of Mx and a 50% chance of Cy, where M is male and C is the croaker, but that's only if x is also male. If x is female, then the chances of the first frog being the croaker (i.e. Cy) doubles to 100%, which means the likelihood of that situation... | The Frog Riddle - Conditional Probabilities | The issue with considering Mm and mM separately is that there's a 50% chance of Mx and a 50% chance of Cy, where M is male and C is the croaker, but that's only if x is also male. If x is female, then | The Frog Riddle - Conditional Probabilities
The issue with considering Mm and mM separately is that there's a 50% chance of Mx and a 50% chance of Cy, where M is male and C is the croaker, but that's only if x is also male. If x is female, then the chances of the first frog being the croaker (i.e. Cy) doubles to 100%, ... | The Frog Riddle - Conditional Probabilities
The issue with considering Mm and mM separately is that there's a 50% chance of Mx and a 50% chance of Cy, where M is male and C is the croaker, but that's only if x is also male. If x is female, then |
41,075 | The Frog Riddle - Conditional Probabilities | Before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Frog 2 is Female
Frog 1 is Female, Frog 2 is Female
Making the assumptions about males and females occurring equally and independently, our sample space is {(M... | The Frog Riddle - Conditional Probabilities | Before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Frog 2 is Female
Frog 1 is Female, Frog | The Frog Riddle - Conditional Probabilities
Before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Frog 2 is Female
Frog 1 is Female, Frog 2 is Female
Making the assumptions about males and females occurring equally... | The Frog Riddle - Conditional Probabilities
Before we hear any croaking, there are 4 equally likely outcomes given 2 frogs:
Frog 1 is Male, Frog 2 is Male
Frog 1 is Female, Frog 2 is Male
Frog 1 is Male, Frog 2 is Female
Frog 1 is Female, Frog |
41,076 | Do the principal components change if we apply PCA more than once (recursively) on data? | PCA at it's heart involves diagonalizing a matrix which means solving for the eigenvalues and eigenvectors of said matrix. The whole purpose of the calculation is to find a diagonal representation of your matrix (i.e. only elements along the diagonal of the matrix). If you solve again, you will find that you are tryi... | Do the principal components change if we apply PCA more than once (recursively) on data? | PCA at it's heart involves diagonalizing a matrix which means solving for the eigenvalues and eigenvectors of said matrix. The whole purpose of the calculation is to find a diagonal representation of | Do the principal components change if we apply PCA more than once (recursively) on data?
PCA at it's heart involves diagonalizing a matrix which means solving for the eigenvalues and eigenvectors of said matrix. The whole purpose of the calculation is to find a diagonal representation of your matrix (i.e. only element... | Do the principal components change if we apply PCA more than once (recursively) on data?
PCA at it's heart involves diagonalizing a matrix which means solving for the eigenvalues and eigenvectors of said matrix. The whole purpose of the calculation is to find a diagonal representation of |
41,077 | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | The t matrix is the one to use in the way you describe. Eqs. 4 through 7 in the Dong & Peng paper that Joe_74 references correspond to the elements of the same names in the mipo object (documentation here), and so t is the accurate variance-covariance matrix for the pooled regression coefficients qbar you're actually u... | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | The t matrix is the one to use in the way you describe. Eqs. 4 through 7 in the Dong & Peng paper that Joe_74 references correspond to the elements of the same names in the mipo object (documentation | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
The t matrix is the one to use in the way you describe. Eqs. 4 through 7 in the Dong & Peng paper that Joe_74 references correspond to the elements of the same names in the mipo object (documentation here), and so t is the a... | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
The t matrix is the one to use in the way you describe. Eqs. 4 through 7 in the Dong & Peng paper that Joe_74 references correspond to the elements of the same names in the mipo object (documentation |
41,078 | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | I recommend you to simply follow Rubin's rule, as spelled out in page 4 of this paper by Dong and Peng, SpringerPlus 2013:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3701793/pdf/40064_2013_Article_296.pdf
Once you have obtained the variance, just proceed to the standard error (square root of the variance), and then u... | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | I recommend you to simply follow Rubin's rule, as spelled out in page 4 of this paper by Dong and Peng, SpringerPlus 2013:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3701793/pdf/40064_2013_Article_2 | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
I recommend you to simply follow Rubin's rule, as spelled out in page 4 of this paper by Dong and Peng, SpringerPlus 2013:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3701793/pdf/40064_2013_Article_296.pdf
Once you have obt... | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
I recommend you to simply follow Rubin's rule, as spelled out in page 4 of this paper by Dong and Peng, SpringerPlus 2013:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3701793/pdf/40064_2013_Article_2 |
41,079 | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | Perhaps something along the lines of extracting each of the fitted models from each of the imputed data sets, invoke predict on each of these and then bootstrap the results. The code below is laughably inefficient but may serve as a starting point for testing the idea. Additionally, there isn't really a need to compute... | compute 95% confidence interval for predictions using a pooled model after multiple imputation? | Perhaps something along the lines of extracting each of the fitted models from each of the imputed data sets, invoke predict on each of these and then bootstrap the results. The code below is laughabl | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
Perhaps something along the lines of extracting each of the fitted models from each of the imputed data sets, invoke predict on each of these and then bootstrap the results. The code below is laughably inefficient but may se... | compute 95% confidence interval for predictions using a pooled model after multiple imputation?
Perhaps something along the lines of extracting each of the fitted models from each of the imputed data sets, invoke predict on each of these and then bootstrap the results. The code below is laughabl |
41,080 | Probability of failure in a finite population | Suppose you sample $n$ members from a population of $N$ (without replacement) and $k$ of them are failures. The definition of confidence tells us to ask this question:
If there are $K$ failures in the population, what is the chance we observe $k$ or fewer failures in the sample?
Without going into the combinatorial ... | Probability of failure in a finite population | Suppose you sample $n$ members from a population of $N$ (without replacement) and $k$ of them are failures. The definition of confidence tells us to ask this question:
If there are $K$ failures in t | Probability of failure in a finite population
Suppose you sample $n$ members from a population of $N$ (without replacement) and $k$ of them are failures. The definition of confidence tells us to ask this question:
If there are $K$ failures in the population, what is the chance we observe $k$ or fewer failures in the ... | Probability of failure in a finite population
Suppose you sample $n$ members from a population of $N$ (without replacement) and $k$ of them are failures. The definition of confidence tells us to ask this question:
If there are $K$ failures in t |
41,081 | Random Forest Learning Curve | I suspect you have trained a series of RF regression models and have plotted explained variance(not error) against training set size. Explained variance is the opposite than a error. The value would be between 0 and 1.
Secondly it does not make much sense to diagnose training explained accuracy for a random forest. Sam... | Random Forest Learning Curve | I suspect you have trained a series of RF regression models and have plotted explained variance(not error) against training set size. Explained variance is the opposite than a error. The value would b | Random Forest Learning Curve
I suspect you have trained a series of RF regression models and have plotted explained variance(not error) against training set size. Explained variance is the opposite than a error. The value would be between 0 and 1.
Secondly it does not make much sense to diagnose training explained accu... | Random Forest Learning Curve
I suspect you have trained a series of RF regression models and have plotted explained variance(not error) against training set size. Explained variance is the opposite than a error. The value would b |
41,082 | How can I control for a variable while conducting Wilcoxon Rank Sum Test? | The generalization of the Wilcoxon-Mann-Whitney 2-sample test and the Kruskal-Wallis $k$-sample test is the proportional odds model. The PO model allows all the modeling flexibility that regression models support, including covariate adjustment. Details may be found in my course notes. | How can I control for a variable while conducting Wilcoxon Rank Sum Test? | The generalization of the Wilcoxon-Mann-Whitney 2-sample test and the Kruskal-Wallis $k$-sample test is the proportional odds model. The PO model allows all the modeling flexibility that regression m | How can I control for a variable while conducting Wilcoxon Rank Sum Test?
The generalization of the Wilcoxon-Mann-Whitney 2-sample test and the Kruskal-Wallis $k$-sample test is the proportional odds model. The PO model allows all the modeling flexibility that regression models support, including covariate adjustment.... | How can I control for a variable while conducting Wilcoxon Rank Sum Test?
The generalization of the Wilcoxon-Mann-Whitney 2-sample test and the Kruskal-Wallis $k$-sample test is the proportional odds model. The PO model allows all the modeling flexibility that regression m |
41,083 | Why is K-Means++ SLOWER than random initialization K-Means? | kmeans in R is pretty good Fortran code.
I haven't looked at the package you got kmeanspp from, but I wouldn't expect it to be of the same quality. If it uses R code to do the initialization that can hurt badly.
So in the end, you haven't been benchmarking kmeans vs. kmeans++, but you have been showing that the quality... | Why is K-Means++ SLOWER than random initialization K-Means? | kmeans in R is pretty good Fortran code.
I haven't looked at the package you got kmeanspp from, but I wouldn't expect it to be of the same quality. If it uses R code to do the initialization that can | Why is K-Means++ SLOWER than random initialization K-Means?
kmeans in R is pretty good Fortran code.
I haven't looked at the package you got kmeanspp from, but I wouldn't expect it to be of the same quality. If it uses R code to do the initialization that can hurt badly.
So in the end, you haven't been benchmarking kme... | Why is K-Means++ SLOWER than random initialization K-Means?
kmeans in R is pretty good Fortran code.
I haven't looked at the package you got kmeanspp from, but I wouldn't expect it to be of the same quality. If it uses R code to do the initialization that can |
41,084 | Why is K-Means++ SLOWER than random initialization K-Means? | Why is K-Means++ SLOWER than random initialization K-Means?
I would argue that it depends. First recall the difference between the two:
K-means ++
K-means++ involves a somewhat intensive initialization step where:
Choose on center uniformly at random among the data points
For each data point, compute the distance $D... | Why is K-Means++ SLOWER than random initialization K-Means? | Why is K-Means++ SLOWER than random initialization K-Means?
I would argue that it depends. First recall the difference between the two:
K-means ++
K-means++ involves a somewhat intensive initializati | Why is K-Means++ SLOWER than random initialization K-Means?
Why is K-Means++ SLOWER than random initialization K-Means?
I would argue that it depends. First recall the difference between the two:
K-means ++
K-means++ involves a somewhat intensive initialization step where:
Choose on center uniformly at random among ... | Why is K-Means++ SLOWER than random initialization K-Means?
Why is K-Means++ SLOWER than random initialization K-Means?
I would argue that it depends. First recall the difference between the two:
K-means ++
K-means++ involves a somewhat intensive initializati |
41,085 | Dirichlet Prior for Multinomial | I do not think this has anything to do with a wrong definition of the Dirichlet prior or posterior: simply, when
$$(x_1,\ldots,x_k)\sim\mathcal{D}(\alpha_1,...,\alpha_k)$$
the mean is given by
$$\mathbb{E}[(x_1,\ldots,x_k)]=\dfrac{(\alpha_1,...,\alpha_k)}{\sum_{i=1}^k\alpha_i}$$
which explains for the discrepancy with ... | Dirichlet Prior for Multinomial | I do not think this has anything to do with a wrong definition of the Dirichlet prior or posterior: simply, when
$$(x_1,\ldots,x_k)\sim\mathcal{D}(\alpha_1,...,\alpha_k)$$
the mean is given by
$$\math | Dirichlet Prior for Multinomial
I do not think this has anything to do with a wrong definition of the Dirichlet prior or posterior: simply, when
$$(x_1,\ldots,x_k)\sim\mathcal{D}(\alpha_1,...,\alpha_k)$$
the mean is given by
$$\mathbb{E}[(x_1,\ldots,x_k)]=\dfrac{(\alpha_1,...,\alpha_k)}{\sum_{i=1}^k\alpha_i}$$
which ex... | Dirichlet Prior for Multinomial
I do not think this has anything to do with a wrong definition of the Dirichlet prior or posterior: simply, when
$$(x_1,\ldots,x_k)\sim\mathcal{D}(\alpha_1,...,\alpha_k)$$
the mean is given by
$$\math |
41,086 | Why doesn't gradient descent terminate on saddle point? | This answer will only consider asymptotic outcomes, that is, whether we get stuck after infinite number of iterations. We will also only consider the classical gradient descent with zero noise. Noisy variants of GD such as SGD always have a non-zero probability of randomly making several consecutive steps away from the... | Why doesn't gradient descent terminate on saddle point? | This answer will only consider asymptotic outcomes, that is, whether we get stuck after infinite number of iterations. We will also only consider the classical gradient descent with zero noise. Noisy | Why doesn't gradient descent terminate on saddle point?
This answer will only consider asymptotic outcomes, that is, whether we get stuck after infinite number of iterations. We will also only consider the classical gradient descent with zero noise. Noisy variants of GD such as SGD always have a non-zero probability of... | Why doesn't gradient descent terminate on saddle point?
This answer will only consider asymptotic outcomes, that is, whether we get stuck after infinite number of iterations. We will also only consider the classical gradient descent with zero noise. Noisy |
41,087 | Why doesn't gradient descent terminate on saddle point? | Old gradient descent will terminate once it touch a point with derivative zero. And so also will terminate in a saddle if the derivative is zero. But in the everyday gradient descent (stochastic) it's pretty hard or almost impossible to terminate in maximum or saddle, because those aren't points with stable equilibrium... | Why doesn't gradient descent terminate on saddle point? | Old gradient descent will terminate once it touch a point with derivative zero. And so also will terminate in a saddle if the derivative is zero. But in the everyday gradient descent (stochastic) it's | Why doesn't gradient descent terminate on saddle point?
Old gradient descent will terminate once it touch a point with derivative zero. And so also will terminate in a saddle if the derivative is zero. But in the everyday gradient descent (stochastic) it's pretty hard or almost impossible to terminate in maximum or sad... | Why doesn't gradient descent terminate on saddle point?
Old gradient descent will terminate once it touch a point with derivative zero. And so also will terminate in a saddle if the derivative is zero. But in the everyday gradient descent (stochastic) it's |
41,088 | Why doesn't gradient descent terminate on saddle point? | Gradient descent can indeed converge to a saddle point, given a worst case initialization (Nesterov 2013, section 1.2.3).
Random initialization can prevent convergence to saddle points
Things are different when gradient descent is initialized randomly (given certain assumptions). Lee et al. (2016) prove that:
If $f: \... | Why doesn't gradient descent terminate on saddle point? | Gradient descent can indeed converge to a saddle point, given a worst case initialization (Nesterov 2013, section 1.2.3).
Random initialization can prevent convergence to saddle points
Things are diff | Why doesn't gradient descent terminate on saddle point?
Gradient descent can indeed converge to a saddle point, given a worst case initialization (Nesterov 2013, section 1.2.3).
Random initialization can prevent convergence to saddle points
Things are different when gradient descent is initialized randomly (given certa... | Why doesn't gradient descent terminate on saddle point?
Gradient descent can indeed converge to a saddle point, given a worst case initialization (Nesterov 2013, section 1.2.3).
Random initialization can prevent convergence to saddle points
Things are diff |
41,089 | Are there any theoretically rigorous justification for why scaling or normalizing data should improve statistical performance? | In the context of kernel methods, this is quite easy to see. Any given data matrix $\mathbf{X}$ corresponds to a kernel matrix $\mathbf{K}$, which in turns corresponds to a certain solution of the training problem which, due to convexity, is unique and guaranteed to be the global optimum.
However, if we change $\mathbf... | Are there any theoretically rigorous justification for why scaling or normalizing data should improv | In the context of kernel methods, this is quite easy to see. Any given data matrix $\mathbf{X}$ corresponds to a kernel matrix $\mathbf{K}$, which in turns corresponds to a certain solution of the tra | Are there any theoretically rigorous justification for why scaling or normalizing data should improve statistical performance?
In the context of kernel methods, this is quite easy to see. Any given data matrix $\mathbf{X}$ corresponds to a kernel matrix $\mathbf{K}$, which in turns corresponds to a certain solution of ... | Are there any theoretically rigorous justification for why scaling or normalizing data should improv
In the context of kernel methods, this is quite easy to see. Any given data matrix $\mathbf{X}$ corresponds to a kernel matrix $\mathbf{K}$, which in turns corresponds to a certain solution of the tra |
41,090 | What purpose does multiplying by 3 serve in Pearson's second coefficient of skewness? | This is a very interesting question.
The answer lies in history. There was one skewness measure, which was very popular initially, say in the beginning of XX-th century: $$\frac{mean-mode}{s.d.}$$. Since the mode is difficult to estimate the next best thing was to use the (approximate) relationship $$mean - mode \appro... | What purpose does multiplying by 3 serve in Pearson's second coefficient of skewness? | This is a very interesting question.
The answer lies in history. There was one skewness measure, which was very popular initially, say in the beginning of XX-th century: $$\frac{mean-mode}{s.d.}$$. Si | What purpose does multiplying by 3 serve in Pearson's second coefficient of skewness?
This is a very interesting question.
The answer lies in history. There was one skewness measure, which was very popular initially, say in the beginning of XX-th century: $$\frac{mean-mode}{s.d.}$$. Since the mode is difficult to estim... | What purpose does multiplying by 3 serve in Pearson's second coefficient of skewness?
This is a very interesting question.
The answer lies in history. There was one skewness measure, which was very popular initially, say in the beginning of XX-th century: $$\frac{mean-mode}{s.d.}$$. Si |
41,091 | How is the determinant of $(X'X)$ related to variance? | This is a result of using Cramer's rule to calculate the inverse of $\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X}$.
Note that the matrix $(\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X})^{-1}$ is the covariance matrix of the parameters $\beta_i$. So
$$
\text{Var}(\beta_1) = (\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X})^{-1}_{1,1}
... | How is the determinant of $(X'X)$ related to variance? | This is a result of using Cramer's rule to calculate the inverse of $\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X}$.
Note that the matrix $(\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X})^{-1}$ is the covariance | How is the determinant of $(X'X)$ related to variance?
This is a result of using Cramer's rule to calculate the inverse of $\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X}$.
Note that the matrix $(\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X})^{-1}$ is the covariance matrix of the parameters $\beta_i$. So
$$
\text{Var}(\beta_1) =... | How is the determinant of $(X'X)$ related to variance?
This is a result of using Cramer's rule to calculate the inverse of $\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X}$.
Note that the matrix $(\mathbf{X}^{\prime}\Sigma^{-1}\mathbf{X})^{-1}$ is the covariance |
41,092 | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients are equal to zero simultaneously? | In your case, you want to know if the coefficients are equal to $0$. A model where the coefficients are $0$ is the same as a model that does not include those variables. Thus, you can perform a nested model test of a reduced model without those variables versus a full model that includes all the variables.
In a lin... | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients | In your case, you want to know if the coefficients are equal to $0$. A model where the coefficients are $0$ is the same as a model that does not include those variables. Thus, you can perform a nest | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients are equal to zero simultaneously?
In your case, you want to know if the coefficients are equal to $0$. A model where the coefficients are $0$ is the same as a model that does not include those variables. Thus, you can ... | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients
In your case, you want to know if the coefficients are equal to $0$. A model where the coefficients are $0$ is the same as a model that does not include those variables. Thus, you can perform a nest |
41,093 | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients are equal to zero simultaneously? | I will illustrate for models with one and two regressors, the generalization to four variables should be obvious.
library(lmtest)
y <- rnorm(100)
x1 <- rnorm(100)
x2 <- rnorm(100)
reg1 <- lm(y~x1)
reg2 <- lm(y~x1+x2)
waldtest(reg1,reg2)
So, load the lmtest package (the next three lines just create some example data),... | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients | I will illustrate for models with one and two regressors, the generalization to four variables should be obvious.
library(lmtest)
y <- rnorm(100)
x1 <- rnorm(100)
x2 <- rnorm(100)
reg1 <- lm(y~x1)
re | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients are equal to zero simultaneously?
I will illustrate for models with one and two regressors, the generalization to four variables should be obvious.
library(lmtest)
y <- rnorm(100)
x1 <- rnorm(100)
x2 <- rnorm(100)
reg1 ... | In a multiple linear regression model, how do I test the null hypothesis that multiple coefficients
I will illustrate for models with one and two regressors, the generalization to four variables should be obvious.
library(lmtest)
y <- rnorm(100)
x1 <- rnorm(100)
x2 <- rnorm(100)
reg1 <- lm(y~x1)
re |
41,094 | Convex optimization: why so much effort on it? | To get started I recommend reading the introduction to this book by Stephen Boyd. He is doing a lot of cutting edge research in convex optimization, and is one of the biggest names in the field.
Often the issue of "making" your problem convex is reparameterization of the problem formulation. But sometimes the function ... | Convex optimization: why so much effort on it? | To get started I recommend reading the introduction to this book by Stephen Boyd. He is doing a lot of cutting edge research in convex optimization, and is one of the biggest names in the field.
Often | Convex optimization: why so much effort on it?
To get started I recommend reading the introduction to this book by Stephen Boyd. He is doing a lot of cutting edge research in convex optimization, and is one of the biggest names in the field.
Often the issue of "making" your problem convex is reparameterization of the p... | Convex optimization: why so much effort on it?
To get started I recommend reading the introduction to this book by Stephen Boyd. He is doing a lot of cutting edge research in convex optimization, and is one of the biggest names in the field.
Often |
41,095 | Convex optimization: why so much effort on it? | The reason so much time is spent developing convex programming is that it already works so well for so many problems, so improving it further is worthwhile.
Also, simple gradient descent does not work on many large non-smooth (but convex) problems that are very interesting. Stuff that is easy to deal efficiently with ... | Convex optimization: why so much effort on it? | The reason so much time is spent developing convex programming is that it already works so well for so many problems, so improving it further is worthwhile.
Also, simple gradient descent does not wor | Convex optimization: why so much effort on it?
The reason so much time is spent developing convex programming is that it already works so well for so many problems, so improving it further is worthwhile.
Also, simple gradient descent does not work on many large non-smooth (but convex) problems that are very interestin... | Convex optimization: why so much effort on it?
The reason so much time is spent developing convex programming is that it already works so well for so many problems, so improving it further is worthwhile.
Also, simple gradient descent does not wor |
41,096 | What is a predictive distribution? [duplicate] | Some want to close this question because they think Predictive Distributions is a Bayesian concept only, so already answered. But there are also frequentist tries on defining predictive distributions, one is the paper
by A C Davison: "Approximate Predictive Likelihood" (Biometrika) http://biomet.oxfordjournals.org/c... | What is a predictive distribution? [duplicate] | Some want to close this question because they think Predictive Distributions is a Bayesian concept only, so already answered. But there are also frequentist tries on defining predictive distributions | What is a predictive distribution? [duplicate]
Some want to close this question because they think Predictive Distributions is a Bayesian concept only, so already answered. But there are also frequentist tries on defining predictive distributions, one is the paper
by A C Davison: "Approximate Predictive Likelihood" (... | What is a predictive distribution? [duplicate]
Some want to close this question because they think Predictive Distributions is a Bayesian concept only, so already answered. But there are also frequentist tries on defining predictive distributions |
41,097 | Generalisation of the notion of correlation for $\alpha$-stable distributions | I have found something that could be useful. An alternative to the traditional correlation for $\alpha$-stable distributions with $\alpha > 1$ is the signed symmetric covariation coefficient.
Definition. Let $(X_{1},X_{2})$ be a bivariate symmetric $\alpha$-stable random vector with $\alpha > 1$. The signed symmetric c... | Generalisation of the notion of correlation for $\alpha$-stable distributions | I have found something that could be useful. An alternative to the traditional correlation for $\alpha$-stable distributions with $\alpha > 1$ is the signed symmetric covariation coefficient.
Definiti | Generalisation of the notion of correlation for $\alpha$-stable distributions
I have found something that could be useful. An alternative to the traditional correlation for $\alpha$-stable distributions with $\alpha > 1$ is the signed symmetric covariation coefficient.
Definition. Let $(X_{1},X_{2})$ be a bivariate sym... | Generalisation of the notion of correlation for $\alpha$-stable distributions
I have found something that could be useful. An alternative to the traditional correlation for $\alpha$-stable distributions with $\alpha > 1$ is the signed symmetric covariation coefficient.
Definiti |
41,098 | Generalisation of the notion of correlation for $\alpha$-stable distributions | I found this idea in the book Gilchrist: "Statistical Modelling with Quantile Functions"; this are based on medians:
The comedian (don't laugh) of $X$ and $Y$ is defined by
$$
\text{coMED}(X,Y) = M[(X-M(X))(Y-M(Y))]
$$
where $M(X)$ is the median of $X$. Then one would have to standardize this by some measures of varia... | Generalisation of the notion of correlation for $\alpha$-stable distributions | I found this idea in the book Gilchrist: "Statistical Modelling with Quantile Functions"; this are based on medians:
The comedian (don't laugh) of $X$ and $Y$ is defined by
$$
\text{coMED}(X,Y) = M[(X | Generalisation of the notion of correlation for $\alpha$-stable distributions
I found this idea in the book Gilchrist: "Statistical Modelling with Quantile Functions"; this are based on medians:
The comedian (don't laugh) of $X$ and $Y$ is defined by
$$
\text{coMED}(X,Y) = M[(X-M(X))(Y-M(Y))]
$$
where $M(X)$ is the med... | Generalisation of the notion of correlation for $\alpha$-stable distributions
I found this idea in the book Gilchrist: "Statistical Modelling with Quantile Functions"; this are based on medians:
The comedian (don't laugh) of $X$ and $Y$ is defined by
$$
\text{coMED}(X,Y) = M[(X |
41,099 | Training a random forest in R with a fixed maximum false positive rate | The default RF classification aggregate trees by majority vote. Either you must modify the distribution of class votes of trees(see example A) or you must change the aggregation rule (see example B). Option A could be achieved by stratification/downsampling or classweight. I mainly mention because it is possible, as it... | Training a random forest in R with a fixed maximum false positive rate | The default RF classification aggregate trees by majority vote. Either you must modify the distribution of class votes of trees(see example A) or you must change the aggregation rule (see example B). | Training a random forest in R with a fixed maximum false positive rate
The default RF classification aggregate trees by majority vote. Either you must modify the distribution of class votes of trees(see example A) or you must change the aggregation rule (see example B). Option A could be achieved by stratification/down... | Training a random forest in R with a fixed maximum false positive rate
The default RF classification aggregate trees by majority vote. Either you must modify the distribution of class votes of trees(see example A) or you must change the aggregation rule (see example B). |
41,100 | How to visualize a significant interaction between two linear predictors using the rms package? | The rms package allows you to model interactions between continuous variables very flexibly. This is a demonstration of modeling crossed regression splines with that data:
(m <- cph(Surv(time, status) ~ rcs(bili,3) * rcs(alk.phos,3), data=d))
bplot(Predict(m, bili=seq(.5,25,by=.5), alk.phos=seq(300,10000, by=500), fun=... | How to visualize a significant interaction between two linear predictors using the rms package? | The rms package allows you to model interactions between continuous variables very flexibly. This is a demonstration of modeling crossed regression splines with that data:
(m <- cph(Surv(time, status) | How to visualize a significant interaction between two linear predictors using the rms package?
The rms package allows you to model interactions between continuous variables very flexibly. This is a demonstration of modeling crossed regression splines with that data:
(m <- cph(Surv(time, status) ~ rcs(bili,3) * rcs(alk... | How to visualize a significant interaction between two linear predictors using the rms package?
The rms package allows you to model interactions between continuous variables very flexibly. This is a demonstration of modeling crossed regression splines with that data:
(m <- cph(Surv(time, status) |
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