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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39,101 | Kruskal-Wallis or Fligner test to check homogeneity of variances? | The straightforward answer seems to be Levene's Test. Also described at Wikipedia. Levene's is applicable in your case because it is less sensitive to departures from normality than an alternative, the Bartlett Test. Levene's is parametric but suitable even with some degree of non-normality. If the distribution rad... | Kruskal-Wallis or Fligner test to check homogeneity of variances? | The straightforward answer seems to be Levene's Test. Also described at Wikipedia. Levene's is applicable in your case because it is less sensitive to departures from normality than an alternative, | Kruskal-Wallis or Fligner test to check homogeneity of variances?
The straightforward answer seems to be Levene's Test. Also described at Wikipedia. Levene's is applicable in your case because it is less sensitive to departures from normality than an alternative, the Bartlett Test. Levene's is parametric but suitabl... | Kruskal-Wallis or Fligner test to check homogeneity of variances?
The straightforward answer seems to be Levene's Test. Also described at Wikipedia. Levene's is applicable in your case because it is less sensitive to departures from normality than an alternative, |
39,102 | Inference with Gaussian Random Variable | The law of iterated expectations can help here. We have:
$$Var[Y]=E(Var[Y|X])+Var[E(Y|X)]$$
Now conditional on $X$ the expected value of $Y$ is $2X+8$, and its variance is $1$. So we have:
$$Var[Y]=E(1)+Var[2X+8]=1+4 Var[X]=1+\frac{4}{\alpha}$$ | Inference with Gaussian Random Variable | The law of iterated expectations can help here. We have:
$$Var[Y]=E(Var[Y|X])+Var[E(Y|X)]$$
Now conditional on $X$ the expected value of $Y$ is $2X+8$, and its variance is $1$. So we have:
$$Var[Y]= | Inference with Gaussian Random Variable
The law of iterated expectations can help here. We have:
$$Var[Y]=E(Var[Y|X])+Var[E(Y|X)]$$
Now conditional on $X$ the expected value of $Y$ is $2X+8$, and its variance is $1$. So we have:
$$Var[Y]=E(1)+Var[2X+8]=1+4 Var[X]=1+\frac{4}{\alpha}$$ | Inference with Gaussian Random Variable
The law of iterated expectations can help here. We have:
$$Var[Y]=E(Var[Y|X])+Var[E(Y|X)]$$
Now conditional on $X$ the expected value of $Y$ is $2X+8$, and its variance is $1$. So we have:
$$Var[Y]= |
39,103 | Inference with Gaussian Random Variable | Solution to this homework is straightforward application of simple algebra and independence of $X$ and $N_y$: $\mathbb{E} (2 X + N_y)^2 = 4 \mathbb{E} X^2 + 4 \mathbb{E} X \mathbb{E} N_y + \mathbb{E} N_y^2 = 4 Var X + 0 + Var N_y = \frac{4}{\alpha} + 1$. | Inference with Gaussian Random Variable | Solution to this homework is straightforward application of simple algebra and independence of $X$ and $N_y$: $\mathbb{E} (2 X + N_y)^2 = 4 \mathbb{E} X^2 + 4 \mathbb{E} X \mathbb{E} N_y + \mathbb{E} | Inference with Gaussian Random Variable
Solution to this homework is straightforward application of simple algebra and independence of $X$ and $N_y$: $\mathbb{E} (2 X + N_y)^2 = 4 \mathbb{E} X^2 + 4 \mathbb{E} X \mathbb{E} N_y + \mathbb{E} N_y^2 = 4 Var X + 0 + Var N_y = \frac{4}{\alpha} + 1$. | Inference with Gaussian Random Variable
Solution to this homework is straightforward application of simple algebra and independence of $X$ and $N_y$: $\mathbb{E} (2 X + N_y)^2 = 4 \mathbb{E} X^2 + 4 \mathbb{E} X \mathbb{E} N_y + \mathbb{E} |
39,104 | Visualize movie/actor relationships | N.B.: This was previously a (long) comment that I've converted to an answer. Hopefully I'll be able to post an example of what I describe below within a day or two.
Why not try something like a heatmap? Have movies as rows and actors as columns. Maybe sort each of them in terms of the number of actors in the movie and... | Visualize movie/actor relationships | N.B.: This was previously a (long) comment that I've converted to an answer. Hopefully I'll be able to post an example of what I describe below within a day or two.
Why not try something like a heatm | Visualize movie/actor relationships
N.B.: This was previously a (long) comment that I've converted to an answer. Hopefully I'll be able to post an example of what I describe below within a day or two.
Why not try something like a heatmap? Have movies as rows and actors as columns. Maybe sort each of them in terms of t... | Visualize movie/actor relationships
N.B.: This was previously a (long) comment that I've converted to an answer. Hopefully I'll be able to post an example of what I describe below within a day or two.
Why not try something like a heatm |
39,105 | Visualize movie/actor relationships | checkout Gephi, this software has some very good layout algorithms to handle the spaghetti problem: http://gephi.org/features/
Especially, try the ForceAtlas layout: http://forum.gephi.org/viewtopic.php?f=26&t=926
The software let you control the parameters in real time, and you can move the nodes manually.
(disclamer:... | Visualize movie/actor relationships | checkout Gephi, this software has some very good layout algorithms to handle the spaghetti problem: http://gephi.org/features/
Especially, try the ForceAtlas layout: http://forum.gephi.org/viewtopic.p | Visualize movie/actor relationships
checkout Gephi, this software has some very good layout algorithms to handle the spaghetti problem: http://gephi.org/features/
Especially, try the ForceAtlas layout: http://forum.gephi.org/viewtopic.php?f=26&t=926
The software let you control the parameters in real time, and you can ... | Visualize movie/actor relationships
checkout Gephi, this software has some very good layout algorithms to handle the spaghetti problem: http://gephi.org/features/
Especially, try the ForceAtlas layout: http://forum.gephi.org/viewtopic.p |
39,106 | Visualize movie/actor relationships | Graphviz can optimise the layout, see something similar here. | Visualize movie/actor relationships | Graphviz can optimise the layout, see something similar here. | Visualize movie/actor relationships
Graphviz can optimise the layout, see something similar here. | Visualize movie/actor relationships
Graphviz can optimise the layout, see something similar here. |
39,107 | Visualize movie/actor relationships | I wouldn't know how you'd go about constructing this but I liked the method using hyperbolic geometry
http://www.newscientist.com/data/images/ns/cms/dn19420/dn19420-1_800.jpg
http://www.newscientist.com/article/dn19420-escherlike-internet-map-could-speed-online-traffic.html | Visualize movie/actor relationships | I wouldn't know how you'd go about constructing this but I liked the method using hyperbolic geometry
http://www.newscientist.com/data/images/ns/cms/dn19420/dn19420-1_800.jpg
http://www.newscientist.c | Visualize movie/actor relationships
I wouldn't know how you'd go about constructing this but I liked the method using hyperbolic geometry
http://www.newscientist.com/data/images/ns/cms/dn19420/dn19420-1_800.jpg
http://www.newscientist.com/article/dn19420-escherlike-internet-map-could-speed-online-traffic.html | Visualize movie/actor relationships
I wouldn't know how you'd go about constructing this but I liked the method using hyperbolic geometry
http://www.newscientist.com/data/images/ns/cms/dn19420/dn19420-1_800.jpg
http://www.newscientist.c |
39,108 | Understanding multiple regression output | It seems like you need an introduction to regression. People made book recommendations here. Free book recommendations here.
It's hard to make sure you're doing the analysis right when we don't know what the variables are or what the goal is. But based on the output, I can tell you that your second regression specif... | Understanding multiple regression output | It seems like you need an introduction to regression. People made book recommendations here. Free book recommendations here.
It's hard to make sure you're doing the analysis right when we don't know | Understanding multiple regression output
It seems like you need an introduction to regression. People made book recommendations here. Free book recommendations here.
It's hard to make sure you're doing the analysis right when we don't know what the variables are or what the goal is. But based on the output, I can te... | Understanding multiple regression output
It seems like you need an introduction to regression. People made book recommendations here. Free book recommendations here.
It's hard to make sure you're doing the analysis right when we don't know |
39,109 | Understanding multiple regression output | The two together don't tell you anything more than the second one would alone! The main effects are uninteresting and misleading when there is interaction present. The second model tells you all you need to know. Here are a couple of plots, with R code, to help you understand what that second model looks like...
libra... | Understanding multiple regression output | The two together don't tell you anything more than the second one would alone! The main effects are uninteresting and misleading when there is interaction present. The second model tells you all you n | Understanding multiple regression output
The two together don't tell you anything more than the second one would alone! The main effects are uninteresting and misleading when there is interaction present. The second model tells you all you need to know. Here are a couple of plots, with R code, to help you understand w... | Understanding multiple regression output
The two together don't tell you anything more than the second one would alone! The main effects are uninteresting and misleading when there is interaction present. The second model tells you all you n |
39,110 | Understanding multiple regression output | You may be interested by this introduction to the linear model (basis of almost any statistical analyses), and linear regression in particular:
it thoroughly explains lots of the mathematical aspects of linear regression, by detailing all important equations (which is usually left for exercise anywhere else on the Int... | Understanding multiple regression output | You may be interested by this introduction to the linear model (basis of almost any statistical analyses), and linear regression in particular:
it thoroughly explains lots of the mathematical aspects | Understanding multiple regression output
You may be interested by this introduction to the linear model (basis of almost any statistical analyses), and linear regression in particular:
it thoroughly explains lots of the mathematical aspects of linear regression, by detailing all important equations (which is usually l... | Understanding multiple regression output
You may be interested by this introduction to the linear model (basis of almost any statistical analyses), and linear regression in particular:
it thoroughly explains lots of the mathematical aspects |
39,111 | Understanding multiple regression output | If you want a book specifically on this sort of regression - as opposed to data analysis in general - I recommend Regression Analysis by Example by Chatterjee and Price. Good, not technical, but it doesn't oversimplify. | Understanding multiple regression output | If you want a book specifically on this sort of regression - as opposed to data analysis in general - I recommend Regression Analysis by Example by Chatterjee and Price. Good, not technical, but it d | Understanding multiple regression output
If you want a book specifically on this sort of regression - as opposed to data analysis in general - I recommend Regression Analysis by Example by Chatterjee and Price. Good, not technical, but it doesn't oversimplify. | Understanding multiple regression output
If you want a book specifically on this sort of regression - as opposed to data analysis in general - I recommend Regression Analysis by Example by Chatterjee and Price. Good, not technical, but it d |
39,112 | How to generate user-friendly summaries of cluster analysis? | I like a 2D plot that shows the clusters and the actual data points so readers can get an idea of the quality of the clustering. If there are more than two factors, you can put the principle components on the axes, as in my example:
The equivalent 3D plots are only good if the viewer can interact with it get a sense o... | How to generate user-friendly summaries of cluster analysis? | I like a 2D plot that shows the clusters and the actual data points so readers can get an idea of the quality of the clustering. If there are more than two factors, you can put the principle component | How to generate user-friendly summaries of cluster analysis?
I like a 2D plot that shows the clusters and the actual data points so readers can get an idea of the quality of the clustering. If there are more than two factors, you can put the principle components on the axes, as in my example:
The equivalent 3D plots a... | How to generate user-friendly summaries of cluster analysis?
I like a 2D plot that shows the clusters and the actual data points so readers can get an idea of the quality of the clustering. If there are more than two factors, you can put the principle component |
39,113 | How to generate user-friendly summaries of cluster analysis? | Bubble Charts are a good visual device that you can use to represent your cluster. Pick your 4 most important variables and plot each cluster using the x and y axis, size and color of bubble to represent the 4 factors. If you have many variables you can perform a principal components analysis first to reduce them to ... | How to generate user-friendly summaries of cluster analysis? | Bubble Charts are a good visual device that you can use to represent your cluster. Pick your 4 most important variables and plot each cluster using the x and y axis, size and color of bubble to repre | How to generate user-friendly summaries of cluster analysis?
Bubble Charts are a good visual device that you can use to represent your cluster. Pick your 4 most important variables and plot each cluster using the x and y axis, size and color of bubble to represent the 4 factors. If you have many variables you can per... | How to generate user-friendly summaries of cluster analysis?
Bubble Charts are a good visual device that you can use to represent your cluster. Pick your 4 most important variables and plot each cluster using the x and y axis, size and color of bubble to repre |
39,114 | How to generate user-friendly summaries of cluster analysis? | The best method I have found for a non-technical audience is to present a table or plots of the centroids of each cluster along with a description of that cluster. It helps in the business world (not sure your domain) to give a name to each cluster describing it's principle characteristics. Example when clustering cust... | How to generate user-friendly summaries of cluster analysis? | The best method I have found for a non-technical audience is to present a table or plots of the centroids of each cluster along with a description of that cluster. It helps in the business world (not | How to generate user-friendly summaries of cluster analysis?
The best method I have found for a non-technical audience is to present a table or plots of the centroids of each cluster along with a description of that cluster. It helps in the business world (not sure your domain) to give a name to each cluster describing... | How to generate user-friendly summaries of cluster analysis?
The best method I have found for a non-technical audience is to present a table or plots of the centroids of each cluster along with a description of that cluster. It helps in the business world (not |
39,115 | Predicting daily electricity load - fitting time series | I've played around with electrical demand models, and I can tell you that it's a good idea to start "zoomed out". Each region has its own characteristics, but the general idea is the same.
Electric demand is a function of many variables. Starting with the slowest moving terms.
General Economic Activity is the slowe... | Predicting daily electricity load - fitting time series | I've played around with electrical demand models, and I can tell you that it's a good idea to start "zoomed out". Each region has its own characteristics, but the general idea is the same.
Electric | Predicting daily electricity load - fitting time series
I've played around with electrical demand models, and I can tell you that it's a good idea to start "zoomed out". Each region has its own characteristics, but the general idea is the same.
Electric demand is a function of many variables. Starting with the slowe... | Predicting daily electricity load - fitting time series
I've played around with electrical demand models, and I can tell you that it's a good idea to start "zoomed out". Each region has its own characteristics, but the general idea is the same.
Electric |
39,116 | Predicting daily electricity load - fitting time series | @Grega:
I ran out of room in the comment area, so I started another answer.
From your comment, that's the problem. Each system/grid has its own "signature". It's a combination of system dynamics, age, weather, local economics, cultures, traditions, etc. In Japan, its worse. Some sections of the country run at 50Hz... | Predicting daily electricity load - fitting time series | @Grega:
I ran out of room in the comment area, so I started another answer.
From your comment, that's the problem. Each system/grid has its own "signature". It's a combination of system dynamics, a | Predicting daily electricity load - fitting time series
@Grega:
I ran out of room in the comment area, so I started another answer.
From your comment, that's the problem. Each system/grid has its own "signature". It's a combination of system dynamics, age, weather, local economics, cultures, traditions, etc. In Jap... | Predicting daily electricity load - fitting time series
@Grega:
I ran out of room in the comment area, so I started another answer.
From your comment, that's the problem. Each system/grid has its own "signature". It's a combination of system dynamics, a |
39,117 | Determining trend significance in a time series | What you are describing is commonly referred to as auto correlated errors. I would suggest you look up resources on ARIMA modelling. ARIMA modelling will allow you to model the correlation in your error term, and hence allow you to assess your trend variable independent of this auto correlation (or other independent va... | Determining trend significance in a time series | What you are describing is commonly referred to as auto correlated errors. I would suggest you look up resources on ARIMA modelling. ARIMA modelling will allow you to model the correlation in your err | Determining trend significance in a time series
What you are describing is commonly referred to as auto correlated errors. I would suggest you look up resources on ARIMA modelling. ARIMA modelling will allow you to model the correlation in your error term, and hence allow you to assess your trend variable independent o... | Determining trend significance in a time series
What you are describing is commonly referred to as auto correlated errors. I would suggest you look up resources on ARIMA modelling. ARIMA modelling will allow you to model the correlation in your err |
39,118 | Determining trend significance in a time series | To add to the existing answers, if you are using R a simple way to proceed is to allow the ARMA errors to be modelled automatically using auto.arima(). If x is your time series, then you can proceed as follows.
t <- 1:length(x)
auto.arima(x,xreg=t,d=0)
This will fit the model $x_t = a + bt + e_t$ where $e_t\sim\text{A... | Determining trend significance in a time series | To add to the existing answers, if you are using R a simple way to proceed is to allow the ARMA errors to be modelled automatically using auto.arima(). If x is your time series, then you can proceed a | Determining trend significance in a time series
To add to the existing answers, if you are using R a simple way to proceed is to allow the ARMA errors to be modelled automatically using auto.arima(). If x is your time series, then you can proceed as follows.
t <- 1:length(x)
auto.arima(x,xreg=t,d=0)
This will fit the ... | Determining trend significance in a time series
To add to the existing answers, if you are using R a simple way to proceed is to allow the ARMA errors to be modelled automatically using auto.arima(). If x is your time series, then you can proceed a |
39,119 | Determining trend significance in a time series | Generalised least squares (GLS) is one potential option here. The OLS estimates of the parameters are given by:
$$\hat{\beta} = (X^{T}\Sigma^{-1}X)^{-1}X^{T}\Sigma^{-1}y$$
Normally we leave out $\Sigma$ as in OLS it is defined as $\sigma^2 \mathbf{I}$, i.e. an identity matrix multiplied by the estimated residual stand... | Determining trend significance in a time series | Generalised least squares (GLS) is one potential option here. The OLS estimates of the parameters are given by:
$$\hat{\beta} = (X^{T}\Sigma^{-1}X)^{-1}X^{T}\Sigma^{-1}y$$
Normally we leave out $\Sig | Determining trend significance in a time series
Generalised least squares (GLS) is one potential option here. The OLS estimates of the parameters are given by:
$$\hat{\beta} = (X^{T}\Sigma^{-1}X)^{-1}X^{T}\Sigma^{-1}y$$
Normally we leave out $\Sigma$ as in OLS it is defined as $\sigma^2 \mathbf{I}$, i.e. an identity m... | Determining trend significance in a time series
Generalised least squares (GLS) is one potential option here. The OLS estimates of the parameters are given by:
$$\hat{\beta} = (X^{T}\Sigma^{-1}X)^{-1}X^{T}\Sigma^{-1}y$$
Normally we leave out $\Sig |
39,120 | Determining trend significance in a time series | Along the lines of a previous answer, if all assumptions for OLS are met except for the fact that errors are correlated, maybe something as simple as a Cochrane-Orcutt correction would be enough to solve your problem. | Determining trend significance in a time series | Along the lines of a previous answer, if all assumptions for OLS are met except for the fact that errors are correlated, maybe something as simple as a Cochrane-Orcutt correction would be enough to so | Determining trend significance in a time series
Along the lines of a previous answer, if all assumptions for OLS are met except for the fact that errors are correlated, maybe something as simple as a Cochrane-Orcutt correction would be enough to solve your problem. | Determining trend significance in a time series
Along the lines of a previous answer, if all assumptions for OLS are met except for the fact that errors are correlated, maybe something as simple as a Cochrane-Orcutt correction would be enough to so |
39,121 | Fuzzy textbooks | here is the best book I would recomend for the subject:
http://www.amazon.com/Fuzzy-Sets-Logic-Theory-Applications/dp/0131011715
Here is an easy to read book:
http://www.amazon.com/Fuzzy-Logic-Revolutionary-Computer-Technology/dp/0671875353
Besides here are a list of links that might help:
http://www.seattlerobotics.or... | Fuzzy textbooks | here is the best book I would recomend for the subject:
http://www.amazon.com/Fuzzy-Sets-Logic-Theory-Applications/dp/0131011715
Here is an easy to read book:
http://www.amazon.com/Fuzzy-Logic-Revolut | Fuzzy textbooks
here is the best book I would recomend for the subject:
http://www.amazon.com/Fuzzy-Sets-Logic-Theory-Applications/dp/0131011715
Here is an easy to read book:
http://www.amazon.com/Fuzzy-Logic-Revolutionary-Computer-Technology/dp/0671875353
Besides here are a list of links that might help:
http://www.se... | Fuzzy textbooks
here is the best book I would recomend for the subject:
http://www.amazon.com/Fuzzy-Sets-Logic-Theory-Applications/dp/0131011715
Here is an easy to read book:
http://www.amazon.com/Fuzzy-Logic-Revolut |
39,122 | Fuzzy textbooks | I have no experience with fuzzy things (well, apart from Fuzzy Felt) but this book looks interesting:
Buckley, James J. Fuzzy probability and statistics. Springer, 2006. ISBN 9783540308416. | Fuzzy textbooks | I have no experience with fuzzy things (well, apart from Fuzzy Felt) but this book looks interesting:
Buckley, James J. Fuzzy probability and statistics. Springer, 2006. ISBN 9783540308416. | Fuzzy textbooks
I have no experience with fuzzy things (well, apart from Fuzzy Felt) but this book looks interesting:
Buckley, James J. Fuzzy probability and statistics. Springer, 2006. ISBN 9783540308416. | Fuzzy textbooks
I have no experience with fuzzy things (well, apart from Fuzzy Felt) but this book looks interesting:
Buckley, James J. Fuzzy probability and statistics. Springer, 2006. ISBN 9783540308416. |
39,123 | Fuzzy textbooks | If you are looking for exhaustiveness, Springer has the Handbook of Fuzzy Systems series which covers most topics. I particularly recommend Fundamentals of Fuzzy Sets edited by Dubois & Prade which was particularly helpful to me when I was taking a graduate class in uncertainty modelling & possibility theory (with one ... | Fuzzy textbooks | If you are looking for exhaustiveness, Springer has the Handbook of Fuzzy Systems series which covers most topics. I particularly recommend Fundamentals of Fuzzy Sets edited by Dubois & Prade which wa | Fuzzy textbooks
If you are looking for exhaustiveness, Springer has the Handbook of Fuzzy Systems series which covers most topics. I particularly recommend Fundamentals of Fuzzy Sets edited by Dubois & Prade which was particularly helpful to me when I was taking a graduate class in uncertainty modelling & possibility t... | Fuzzy textbooks
If you are looking for exhaustiveness, Springer has the Handbook of Fuzzy Systems series which covers most topics. I particularly recommend Fundamentals of Fuzzy Sets edited by Dubois & Prade which wa |
39,124 | Combining repeated experiments into one dataset | Just add "experiment" as an effect to your model, that should account for the shift between experiments and let you gain the power of increased N across experiments to detect effects of concentration and time.
In R, if using ANOVA and treating time as a factor (e.g. not numeric), then do:
library(ez)
ezANOVA(
data... | Combining repeated experiments into one dataset | Just add "experiment" as an effect to your model, that should account for the shift between experiments and let you gain the power of increased N across experiments to detect effects of concentration | Combining repeated experiments into one dataset
Just add "experiment" as an effect to your model, that should account for the shift between experiments and let you gain the power of increased N across experiments to detect effects of concentration and time.
In R, if using ANOVA and treating time as a factor (e.g. not n... | Combining repeated experiments into one dataset
Just add "experiment" as an effect to your model, that should account for the shift between experiments and let you gain the power of increased N across experiments to detect effects of concentration |
39,125 | Combining repeated experiments into one dataset | I think a way in which you could analyze the two experiments together is by defining a multilevel/hierarchical model. The individuals are nested within each experiment.
The standard for this approach is Gelman's book (I think).
The Journal of Memory and Language had a special issue on analyzing data in 2008 which cover... | Combining repeated experiments into one dataset | I think a way in which you could analyze the two experiments together is by defining a multilevel/hierarchical model. The individuals are nested within each experiment.
The standard for this approach | Combining repeated experiments into one dataset
I think a way in which you could analyze the two experiments together is by defining a multilevel/hierarchical model. The individuals are nested within each experiment.
The standard for this approach is Gelman's book (I think).
The Journal of Memory and Language had a spe... | Combining repeated experiments into one dataset
I think a way in which you could analyze the two experiments together is by defining a multilevel/hierarchical model. The individuals are nested within each experiment.
The standard for this approach |
39,126 | Negative R2 on Simple Linear Regression (with intercept) | Detailed explanation of the problem:
In the case of X being near-singular (high colinearity/covariance between features), different issues where coming both from scipy.linalg.lstsq() and sklearn.linear_model.LinearRegession()
Source of error 1: As @SextusEmpiricus explained, the matrix being near-singular leads to roun... | Negative R2 on Simple Linear Regression (with intercept) | Detailed explanation of the problem:
In the case of X being near-singular (high colinearity/covariance between features), different issues where coming both from scipy.linalg.lstsq() and sklearn.linea | Negative R2 on Simple Linear Regression (with intercept)
Detailed explanation of the problem:
In the case of X being near-singular (high colinearity/covariance between features), different issues where coming both from scipy.linalg.lstsq() and sklearn.linear_model.LinearRegession()
Source of error 1: As @SextusEmpiricu... | Negative R2 on Simple Linear Regression (with intercept)
Detailed explanation of the problem:
In the case of X being near-singular (high colinearity/covariance between features), different issues where coming both from scipy.linalg.lstsq() and sklearn.linea |
39,127 | Negative R2 on Simple Linear Regression (with intercept) | I can reproduce it with np.random.seed(15) and dig a bit deeper. It seems a lot like a computational round-off error due to the high collinearity.
The steps I took
I manually added a column with ones to the matrix X
#X = df_Xy[l_com]
X = pd.concat([pd.DataFrame(np.repeat(1, 999)),df_Xy[l_com]], axis=1)
and used di... | Negative R2 on Simple Linear Regression (with intercept) | I can reproduce it with np.random.seed(15) and dig a bit deeper. It seems a lot like a computational round-off error due to the high collinearity.
The steps I took
I manually added a column with ones | Negative R2 on Simple Linear Regression (with intercept)
I can reproduce it with np.random.seed(15) and dig a bit deeper. It seems a lot like a computational round-off error due to the high collinearity.
The steps I took
I manually added a column with ones to the matrix X
#X = df_Xy[l_com]
X = pd.concat([pd.DataFrame... | Negative R2 on Simple Linear Regression (with intercept)
I can reproduce it with np.random.seed(15) and dig a bit deeper. It seems a lot like a computational round-off error due to the high collinearity.
The steps I took
I manually added a column with ones |
39,128 | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"? | We could rephrase your question asking whether methods based on full data (i.e. noncensored data) are necessarily more efficient than methods based on observed data (i.e. censored data). This question can be answered in general by semiparametric efficiency theory.
Let $Z$ denote the full data (such as covariates and fa... | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"? | We could rephrase your question asking whether methods based on full data (i.e. noncensored data) are necessarily more efficient than methods based on observed data (i.e. censored data). This question | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"?
We could rephrase your question asking whether methods based on full data (i.e. noncensored data) are necessarily more efficient than methods based on observed data (i.e. censored data). This question can be answered in general by sem... | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"?
We could rephrase your question asking whether methods based on full data (i.e. noncensored data) are necessarily more efficient than methods based on observed data (i.e. censored data). This question |
39,129 | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"? | This comes down to the central role of events in survival models. Consider, for example, the Nelson-Aalen non-parametric estimate of a cumulative hazard function $\Lambda(t)$ and the variance of the estimate (Equations 2.2 and 2.4 of Therneau and Grambsch):
$$ \hat \Lambda(t) = \sum_{i:t_i\le t}\frac{\Delta \bar N (t_i... | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"? | This comes down to the central role of events in survival models. Consider, for example, the Nelson-Aalen non-parametric estimate of a cumulative hazard function $\Lambda(t)$ and the variance of the e | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"?
This comes down to the central role of events in survival models. Consider, for example, the Nelson-Aalen non-parametric estimate of a cumulative hazard function $\Lambda(t)$ and the variance of the estimate (Equations 2.2 and 2.4 of ... | Is "Uncensored Data" necessarily more "Informative" when compared to "Censored Data"?
This comes down to the central role of events in survival models. Consider, for example, the Nelson-Aalen non-parametric estimate of a cumulative hazard function $\Lambda(t)$ and the variance of the e |
39,130 | How do I interpret or explain loess plot? | Bootstrapping or a permutation test (as suggested by Stephan Kolassa) will help you assess the significance of the apparent (but complex) association in the plot.
You need to adopt a reasonable measure of what the Loess fit has accomplished. One is the mean squared residual. (Many others are possible, but a discussio... | How do I interpret or explain loess plot? | Bootstrapping or a permutation test (as suggested by Stephan Kolassa) will help you assess the significance of the apparent (but complex) association in the plot.
You need to adopt a reasonable measur | How do I interpret or explain loess plot?
Bootstrapping or a permutation test (as suggested by Stephan Kolassa) will help you assess the significance of the apparent (but complex) association in the plot.
You need to adopt a reasonable measure of what the Loess fit has accomplished. One is the mean squared residual. ... | How do I interpret or explain loess plot?
Bootstrapping or a permutation test (as suggested by Stephan Kolassa) will help you assess the significance of the apparent (but complex) association in the plot.
You need to adopt a reasonable measur |
39,131 | How do I interpret or explain loess plot? | First off, it's extremely good practice to also show the original data, which puts the loess plot into context. Here, the context is that there is still a lot of variation in the data. For instance, the initial dip looks rather strange and could easily be due to noise - and we only see this because we see the full poin... | How do I interpret or explain loess plot? | First off, it's extremely good practice to also show the original data, which puts the loess plot into context. Here, the context is that there is still a lot of variation in the data. For instance, t | How do I interpret or explain loess plot?
First off, it's extremely good practice to also show the original data, which puts the loess plot into context. Here, the context is that there is still a lot of variation in the data. For instance, the initial dip looks rather strange and could easily be due to noise - and we ... | How do I interpret or explain loess plot?
First off, it's extremely good practice to also show the original data, which puts the loess plot into context. Here, the context is that there is still a lot of variation in the data. For instance, t |
39,132 | matrix-calculus - Understanding numerator/denominator layouts | If you think of $L$ as a column vector, then I think both your sources agree that $\frac{dJ}{dL}$ should be a row vector.
But what if you really want $L$ as a row vector. Surely, the math shouldn't "care" about how you arrange your collection of numbers. One way to clarify this is by designating dimensions of your obj... | matrix-calculus - Understanding numerator/denominator layouts | If you think of $L$ as a column vector, then I think both your sources agree that $\frac{dJ}{dL}$ should be a row vector.
But what if you really want $L$ as a row vector. Surely, the math shouldn't " | matrix-calculus - Understanding numerator/denominator layouts
If you think of $L$ as a column vector, then I think both your sources agree that $\frac{dJ}{dL}$ should be a row vector.
But what if you really want $L$ as a row vector. Surely, the math shouldn't "care" about how you arrange your collection of numbers. On... | matrix-calculus - Understanding numerator/denominator layouts
If you think of $L$ as a column vector, then I think both your sources agree that $\frac{dJ}{dL}$ should be a row vector.
But what if you really want $L$ as a row vector. Surely, the math shouldn't " |
39,133 | matrix-calculus - Understanding numerator/denominator layouts | Unfortunately, I didn't come across a resource that doesn't leave gaps. It's a disputed area. Even the chain rule may sometimes not make a lot sense, e.g. some terms might be 3D tensors that the matrix multiplication is not well-defined because of matrices differentiated by vectors or vice versa.
Having said that, thes... | matrix-calculus - Understanding numerator/denominator layouts | Unfortunately, I didn't come across a resource that doesn't leave gaps. It's a disputed area. Even the chain rule may sometimes not make a lot sense, e.g. some terms might be 3D tensors that the matri | matrix-calculus - Understanding numerator/denominator layouts
Unfortunately, I didn't come across a resource that doesn't leave gaps. It's a disputed area. Even the chain rule may sometimes not make a lot sense, e.g. some terms might be 3D tensors that the matrix multiplication is not well-defined because of matrices d... | matrix-calculus - Understanding numerator/denominator layouts
Unfortunately, I didn't come across a resource that doesn't leave gaps. It's a disputed area. Even the chain rule may sometimes not make a lot sense, e.g. some terms might be 3D tensors that the matri |
39,134 | Why getting very high values for MSE/MAE/MAPE when R2 score is very good | I don’t see how you tell from those metrics that the results are “very bad”. Compare the metrics to things like mean, range, or standard deviations, in all the cases MSE or RMSE (square root of MSE) is much smaller than the variability of the data.
The metrics don’t have an absolute numeric value, so you need some kind... | Why getting very high values for MSE/MAE/MAPE when R2 score is very good | I don’t see how you tell from those metrics that the results are “very bad”. Compare the metrics to things like mean, range, or standard deviations, in all the cases MSE or RMSE (square root of MSE) i | Why getting very high values for MSE/MAE/MAPE when R2 score is very good
I don’t see how you tell from those metrics that the results are “very bad”. Compare the metrics to things like mean, range, or standard deviations, in all the cases MSE or RMSE (square root of MSE) is much smaller than the variability of the data... | Why getting very high values for MSE/MAE/MAPE when R2 score is very good
I don’t see how you tell from those metrics that the results are “very bad”. Compare the metrics to things like mean, range, or standard deviations, in all the cases MSE or RMSE (square root of MSE) i |
39,135 | Why getting very high values for MSE/MAE/MAPE when R2 score is very good | The quick answer is that $R^2$ measures a reduction in variance, compared to always guessing $\bar y$, no matter the predictors. What your results tell me is that the variance from always guessing $\bar y$ is so gigantic that even a huge $R^2$ value like $0.9$ or $0.99$ still does not let you get as accurate as you wan... | Why getting very high values for MSE/MAE/MAPE when R2 score is very good | The quick answer is that $R^2$ measures a reduction in variance, compared to always guessing $\bar y$, no matter the predictors. What your results tell me is that the variance from always guessing $\b | Why getting very high values for MSE/MAE/MAPE when R2 score is very good
The quick answer is that $R^2$ measures a reduction in variance, compared to always guessing $\bar y$, no matter the predictors. What your results tell me is that the variance from always guessing $\bar y$ is so gigantic that even a huge $R^2$ val... | Why getting very high values for MSE/MAE/MAPE when R2 score is very good
The quick answer is that $R^2$ measures a reduction in variance, compared to always guessing $\bar y$, no matter the predictors. What your results tell me is that the variance from always guessing $\b |
39,136 | "Histomancy": What does McElreath propose we do instead? | McElreath appears to be maligning the oft repeated practice of requiring regressors (to the right of the equality sign, sometimes 'independent' or 'predictor' variables) and regressands (to the left of the equality sign, sometimes 'dependent' or 'outcome' variables) to be normally distributed (i.e. "Gaussian") in the c... | "Histomancy": What does McElreath propose we do instead? | McElreath appears to be maligning the oft repeated practice of requiring regressors (to the right of the equality sign, sometimes 'independent' or 'predictor' variables) and regressands (to the left o | "Histomancy": What does McElreath propose we do instead?
McElreath appears to be maligning the oft repeated practice of requiring regressors (to the right of the equality sign, sometimes 'independent' or 'predictor' variables) and regressands (to the left of the equality sign, sometimes 'dependent' or 'outcome' variabl... | "Histomancy": What does McElreath propose we do instead?
McElreath appears to be maligning the oft repeated practice of requiring regressors (to the right of the equality sign, sometimes 'independent' or 'predictor' variables) and regressands (to the left o |
39,137 | Train and Validation vs. Train, Test, and Validation | In your two-way split, as you also mentioned, your validation set is actually your test set. In your way, you haven't mentioned about hyperparameter optimisation (HPO), but it's a key step in many machine learning algorithms. When you need HPO, you'll either need to have a separate validation set to tune the HPs or tun... | Train and Validation vs. Train, Test, and Validation | In your two-way split, as you also mentioned, your validation set is actually your test set. In your way, you haven't mentioned about hyperparameter optimisation (HPO), but it's a key step in many mac | Train and Validation vs. Train, Test, and Validation
In your two-way split, as you also mentioned, your validation set is actually your test set. In your way, you haven't mentioned about hyperparameter optimisation (HPO), but it's a key step in many machine learning algorithms. When you need HPO, you'll either need to ... | Train and Validation vs. Train, Test, and Validation
In your two-way split, as you also mentioned, your validation set is actually your test set. In your way, you haven't mentioned about hyperparameter optimisation (HPO), but it's a key step in many mac |
39,138 | Train and Validation vs. Train, Test, and Validation | The two methods you are describing are essentially the same thing. When you describe using cross validation, this is analogous to using a train test split just repeated multiple times. Train/validation/test and train/test with cross validation on the training set are exactly the same but using cross validation repeats ... | Train and Validation vs. Train, Test, and Validation | The two methods you are describing are essentially the same thing. When you describe using cross validation, this is analogous to using a train test split just repeated multiple times. Train/validatio | Train and Validation vs. Train, Test, and Validation
The two methods you are describing are essentially the same thing. When you describe using cross validation, this is analogous to using a train test split just repeated multiple times. Train/validation/test and train/test with cross validation on the training set are... | Train and Validation vs. Train, Test, and Validation
The two methods you are describing are essentially the same thing. When you describe using cross validation, this is analogous to using a train test split just repeated multiple times. Train/validatio |
39,139 | Train and Validation vs. Train, Test, and Validation | In general, you need a split of your data set into test/training whenever there is danger of using information from the test set on the trainig set. The information might flow through you as a modeller. Let this sink in for a moment.
For example, if you build one model with one method, use a 80/20 split or cross valida... | Train and Validation vs. Train, Test, and Validation | In general, you need a split of your data set into test/training whenever there is danger of using information from the test set on the trainig set. The information might flow through you as a modelle | Train and Validation vs. Train, Test, and Validation
In general, you need a split of your data set into test/training whenever there is danger of using information from the test set on the trainig set. The information might flow through you as a modeller. Let this sink in for a moment.
For example, if you build one mod... | Train and Validation vs. Train, Test, and Validation
In general, you need a split of your data set into test/training whenever there is danger of using information from the test set on the trainig set. The information might flow through you as a modelle |
39,140 | Train and Validation vs. Train, Test, and Validation | “All models are wrong, but some are useful”. George E. P. Box
When we build a model, our main materials is data. But it is ultimate goal that probably will never be achieved is to use our model to different data and performs like we trained with our trained data. But we can obviously try and then comes this split.
Now ... | Train and Validation vs. Train, Test, and Validation | “All models are wrong, but some are useful”. George E. P. Box
When we build a model, our main materials is data. But it is ultimate goal that probably will never be achieved is to use our model to dif | Train and Validation vs. Train, Test, and Validation
“All models are wrong, but some are useful”. George E. P. Box
When we build a model, our main materials is data. But it is ultimate goal that probably will never be achieved is to use our model to different data and performs like we trained with our trained data. But... | Train and Validation vs. Train, Test, and Validation
“All models are wrong, but some are useful”. George E. P. Box
When we build a model, our main materials is data. But it is ultimate goal that probably will never be achieved is to use our model to dif |
39,141 | Mean squared error of OLS smaller than Ridge? | That is correct because $b_{OLS}$ is the minimizer of MSE by definition. The problem ($X^TX$ is invertible here) has only one minimum and any value other than $b_{OLS}$ will have higher MSE on the training dataset. | Mean squared error of OLS smaller than Ridge? | That is correct because $b_{OLS}$ is the minimizer of MSE by definition. The problem ($X^TX$ is invertible here) has only one minimum and any value other than $b_{OLS}$ will have higher MSE on the tra | Mean squared error of OLS smaller than Ridge?
That is correct because $b_{OLS}$ is the minimizer of MSE by definition. The problem ($X^TX$ is invertible here) has only one minimum and any value other than $b_{OLS}$ will have higher MSE on the training dataset. | Mean squared error of OLS smaller than Ridge?
That is correct because $b_{OLS}$ is the minimizer of MSE by definition. The problem ($X^TX$ is invertible here) has only one minimum and any value other than $b_{OLS}$ will have higher MSE on the tra |
39,142 | Mean squared error of OLS smaller than Ridge? | like gunes said, the hastie quote applies to out-of-sample (test) MSE, whereas in your question you are showing us in-sample (training) MSE, which Hastie is not referring to.
For your in-sample case, maybe check mean absolute error instead, MAE, which will put the OLS and ridge on equal footing. Otherwise OLS has the u... | Mean squared error of OLS smaller than Ridge? | like gunes said, the hastie quote applies to out-of-sample (test) MSE, whereas in your question you are showing us in-sample (training) MSE, which Hastie is not referring to.
For your in-sample case, | Mean squared error of OLS smaller than Ridge?
like gunes said, the hastie quote applies to out-of-sample (test) MSE, whereas in your question you are showing us in-sample (training) MSE, which Hastie is not referring to.
For your in-sample case, maybe check mean absolute error instead, MAE, which will put the OLS and r... | Mean squared error of OLS smaller than Ridge?
like gunes said, the hastie quote applies to out-of-sample (test) MSE, whereas in your question you are showing us in-sample (training) MSE, which Hastie is not referring to.
For your in-sample case, |
39,143 | Mean squared error of OLS smaller than Ridge? | Ordinary least squares (OLS) minimizes the residual sum of squares (RSS)
$$
RSS=\sum_{i}\left( \varepsilon _{i}\right) ^{2}=\varepsilon ^{\prime
}\varepsilon =\sum_{i}\left( y_{i}-\hat{y}_{i}\right) ^{2}
$$
The mean squared deviation (in the version you are using it) equals
$$
MSE=\frac{RSS}{n}
$$
where $n$ is the numb... | Mean squared error of OLS smaller than Ridge? | Ordinary least squares (OLS) minimizes the residual sum of squares (RSS)
$$
RSS=\sum_{i}\left( \varepsilon _{i}\right) ^{2}=\varepsilon ^{\prime
}\varepsilon =\sum_{i}\left( y_{i}-\hat{y}_{i}\right) ^ | Mean squared error of OLS smaller than Ridge?
Ordinary least squares (OLS) minimizes the residual sum of squares (RSS)
$$
RSS=\sum_{i}\left( \varepsilon _{i}\right) ^{2}=\varepsilon ^{\prime
}\varepsilon =\sum_{i}\left( y_{i}-\hat{y}_{i}\right) ^{2}
$$
The mean squared deviation (in the version you are using it) equals... | Mean squared error of OLS smaller than Ridge?
Ordinary least squares (OLS) minimizes the residual sum of squares (RSS)
$$
RSS=\sum_{i}\left( \varepsilon _{i}\right) ^{2}=\varepsilon ^{\prime
}\varepsilon =\sum_{i}\left( y_{i}-\hat{y}_{i}\right) ^ |
39,144 | Mean squared error of OLS smaller than Ridge? | As others have pointed out, the reason $β_{λ=0}$ (OLS) appears to have lower MSE than $β_{λ>0}$ (ridge) in your example is that you computed both values of $β$ from a matrix of
four (more generally, $N$) observations of two (more generally, $P$) predictors $X$ and corresponding four response values $Y$ and then compute... | Mean squared error of OLS smaller than Ridge? | As others have pointed out, the reason $β_{λ=0}$ (OLS) appears to have lower MSE than $β_{λ>0}$ (ridge) in your example is that you computed both values of $β$ from a matrix of
four (more generally, $ | Mean squared error of OLS smaller than Ridge?
As others have pointed out, the reason $β_{λ=0}$ (OLS) appears to have lower MSE than $β_{λ>0}$ (ridge) in your example is that you computed both values of $β$ from a matrix of
four (more generally, $N$) observations of two (more generally, $P$) predictors $X$ and correspon... | Mean squared error of OLS smaller than Ridge?
As others have pointed out, the reason $β_{λ=0}$ (OLS) appears to have lower MSE than $β_{λ>0}$ (ridge) in your example is that you computed both values of $β$ from a matrix of
four (more generally, $ |
39,145 | Mean squared error of OLS smaller than Ridge? | The result that gunes underscore, efficiency of OLS estimators, is valid among unbiased estimators. The RIDGE estimator induce bias in the estimates but can achieve lower MSE. See the start of the story that bring at the theorem that you cited (bias-variance tradeoff). At practical level RIDGE estimator is useful in pr... | Mean squared error of OLS smaller than Ridge? | The result that gunes underscore, efficiency of OLS estimators, is valid among unbiased estimators. The RIDGE estimator induce bias in the estimates but can achieve lower MSE. See the start of the sto | Mean squared error of OLS smaller than Ridge?
The result that gunes underscore, efficiency of OLS estimators, is valid among unbiased estimators. The RIDGE estimator induce bias in the estimates but can achieve lower MSE. See the start of the story that bring at the theorem that you cited (bias-variance tradeoff). At p... | Mean squared error of OLS smaller than Ridge?
The result that gunes underscore, efficiency of OLS estimators, is valid among unbiased estimators. The RIDGE estimator induce bias in the estimates but can achieve lower MSE. See the start of the sto |
39,146 | Why do we use parametric distributions instead of empirical distributions? | An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are needed for filling the gaps (interpolating) between the observed values and other assumptions are needed for extrapolating o... | Why do we use parametric distributions instead of empirical distributions? | An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are need | Why do we use parametric distributions instead of empirical distributions?
An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are needed for filling the gaps (interpolating) betwe... | Why do we use parametric distributions instead of empirical distributions?
An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are need |
39,147 | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the probability of 2X>Y [closed] | As to what the question is asking: you have two random variables, $X$ and $Y$. Both are uniformly distributed in some interval, each with mean 0 and standard deviation 1. (From this information, we can calculate the interval in which $X$ and $Y$ actually live.)
We thus draw $X$ and $Y$ randomly and independently from t... | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the | As to what the question is asking: you have two random variables, $X$ and $Y$. Both are uniformly distributed in some interval, each with mean 0 and standard deviation 1. (From this information, we ca | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the probability of 2X>Y [closed]
As to what the question is asking: you have two random variables, $X$ and $Y$. Both are uniformly distributed in some interval, each with mean 0 and standard deviation 1. (From this informati... | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the
As to what the question is asking: you have two random variables, $X$ and $Y$. Both are uniformly distributed in some interval, each with mean 0 and standard deviation 1. (From this information, we ca |
39,148 | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the probability of 2X>Y [closed] | This question actually does not require specification of the standard deviation, since the answer is the same for any IID uniform random variables with zero mean. Moreover, the result holds for any event based on cutting the domain into two equal parts via a straight line segment through the origin (hat tip to Stephen... | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the | This question actually does not require specification of the standard deviation, since the answer is the same for any IID uniform random variables with zero mean. Moreover, the result holds for any e | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the probability of 2X>Y [closed]
This question actually does not require specification of the standard deviation, since the answer is the same for any IID uniform random variables with zero mean. Moreover, the result holds ... | Given uniform distributions of X and Y and the mean 0 and standard deviation 1 for both, what’s the
This question actually does not require specification of the standard deviation, since the answer is the same for any IID uniform random variables with zero mean. Moreover, the result holds for any e |
39,149 | Overdispersion in fitted generalized linear model with insignificant regression coefficients | Yes, that is true.
There are only two commonly-used generalized linear model families for which the concept of overdispersion is relevant. These are Poisson regression or binomial regression when the number of trials is greater than one. If the data is genuinely overdispersed then switching from one of these glm regres... | Overdispersion in fitted generalized linear model with insignificant regression coefficients | Yes, that is true.
There are only two commonly-used generalized linear model families for which the concept of overdispersion is relevant. These are Poisson regression or binomial regression when the | Overdispersion in fitted generalized linear model with insignificant regression coefficients
Yes, that is true.
There are only two commonly-used generalized linear model families for which the concept of overdispersion is relevant. These are Poisson regression or binomial regression when the number of trials is greater... | Overdispersion in fitted generalized linear model with insignificant regression coefficients
Yes, that is true.
There are only two commonly-used generalized linear model families for which the concept of overdispersion is relevant. These are Poisson regression or binomial regression when the |
39,150 | Overdispersion in fitted generalized linear model with insignificant regression coefficients | Just to add to @GordonSmyth's answer, when you are fitting a quasipoisson or quasibinomial, the variance-covariance matrix is scaled by the dispersion value. This means the standard error of your coefficients are multiplied by sqrt(dispersion). So
For example, we fit a poisson:
library(pscl)
fm_pois <- glm(art ~ ., da... | Overdispersion in fitted generalized linear model with insignificant regression coefficients | Just to add to @GordonSmyth's answer, when you are fitting a quasipoisson or quasibinomial, the variance-covariance matrix is scaled by the dispersion value. This means the standard error of your coef | Overdispersion in fitted generalized linear model with insignificant regression coefficients
Just to add to @GordonSmyth's answer, when you are fitting a quasipoisson or quasibinomial, the variance-covariance matrix is scaled by the dispersion value. This means the standard error of your coefficients are multiplied by ... | Overdispersion in fitted generalized linear model with insignificant regression coefficients
Just to add to @GordonSmyth's answer, when you are fitting a quasipoisson or quasibinomial, the variance-covariance matrix is scaled by the dispersion value. This means the standard error of your coef |
39,151 | Feature selection for Logistic Regression | My questions is this right approach to do feature selection when data volume is high?
Simply, no.
Basing feature selection on p values is a bad idea, especially when data are large. First, p-values tell you nothing about the effect of the variable. I can always construct a model with a highly significant feature but w... | Feature selection for Logistic Regression | My questions is this right approach to do feature selection when data volume is high?
Simply, no.
Basing feature selection on p values is a bad idea, especially when data are large. First, p-values t | Feature selection for Logistic Regression
My questions is this right approach to do feature selection when data volume is high?
Simply, no.
Basing feature selection on p values is a bad idea, especially when data are large. First, p-values tell you nothing about the effect of the variable. I can always construct a mod... | Feature selection for Logistic Regression
My questions is this right approach to do feature selection when data volume is high?
Simply, no.
Basing feature selection on p values is a bad idea, especially when data are large. First, p-values t |
39,152 | Feature selection for Logistic Regression | From computational perspective, 1M data points and 12 features for logistic regression is nothing, i.e., the computer can return results in seconds.
try this example in R, and you will see how fast we can fit.
d=data.frame(matrix(runif(1e6*12),ncol=12))
d$y=sample(c(0,1),1e6, replace = T)
fit = glm(y~.,d,family='binomi... | Feature selection for Logistic Regression | From computational perspective, 1M data points and 12 features for logistic regression is nothing, i.e., the computer can return results in seconds.
try this example in R, and you will see how fast we | Feature selection for Logistic Regression
From computational perspective, 1M data points and 12 features for logistic regression is nothing, i.e., the computer can return results in seconds.
try this example in R, and you will see how fast we can fit.
d=data.frame(matrix(runif(1e6*12),ncol=12))
d$y=sample(c(0,1),1e6, r... | Feature selection for Logistic Regression
From computational perspective, 1M data points and 12 features for logistic regression is nothing, i.e., the computer can return results in seconds.
try this example in R, and you will see how fast we |
39,153 | Feature selection for Logistic Regression | I agree with the others that p values are not useful here and that regularized regression (ridge, elastic net, lasso) are potential way to go (elastic net might be more useful if the variables are correlated - but which one is best is an empirical question).
I would also decide whether theoretical or potential interact... | Feature selection for Logistic Regression | I agree with the others that p values are not useful here and that regularized regression (ridge, elastic net, lasso) are potential way to go (elastic net might be more useful if the variables are cor | Feature selection for Logistic Regression
I agree with the others that p values are not useful here and that regularized regression (ridge, elastic net, lasso) are potential way to go (elastic net might be more useful if the variables are correlated - but which one is best is an empirical question).
I would also decide... | Feature selection for Logistic Regression
I agree with the others that p values are not useful here and that regularized regression (ridge, elastic net, lasso) are potential way to go (elastic net might be more useful if the variables are cor |
39,154 | Feature selection for Logistic Regression | As with any regression it is best to either be well versed in the subject matter or work with a Subject Matter Expert (SME) to help determine which variables make sense.
A significant step in the process is to look at the stepwise results and see when the point of diminishing returns is reached. In other words, look at... | Feature selection for Logistic Regression | As with any regression it is best to either be well versed in the subject matter or work with a Subject Matter Expert (SME) to help determine which variables make sense.
A significant step in the proc | Feature selection for Logistic Regression
As with any regression it is best to either be well versed in the subject matter or work with a Subject Matter Expert (SME) to help determine which variables make sense.
A significant step in the process is to look at the stepwise results and see when the point of diminishing r... | Feature selection for Logistic Regression
As with any regression it is best to either be well versed in the subject matter or work with a Subject Matter Expert (SME) to help determine which variables make sense.
A significant step in the proc |
39,155 | Feature selection for Logistic Regression | Do you have a stepwise option on your Logistic Regression? That would be preferred.
While all 12 features may yield a significant p-value individually, they may not all be significant when considered in combination with one or more other features. You need to find the best subset.
In any case, it is not the p-values yo... | Feature selection for Logistic Regression | Do you have a stepwise option on your Logistic Regression? That would be preferred.
While all 12 features may yield a significant p-value individually, they may not all be significant when considered | Feature selection for Logistic Regression
Do you have a stepwise option on your Logistic Regression? That would be preferred.
While all 12 features may yield a significant p-value individually, they may not all be significant when considered in combination with one or more other features. You need to find the best subs... | Feature selection for Logistic Regression
Do you have a stepwise option on your Logistic Regression? That would be preferred.
While all 12 features may yield a significant p-value individually, they may not all be significant when considered |
39,156 | Where does the underlying statistical model for inference for a proportion come from? | The answer to your question can be very simple or very deep, even with some philosophical idea involved (e.g., Bayesian and frequentist).
I will try to answer it using frequentist's point of view and with Maximize Likelihood Estimation.
We will start with the coin flip example. Let's assume each coin has its own attri... | Where does the underlying statistical model for inference for a proportion come from? | The answer to your question can be very simple or very deep, even with some philosophical idea involved (e.g., Bayesian and frequentist).
I will try to answer it using frequentist's point of view and | Where does the underlying statistical model for inference for a proportion come from?
The answer to your question can be very simple or very deep, even with some philosophical idea involved (e.g., Bayesian and frequentist).
I will try to answer it using frequentist's point of view and with Maximize Likelihood Estimati... | Where does the underlying statistical model for inference for a proportion come from?
The answer to your question can be very simple or very deep, even with some philosophical idea involved (e.g., Bayesian and frequentist).
I will try to answer it using frequentist's point of view and |
39,157 | Where does the underlying statistical model for inference for a proportion come from? | Since Haitao gave an explanation of a frequentist approach, I will supply a Bayesian one.
In a Bayesian setting, we still generally believe that there is a "true" $p$. We want to understand how probable different values of $p$ are given the data we have observed. In the coin flip example, $p$ is the probability of obse... | Where does the underlying statistical model for inference for a proportion come from? | Since Haitao gave an explanation of a frequentist approach, I will supply a Bayesian one.
In a Bayesian setting, we still generally believe that there is a "true" $p$. We want to understand how probab | Where does the underlying statistical model for inference for a proportion come from?
Since Haitao gave an explanation of a frequentist approach, I will supply a Bayesian one.
In a Bayesian setting, we still generally believe that there is a "true" $p$. We want to understand how probable different values of $p$ are giv... | Where does the underlying statistical model for inference for a proportion come from?
Since Haitao gave an explanation of a frequentist approach, I will supply a Bayesian one.
In a Bayesian setting, we still generally believe that there is a "true" $p$. We want to understand how probab |
39,158 | Where does the underlying statistical model for inference for a proportion come from? | This is saying that we're assuming there is some p such that the event happens with probability p. At this point, we are not making any assumptions about p other than it exists. We later make an estimate of p, but p itself is not obtained; it remains an unknown theoretical quantity. | Where does the underlying statistical model for inference for a proportion come from? | This is saying that we're assuming there is some p such that the event happens with probability p. At this point, we are not making any assumptions about p other than it exists. We later make an estim | Where does the underlying statistical model for inference for a proportion come from?
This is saying that we're assuming there is some p such that the event happens with probability p. At this point, we are not making any assumptions about p other than it exists. We later make an estimate of p, but p itself is not obta... | Where does the underlying statistical model for inference for a proportion come from?
This is saying that we're assuming there is some p such that the event happens with probability p. At this point, we are not making any assumptions about p other than it exists. We later make an estim |
39,159 | Does Bayes theorem apply to joint distributions of discrete and continuous random variables? | Yes.
A joint distribution $f_{X,Z}(x, z)$ of continuous variable $X \sim f_X$, and discrete variable $Z \sim p_Z$, is defined as any non-negative function of $x$ and $z$ that satisfies
$$
\int f_{X,Z}(x, z) dx = p_Z(z),
$$
$$
\sum_z f_{X,Z}(x, z) = f_X(x).
$$
For a given distribution $f_{X,Z}$, the conditional distribu... | Does Bayes theorem apply to joint distributions of discrete and continuous random variables? | Yes.
A joint distribution $f_{X,Z}(x, z)$ of continuous variable $X \sim f_X$, and discrete variable $Z \sim p_Z$, is defined as any non-negative function of $x$ and $z$ that satisfies
$$
\int f_{X,Z} | Does Bayes theorem apply to joint distributions of discrete and continuous random variables?
Yes.
A joint distribution $f_{X,Z}(x, z)$ of continuous variable $X \sim f_X$, and discrete variable $Z \sim p_Z$, is defined as any non-negative function of $x$ and $z$ that satisfies
$$
\int f_{X,Z}(x, z) dx = p_Z(z),
$$
$$
\... | Does Bayes theorem apply to joint distributions of discrete and continuous random variables?
Yes.
A joint distribution $f_{X,Z}(x, z)$ of continuous variable $X \sim f_X$, and discrete variable $Z \sim p_Z$, is defined as any non-negative function of $x$ and $z$ that satisfies
$$
\int f_{X,Z} |
39,160 | Does Bayes theorem apply to joint distributions of discrete and continuous random variables? | [This is just rough intuition, avoiding measure theory]
For continuous random variables ${ X }$ and ${ Y ,}$ their joint density ${ f _{X, Y} }$ (if it exists) is a map ${ f _{ {\color{purple}{X}}, {\color{blue}{Y}} } : {\color{purple}{\mathbb{R}}} \times {\color{blue}{\mathbb{R}}} \to \mathbb{R} _{\geq 0} }$ such tha... | Does Bayes theorem apply to joint distributions of discrete and continuous random variables? | [This is just rough intuition, avoiding measure theory]
For continuous random variables ${ X }$ and ${ Y ,}$ their joint density ${ f _{X, Y} }$ (if it exists) is a map ${ f _{ {\color{purple}{X}}, {\ | Does Bayes theorem apply to joint distributions of discrete and continuous random variables?
[This is just rough intuition, avoiding measure theory]
For continuous random variables ${ X }$ and ${ Y ,}$ their joint density ${ f _{X, Y} }$ (if it exists) is a map ${ f _{ {\color{purple}{X}}, {\color{blue}{Y}} } : {\color... | Does Bayes theorem apply to joint distributions of discrete and continuous random variables?
[This is just rough intuition, avoiding measure theory]
For continuous random variables ${ X }$ and ${ Y ,}$ their joint density ${ f _{X, Y} }$ (if it exists) is a map ${ f _{ {\color{purple}{X}}, {\ |
39,161 | What are the assumptions in bayesian statistics? | Let me use the linear regression example, that you mentioned. The simple linear regression model is
$$
y_i = \alpha + \beta x_i + \varepsilon_i
$$
with noise being independent, normally distributed random variables $\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$. This is equivalent of stating the model in terms of normal... | What are the assumptions in bayesian statistics? | Let me use the linear regression example, that you mentioned. The simple linear regression model is
$$
y_i = \alpha + \beta x_i + \varepsilon_i
$$
with noise being independent, normally distributed ra | What are the assumptions in bayesian statistics?
Let me use the linear regression example, that you mentioned. The simple linear regression model is
$$
y_i = \alpha + \beta x_i + \varepsilon_i
$$
with noise being independent, normally distributed random variables $\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$. This is e... | What are the assumptions in bayesian statistics?
Let me use the linear regression example, that you mentioned. The simple linear regression model is
$$
y_i = \alpha + \beta x_i + \varepsilon_i
$$
with noise being independent, normally distributed ra |
39,162 | What are the assumptions in bayesian statistics? | Assumptions in bayesian statistics are generally stronger than those, because you need, in every model, to specify the full distribution of your data and parameters.
In many cases, gaussian distribution is used, because of its relation to expected value and arithmetic mean, without really believing in the assumption of... | What are the assumptions in bayesian statistics? | Assumptions in bayesian statistics are generally stronger than those, because you need, in every model, to specify the full distribution of your data and parameters.
In many cases, gaussian distributi | What are the assumptions in bayesian statistics?
Assumptions in bayesian statistics are generally stronger than those, because you need, in every model, to specify the full distribution of your data and parameters.
In many cases, gaussian distribution is used, because of its relation to expected value and arithmetic me... | What are the assumptions in bayesian statistics?
Assumptions in bayesian statistics are generally stronger than those, because you need, in every model, to specify the full distribution of your data and parameters.
In many cases, gaussian distributi |
39,163 | Clarifications on Poisson Regression | In Poisson regression we use exponential link function. This means that
$$
\mathbb{E}[Y | X] = e^{W^TX}.
$$
Note that the expression above contains the expectation, not a probability. The expression is known as intensity and is usually denoted with $\lambda(X)$. Conditional on $X$, variable $Y$ has Poisson distribution... | Clarifications on Poisson Regression | In Poisson regression we use exponential link function. This means that
$$
\mathbb{E}[Y | X] = e^{W^TX}.
$$
Note that the expression above contains the expectation, not a probability. The expression i | Clarifications on Poisson Regression
In Poisson regression we use exponential link function. This means that
$$
\mathbb{E}[Y | X] = e^{W^TX}.
$$
Note that the expression above contains the expectation, not a probability. The expression is known as intensity and is usually denoted with $\lambda(X)$. Conditional on $X$, ... | Clarifications on Poisson Regression
In Poisson regression we use exponential link function. This means that
$$
\mathbb{E}[Y | X] = e^{W^TX}.
$$
Note that the expression above contains the expectation, not a probability. The expression i |
39,164 | When does the sum of the medians = the median of the sum | Actually my comment is not entirely correct, allow me to clear up;
The median of a series of numbers $X$ is calculated by ordering all the numbers from smallest to largest, then finding the number in the middle. This means that when you change the numbers in $X$ you also change the ordering, hence the median ch... | When does the sum of the medians = the median of the sum | Actually my comment is not entirely correct, allow me to clear up;
The median of a series of numbers $X$ is calculated by ordering all the numbers from smallest to largest, then finding the number in | When does the sum of the medians = the median of the sum
Actually my comment is not entirely correct, allow me to clear up;
The median of a series of numbers $X$ is calculated by ordering all the numbers from smallest to largest, then finding the number in the middle. This means that when you change the numbers in ... | When does the sum of the medians = the median of the sum
Actually my comment is not entirely correct, allow me to clear up;
The median of a series of numbers $X$ is calculated by ordering all the numbers from smallest to largest, then finding the number in |
39,165 | When does the sum of the medians = the median of the sum | For continuous variables the following are equivalent
$$\text{M}(X + Y) = \text{M}(X) + \text{M}(Y) \\ \iff \\ \mathbb{P}[(X-\text{M}(X)) > -(Y -\text{M}(Y))] =
\mathbb{P}[(X-\text{M}(X)) < -(Y -\text{M}(Y))] $$
You can imagine this geometrically from the joint distribution of X and Y. Half the mass needs to be on e... | When does the sum of the medians = the median of the sum | For continuous variables the following are equivalent
$$\text{M}(X + Y) = \text{M}(X) + \text{M}(Y) \\ \iff \\ \mathbb{P}[(X-\text{M}(X)) > -(Y -\text{M}(Y))] =
\mathbb{P}[(X-\text{M}(X)) < -(Y -\t | When does the sum of the medians = the median of the sum
For continuous variables the following are equivalent
$$\text{M}(X + Y) = \text{M}(X) + \text{M}(Y) \\ \iff \\ \mathbb{P}[(X-\text{M}(X)) > -(Y -\text{M}(Y))] =
\mathbb{P}[(X-\text{M}(X)) < -(Y -\text{M}(Y))] $$
You can imagine this geometrically from the join... | When does the sum of the medians = the median of the sum
For continuous variables the following are equivalent
$$\text{M}(X + Y) = \text{M}(X) + \text{M}(Y) \\ \iff \\ \mathbb{P}[(X-\text{M}(X)) > -(Y -\text{M}(Y))] =
\mathbb{P}[(X-\text{M}(X)) < -(Y -\t |
39,166 | When does the sum of the medians = the median of the sum | Comment: This is parallel with other comments, but it might give
you a quick way to check whether one variable increases precisely when
the other does.
If Spearman correlation between x and y is $1,$ I believe the sum of medians
is the median of the sum. In R:
x = rexp(100); y = sqrt(x)
median(x+y)
[1] 1.598729
media... | When does the sum of the medians = the median of the sum | Comment: This is parallel with other comments, but it might give
you a quick way to check whether one variable increases precisely when
the other does.
If Spearman correlation between x and y is $1,$ | When does the sum of the medians = the median of the sum
Comment: This is parallel with other comments, but it might give
you a quick way to check whether one variable increases precisely when
the other does.
If Spearman correlation between x and y is $1,$ I believe the sum of medians
is the median of the sum. In R:
x... | When does the sum of the medians = the median of the sum
Comment: This is parallel with other comments, but it might give
you a quick way to check whether one variable increases precisely when
the other does.
If Spearman correlation between x and y is $1,$ |
39,167 | How does one most easily overfit? | As long as all the observations are unique, then K-nearest neighbors with K set to 1 and with any arbitrary valid distance metric will give a classifier which perfectly fits the training set (since the nearest neighbor of every point in the training set is trivially, itself). And it's probably the most efficient since ... | How does one most easily overfit? | As long as all the observations are unique, then K-nearest neighbors with K set to 1 and with any arbitrary valid distance metric will give a classifier which perfectly fits the training set (since th | How does one most easily overfit?
As long as all the observations are unique, then K-nearest neighbors with K set to 1 and with any arbitrary valid distance metric will give a classifier which perfectly fits the training set (since the nearest neighbor of every point in the training set is trivially, itself). And it's ... | How does one most easily overfit?
As long as all the observations are unique, then K-nearest neighbors with K set to 1 and with any arbitrary valid distance metric will give a classifier which perfectly fits the training set (since th |
39,168 | How does one most easily overfit? | You can't.
At least not in general, to the degree you want, if you want a perfect fit with arbitrary data and arbitrary dimensionality.
As an example, suppose we have $n_1=0$ predictor dimensions (i.e., none at all) and $n_2=2$ observations classified into $n_3=2$ buckets. The two observations are classified into two d... | How does one most easily overfit? | You can't.
At least not in general, to the degree you want, if you want a perfect fit with arbitrary data and arbitrary dimensionality.
As an example, suppose we have $n_1=0$ predictor dimensions (i.e | How does one most easily overfit?
You can't.
At least not in general, to the degree you want, if you want a perfect fit with arbitrary data and arbitrary dimensionality.
As an example, suppose we have $n_1=0$ predictor dimensions (i.e., none at all) and $n_2=2$ observations classified into $n_3=2$ buckets. The two obse... | How does one most easily overfit?
You can't.
At least not in general, to the degree you want, if you want a perfect fit with arbitrary data and arbitrary dimensionality.
As an example, suppose we have $n_1=0$ predictor dimensions (i.e |
39,169 | lm and glm function in R | If you take a look at the R help documentation you will note that there is no family argument for the lm function. By definition, lm models (ordinary linear regression) in R are fit using ordinary least squares regression (OLS) which assumes the error terms of your model are normally distributed (i.e. family = gaussia... | lm and glm function in R | If you take a look at the R help documentation you will note that there is no family argument for the lm function. By definition, lm models (ordinary linear regression) in R are fit using ordinary le | lm and glm function in R
If you take a look at the R help documentation you will note that there is no family argument for the lm function. By definition, lm models (ordinary linear regression) in R are fit using ordinary least squares regression (OLS) which assumes the error terms of your model are normally distribut... | lm and glm function in R
If you take a look at the R help documentation you will note that there is no family argument for the lm function. By definition, lm models (ordinary linear regression) in R are fit using ordinary le |
39,170 | lm and glm function in R | Linear regression (lm in R) does not have link function and assumes normal distribution. It is generalized linear model (glm in R) that generalizes linear model beyond what linear regression assumes and allows for such modifications. In your case, the family parameter was passed to the ... method and passed further to ... | lm and glm function in R | Linear regression (lm in R) does not have link function and assumes normal distribution. It is generalized linear model (glm in R) that generalizes linear model beyond what linear regression assumes a | lm and glm function in R
Linear regression (lm in R) does not have link function and assumes normal distribution. It is generalized linear model (glm in R) that generalizes linear model beyond what linear regression assumes and allows for such modifications. In your case, the family parameter was passed to the ... meth... | lm and glm function in R
Linear regression (lm in R) does not have link function and assumes normal distribution. It is generalized linear model (glm in R) that generalizes linear model beyond what linear regression assumes a |
39,171 | Can the discrete variable be a negative number? | Your intuition is correct -- a discrete variable can take on negative values.
The example is just an example: a person can't have $-2$ children, but the difference in scores between Home and Away sports teams can be $-2$ when the Home team is behind by two points.
Discrete variables with negative values exist all over ... | Can the discrete variable be a negative number? | Your intuition is correct -- a discrete variable can take on negative values.
The example is just an example: a person can't have $-2$ children, but the difference in scores between Home and Away spor | Can the discrete variable be a negative number?
Your intuition is correct -- a discrete variable can take on negative values.
The example is just an example: a person can't have $-2$ children, but the difference in scores between Home and Away sports teams can be $-2$ when the Home team is behind by two points.
Discret... | Can the discrete variable be a negative number?
Your intuition is correct -- a discrete variable can take on negative values.
The example is just an example: a person can't have $-2$ children, but the difference in scores between Home and Away spor |
39,172 | Can the discrete variable be a negative number? | The difference between continuous and discrete variables is not a mathematical essential one like the difference between natural and real numbers. It's just a matter of practicality: we use different tools to address each one because we are interested on answering different questions.
Basically, in discrete variables w... | Can the discrete variable be a negative number? | The difference between continuous and discrete variables is not a mathematical essential one like the difference between natural and real numbers. It's just a matter of practicality: we use different | Can the discrete variable be a negative number?
The difference between continuous and discrete variables is not a mathematical essential one like the difference between natural and real numbers. It's just a matter of practicality: we use different tools to address each one because we are interested on answering differe... | Can the discrete variable be a negative number?
The difference between continuous and discrete variables is not a mathematical essential one like the difference between natural and real numbers. It's just a matter of practicality: we use different |
39,173 | Sufficient Statistic for $\beta$ in OLS | Sometimes the simplest way to look at sufficiency is by looking directly at the log-likelihood and using the factorisation theorem. For a linear regression model with Gaussian error term the log-likelihood function can be written as:
$$\begin{equation} \begin{aligned}
\ell_{\mathbf{y}, \mathbf{x}}(\boldsymbol{\beta}, ... | Sufficient Statistic for $\beta$ in OLS | Sometimes the simplest way to look at sufficiency is by looking directly at the log-likelihood and using the factorisation theorem. For a linear regression model with Gaussian error term the log-like | Sufficient Statistic for $\beta$ in OLS
Sometimes the simplest way to look at sufficiency is by looking directly at the log-likelihood and using the factorisation theorem. For a linear regression model with Gaussian error term the log-likelihood function can be written as:
$$\begin{equation} \begin{aligned}
\ell_{\mat... | Sufficient Statistic for $\beta$ in OLS
Sometimes the simplest way to look at sufficiency is by looking directly at the log-likelihood and using the factorisation theorem. For a linear regression model with Gaussian error term the log-like |
39,174 | Sufficient Statistic for $\beta$ in OLS | I was looking for some detailed proof for the simple linear model. By the factorisation theorem we have the following solution
Considering the classical model
$$y_i=\beta_0+\beta_1 x_i+\varepsilon_i,\ \mbox{ em que}\ \varepsilon_i\sim \mathrm{N}(0;\sigma^2),\ \hbox{Cov}(\varepsilon_i,\varepsilon_j)=0, i\neq j,\ i,j=1,... | Sufficient Statistic for $\beta$ in OLS | I was looking for some detailed proof for the simple linear model. By the factorisation theorem we have the following solution
Considering the classical model
$$y_i=\beta_0+\beta_1 x_i+\varepsilon_i, | Sufficient Statistic for $\beta$ in OLS
I was looking for some detailed proof for the simple linear model. By the factorisation theorem we have the following solution
Considering the classical model
$$y_i=\beta_0+\beta_1 x_i+\varepsilon_i,\ \mbox{ em que}\ \varepsilon_i\sim \mathrm{N}(0;\sigma^2),\ \hbox{Cov}(\varepsi... | Sufficient Statistic for $\beta$ in OLS
I was looking for some detailed proof for the simple linear model. By the factorisation theorem we have the following solution
Considering the classical model
$$y_i=\beta_0+\beta_1 x_i+\varepsilon_i, |
39,175 | How can change in cost function be positive? | If the learning rate is too large, you can "overshoot". Imagine you're using gradient descent to minimize a 1-dimensional, convex parabola. If you take a small step, you'll (probably) end up closer to the minimum than you were before. But if you take a large step, it's possible that you'll end up on the opposite side o... | How can change in cost function be positive? | If the learning rate is too large, you can "overshoot". Imagine you're using gradient descent to minimize a 1-dimensional, convex parabola. If you take a small step, you'll (probably) end up closer to | How can change in cost function be positive?
If the learning rate is too large, you can "overshoot". Imagine you're using gradient descent to minimize a 1-dimensional, convex parabola. If you take a small step, you'll (probably) end up closer to the minimum than you were before. But if you take a large step, it's possi... | How can change in cost function be positive?
If the learning rate is too large, you can "overshoot". Imagine you're using gradient descent to minimize a 1-dimensional, convex parabola. If you take a small step, you'll (probably) end up closer to |
39,176 | Statistics without hypothesis testing | Let me take the liberty to rephrase the question as "What are the arguments that Andrew Gelman puts forward against hypothesis testing?"
In the paper that is linked in the post, the authors take issue with using a mechanical procedure for model selection, or, as they phrase it:
[Raftery] promises the impossible: The s... | Statistics without hypothesis testing | Let me take the liberty to rephrase the question as "What are the arguments that Andrew Gelman puts forward against hypothesis testing?"
In the paper that is linked in the post, the authors take issue | Statistics without hypothesis testing
Let me take the liberty to rephrase the question as "What are the arguments that Andrew Gelman puts forward against hypothesis testing?"
In the paper that is linked in the post, the authors take issue with using a mechanical procedure for model selection, or, as they phrase it:
[R... | Statistics without hypothesis testing
Let me take the liberty to rephrase the question as "What are the arguments that Andrew Gelman puts forward against hypothesis testing?"
In the paper that is linked in the post, the authors take issue |
39,177 | Statistics without hypothesis testing | The Neyman-Pearson decision-theoretic approach to hypothesis testing (reject/accept) is closely aligned with Popper's Falsification. This method is not invalid, it just has not accommodated the growing human greed for consumption of knowledge, products, and professional gain.
The validity of Popper's approach to scienc... | Statistics without hypothesis testing | The Neyman-Pearson decision-theoretic approach to hypothesis testing (reject/accept) is closely aligned with Popper's Falsification. This method is not invalid, it just has not accommodated the growin | Statistics without hypothesis testing
The Neyman-Pearson decision-theoretic approach to hypothesis testing (reject/accept) is closely aligned with Popper's Falsification. This method is not invalid, it just has not accommodated the growing human greed for consumption of knowledge, products, and professional gain.
The v... | Statistics without hypothesis testing
The Neyman-Pearson decision-theoretic approach to hypothesis testing (reject/accept) is closely aligned with Popper's Falsification. This method is not invalid, it just has not accommodated the growin |
39,178 | How does one graph the PDF of a variable having a mixed discrete-continuous distribution? | The two are not on the same "scale" (probability is p(x) for the discrete and f(x)dx for the continuous, so p and f are very different things); strictly speaking the way to draw the distribution for a mixed variable would be to draw the cdf.
You could also draw the discrete and continuous parts separately, as you sugg... | How does one graph the PDF of a variable having a mixed discrete-continuous distribution? | The two are not on the same "scale" (probability is p(x) for the discrete and f(x)dx for the continuous, so p and f are very different things); strictly speaking the way to draw the distribution for a | How does one graph the PDF of a variable having a mixed discrete-continuous distribution?
The two are not on the same "scale" (probability is p(x) for the discrete and f(x)dx for the continuous, so p and f are very different things); strictly speaking the way to draw the distribution for a mixed variable would be to dr... | How does one graph the PDF of a variable having a mixed discrete-continuous distribution?
The two are not on the same "scale" (probability is p(x) for the discrete and f(x)dx for the continuous, so p and f are very different things); strictly speaking the way to draw the distribution for a |
39,179 | How does one graph the PDF of a variable having a mixed discrete-continuous distribution? | Following on the back and forth with @Glen_b in the commentary to his answer, I created a "mixed" histogram of a mixed discrete-continuous distribution. The variable mixednorm has a 20% chance of producing data from a normal distribution with a mean of 2 and a standard deviation of .8, and an 80% chance of producing a ... | How does one graph the PDF of a variable having a mixed discrete-continuous distribution? | Following on the back and forth with @Glen_b in the commentary to his answer, I created a "mixed" histogram of a mixed discrete-continuous distribution. The variable mixednorm has a 20% chance of prod | How does one graph the PDF of a variable having a mixed discrete-continuous distribution?
Following on the back and forth with @Glen_b in the commentary to his answer, I created a "mixed" histogram of a mixed discrete-continuous distribution. The variable mixednorm has a 20% chance of producing data from a normal distr... | How does one graph the PDF of a variable having a mixed discrete-continuous distribution?
Following on the back and forth with @Glen_b in the commentary to his answer, I created a "mixed" histogram of a mixed discrete-continuous distribution. The variable mixednorm has a 20% chance of prod |
39,180 | How is bagging different from cross-validation? | To add to @juampa's answer,
The big difference between bagging and validation techniques is that bagging averages models (or predictions of an ensemble of models) in order to reduce the variance the prediction is subject to while resampling validation such as cross validation and out-of-bootstrap validation evaluate a ... | How is bagging different from cross-validation? | To add to @juampa's answer,
The big difference between bagging and validation techniques is that bagging averages models (or predictions of an ensemble of models) in order to reduce the variance the p | How is bagging different from cross-validation?
To add to @juampa's answer,
The big difference between bagging and validation techniques is that bagging averages models (or predictions of an ensemble of models) in order to reduce the variance the prediction is subject to while resampling validation such as cross valida... | How is bagging different from cross-validation?
To add to @juampa's answer,
The big difference between bagging and validation techniques is that bagging averages models (or predictions of an ensemble of models) in order to reduce the variance the p |
39,181 | How is bagging different from cross-validation? | Bagging is useful when you work with classifiers that can achieve high accuracy because they tend to overfit (for example decision trees and neural networks) but because of that, show high variance, specially in the small data domain. Then you generate bootstraps samples of your data, train classifiers on each of them,... | How is bagging different from cross-validation? | Bagging is useful when you work with classifiers that can achieve high accuracy because they tend to overfit (for example decision trees and neural networks) but because of that, show high variance, s | How is bagging different from cross-validation?
Bagging is useful when you work with classifiers that can achieve high accuracy because they tend to overfit (for example decision trees and neural networks) but because of that, show high variance, specially in the small data domain. Then you generate bootstraps samples ... | How is bagging different from cross-validation?
Bagging is useful when you work with classifiers that can achieve high accuracy because they tend to overfit (for example decision trees and neural networks) but because of that, show high variance, s |
39,182 | What are the real hyperparameters of a neural network? | Yes. Essentially, any parameter that you can initialize (before training the neural network model) can be seen as a hyperparameter.
This includes the optimizer's hyperparameters (e.g., SGD, Adam, etc.): learning rate, decay rates, step size, and batch-size; as well as model's hyperparameter (CNN): number of layers, num... | What are the real hyperparameters of a neural network? | Yes. Essentially, any parameter that you can initialize (before training the neural network model) can be seen as a hyperparameter.
This includes the optimizer's hyperparameters (e.g., SGD, Adam, etc. | What are the real hyperparameters of a neural network?
Yes. Essentially, any parameter that you can initialize (before training the neural network model) can be seen as a hyperparameter.
This includes the optimizer's hyperparameters (e.g., SGD, Adam, etc.): learning rate, decay rates, step size, and batch-size; as well... | What are the real hyperparameters of a neural network?
Yes. Essentially, any parameter that you can initialize (before training the neural network model) can be seen as a hyperparameter.
This includes the optimizer's hyperparameters (e.g., SGD, Adam, etc. |
39,183 | What are the real hyperparameters of a neural network? | Hyperparameters for a deep neural network:
- Number of iterations
- Number of layers LL in the neural network
- Number of hidden units in each layer
- Learning rate α
- Step size
- Choice of the activation function
- Losss function
- Mini-batch Size
- Momentum
- Regularization
- Drop out rate
- Weight Decay | What are the real hyperparameters of a neural network? | Hyperparameters for a deep neural network:
- Number of iterations
- Number of layers LL in the neural network
- Number of hidden units in each layer
- Learning rate α
- Step size
- Choice of the activ | What are the real hyperparameters of a neural network?
Hyperparameters for a deep neural network:
- Number of iterations
- Number of layers LL in the neural network
- Number of hidden units in each layer
- Learning rate α
- Step size
- Choice of the activation function
- Losss function
- Mini-batch Size
- Momentum
- Re... | What are the real hyperparameters of a neural network?
Hyperparameters for a deep neural network:
- Number of iterations
- Number of layers LL in the neural network
- Number of hidden units in each layer
- Learning rate α
- Step size
- Choice of the activ |
39,184 | PDF of cosine of a uniform random variable | First note that $\cos$ is an even function; $\cos(-X)=\cos(X)$. Consequently it's the same as taking $\cos(W)$ where $W=|X|$ (or indeed you could work instead with $\cos(-W)$). Now $W$ is uniform on $[0,\pi)$. This is easier because the $\cos$ function is now monotonic over the values taken by the new variable and is n... | PDF of cosine of a uniform random variable | First note that $\cos$ is an even function; $\cos(-X)=\cos(X)$. Consequently it's the same as taking $\cos(W)$ where $W=|X|$ (or indeed you could work instead with $\cos(-W)$). Now $W$ is uniform on $ | PDF of cosine of a uniform random variable
First note that $\cos$ is an even function; $\cos(-X)=\cos(X)$. Consequently it's the same as taking $\cos(W)$ where $W=|X|$ (or indeed you could work instead with $\cos(-W)$). Now $W$ is uniform on $[0,\pi)$. This is easier because the $\cos$ function is now monotonic over th... | PDF of cosine of a uniform random variable
First note that $\cos$ is an even function; $\cos(-X)=\cos(X)$. Consequently it's the same as taking $\cos(W)$ where $W=|X|$ (or indeed you could work instead with $\cos(-W)$). Now $W$ is uniform on $ |
39,185 | What are the motivations for the use of the logistic function as a model for binary classification? | There are several reasons for choosing the logistic function as a "default" method for estimating probabilities from one or more variables. Here are a few:
Historical, e.g. dose-response curves
When used with a regression specification on the right hand side of a model, the regression effects are interpretable in tha... | What are the motivations for the use of the logistic function as a model for binary classification? | There are several reasons for choosing the logistic function as a "default" method for estimating probabilities from one or more variables. Here are a few:
Historical, e.g. dose-response curves
When | What are the motivations for the use of the logistic function as a model for binary classification?
There are several reasons for choosing the logistic function as a "default" method for estimating probabilities from one or more variables. Here are a few:
Historical, e.g. dose-response curves
When used with a regress... | What are the motivations for the use of the logistic function as a model for binary classification?
There are several reasons for choosing the logistic function as a "default" method for estimating probabilities from one or more variables. Here are a few:
Historical, e.g. dose-response curves
When |
39,186 | What are the motivations for the use of the logistic function as a model for binary classification? | A technical feature of the logistic link function is that when combined with ML estimation (or posterior mode with flat prior) the gradient equation becomes
$$\sum_i x_i (p_i - y_i) = 0 $$
That is, your residuals, $p_i - y_i$, are uncorrelated with the covariates, $x_i$ (note: $y_i $ is binary $0$ or $1$). This is anal... | What are the motivations for the use of the logistic function as a model for binary classification? | A technical feature of the logistic link function is that when combined with ML estimation (or posterior mode with flat prior) the gradient equation becomes
$$\sum_i x_i (p_i - y_i) = 0 $$
That is, yo | What are the motivations for the use of the logistic function as a model for binary classification?
A technical feature of the logistic link function is that when combined with ML estimation (or posterior mode with flat prior) the gradient equation becomes
$$\sum_i x_i (p_i - y_i) = 0 $$
That is, your residuals, $p_i -... | What are the motivations for the use of the logistic function as a model for binary classification?
A technical feature of the logistic link function is that when combined with ML estimation (or posterior mode with flat prior) the gradient equation becomes
$$\sum_i x_i (p_i - y_i) = 0 $$
That is, yo |
39,187 | How can I quickly detect cheating variables in large data? | This is sometimes referred to as "Data Leakage." There's a nice paper on this here:
Leakage in Data Mining:
Formulation, Detection, and Avoidance
The above paper has plenty of amusing (and horrifying) examples of data leakage, for example, a cancer prediction competition where it turned out that patient ID numbers had... | How can I quickly detect cheating variables in large data? | This is sometimes referred to as "Data Leakage." There's a nice paper on this here:
Leakage in Data Mining:
Formulation, Detection, and Avoidance
The above paper has plenty of amusing (and horrifying | How can I quickly detect cheating variables in large data?
This is sometimes referred to as "Data Leakage." There's a nice paper on this here:
Leakage in Data Mining:
Formulation, Detection, and Avoidance
The above paper has plenty of amusing (and horrifying) examples of data leakage, for example, a cancer prediction ... | How can I quickly detect cheating variables in large data?
This is sometimes referred to as "Data Leakage." There's a nice paper on this here:
Leakage in Data Mining:
Formulation, Detection, and Avoidance
The above paper has plenty of amusing (and horrifying |
39,188 | How can I quickly detect cheating variables in large data? | One way of detecting cheating variables is building a tree model and look at first few splits. Here is a simulated example.
cheating_variable=runif(1e3)
x=matrix(runif(1e5),nrow = 1e3)
y=rbinom(1e3,1,cheating_variable)
d=data.frame(x=cbind(x,cheating_variable),y=y)
library(rpart)
library(partykit)
tree_fit=rpart(y~.,... | How can I quickly detect cheating variables in large data? | One way of detecting cheating variables is building a tree model and look at first few splits. Here is a simulated example.
cheating_variable=runif(1e3)
x=matrix(runif(1e5),nrow = 1e3)
y=rbinom(1e3,1, | How can I quickly detect cheating variables in large data?
One way of detecting cheating variables is building a tree model and look at first few splits. Here is a simulated example.
cheating_variable=runif(1e3)
x=matrix(runif(1e5),nrow = 1e3)
y=rbinom(1e3,1,cheating_variable)
d=data.frame(x=cbind(x,cheating_variable)... | How can I quickly detect cheating variables in large data?
One way of detecting cheating variables is building a tree model and look at first few splits. Here is a simulated example.
cheating_variable=runif(1e3)
x=matrix(runif(1e5),nrow = 1e3)
y=rbinom(1e3,1, |
39,189 | Why doesn't homoskedacticity bias an estimator? | What heteroskedasticity describes is that the variation of the errors may depend on the values of the regressors. That is, that for certain values of $x$ we expect that, while we still expect zero errors on average, any given error tends to be further away from the true regression line in either direction.
The situatio... | Why doesn't homoskedacticity bias an estimator? | What heteroskedasticity describes is that the variation of the errors may depend on the values of the regressors. That is, that for certain values of $x$ we expect that, while we still expect zero err | Why doesn't homoskedacticity bias an estimator?
What heteroskedasticity describes is that the variation of the errors may depend on the values of the regressors. That is, that for certain values of $x$ we expect that, while we still expect zero errors on average, any given error tends to be further away from the true r... | Why doesn't homoskedacticity bias an estimator?
What heteroskedasticity describes is that the variation of the errors may depend on the values of the regressors. That is, that for certain values of $x$ we expect that, while we still expect zero err |
39,190 | Why doesn't homoskedacticity bias an estimator? | Consider a heteroskedastic linear regression with model form:
$$\boldsymbol{Y} = \boldsymbol{x} \boldsymbol{\beta} + \boldsymbol{\varepsilon} \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \boldsymbol{\varepsilon} \sim \text{N}(\boldsymbol{0}, \sigma^2 \text{diag}(\boldsymbol... | Why doesn't homoskedacticity bias an estimator? | Consider a heteroskedastic linear regression with model form:
$$\boldsymbol{Y} = \boldsymbol{x} \boldsymbol{\beta} + \boldsymbol{\varepsilon} \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \tex | Why doesn't homoskedacticity bias an estimator?
Consider a heteroskedastic linear regression with model form:
$$\boldsymbol{Y} = \boldsymbol{x} \boldsymbol{\beta} + \boldsymbol{\varepsilon} \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \boldsymbol{\varepsilon} \sim \text{N}(... | Why doesn't homoskedacticity bias an estimator?
Consider a heteroskedastic linear regression with model form:
$$\boldsymbol{Y} = \boldsymbol{x} \boldsymbol{\beta} + \boldsymbol{\varepsilon} \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \tex |
39,191 | Why doesn't homoskedacticity bias an estimator? | I would like to go a bit theoretical on this. One of the most important assumptions for the unbiasedness of estimators is:
\begin{equation}
E(u|x) = 0
\end{equation}
This assumption implies that we expect no variance between a parameter's true value and its estimator. This can also be expressed as:
\begin{equation}
E(\... | Why doesn't homoskedacticity bias an estimator? | I would like to go a bit theoretical on this. One of the most important assumptions for the unbiasedness of estimators is:
\begin{equation}
E(u|x) = 0
\end{equation}
This assumption implies that we ex | Why doesn't homoskedacticity bias an estimator?
I would like to go a bit theoretical on this. One of the most important assumptions for the unbiasedness of estimators is:
\begin{equation}
E(u|x) = 0
\end{equation}
This assumption implies that we expect no variance between a parameter's true value and its estimator. Thi... | Why doesn't homoskedacticity bias an estimator?
I would like to go a bit theoretical on this. One of the most important assumptions for the unbiasedness of estimators is:
\begin{equation}
E(u|x) = 0
\end{equation}
This assumption implies that we ex |
39,192 | Statistical Significance of multiple classifiers by using p-value | This is common practice but quite controversial among statisticians (Dietterich 1998)
(Kohavi 1995)
For a t-test you need 30 or more samples if you cannot assume a normal distribution (which you really cannot). The common approach to this is to have 3 repetitions of a 10 fold CV. The folds should be shuffled by a rando... | Statistical Significance of multiple classifiers by using p-value | This is common practice but quite controversial among statisticians (Dietterich 1998)
(Kohavi 1995)
For a t-test you need 30 or more samples if you cannot assume a normal distribution (which you reall | Statistical Significance of multiple classifiers by using p-value
This is common practice but quite controversial among statisticians (Dietterich 1998)
(Kohavi 1995)
For a t-test you need 30 or more samples if you cannot assume a normal distribution (which you really cannot). The common approach to this is to have 3 re... | Statistical Significance of multiple classifiers by using p-value
This is common practice but quite controversial among statisticians (Dietterich 1998)
(Kohavi 1995)
For a t-test you need 30 or more samples if you cannot assume a normal distribution (which you reall |
39,193 | Statistical Significance of multiple classifiers by using p-value | I agree with Ramalho that it is a good idea to use statistical hypothesis testing for interpretation of model comparison.
Here are some points, that should give you a start:
You can and should set up your performance measurements so that all algorithms use exactly the same splits. This gives you a paired test setup.... | Statistical Significance of multiple classifiers by using p-value | I agree with Ramalho that it is a good idea to use statistical hypothesis testing for interpretation of model comparison.
Here are some points, that should give you a start:
You can and should set | Statistical Significance of multiple classifiers by using p-value
I agree with Ramalho that it is a good idea to use statistical hypothesis testing for interpretation of model comparison.
Here are some points, that should give you a start:
You can and should set up your performance measurements so that all algorithm... | Statistical Significance of multiple classifiers by using p-value
I agree with Ramalho that it is a good idea to use statistical hypothesis testing for interpretation of model comparison.
Here are some points, that should give you a start:
You can and should set |
39,194 | Statistical Significance of multiple classifiers by using p-value | You might consider adding the best naive classifier you can think of, assign classes based on some simple historical average, or assign randomly based on expected distribution. (Perhaps that's the 'random dummy'.)
Then do a chi-square test on the confusion matrix of each test vs. the naive classifier to determine the l... | Statistical Significance of multiple classifiers by using p-value | You might consider adding the best naive classifier you can think of, assign classes based on some simple historical average, or assign randomly based on expected distribution. (Perhaps that's the 'ra | Statistical Significance of multiple classifiers by using p-value
You might consider adding the best naive classifier you can think of, assign classes based on some simple historical average, or assign randomly based on expected distribution. (Perhaps that's the 'random dummy'.)
Then do a chi-square test on the confusi... | Statistical Significance of multiple classifiers by using p-value
You might consider adding the best naive classifier you can think of, assign classes based on some simple historical average, or assign randomly based on expected distribution. (Perhaps that's the 'ra |
39,195 | Statistical Significance of multiple classifiers by using p-value | Yes you can apply statistical hypothesis testing.
You evaluate each classifier with cross-validation (like you did) but instead store the (accuracy/other metric) of that classifier during each fold. After you have done that you can use hypothesis testing to test if the mean acc of that classifier is significantly bette... | Statistical Significance of multiple classifiers by using p-value | Yes you can apply statistical hypothesis testing.
You evaluate each classifier with cross-validation (like you did) but instead store the (accuracy/other metric) of that classifier during each fold. A | Statistical Significance of multiple classifiers by using p-value
Yes you can apply statistical hypothesis testing.
You evaluate each classifier with cross-validation (like you did) but instead store the (accuracy/other metric) of that classifier during each fold. After you have done that you can use hypothesis testing... | Statistical Significance of multiple classifiers by using p-value
Yes you can apply statistical hypothesis testing.
You evaluate each classifier with cross-validation (like you did) but instead store the (accuracy/other metric) of that classifier during each fold. A |
39,196 | When should I use `scipy.stats.wilcoxon` instead of `scipy.stats.ranksums`? [duplicate] | Frank Wilcoxon's 1945 paper [1] described two tests -- for "Unpaired Experiments" and "Paired Comparisons" which have come to be called the (Wilcoxon) rank sum test and the (Wilcoxon) signed rank test respectively.
So the first test is for independent (unpaired) samples and the second is for paired samples*.
* It can a... | When should I use `scipy.stats.wilcoxon` instead of `scipy.stats.ranksums`? [duplicate] | Frank Wilcoxon's 1945 paper [1] described two tests -- for "Unpaired Experiments" and "Paired Comparisons" which have come to be called the (Wilcoxon) rank sum test and the (Wilcoxon) signed rank test | When should I use `scipy.stats.wilcoxon` instead of `scipy.stats.ranksums`? [duplicate]
Frank Wilcoxon's 1945 paper [1] described two tests -- for "Unpaired Experiments" and "Paired Comparisons" which have come to be called the (Wilcoxon) rank sum test and the (Wilcoxon) signed rank test respectively.
So the first test... | When should I use `scipy.stats.wilcoxon` instead of `scipy.stats.ranksums`? [duplicate]
Frank Wilcoxon's 1945 paper [1] described two tests -- for "Unpaired Experiments" and "Paired Comparisons" which have come to be called the (Wilcoxon) rank sum test and the (Wilcoxon) signed rank test |
39,197 | Logistic regression diagnostic plots in R | This question is related to: Interpretation of plot(glm.model), which it may benefit you to read. Regarding your specific questions:
What constitutes a predicted value in logistic regression is a tricky subject. That's because the prediction can be made on several different scales. I think the most intuitive pred... | Logistic regression diagnostic plots in R | This question is related to: Interpretation of plot(glm.model), which it may benefit you to read. Regarding your specific questions:
What constitutes a predicted value in logistic regression is a | Logistic regression diagnostic plots in R
This question is related to: Interpretation of plot(glm.model), which it may benefit you to read. Regarding your specific questions:
What constitutes a predicted value in logistic regression is a tricky subject. That's because the prediction can be made on several differen... | Logistic regression diagnostic plots in R
This question is related to: Interpretation of plot(glm.model), which it may benefit you to read. Regarding your specific questions:
What constitutes a predicted value in logistic regression is a |
39,198 | Why are the cluster analysis results using raw data the same as the ones using PCA scores? | This is because PCA scores are simply original data in a rotated coordinate frame.
Below on the left I show some example 2D data (100 points in 2D) and on the right the corresponding PCA scores. The data cloud simply gets rotated clockwise by approximately 45 degrees.
If it is not completely clear to you how one gets ... | Why are the cluster analysis results using raw data the same as the ones using PCA scores? | This is because PCA scores are simply original data in a rotated coordinate frame.
Below on the left I show some example 2D data (100 points in 2D) and on the right the corresponding PCA scores. The d | Why are the cluster analysis results using raw data the same as the ones using PCA scores?
This is because PCA scores are simply original data in a rotated coordinate frame.
Below on the left I show some example 2D data (100 points in 2D) and on the right the corresponding PCA scores. The data cloud simply gets rotated... | Why are the cluster analysis results using raw data the same as the ones using PCA scores?
This is because PCA scores are simply original data in a rotated coordinate frame.
Below on the left I show some example 2D data (100 points in 2D) and on the right the corresponding PCA scores. The d |
39,199 | Relationship between Poisson, binomial, negative binomial distributions and normal distribution | The binomial distribution is the distribution of the number of successes in a fixed (i.e. not random) number of independent trials with the same probability of success on each trial. It support is the set $\{0,1,2,\ldots,n\}$, which is finite, where $n$ is the number of trials.
The negative binomial distribution is th... | Relationship between Poisson, binomial, negative binomial distributions and normal distribution | The binomial distribution is the distribution of the number of successes in a fixed (i.e. not random) number of independent trials with the same probability of success on each trial. It support is th | Relationship between Poisson, binomial, negative binomial distributions and normal distribution
The binomial distribution is the distribution of the number of successes in a fixed (i.e. not random) number of independent trials with the same probability of success on each trial. It support is the set $\{0,1,2,\ldots,n\... | Relationship between Poisson, binomial, negative binomial distributions and normal distribution
The binomial distribution is the distribution of the number of successes in a fixed (i.e. not random) number of independent trials with the same probability of success on each trial. It support is th |
39,200 | Covariance matrix of complex random variables | Here is a geometric interpretation.
First, take two vectors in $\mathbb{R}^2$
$$\vec{\mathbb{z}}=[x,y] \,, \vec{\mathbb{w}}=[u,v]$$
For these vectors, there are two standard types of "products", the dot product
$$\vec{\mathbb{z}}\dot{}\vec{\mathbb{w}}=xu+yv$$
and the cross product*
$$\vec{\mathbb{z}}\times\vec{\mathbb{... | Covariance matrix of complex random variables | Here is a geometric interpretation.
First, take two vectors in $\mathbb{R}^2$
$$\vec{\mathbb{z}}=[x,y] \,, \vec{\mathbb{w}}=[u,v]$$
For these vectors, there are two standard types of "products", the d | Covariance matrix of complex random variables
Here is a geometric interpretation.
First, take two vectors in $\mathbb{R}^2$
$$\vec{\mathbb{z}}=[x,y] \,, \vec{\mathbb{w}}=[u,v]$$
For these vectors, there are two standard types of "products", the dot product
$$\vec{\mathbb{z}}\dot{}\vec{\mathbb{w}}=xu+yv$$
and the cross ... | Covariance matrix of complex random variables
Here is a geometric interpretation.
First, take two vectors in $\mathbb{R}^2$
$$\vec{\mathbb{z}}=[x,y] \,, \vec{\mathbb{w}}=[u,v]$$
For these vectors, there are two standard types of "products", the d |
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