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 |
|---|---|---|---|---|---|---|
51,901 | If X and Y are correlated, but Y and Z are independent, is X and Z always independent? | No.
Counterexample: Suppose $Y, Z \sim {\rm Bernoulli}(0.5)$ are IID. Now let $X = YZ$. $X$ clearly is not independent of either $Y$ or $Z$. | If X and Y are correlated, but Y and Z are independent, is X and Z always independent? | No.
Counterexample: Suppose $Y, Z \sim {\rm Bernoulli}(0.5)$ are IID. Now let $X = YZ$. $X$ clearly is not independent of either $Y$ or $Z$. | If X and Y are correlated, but Y and Z are independent, is X and Z always independent?
No.
Counterexample: Suppose $Y, Z \sim {\rm Bernoulli}(0.5)$ are IID. Now let $X = YZ$. $X$ clearly is not independent of either $Y$ or $Z$. | If X and Y are correlated, but Y and Z are independent, is X and Z always independent?
No.
Counterexample: Suppose $Y, Z \sim {\rm Bernoulli}(0.5)$ are IID. Now let $X = YZ$. $X$ clearly is not independent of either $Y$ or $Z$. |
51,902 | Explanatory variables may bias predictions | The point is that you don't code levels of categorical variables as 1,2,3 even if you call them that. You code them using dummy variables, e.g.
$$\begin{array}{l c c}
&x_1&x_2\\
\text{level 1} &0 & 0\\
\text{level 2} &1 & 0\\
\text{level 3} &0 & 1\\
\end{array}$$
so that when the linear predictor is given by
$$\eta = ... | Explanatory variables may bias predictions | The point is that you don't code levels of categorical variables as 1,2,3 even if you call them that. You code them using dummy variables, e.g.
$$\begin{array}{l c c}
&x_1&x_2\\
\text{level 1} &0 & 0 | Explanatory variables may bias predictions
The point is that you don't code levels of categorical variables as 1,2,3 even if you call them that. You code them using dummy variables, e.g.
$$\begin{array}{l c c}
&x_1&x_2\\
\text{level 1} &0 & 0\\
\text{level 2} &1 & 0\\
\text{level 3} &0 & 1\\
\end{array}$$
so that when... | Explanatory variables may bias predictions
The point is that you don't code levels of categorical variables as 1,2,3 even if you call them that. You code them using dummy variables, e.g.
$$\begin{array}{l c c}
&x_1&x_2\\
\text{level 1} &0 & 0 |
51,903 | Explanatory variables may bias predictions | If your software treats a factor variable with factor levels 1, 2, 3 as numeric, your model and your predictions will be garbage, simply because the difference (on the link scale) between groups 1 and 3 will be fitted to be double the difference between groups 1 and 2, which will usually not make sense.
So: tell your s... | Explanatory variables may bias predictions | If your software treats a factor variable with factor levels 1, 2, 3 as numeric, your model and your predictions will be garbage, simply because the difference (on the link scale) between groups 1 and | Explanatory variables may bias predictions
If your software treats a factor variable with factor levels 1, 2, 3 as numeric, your model and your predictions will be garbage, simply because the difference (on the link scale) between groups 1 and 3 will be fitted to be double the difference between groups 1 and 2, which w... | Explanatory variables may bias predictions
If your software treats a factor variable with factor levels 1, 2, 3 as numeric, your model and your predictions will be garbage, simply because the difference (on the link scale) between groups 1 and |
51,904 | How to test a curvilinear relationship in a logistic regression | "curvilinear" could mean anything geometrically not a straight line on the scale being used. So, that could mean many things, including behaviour best tackled with powers of another variable, exponentials, logarithms, trigonometric and hyperbolic functions, etc., etc.
Using logistic regression does not change what is ... | How to test a curvilinear relationship in a logistic regression | "curvilinear" could mean anything geometrically not a straight line on the scale being used. So, that could mean many things, including behaviour best tackled with powers of another variable, exponent | How to test a curvilinear relationship in a logistic regression
"curvilinear" could mean anything geometrically not a straight line on the scale being used. So, that could mean many things, including behaviour best tackled with powers of another variable, exponentials, logarithms, trigonometric and hyperbolic functions... | How to test a curvilinear relationship in a logistic regression
"curvilinear" could mean anything geometrically not a straight line on the scale being used. So, that could mean many things, including behaviour best tackled with powers of another variable, exponent |
51,905 | How to test a curvilinear relationship in a logistic regression | In addition to @Nick's excellent answer, let me just add some practical things about the modelling of nonlinear relationships that I've come across in my work. In epidemiology, for example, we are often faced with nonlinear dose-response relationships. An example would be the relationship between number of cigarettes s... | How to test a curvilinear relationship in a logistic regression | In addition to @Nick's excellent answer, let me just add some practical things about the modelling of nonlinear relationships that I've come across in my work. In epidemiology, for example, we are oft | How to test a curvilinear relationship in a logistic regression
In addition to @Nick's excellent answer, let me just add some practical things about the modelling of nonlinear relationships that I've come across in my work. In epidemiology, for example, we are often faced with nonlinear dose-response relationships. An ... | How to test a curvilinear relationship in a logistic regression
In addition to @Nick's excellent answer, let me just add some practical things about the modelling of nonlinear relationships that I've come across in my work. In epidemiology, for example, we are oft |
51,906 | Why can't I trim the the dependent variable in a regression? Or can I? | There is a simulated data set called outliers in the TeachingDemos package for R. If you remove the "outliers" using a common rule of thumb, then relook at the data and remove the points that are now outliers and continue until you have no "outliers" you end up trowing away 75% of the data as "outliers". Are they rea... | Why can't I trim the the dependent variable in a regression? Or can I? | There is a simulated data set called outliers in the TeachingDemos package for R. If you remove the "outliers" using a common rule of thumb, then relook at the data and remove the points that are now | Why can't I trim the the dependent variable in a regression? Or can I?
There is a simulated data set called outliers in the TeachingDemos package for R. If you remove the "outliers" using a common rule of thumb, then relook at the data and remove the points that are now outliers and continue until you have no "outlier... | Why can't I trim the the dependent variable in a regression? Or can I?
There is a simulated data set called outliers in the TeachingDemos package for R. If you remove the "outliers" using a common rule of thumb, then relook at the data and remove the points that are now |
51,907 | Why can't I trim the the dependent variable in a regression? Or can I? | Rejecting to predict particular $X$s that do not fit your specification can be seen as part of the model:
$ Y = \begin{cases}
f (X) & \text{if } X_{min } \leq X \leq X_{max}\\
\text{refuse prediction, e.g. NA} & \text{else}
\end{cases}
$
It is not sensible to formulate such a condition on $Y$. You model the ... | Why can't I trim the the dependent variable in a regression? Or can I? | Rejecting to predict particular $X$s that do not fit your specification can be seen as part of the model:
$ Y = \begin{cases}
f (X) & \text{if } X_{min } \leq X \leq X_{max}\\
\text{refuse p | Why can't I trim the the dependent variable in a regression? Or can I?
Rejecting to predict particular $X$s that do not fit your specification can be seen as part of the model:
$ Y = \begin{cases}
f (X) & \text{if } X_{min } \leq X \leq X_{max}\\
\text{refuse prediction, e.g. NA} & \text{else}
\end{cases}
$
... | Why can't I trim the the dependent variable in a regression? Or can I?
Rejecting to predict particular $X$s that do not fit your specification can be seen as part of the model:
$ Y = \begin{cases}
f (X) & \text{if } X_{min } \leq X \leq X_{max}\\
\text{refuse p |
51,908 | Why can't I trim the the dependent variable in a regression? Or can I? | Regression is carried out conditional on the observed values of the independent variables. So if you only want to model over a certain range and you're not confident in the assumption of a linear relationship with the response (or another assumption such as constant variance) outside this range it can be reasonable to ... | Why can't I trim the the dependent variable in a regression? Or can I? | Regression is carried out conditional on the observed values of the independent variables. So if you only want to model over a certain range and you're not confident in the assumption of a linear rela | Why can't I trim the the dependent variable in a regression? Or can I?
Regression is carried out conditional on the observed values of the independent variables. So if you only want to model over a certain range and you're not confident in the assumption of a linear relationship with the response (or another assumption... | Why can't I trim the the dependent variable in a regression? Or can I?
Regression is carried out conditional on the observed values of the independent variables. So if you only want to model over a certain range and you're not confident in the assumption of a linear rela |
51,909 | Why can't I trim the the dependent variable in a regression? Or can I? | I don't know about financial economics, but there are instances when our understanding is so embarrassingly inadequate that discarding data is the only thing we know how to do. For example, the field of energy intake/consumption and calories sometimes entails using data driven approaches to discard extreme observation... | Why can't I trim the the dependent variable in a regression? Or can I? | I don't know about financial economics, but there are instances when our understanding is so embarrassingly inadequate that discarding data is the only thing we know how to do. For example, the field | Why can't I trim the the dependent variable in a regression? Or can I?
I don't know about financial economics, but there are instances when our understanding is so embarrassingly inadequate that discarding data is the only thing we know how to do. For example, the field of energy intake/consumption and calories someti... | Why can't I trim the the dependent variable in a regression? Or can I?
I don't know about financial economics, but there are instances when our understanding is so embarrassingly inadequate that discarding data is the only thing we know how to do. For example, the field |
51,910 | How to reduce the number of variables in cluster analysis? | The problem with dimensionality reduction and number of variables >> number of observations is that the $k$ observations that you have define an at most $k-1$ dimensional hyperplane on which the objects perfectly are located on.
So yes, anything more than 9 dimensions still has proven redundancies.
Many dimension reduc... | How to reduce the number of variables in cluster analysis? | The problem with dimensionality reduction and number of variables >> number of observations is that the $k$ observations that you have define an at most $k-1$ dimensional hyperplane on which the objec | How to reduce the number of variables in cluster analysis?
The problem with dimensionality reduction and number of variables >> number of observations is that the $k$ observations that you have define an at most $k-1$ dimensional hyperplane on which the objects perfectly are located on.
So yes, anything more than 9 dim... | How to reduce the number of variables in cluster analysis?
The problem with dimensionality reduction and number of variables >> number of observations is that the $k$ observations that you have define an at most $k-1$ dimensional hyperplane on which the objec |
51,911 | How to reduce the number of variables in cluster analysis? | Here's something else you can try.
This data is similar to what they see in genomics, so you could look to that field for ideas of analysis. In genomics there are lots of variables (20,000+), many of which are highly correlated with each other, and a relatively small number of rows.
If this were a genomics problem, ... | How to reduce the number of variables in cluster analysis? | Here's something else you can try.
This data is similar to what they see in genomics, so you could look to that field for ideas of analysis. In genomics there are lots of variables (20,000+), many | How to reduce the number of variables in cluster analysis?
Here's something else you can try.
This data is similar to what they see in genomics, so you could look to that field for ideas of analysis. In genomics there are lots of variables (20,000+), many of which are highly correlated with each other, and a relativ... | How to reduce the number of variables in cluster analysis?
Here's something else you can try.
This data is similar to what they see in genomics, so you could look to that field for ideas of analysis. In genomics there are lots of variables (20,000+), many |
51,912 | How to reduce the number of variables in cluster analysis? | You could try transposing the data and computing principal components to see which cases load on which components. It might be necessary to rotate the results to get clearer clusters, but ideally you could end up with three good components with each representing the groups you are expecting. Even if that doesn't work, ... | How to reduce the number of variables in cluster analysis? | You could try transposing the data and computing principal components to see which cases load on which components. It might be necessary to rotate the results to get clearer clusters, but ideally you | How to reduce the number of variables in cluster analysis?
You could try transposing the data and computing principal components to see which cases load on which components. It might be necessary to rotate the results to get clearer clusters, but ideally you could end up with three good components with each representin... | How to reduce the number of variables in cluster analysis?
You could try transposing the data and computing principal components to see which cases load on which components. It might be necessary to rotate the results to get clearer clusters, but ideally you |
51,913 | Based only on these sensitivity and specificity values, what is the best decision method? | To make an optimal decision you need to know all relevant data about an individual (used to estimate the probability of an outcome), and the utility (cost, loss function) of making each decision. Sensitivity and specificity do not provide this information. That's why direct probability models such as the binary logis... | Based only on these sensitivity and specificity values, what is the best decision method? | To make an optimal decision you need to know all relevant data about an individual (used to estimate the probability of an outcome), and the utility (cost, loss function) of making each decision. Sen | Based only on these sensitivity and specificity values, what is the best decision method?
To make an optimal decision you need to know all relevant data about an individual (used to estimate the probability of an outcome), and the utility (cost, loss function) of making each decision. Sensitivity and specificity do no... | Based only on these sensitivity and specificity values, what is the best decision method?
To make an optimal decision you need to know all relevant data about an individual (used to estimate the probability of an outcome), and the utility (cost, loss function) of making each decision. Sen |
51,914 | Based only on these sensitivity and specificity values, what is the best decision method? | As Nick has already pointed out the answer depends on context. However, if you only had to judge based on the values of sensitivity and specificity values that you provided, then a good strategy is to plot the sensitivity on the y-axis and $(100\%-$ specificity$)$ on the x-axis and look for the highest leftmost point. ... | Based only on these sensitivity and specificity values, what is the best decision method? | As Nick has already pointed out the answer depends on context. However, if you only had to judge based on the values of sensitivity and specificity values that you provided, then a good strategy is to | Based only on these sensitivity and specificity values, what is the best decision method?
As Nick has already pointed out the answer depends on context. However, if you only had to judge based on the values of sensitivity and specificity values that you provided, then a good strategy is to plot the sensitivity on the y... | Based only on these sensitivity and specificity values, what is the best decision method?
As Nick has already pointed out the answer depends on context. However, if you only had to judge based on the values of sensitivity and specificity values that you provided, then a good strategy is to |
51,915 | Why are the correlations in two groups less than the correlation when the groups are combined? | Here are just a couple of ideas:
Range restriction is one explanation. Check out this simulation; and this explanation.
Correlated group mean differences is another related idea. Say group 1 has a mean two standard deviations higher than group 2 on both X and Y, but that there is no correlation between X and Y withi... | Why are the correlations in two groups less than the correlation when the groups are combined? | Here are just a couple of ideas:
Range restriction is one explanation. Check out this simulation; and this explanation.
Correlated group mean differences is another related idea. Say group 1 has a | Why are the correlations in two groups less than the correlation when the groups are combined?
Here are just a couple of ideas:
Range restriction is one explanation. Check out this simulation; and this explanation.
Correlated group mean differences is another related idea. Say group 1 has a mean two standard deviati... | Why are the correlations in two groups less than the correlation when the groups are combined?
Here are just a couple of ideas:
Range restriction is one explanation. Check out this simulation; and this explanation.
Correlated group mean differences is another related idea. Say group 1 has a |
51,916 | Why are the correlations in two groups less than the correlation when the groups are combined? | Sounds like Simpson's Paradox. | Why are the correlations in two groups less than the correlation when the groups are combined? | Sounds like Simpson's Paradox. | Why are the correlations in two groups less than the correlation when the groups are combined?
Sounds like Simpson's Paradox. | Why are the correlations in two groups less than the correlation when the groups are combined?
Sounds like Simpson's Paradox. |
51,917 | Why are the correlations in two groups less than the correlation when the groups are combined? | During this analysis I ran into a situation where the $R^2$ for two groups was smaller in each individual group as opposed to when they are grouped together. Is there any straight forward explanation for how this could happen?
Yes, this simply means that knowing to which group the observation belongs to explains part ... | Why are the correlations in two groups less than the correlation when the groups are combined? | During this analysis I ran into a situation where the $R^2$ for two groups was smaller in each individual group as opposed to when they are grouped together. Is there any straight forward explanation | Why are the correlations in two groups less than the correlation when the groups are combined?
During this analysis I ran into a situation where the $R^2$ for two groups was smaller in each individual group as opposed to when they are grouped together. Is there any straight forward explanation for how this could happen... | Why are the correlations in two groups less than the correlation when the groups are combined?
During this analysis I ran into a situation where the $R^2$ for two groups was smaller in each individual group as opposed to when they are grouped together. Is there any straight forward explanation |
51,918 | Why are the correlations in two groups less than the correlation when the groups are combined? | The phenomenon relates to the geometry of linear regression analysis
This result is quite common in regression, and it reflects the fact that each new explanatory variables generally gives some additional information about the response variable, so that a model that combines variables from two other models will give a ... | Why are the correlations in two groups less than the correlation when the groups are combined? | The phenomenon relates to the geometry of linear regression analysis
This result is quite common in regression, and it reflects the fact that each new explanatory variables generally gives some additi | Why are the correlations in two groups less than the correlation when the groups are combined?
The phenomenon relates to the geometry of linear regression analysis
This result is quite common in regression, and it reflects the fact that each new explanatory variables generally gives some additional information about th... | Why are the correlations in two groups less than the correlation when the groups are combined?
The phenomenon relates to the geometry of linear regression analysis
This result is quite common in regression, and it reflects the fact that each new explanatory variables generally gives some additi |
51,919 | R packages for seasonality analysis [closed] | You don't need to install any packages because this is possible with base-R functions. Have a look at the arima function.
This is a basic function of Box-Jenkins analysis, so you should consider reading one of the R time series text-books for an overview; my favorite is Shumway and Stoffer. "Time Series Analysis and... | R packages for seasonality analysis [closed] | You don't need to install any packages because this is possible with base-R functions. Have a look at the arima function.
This is a basic function of Box-Jenkins analysis, so you should consider re | R packages for seasonality analysis [closed]
You don't need to install any packages because this is possible with base-R functions. Have a look at the arima function.
This is a basic function of Box-Jenkins analysis, so you should consider reading one of the R time series text-books for an overview; my favorite is S... | R packages for seasonality analysis [closed]
You don't need to install any packages because this is possible with base-R functions. Have a look at the arima function.
This is a basic function of Box-Jenkins analysis, so you should consider re |
51,920 | R packages for seasonality analysis [closed] | Try using the stl() function for time series decomposition. It provides a very flexible method for extracting a seasonal component from a time series. | R packages for seasonality analysis [closed] | Try using the stl() function for time series decomposition. It provides a very flexible method for extracting a seasonal component from a time series. | R packages for seasonality analysis [closed]
Try using the stl() function for time series decomposition. It provides a very flexible method for extracting a seasonal component from a time series. | R packages for seasonality analysis [closed]
Try using the stl() function for time series decomposition. It provides a very flexible method for extracting a seasonal component from a time series. |
51,921 | R packages for seasonality analysis [closed] | I build/published an R package named seas for my M.Sc. work a few years ago. The package is good for discretizing a time-series over years into seasonal divisions, such as months or 11-day periods. These divisions can then be applied to continuous variables (e.g., temperature, water levels) or discontinuous variables (... | R packages for seasonality analysis [closed] | I build/published an R package named seas for my M.Sc. work a few years ago. The package is good for discretizing a time-series over years into seasonal divisions, such as months or 11-day periods. Th | R packages for seasonality analysis [closed]
I build/published an R package named seas for my M.Sc. work a few years ago. The package is good for discretizing a time-series over years into seasonal divisions, such as months or 11-day periods. These divisions can then be applied to continuous variables (e.g., temperatur... | R packages for seasonality analysis [closed]
I build/published an R package named seas for my M.Sc. work a few years ago. The package is good for discretizing a time-series over years into seasonal divisions, such as months or 11-day periods. Th |
51,922 | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)? | The conclusion does not follow. The family of distributions described at How is the kurtosis of a distribution related to the geometry of the density function? gives a counterexample. These are densities $f_{k,s}$ all of which have identical moments. If we were, then, to shift one of them by some positive amount, al... | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than | The conclusion does not follow. The family of distributions described at How is the kurtosis of a distribution related to the geometry of the density function? gives a counterexample. These are dens | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)?
The conclusion does not follow. The family of distributions described at How is the kurtosis of a distribution related to the geometry of the density function? gives a counterexample. These are densities $f_{k... | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than
The conclusion does not follow. The family of distributions described at How is the kurtosis of a distribution related to the geometry of the density function? gives a counterexample. These are dens |
51,923 | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)? | The higher moments say relatively little about the behaviour of the distribution at small values. The tails have a lot of influence on a distribution.
You can see this in particular with distributions that have infinite higher moments.
For example compare a Frechet distribution with location = 1, scale = 1 and shape = ... | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than | The higher moments say relatively little about the behaviour of the distribution at small values. The tails have a lot of influence on a distribution.
You can see this in particular with distributions | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)?
The higher moments say relatively little about the behaviour of the distribution at small values. The tails have a lot of influence on a distribution.
You can see this in particular with distributions that have ... | If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than
The higher moments say relatively little about the behaviour of the distribution at small values. The tails have a lot of influence on a distribution.
You can see this in particular with distributions |
51,924 | Estimating the parameter $\beta$ | To estimate the $\beta$ by the maximum likelihood method, let $Y_1,\ldots, Y_n$ be the sample of lifetimes, with $Y_i\sim \text{Exp}(\beta/s_i)$, s.t. $E(Y_i) = \beta/s_i$ and independently for each $i$, where $s_i$ is the monitor brightness.
Then the likelihood function for $\beta$ is given by
$$
L(\beta) = \prod_{i=1... | Estimating the parameter $\beta$ | To estimate the $\beta$ by the maximum likelihood method, let $Y_1,\ldots, Y_n$ be the sample of lifetimes, with $Y_i\sim \text{Exp}(\beta/s_i)$, s.t. $E(Y_i) = \beta/s_i$ and independently for each $ | Estimating the parameter $\beta$
To estimate the $\beta$ by the maximum likelihood method, let $Y_1,\ldots, Y_n$ be the sample of lifetimes, with $Y_i\sim \text{Exp}(\beta/s_i)$, s.t. $E(Y_i) = \beta/s_i$ and independently for each $i$, where $s_i$ is the monitor brightness.
Then the likelihood function for $\beta$ is ... | Estimating the parameter $\beta$
To estimate the $\beta$ by the maximum likelihood method, let $Y_1,\ldots, Y_n$ be the sample of lifetimes, with $Y_i\sim \text{Exp}(\beta/s_i)$, s.t. $E(Y_i) = \beta/s_i$ and independently for each $ |
51,925 | Estimating the parameter $\beta$ | $
L(\beta) = P(\mathrm{Data} \mid \beta) = \prod_i f(x_i)
$ and $\lambda = \frac{1}{\mu} = \frac{s}{\beta}$
Log is increasing so maximizing log is same as maximizing likelihood:
$\ell(\beta) = \sum \log f(x_i).$
$
\log f(x_i) = \log \lambda e^{-\lambda x_i} = \log s - \log \beta - \frac{s}{\beta}x_i$
and
\frac{d}{d ... | Estimating the parameter $\beta$ | $
L(\beta) = P(\mathrm{Data} \mid \beta) = \prod_i f(x_i)
$ and $\lambda = \frac{1}{\mu} = \frac{s}{\beta}$
Log is increasing so maximizing log is same as maximizing likelihood:
$\ell(\beta) = \sum \l | Estimating the parameter $\beta$
$
L(\beta) = P(\mathrm{Data} \mid \beta) = \prod_i f(x_i)
$ and $\lambda = \frac{1}{\mu} = \frac{s}{\beta}$
Log is increasing so maximizing log is same as maximizing likelihood:
$\ell(\beta) = \sum \log f(x_i).$
$
\log f(x_i) = \log \lambda e^{-\lambda x_i} = \log s - \log \beta - \fra... | Estimating the parameter $\beta$
$
L(\beta) = P(\mathrm{Data} \mid \beta) = \prod_i f(x_i)
$ and $\lambda = \frac{1}{\mu} = \frac{s}{\beta}$
Log is increasing so maximizing log is same as maximizing likelihood:
$\ell(\beta) = \sum \l |
51,926 | Estimating the parameter $\beta$ | Assise from using a direct derivation, leading to a closed form expression, you can do this with a generalized linear model a description is given in the question
Fitting exponential (regression) model by MLE?
With that approach you have more flexibility (e.g. change the function $\mu(s)$)
Example
In R you would use th... | Estimating the parameter $\beta$ | Assise from using a direct derivation, leading to a closed form expression, you can do this with a generalized linear model a description is given in the question
Fitting exponential (regression) mode | Estimating the parameter $\beta$
Assise from using a direct derivation, leading to a closed form expression, you can do this with a generalized linear model a description is given in the question
Fitting exponential (regression) model by MLE?
With that approach you have more flexibility (e.g. change the function $\mu(s... | Estimating the parameter $\beta$
Assise from using a direct derivation, leading to a closed form expression, you can do this with a generalized linear model a description is given in the question
Fitting exponential (regression) mode |
51,927 | How to apply the formula for Shannon entropy to a 4-sided die? | The distribution of each outcome would be $[1/4,1/4/,1/4,1/4]$, i.e. $p_i=1/4$ for each $i\in \{1,2,3,4\}$. This yields $$H(p)=-4\times \frac{1}{4}\times\log_2\frac{1}{4}=2$$
As pointed out in the comments, the answer for an N-sided fair die would be $\log N$:
$$H(p)=-\sum_{i=1}^N\frac{1}{N} \log\frac{1}{N}=-\log \frac... | How to apply the formula for Shannon entropy to a 4-sided die? | The distribution of each outcome would be $[1/4,1/4/,1/4,1/4]$, i.e. $p_i=1/4$ for each $i\in \{1,2,3,4\}$. This yields $$H(p)=-4\times \frac{1}{4}\times\log_2\frac{1}{4}=2$$
As pointed out in the com | How to apply the formula for Shannon entropy to a 4-sided die?
The distribution of each outcome would be $[1/4,1/4/,1/4,1/4]$, i.e. $p_i=1/4$ for each $i\in \{1,2,3,4\}$. This yields $$H(p)=-4\times \frac{1}{4}\times\log_2\frac{1}{4}=2$$
As pointed out in the comments, the answer for an N-sided fair die would be $\log ... | How to apply the formula for Shannon entropy to a 4-sided die?
The distribution of each outcome would be $[1/4,1/4/,1/4,1/4]$, i.e. $p_i=1/4$ for each $i\in \{1,2,3,4\}$. This yields $$H(p)=-4\times \frac{1}{4}\times\log_2\frac{1}{4}=2$$
As pointed out in the com |
51,928 | How to apply the formula for Shannon entropy to a 4-sided die? | I think it's worth mentioning that not only does the formula reduce to $\log N$ ($\log_2N$ if we're specifically talking about entropy measured in bits) in the case of $N$ different equally likely possibilities, but the formula for different probabilities is derived from the equally likely possibilities case, rather th... | How to apply the formula for Shannon entropy to a 4-sided die? | I think it's worth mentioning that not only does the formula reduce to $\log N$ ($\log_2N$ if we're specifically talking about entropy measured in bits) in the case of $N$ different equally likely pos | How to apply the formula for Shannon entropy to a 4-sided die?
I think it's worth mentioning that not only does the formula reduce to $\log N$ ($\log_2N$ if we're specifically talking about entropy measured in bits) in the case of $N$ different equally likely possibilities, but the formula for different probabilities i... | How to apply the formula for Shannon entropy to a 4-sided die?
I think it's worth mentioning that not only does the formula reduce to $\log N$ ($\log_2N$ if we're specifically talking about entropy measured in bits) in the case of $N$ different equally likely pos |
51,929 | How to apply the formula for Shannon entropy to a 4-sided die? | The sum is over all possible outcomes, and $p_i$ are the probabilities for each. There are 4 outcomes, so the sum will run over each one of those - that is, $i$ will stand for a face of the die, and what will go in for $p_i$ will be the probability for the die to land with that face sticking up. For a fair die, that wi... | How to apply the formula for Shannon entropy to a 4-sided die? | The sum is over all possible outcomes, and $p_i$ are the probabilities for each. There are 4 outcomes, so the sum will run over each one of those - that is, $i$ will stand for a face of the die, and w | How to apply the formula for Shannon entropy to a 4-sided die?
The sum is over all possible outcomes, and $p_i$ are the probabilities for each. There are 4 outcomes, so the sum will run over each one of those - that is, $i$ will stand for a face of the die, and what will go in for $p_i$ will be the probability for the ... | How to apply the formula for Shannon entropy to a 4-sided die?
The sum is over all possible outcomes, and $p_i$ are the probabilities for each. There are 4 outcomes, so the sum will run over each one of those - that is, $i$ will stand for a face of the die, and w |
51,930 | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)? | Your constant prediction of $0.5$ is a great benchmark. Meteorological forecasters would call it a "climatological forecast", i.e., one that only relies on the overall and unconditional distribution of your target variable.
Any other model should improve on this benchmark. If it doesn't, you are better off with the sim... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func | Your constant prediction of $0.5$ is a great benchmark. Meteorological forecasters would call it a "climatological forecast", i.e., one that only relies on the overall and unconditional distribution o | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)?
Your constant prediction of $0.5$ is a great benchmark. Meteorological forecasters would call it a "climatological forecast", i.e., one that only relies on the overall and unconditional distribution of your target... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func
Your constant prediction of $0.5$ is a great benchmark. Meteorological forecasters would call it a "climatological forecast", i.e., one that only relies on the overall and unconditional distribution o |
51,931 | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)? | It’s hard to say what constitutes acceptable performance. For instance, it might sound like awesome performance if your classifier gets $90\%$ of its predictions right, but if the data are the MNIST handwritten digits, such performance is rather pedestrian.
(Note, however, that “classification accuracy” is more problem... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func | It’s hard to say what constitutes acceptable performance. For instance, it might sound like awesome performance if your classifier gets $90\%$ of its predictions right, but if the data are the MNIST h | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)?
It’s hard to say what constitutes acceptable performance. For instance, it might sound like awesome performance if your classifier gets $90\%$ of its predictions right, but if the data are the MNIST handwritten di... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func
It’s hard to say what constitutes acceptable performance. For instance, it might sound like awesome performance if your classifier gets $90\%$ of its predictions right, but if the data are the MNIST h |
51,932 | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)? | In addition to the other answers, I'll add that usually with machine learning we fit the model by finding the weights that minimise the loss, but we then usually evaluate the model using a different metric that has more "intuitive" meaning (and to be robust this should be on a different dataset that wasn't used to trai... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func | In addition to the other answers, I'll add that usually with machine learning we fit the model by finding the weights that minimise the loss, but we then usually evaluate the model using a different m | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss function)?
In addition to the other answers, I'll add that usually with machine learning we fit the model by finding the weights that minimise the loss, but we then usually evaluate the model using a different metric that ha... | What is an acceptable value of square loss in machine learning (using mxnet gluon's square loss func
In addition to the other answers, I'll add that usually with machine learning we fit the model by finding the weights that minimise the loss, but we then usually evaluate the model using a different m |
51,933 | GAM and multiple continuous-continuous interactions/tensor smooths | Nice question! Here are some hints to get you started. Your questions are not numbered well (question 2 appears twice), so you may want to renumber them to avoid confusion.
Question 1
The general rule is that you would use an isotropic smooth s(x, cov1) if x and cov1 had the same units and you would use an anisotropic... | GAM and multiple continuous-continuous interactions/tensor smooths | Nice question! Here are some hints to get you started. Your questions are not numbered well (question 2 appears twice), so you may want to renumber them to avoid confusion.
Question 1
The general rul | GAM and multiple continuous-continuous interactions/tensor smooths
Nice question! Here are some hints to get you started. Your questions are not numbered well (question 2 appears twice), so you may want to renumber them to avoid confusion.
Question 1
The general rule is that you would use an isotropic smooth s(x, cov1... | GAM and multiple continuous-continuous interactions/tensor smooths
Nice question! Here are some hints to get you started. Your questions are not numbered well (question 2 appears twice), so you may want to renumber them to avoid confusion.
Question 1
The general rul |
51,934 | GAM and multiple continuous-continuous interactions/tensor smooths | Q1
Almost invariably yes, you should use te() for smooth interactions. The only situation where you might not want to do that is where you are smoothing in multiple dimensions but everything is in the same units, like space. In such circumstances an isotropic smooth may make sense via s().
Q2
My personal belief is that... | GAM and multiple continuous-continuous interactions/tensor smooths | Q1
Almost invariably yes, you should use te() for smooth interactions. The only situation where you might not want to do that is where you are smoothing in multiple dimensions but everything is in the | GAM and multiple continuous-continuous interactions/tensor smooths
Q1
Almost invariably yes, you should use te() for smooth interactions. The only situation where you might not want to do that is where you are smoothing in multiple dimensions but everything is in the same units, like space. In such circumstances an iso... | GAM and multiple continuous-continuous interactions/tensor smooths
Q1
Almost invariably yes, you should use te() for smooth interactions. The only situation where you might not want to do that is where you are smoothing in multiple dimensions but everything is in the |
51,935 | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$? | I will describe every possible solution. This gives you maximal freedom to craft distributions that meet your needs.
Basically, sketch any curve you like for the density function $f$ that meets your requirements. Separately scale the heights of the left and right halves of it (on either side of $c$) to make their mas... | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$? | I will describe every possible solution. This gives you maximal freedom to craft distributions that meet your needs.
Basically, sketch any curve you like for the density function $f$ that meets your | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?
I will describe every possible solution. This gives you maximal freedom to craft distributions that meet your needs.
Basically, sketch any curve you like for the density function $f$ that meets your requirements. Separately scal... | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?
I will describe every possible solution. This gives you maximal freedom to craft distributions that meet your needs.
Basically, sketch any curve you like for the density function $f$ that meets your |
51,936 | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$? | All you need to do is come up with a two-parameter family of distributions, express the mean and mode in terms of those parameters, and then solve for them both being $c$. Working off your triangle idea, we can have a pentagon instead (straight lines from $a$ to $c$ and from $c$ to $b$), giving us three parameters ($y_... | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$? | All you need to do is come up with a two-parameter family of distributions, express the mean and mode in terms of those parameters, and then solve for them both being $c$. Working off your triangle id | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?
All you need to do is come up with a two-parameter family of distributions, express the mean and mode in terms of those parameters, and then solve for them both being $c$. Working off your triangle idea, we can have a pentagon ins... | How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?
All you need to do is come up with a two-parameter family of distributions, express the mean and mode in terms of those parameters, and then solve for them both being $c$. Working off your triangle id |
51,937 | Why is betareg() giving "invalid dependent variable" error? | The error message posted by R tells you what the problem is:
invalid dependent variable, all observations must be in (0,1)
In other words, your dependent variable should take values that are strictly greater than 0 and strictly less than 1; they cannot be equal to 0 or equal to 1.
In your case, you have several respons... | Why is betareg() giving "invalid dependent variable" error? | The error message posted by R tells you what the problem is:
invalid dependent variable, all observations must be in (0,1)
In other words, your dependent variable should take values that are strictly | Why is betareg() giving "invalid dependent variable" error?
The error message posted by R tells you what the problem is:
invalid dependent variable, all observations must be in (0,1)
In other words, your dependent variable should take values that are strictly greater than 0 and strictly less than 1; they cannot be equa... | Why is betareg() giving "invalid dependent variable" error?
The error message posted by R tells you what the problem is:
invalid dependent variable, all observations must be in (0,1)
In other words, your dependent variable should take values that are strictly |
51,938 | Why is betareg() giving "invalid dependent variable" error? | As the error message tells you betareg() requires responses to be greater than $0$ and less than $1$, i.e., the open interval $(0, 1)$. That is the support of the beta distribution which is modeled by beta regression. Your data has a very high share of $1$ observations (40 out of 71, excluding the NA) and consequently ... | Why is betareg() giving "invalid dependent variable" error? | As the error message tells you betareg() requires responses to be greater than $0$ and less than $1$, i.e., the open interval $(0, 1)$. That is the support of the beta distribution which is modeled by | Why is betareg() giving "invalid dependent variable" error?
As the error message tells you betareg() requires responses to be greater than $0$ and less than $1$, i.e., the open interval $(0, 1)$. That is the support of the beta distribution which is modeled by beta regression. Your data has a very high share of $1$ obs... | Why is betareg() giving "invalid dependent variable" error?
As the error message tells you betareg() requires responses to be greater than $0$ and less than $1$, i.e., the open interval $(0, 1)$. That is the support of the beta distribution which is modeled by |
51,939 | Can I resample two datasets and then perform a t-test? | No, you will assure yourself of eventually rejecting the null hypothesis of equality for a large enough sample size (1000 ought to do the trick unless the difference between sample means is tiny tiny tiny). All this would be doing is confirming your observation that the sample means are different, which you already kno... | Can I resample two datasets and then perform a t-test? | No, you will assure yourself of eventually rejecting the null hypothesis of equality for a large enough sample size (1000 ought to do the trick unless the difference between sample means is tiny tiny | Can I resample two datasets and then perform a t-test?
No, you will assure yourself of eventually rejecting the null hypothesis of equality for a large enough sample size (1000 ought to do the trick unless the difference between sample means is tiny tiny tiny). All this would be doing is confirming your observation tha... | Can I resample two datasets and then perform a t-test?
No, you will assure yourself of eventually rejecting the null hypothesis of equality for a large enough sample size (1000 ought to do the trick unless the difference between sample means is tiny tiny |
51,940 | Can I resample two datasets and then perform a t-test? | But...why? Your data is as ideal as could be. It satisfies nearly every assumption of sophomore stats. People only write about this kind of problem.
Resampling opens you up to simulation noise in which you could falsely reject/fail to reject simply because of simulation error. The statistical significance would not... | Can I resample two datasets and then perform a t-test? | But...why? Your data is as ideal as could be. It satisfies nearly every assumption of sophomore stats. People only write about this kind of problem.
Resampling opens you up to simulation noise in w | Can I resample two datasets and then perform a t-test?
But...why? Your data is as ideal as could be. It satisfies nearly every assumption of sophomore stats. People only write about this kind of problem.
Resampling opens you up to simulation noise in which you could falsely reject/fail to reject simply because of si... | Can I resample two datasets and then perform a t-test?
But...why? Your data is as ideal as could be. It satisfies nearly every assumption of sophomore stats. People only write about this kind of problem.
Resampling opens you up to simulation noise in w |
51,941 | How to obtain the variance of fixed effects? | I second @IsabellaGhement's suggestion that you strongly consider a binomial model for the incidence (you'll need to know the 'denominator' — the total number of individuals used to compute the incidence).
$R^2$ measures do exist for linear mixed models, although there are several different, all slightly different,defi... | How to obtain the variance of fixed effects? | I second @IsabellaGhement's suggestion that you strongly consider a binomial model for the incidence (you'll need to know the 'denominator' — the total number of individuals used to compute the incide | How to obtain the variance of fixed effects?
I second @IsabellaGhement's suggestion that you strongly consider a binomial model for the incidence (you'll need to know the 'denominator' — the total number of individuals used to compute the incidence).
$R^2$ measures do exist for linear mixed models, although there are s... | How to obtain the variance of fixed effects?
I second @IsabellaGhement's suggestion that you strongly consider a binomial model for the incidence (you'll need to know the 'denominator' — the total number of individuals used to compute the incide |
51,942 | How to obtain the variance of fixed effects? | Have you checked the model diagnostics for your linear mixed effects model? Incidence is a so-called "discrete proportion" and may be better modelled by a generalized linear mixed effects model (GLMM) with binomial family via the glmer() function in the lme4 package of R.
It seems like what you need is to report R-squa... | How to obtain the variance of fixed effects? | Have you checked the model diagnostics for your linear mixed effects model? Incidence is a so-called "discrete proportion" and may be better modelled by a generalized linear mixed effects model (GLMM) | How to obtain the variance of fixed effects?
Have you checked the model diagnostics for your linear mixed effects model? Incidence is a so-called "discrete proportion" and may be better modelled by a generalized linear mixed effects model (GLMM) with binomial family via the glmer() function in the lme4 package of R.
It... | How to obtain the variance of fixed effects?
Have you checked the model diagnostics for your linear mixed effects model? Incidence is a so-called "discrete proportion" and may be better modelled by a generalized linear mixed effects model (GLMM) |
51,943 | Is it true that the word prior should be used only with latent random variables? | The term prior (as well as posterior) is usually reserved for distributions defined in a Bayesian framework on objects that are not considered as random variables by other inferential approaches, namely parameters. Latent variable models are most often defined outside the Bayesian/non-Bayesian dichotomy and the distrib... | Is it true that the word prior should be used only with latent random variables? | The term prior (as well as posterior) is usually reserved for distributions defined in a Bayesian framework on objects that are not considered as random variables by other inferential approaches, name | Is it true that the word prior should be used only with latent random variables?
The term prior (as well as posterior) is usually reserved for distributions defined in a Bayesian framework on objects that are not considered as random variables by other inferential approaches, namely parameters. Latent variable models a... | Is it true that the word prior should be used only with latent random variables?
The term prior (as well as posterior) is usually reserved for distributions defined in a Bayesian framework on objects that are not considered as random variables by other inferential approaches, name |
51,944 | Is it true that the word prior should be used only with latent random variables? | Before answering your question, let's first explain some basic Bayesian mindset.
In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether they are observed or hidden. Say for example if you believe $X$ follows a distribution defined by $\theta$, denote
$$
X ... | Is it true that the word prior should be used only with latent random variables? | Before answering your question, let's first explain some basic Bayesian mindset.
In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether | Is it true that the word prior should be used only with latent random variables?
Before answering your question, let's first explain some basic Bayesian mindset.
In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether they are observed or hidden. Say for ex... | Is it true that the word prior should be used only with latent random variables?
Before answering your question, let's first explain some basic Bayesian mindset.
In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether |
51,945 | Understanding correlation on clearly correlated but not reversible data | Correlation does not have a "from" and a "to". It is invariant $Cor(A, B) = Cor(B, A)$.
The terms "from" and "to" can make sense in the context of regression, where we speak of "independent" and "dependent" variables or "predictor" and "predicted". Pearson correlation is closely related to linear regression. In Linear ... | Understanding correlation on clearly correlated but not reversible data | Correlation does not have a "from" and a "to". It is invariant $Cor(A, B) = Cor(B, A)$.
The terms "from" and "to" can make sense in the context of regression, where we speak of "independent" and "depe | Understanding correlation on clearly correlated but not reversible data
Correlation does not have a "from" and a "to". It is invariant $Cor(A, B) = Cor(B, A)$.
The terms "from" and "to" can make sense in the context of regression, where we speak of "independent" and "dependent" variables or "predictor" and "predicted".... | Understanding correlation on clearly correlated but not reversible data
Correlation does not have a "from" and a "to". It is invariant $Cor(A, B) = Cor(B, A)$.
The terms "from" and "to" can make sense in the context of regression, where we speak of "independent" and "depe |
51,946 | Understanding correlation on clearly correlated but not reversible data | This is simply a case where dec is a function of num ---i.e., the value of dec is fully determined by the value of num. That is all it is called --- a function. Functions of random variables are often correlated with the initial random variables, so this is not an unusual situation. The correlation indicates that th... | Understanding correlation on clearly correlated but not reversible data | This is simply a case where dec is a function of num ---i.e., the value of dec is fully determined by the value of num. That is all it is called --- a function. Functions of random variables are oft | Understanding correlation on clearly correlated but not reversible data
This is simply a case where dec is a function of num ---i.e., the value of dec is fully determined by the value of num. That is all it is called --- a function. Functions of random variables are often correlated with the initial random variables,... | Understanding correlation on clearly correlated but not reversible data
This is simply a case where dec is a function of num ---i.e., the value of dec is fully determined by the value of num. That is all it is called --- a function. Functions of random variables are oft |
51,947 | Understanding correlation on clearly correlated but not reversible data | As Bernhard mentioned, correlation does not have a "from - to" concept. It describes the relationship between to variables.
Another useful idea to think about is that if we change (or filter on) one variable, how would another variable change.
Think about the relationship between human height and weight, if we focus o... | Understanding correlation on clearly correlated but not reversible data | As Bernhard mentioned, correlation does not have a "from - to" concept. It describes the relationship between to variables.
Another useful idea to think about is that if we change (or filter on) one v | Understanding correlation on clearly correlated but not reversible data
As Bernhard mentioned, correlation does not have a "from - to" concept. It describes the relationship between to variables.
Another useful idea to think about is that if we change (or filter on) one variable, how would another variable change.
Thi... | Understanding correlation on clearly correlated but not reversible data
As Bernhard mentioned, correlation does not have a "from - to" concept. It describes the relationship between to variables.
Another useful idea to think about is that if we change (or filter on) one v |
51,948 | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$ | The answers have already been provided in the comments. Just so this question has an answer attached...
$Q$ is defined in the optimization problem itself as the variable we're minimizing with respect to. The expression
$$\underset{Q^T Q = I}{\text{argmin }} f(Q)$$
could also be written as:
$$\underset{Q}{\text{argmin }... | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$ | The answers have already been provided in the comments. Just so this question has an answer attached...
$Q$ is defined in the optimization problem itself as the variable we're minimizing with respect | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$
The answers have already been provided in the comments. Just so this question has an answer attached...
$Q$ is defined in the optimization problem itself as the variable we're minimizing with respect to. The expression
$$\underset{Q^T Q =... | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$
The answers have already been provided in the comments. Just so this question has an answer attached...
$Q$ is defined in the optimization problem itself as the variable we're minimizing with respect |
51,949 | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$ | The commenters essentially answered this question, but I will memorialize it here.
The argmin (or argmax) notation can be a bit confusing, because it often introduces a dummy variable (much like the dx or dt in an integral). As Matthew Drury's comment indicates, the $\mathbf{Q}$ is the dummy variable here (so it won't... | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$ | The commenters essentially answered this question, but I will memorialize it here.
The argmin (or argmax) notation can be a bit confusing, because it often introduces a dummy variable (much like the d | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$
The commenters essentially answered this question, but I will memorialize it here.
The argmin (or argmax) notation can be a bit confusing, because it often introduces a dummy variable (much like the dx or dt in an integral). As Matthew D... | What does this notation mean: $F$ at the matrix norm and $Q$ under the $\arg\min$
The commenters essentially answered this question, but I will memorialize it here.
The argmin (or argmax) notation can be a bit confusing, because it often introduces a dummy variable (much like the d |
51,950 | Difference between invertible NN and flow-based NN | After some more reading I came to following conclusion:
Invertible NN are just neural networks that represent bijective functions $f$.
Normalizing flows are invertible NN $f$ that also have a tractable determinant of the Jacobian $D_x f$ as well as a tractable inverse $f^{-1}$. This allows for following interpretation... | Difference between invertible NN and flow-based NN | After some more reading I came to following conclusion:
Invertible NN are just neural networks that represent bijective functions $f$.
Normalizing flows are invertible NN $f$ that also have a tractab | Difference between invertible NN and flow-based NN
After some more reading I came to following conclusion:
Invertible NN are just neural networks that represent bijective functions $f$.
Normalizing flows are invertible NN $f$ that also have a tractable determinant of the Jacobian $D_x f$ as well as a tractable inverse... | Difference between invertible NN and flow-based NN
After some more reading I came to following conclusion:
Invertible NN are just neural networks that represent bijective functions $f$.
Normalizing flows are invertible NN $f$ that also have a tractab |
51,951 | Difference between invertible NN and flow-based NN | An invertible neural network is a general term used for any neural network that’s invertible. A flow neural network is a specific kind of invertible neural network. It’s just that it’s rather difficult to construct invertible neural networks, and flow neural networks offer an easy recipe. | Difference between invertible NN and flow-based NN | An invertible neural network is a general term used for any neural network that’s invertible. A flow neural network is a specific kind of invertible neural network. It’s just that it’s rather difficul | Difference between invertible NN and flow-based NN
An invertible neural network is a general term used for any neural network that’s invertible. A flow neural network is a specific kind of invertible neural network. It’s just that it’s rather difficult to construct invertible neural networks, and flow neural networks o... | Difference between invertible NN and flow-based NN
An invertible neural network is a general term used for any neural network that’s invertible. A flow neural network is a specific kind of invertible neural network. It’s just that it’s rather difficul |
51,952 | Control variables and other independent variables | When you say "control", I suspect you mean that you have a primary variable of interest, and then you have other variables that are potential confounders.
In the presence of a confounder, the effect size of the primary variable may appear higher or lower than it actually is (Simpson's Paradoxon / omitted variable bias... | Control variables and other independent variables | When you say "control", I suspect you mean that you have a primary variable of interest, and then you have other variables that are potential confounders.
In the presence of a confounder, the effect | Control variables and other independent variables
When you say "control", I suspect you mean that you have a primary variable of interest, and then you have other variables that are potential confounders.
In the presence of a confounder, the effect size of the primary variable may appear higher or lower than it actual... | Control variables and other independent variables
When you say "control", I suspect you mean that you have a primary variable of interest, and then you have other variables that are potential confounders.
In the presence of a confounder, the effect |
51,953 | Control variables and other independent variables | While it is common practice to "control" (put another independent variable in regression) for any potential confounders, this isn't always the best case. Sometimes "controlling" for variables introduces confounders into the regression.1 It all depends on the underlying relationship between your variables.
As @Florian ... | Control variables and other independent variables | While it is common practice to "control" (put another independent variable in regression) for any potential confounders, this isn't always the best case. Sometimes "controlling" for variables introduc | Control variables and other independent variables
While it is common practice to "control" (put another independent variable in regression) for any potential confounders, this isn't always the best case. Sometimes "controlling" for variables introduces confounders into the regression.1 It all depends on the underlying ... | Control variables and other independent variables
While it is common practice to "control" (put another independent variable in regression) for any potential confounders, this isn't always the best case. Sometimes "controlling" for variables introduc |
51,954 | Control variables and other independent variables | My suggestion
Model: Run a model only with the independent variables you are interested in.
Model: Add the variables you want to control for into the model.
Reason for the suggestion
If you add all the variables together this might change the coefficients of your predictors, i.e. increase/ decrease or even change the... | Control variables and other independent variables | My suggestion
Model: Run a model only with the independent variables you are interested in.
Model: Add the variables you want to control for into the model.
Reason for the suggestion
If you add all | Control variables and other independent variables
My suggestion
Model: Run a model only with the independent variables you are interested in.
Model: Add the variables you want to control for into the model.
Reason for the suggestion
If you add all the variables together this might change the coefficients of your pred... | Control variables and other independent variables
My suggestion
Model: Run a model only with the independent variables you are interested in.
Model: Add the variables you want to control for into the model.
Reason for the suggestion
If you add all |
51,955 | Control variables and other independent variables | Keep in mind the goal of regression through this simple example:
Let's say you want to determine the relationship between the price of homes in the state you live in and the size of the home. And you are clever, so you also include a control variable which indicates whether the home is in a rural or urban district of... | Control variables and other independent variables | Keep in mind the goal of regression through this simple example:
Let's say you want to determine the relationship between the price of homes in the state you live in and the size of the home. And yo | Control variables and other independent variables
Keep in mind the goal of regression through this simple example:
Let's say you want to determine the relationship between the price of homes in the state you live in and the size of the home. And you are clever, so you also include a control variable which indicates w... | Control variables and other independent variables
Keep in mind the goal of regression through this simple example:
Let's say you want to determine the relationship between the price of homes in the state you live in and the size of the home. And yo |
51,956 | Probability that a draw from a normal distribution is some number greater than another draw from the same distribution | Since it looks like self-study question, I'll start with a hint: Think of $X_1-X_2$ with
$X_1, X_2 \sim N(470, 70^2)$. What distribution does $X_1-X_2$ follow? How to interpret $X_1-X_2$? | Probability that a draw from a normal distribution is some number greater than another draw from the | Since it looks like self-study question, I'll start with a hint: Think of $X_1-X_2$ with
$X_1, X_2 \sim N(470, 70^2)$. What distribution does $X_1-X_2$ follow? How to interpret $X_1-X_2$? | Probability that a draw from a normal distribution is some number greater than another draw from the same distribution
Since it looks like self-study question, I'll start with a hint: Think of $X_1-X_2$ with
$X_1, X_2 \sim N(470, 70^2)$. What distribution does $X_1-X_2$ follow? How to interpret $X_1-X_2$? | Probability that a draw from a normal distribution is some number greater than another draw from the
Since it looks like self-study question, I'll start with a hint: Think of $X_1-X_2$ with
$X_1, X_2 \sim N(470, 70^2)$. What distribution does $X_1-X_2$ follow? How to interpret $X_1-X_2$? |
51,957 | Is there any better alternative to Linear Probability Model? | The first "drawback" you mention is the definition of the risk difference, so there is no avoiding this.
There is at least one way to obtain the risk difference using the logistic regression model. It is the average marginal effects approach. The formula depends on whether the predictor of interest is binary or continu... | Is there any better alternative to Linear Probability Model? | The first "drawback" you mention is the definition of the risk difference, so there is no avoiding this.
There is at least one way to obtain the risk difference using the logistic regression model. It | Is there any better alternative to Linear Probability Model?
The first "drawback" you mention is the definition of the risk difference, so there is no avoiding this.
There is at least one way to obtain the risk difference using the logistic regression model. It is the average marginal effects approach. The formula depe... | Is there any better alternative to Linear Probability Model?
The first "drawback" you mention is the definition of the risk difference, so there is no avoiding this.
There is at least one way to obtain the risk difference using the logistic regression model. It |
51,958 | Is there any better alternative to Linear Probability Model? | Every model has this problem. For example, logistic regression implies the constant log odd ratio.
For binomial distribution, the variance is $p(1-p)$ for one trial. So the different predict value of p implies the different variance. But in the model fitting process this problem is resolve by WLS (weighted least square... | Is there any better alternative to Linear Probability Model? | Every model has this problem. For example, logistic regression implies the constant log odd ratio.
For binomial distribution, the variance is $p(1-p)$ for one trial. So the different predict value of | Is there any better alternative to Linear Probability Model?
Every model has this problem. For example, logistic regression implies the constant log odd ratio.
For binomial distribution, the variance is $p(1-p)$ for one trial. So the different predict value of p implies the different variance. But in the model fitting ... | Is there any better alternative to Linear Probability Model?
Every model has this problem. For example, logistic regression implies the constant log odd ratio.
For binomial distribution, the variance is $p(1-p)$ for one trial. So the different predict value of |
51,959 | A simple Neural Network, finding weights to achieve 100% accuracy | There's three sides to the triangles and three hidden neurons. You want each hidden neurons to check on which side of the triangle side an input is. So:
The first hidden neuron will represent $x_1 > 0.5$
The second will represent $x_2 > 0.5$
The third will represent $x_1 + x_2 < 1$
Then the output will be something l... | A simple Neural Network, finding weights to achieve 100% accuracy | There's three sides to the triangles and three hidden neurons. You want each hidden neurons to check on which side of the triangle side an input is. So:
The first hidden neuron will represent $x_1 > | A simple Neural Network, finding weights to achieve 100% accuracy
There's three sides to the triangles and three hidden neurons. You want each hidden neurons to check on which side of the triangle side an input is. So:
The first hidden neuron will represent $x_1 > 0.5$
The second will represent $x_2 > 0.5$
The third w... | A simple Neural Network, finding weights to achieve 100% accuracy
There's three sides to the triangles and three hidden neurons. You want each hidden neurons to check on which side of the triangle side an input is. So:
The first hidden neuron will represent $x_1 > |
51,960 | Approximating leaky ReLU with a differentiable function | The softplus function is commonly described as a smooth approximation of the standard ReLU:
$$s(x) = \log(1 + e^x)$$
The leaky ReLU (with leak coefficient $\alpha$) is:
$$r_L(x) = \max \{ \alpha x, x\}$$
We can also write this as:
$$r_L(x) = \alpha x + (1-\alpha) \max\{0, x\}$$
Note that $\max\{0, x\}$ is the standard... | Approximating leaky ReLU with a differentiable function | The softplus function is commonly described as a smooth approximation of the standard ReLU:
$$s(x) = \log(1 + e^x)$$
The leaky ReLU (with leak coefficient $\alpha$) is:
$$r_L(x) = \max \{ \alpha x, x | Approximating leaky ReLU with a differentiable function
The softplus function is commonly described as a smooth approximation of the standard ReLU:
$$s(x) = \log(1 + e^x)$$
The leaky ReLU (with leak coefficient $\alpha$) is:
$$r_L(x) = \max \{ \alpha x, x\}$$
We can also write this as:
$$r_L(x) = \alpha x + (1-\alpha)... | Approximating leaky ReLU with a differentiable function
The softplus function is commonly described as a smooth approximation of the standard ReLU:
$$s(x) = \log(1 + e^x)$$
The leaky ReLU (with leak coefficient $\alpha$) is:
$$r_L(x) = \max \{ \alpha x, x |
51,961 | Approximating leaky ReLU with a differentiable function | I stumbled on this on accident, not sure if this would be useful but try this weird function I thought of:
(1/20)x(e^arctan(x))
Edit: let's put a constant of say (1/20) in front to prevent the slope beign greater than one
arctan goes to plus or minus pi/2 at values |x|>3 so we have on the left hand side xe^(-pi/2)= x(s... | Approximating leaky ReLU with a differentiable function | I stumbled on this on accident, not sure if this would be useful but try this weird function I thought of:
(1/20)x(e^arctan(x))
Edit: let's put a constant of say (1/20) in front to prevent the slope b | Approximating leaky ReLU with a differentiable function
I stumbled on this on accident, not sure if this would be useful but try this weird function I thought of:
(1/20)x(e^arctan(x))
Edit: let's put a constant of say (1/20) in front to prevent the slope beign greater than one
arctan goes to plus or minus pi/2 at value... | Approximating leaky ReLU with a differentiable function
I stumbled on this on accident, not sure if this would be useful but try this weird function I thought of:
(1/20)x(e^arctan(x))
Edit: let's put a constant of say (1/20) in front to prevent the slope b |
51,962 | Approximating leaky ReLU with a differentiable function | Leaky RELU is defined as:
LogSumExp is a smooth approximation of the max function. So I would go with: | Approximating leaky ReLU with a differentiable function | Leaky RELU is defined as:
LogSumExp is a smooth approximation of the max function. So I would go with: | Approximating leaky ReLU with a differentiable function
Leaky RELU is defined as:
LogSumExp is a smooth approximation of the max function. So I would go with: | Approximating leaky ReLU with a differentiable function
Leaky RELU is defined as:
LogSumExp is a smooth approximation of the max function. So I would go with: |
51,963 | VIF(collinearity) vs Correlation? | First, I think it is better to use condition indexes rather than VIF to diagnose collinearity. See the work of David Belsley or even (if you want a soporific) my dissertation (that link seems to have vanished; this one should work (I hope).
However, to get to your question: It is possible to have very low correlations... | VIF(collinearity) vs Correlation? | First, I think it is better to use condition indexes rather than VIF to diagnose collinearity. See the work of David Belsley or even (if you want a soporific) my dissertation (that link seems to have | VIF(collinearity) vs Correlation?
First, I think it is better to use condition indexes rather than VIF to diagnose collinearity. See the work of David Belsley or even (if you want a soporific) my dissertation (that link seems to have vanished; this one should work (I hope).
However, to get to your question: It is poss... | VIF(collinearity) vs Correlation?
First, I think it is better to use condition indexes rather than VIF to diagnose collinearity. See the work of David Belsley or even (if you want a soporific) my dissertation (that link seems to have |
51,964 | Doubling &/or halving p-values for one- vs. two-tailed tests | If you do a two-tailed test and computation gives you $p=0.03$, then $p<0.05$. The result is significant. If you do a one-tailed test, you will get a different result, depending on which tail you investigate. It will be either a lot larger or only half as big.
$\alpha=0.05$ is the usual convention, no matter whether y... | Doubling &/or halving p-values for one- vs. two-tailed tests | If you do a two-tailed test and computation gives you $p=0.03$, then $p<0.05$. The result is significant. If you do a one-tailed test, you will get a different result, depending on which tail you inve | Doubling &/or halving p-values for one- vs. two-tailed tests
If you do a two-tailed test and computation gives you $p=0.03$, then $p<0.05$. The result is significant. If you do a one-tailed test, you will get a different result, depending on which tail you investigate. It will be either a lot larger or only half as big... | Doubling &/or halving p-values for one- vs. two-tailed tests
If you do a two-tailed test and computation gives you $p=0.03$, then $p<0.05$. The result is significant. If you do a one-tailed test, you will get a different result, depending on which tail you inve |
51,965 | Doubling &/or halving p-values for one- vs. two-tailed tests | You have to consider how the $p$-value was determined. That will dictate how you should move between the $p$-value you have and the one you want. In general, a $p$-value is the proportion of possible values (e.g., test statistics, mean differences, etc.) that are as far away or further than your value under a given d... | Doubling &/or halving p-values for one- vs. two-tailed tests | You have to consider how the $p$-value was determined. That will dictate how you should move between the $p$-value you have and the one you want. In general, a $p$-value is the proportion of possibl | Doubling &/or halving p-values for one- vs. two-tailed tests
You have to consider how the $p$-value was determined. That will dictate how you should move between the $p$-value you have and the one you want. In general, a $p$-value is the proportion of possible values (e.g., test statistics, mean differences, etc.) th... | Doubling &/or halving p-values for one- vs. two-tailed tests
You have to consider how the $p$-value was determined. That will dictate how you should move between the $p$-value you have and the one you want. In general, a $p$-value is the proportion of possibl |
51,966 | Doubling &/or halving p-values for one- vs. two-tailed tests | In ALL cases, $\alpha$ is some standard value like 0.05, and you do not alter it for the directionality of the test. Do not halve it, double it, etc. What is always true is that $p$ is some rejection area, and you are testing whether the area is smaller than some standard threshold (e.g. 0.05). What changes is how y... | Doubling &/or halving p-values for one- vs. two-tailed tests | In ALL cases, $\alpha$ is some standard value like 0.05, and you do not alter it for the directionality of the test. Do not halve it, double it, etc. What is always true is that $p$ is some rejectio | Doubling &/or halving p-values for one- vs. two-tailed tests
In ALL cases, $\alpha$ is some standard value like 0.05, and you do not alter it for the directionality of the test. Do not halve it, double it, etc. What is always true is that $p$ is some rejection area, and you are testing whether the area is smaller tha... | Doubling &/or halving p-values for one- vs. two-tailed tests
In ALL cases, $\alpha$ is some standard value like 0.05, and you do not alter it for the directionality of the test. Do not halve it, double it, etc. What is always true is that $p$ is some rejectio |
51,967 | A data set with missing values in multiple variables | @Tim gave a nice response. To add to that, the best thinking about dealing with missing values (MVs) began with Donald Rubin and Roderick Little in their book Statistical Analysis with Missing Data, now in its 9th edition. They originated the classifications into MAR, MCAR, etc. To their several books I would add Paul ... | A data set with missing values in multiple variables | @Tim gave a nice response. To add to that, the best thinking about dealing with missing values (MVs) began with Donald Rubin and Roderick Little in their book Statistical Analysis with Missing Data, n | A data set with missing values in multiple variables
@Tim gave a nice response. To add to that, the best thinking about dealing with missing values (MVs) began with Donald Rubin and Roderick Little in their book Statistical Analysis with Missing Data, now in its 9th edition. They originated the classifications into MAR... | A data set with missing values in multiple variables
@Tim gave a nice response. To add to that, the best thinking about dealing with missing values (MVs) began with Donald Rubin and Roderick Little in their book Statistical Analysis with Missing Data, n |
51,968 | A data set with missing values in multiple variables | It all depends on why the data is missing. If the data is Missing Completely At Random, you can discard the incomplete data. If the data is Missing At Random, your best bet is multiple imputation (e.g., check out the mice or mi packages in R, and the various blog posts that describe how to use multiple imputation).
How... | A data set with missing values in multiple variables | It all depends on why the data is missing. If the data is Missing Completely At Random, you can discard the incomplete data. If the data is Missing At Random, your best bet is multiple imputation (e.g | A data set with missing values in multiple variables
It all depends on why the data is missing. If the data is Missing Completely At Random, you can discard the incomplete data. If the data is Missing At Random, your best bet is multiple imputation (e.g., check out the mice or mi packages in R, and the various blog pos... | A data set with missing values in multiple variables
It all depends on why the data is missing. If the data is Missing Completely At Random, you can discard the incomplete data. If the data is Missing At Random, your best bet is multiple imputation (e.g |
51,969 | A data set with missing values in multiple variables | Before you decide on whether you want to impute, you should ask yourself whether the patients not giving you information should be part of your model. For instance, if you want to model for control vs treated, all those missing patients should be dropped (they don't tell you which group they are in). However, if you wa... | A data set with missing values in multiple variables | Before you decide on whether you want to impute, you should ask yourself whether the patients not giving you information should be part of your model. For instance, if you want to model for control vs | A data set with missing values in multiple variables
Before you decide on whether you want to impute, you should ask yourself whether the patients not giving you information should be part of your model. For instance, if you want to model for control vs treated, all those missing patients should be dropped (they don't ... | A data set with missing values in multiple variables
Before you decide on whether you want to impute, you should ask yourself whether the patients not giving you information should be part of your model. For instance, if you want to model for control vs |
51,970 | What would cause a residual plot to be entirely above 0? | To summarize the various comments and answers so far:
If the predictions are on data that was not part of the training sample, there could be a systematic difference between the training data and the prediction data. For example, if you are fitting time-series data and the data contains an upward-curving trend, then ... | What would cause a residual plot to be entirely above 0? | To summarize the various comments and answers so far:
If the predictions are on data that was not part of the training sample, there could be a systematic difference between the training data and th | What would cause a residual plot to be entirely above 0?
To summarize the various comments and answers so far:
If the predictions are on data that was not part of the training sample, there could be a systematic difference between the training data and the prediction data. For example, if you are fitting time-series ... | What would cause a residual plot to be entirely above 0?
To summarize the various comments and answers so far:
If the predictions are on data that was not part of the training sample, there could be a systematic difference between the training data and th |
51,971 | What would cause a residual plot to be entirely above 0? | We will assume that the linear regression fit is through least squares, contains an intercept, and the residual plot is that from the training data.
From the normal equations, we see that the residuals of the regression has sample mean 0. Therefore, it's not possible for the residual plot to be entirely above 0. There ... | What would cause a residual plot to be entirely above 0? | We will assume that the linear regression fit is through least squares, contains an intercept, and the residual plot is that from the training data.
From the normal equations, we see that the residual | What would cause a residual plot to be entirely above 0?
We will assume that the linear regression fit is through least squares, contains an intercept, and the residual plot is that from the training data.
From the normal equations, we see that the residuals of the regression has sample mean 0. Therefore, it's not poss... | What would cause a residual plot to be entirely above 0?
We will assume that the linear regression fit is through least squares, contains an intercept, and the residual plot is that from the training data.
From the normal equations, we see that the residual |
51,972 | What is the distribution of the ratio between independent Beta and Gamma random variables? | For independent $a, d\sim\operatorname{\Gamma}(M,c)$, a remarkable result is that $U=a+d$ and $V=a/(a+d)$ are also independent. In addition, $U\sim\operatorname{\Gamma}(2M, c)$, $V\sim\mathrm{B}(M,M)$. See for example Ch.25, Sec. 2 of Johnson, Kotz, and Balakrishnan's
Continuous Univariate Distributions, Volume 2 for ... | What is the distribution of the ratio between independent Beta and Gamma random variables? | For independent $a, d\sim\operatorname{\Gamma}(M,c)$, a remarkable result is that $U=a+d$ and $V=a/(a+d)$ are also independent. In addition, $U\sim\operatorname{\Gamma}(2M, c)$, $V\sim\mathrm{B}(M,M)$ | What is the distribution of the ratio between independent Beta and Gamma random variables?
For independent $a, d\sim\operatorname{\Gamma}(M,c)$, a remarkable result is that $U=a+d$ and $V=a/(a+d)$ are also independent. In addition, $U\sim\operatorname{\Gamma}(2M, c)$, $V\sim\mathrm{B}(M,M)$. See for example Ch.25, Sec.... | What is the distribution of the ratio between independent Beta and Gamma random variables?
For independent $a, d\sim\operatorname{\Gamma}(M,c)$, a remarkable result is that $U=a+d$ and $V=a/(a+d)$ are also independent. In addition, $U\sim\operatorname{\Gamma}(2M, c)$, $V\sim\mathrm{B}(M,M)$ |
51,973 | What is the distribution of the ratio between independent Beta and Gamma random variables? | Francis has provided the definitive elegant solution. Following his beautiful method, it seems possible to produce a somewhat simpler form for the pdf.
In particular, following Francis' elegant solution, let $V \sim \text{Beta}(m,m)$ with pdf $f(v)$:
.. and let $U \sim \text{Gamma}(2m, c)$ where $U$ is independent of... | What is the distribution of the ratio between independent Beta and Gamma random variables? | Francis has provided the definitive elegant solution. Following his beautiful method, it seems possible to produce a somewhat simpler form for the pdf.
In particular, following Francis' elegant soluti | What is the distribution of the ratio between independent Beta and Gamma random variables?
Francis has provided the definitive elegant solution. Following his beautiful method, it seems possible to produce a somewhat simpler form for the pdf.
In particular, following Francis' elegant solution, let $V \sim \text{Beta}(m... | What is the distribution of the ratio between independent Beta and Gamma random variables?
Francis has provided the definitive elegant solution. Following his beautiful method, it seems possible to produce a somewhat simpler form for the pdf.
In particular, following Francis' elegant soluti |
51,974 | What is Shapley value regression and how does one implement it? | The Shapley Value Regression: Shapley value regression significantly ameliorates the deleterious effects of collinearity on the estimated parameters of a regression equation. The concept of Shapley value was introduced in (cooperative collusive) game theory where agents form collusion and cooperate with each other to r... | What is Shapley value regression and how does one implement it? | The Shapley Value Regression: Shapley value regression significantly ameliorates the deleterious effects of collinearity on the estimated parameters of a regression equation. The concept of Shapley va | What is Shapley value regression and how does one implement it?
The Shapley Value Regression: Shapley value regression significantly ameliorates the deleterious effects of collinearity on the estimated parameters of a regression equation. The concept of Shapley value was introduced in (cooperative collusive) game theor... | What is Shapley value regression and how does one implement it?
The Shapley Value Regression: Shapley value regression significantly ameliorates the deleterious effects of collinearity on the estimated parameters of a regression equation. The concept of Shapley va |
51,975 | What is Shapley value regression and how does one implement it? | There are two good papers to tell you a lot about the Shapley Value Regression:
Lipovetsky, S. (2006). Entropy criterion in logistic regression and Shapley value of predictors. Journal of Modern Applied Statistical Methods, 5(1), 95-106.
Mishra, S.K. (2016). Shapley Value Regression and the Resolution of Multicollinear... | What is Shapley value regression and how does one implement it? | There are two good papers to tell you a lot about the Shapley Value Regression:
Lipovetsky, S. (2006). Entropy criterion in logistic regression and Shapley value of predictors. Journal of Modern Appli | What is Shapley value regression and how does one implement it?
There are two good papers to tell you a lot about the Shapley Value Regression:
Lipovetsky, S. (2006). Entropy criterion in logistic regression and Shapley value of predictors. Journal of Modern Applied Statistical Methods, 5(1), 95-106.
Mishra, S.K. (2016... | What is Shapley value regression and how does one implement it?
There are two good papers to tell you a lot about the Shapley Value Regression:
Lipovetsky, S. (2006). Entropy criterion in logistic regression and Shapley value of predictors. Journal of Modern Appli |
51,976 | What is Shapley value regression and how does one implement it? | In statistics, "Shapely value regression" is called "averaging of the sequential sum-of-squares." Ulrike Grömping is the author of a R package called relaimpo in this package, she named this method which is based on this work lmg that calculates the relative importance when the predictor unlike the common methods has ... | What is Shapley value regression and how does one implement it? | In statistics, "Shapely value regression" is called "averaging of the sequential sum-of-squares." Ulrike Grömping is the author of a R package called relaimpo in this package, she named this method w | What is Shapley value regression and how does one implement it?
In statistics, "Shapely value regression" is called "averaging of the sequential sum-of-squares." Ulrike Grömping is the author of a R package called relaimpo in this package, she named this method which is based on this work lmg that calculates the relat... | What is Shapley value regression and how does one implement it?
In statistics, "Shapely value regression" is called "averaging of the sequential sum-of-squares." Ulrike Grömping is the author of a R package called relaimpo in this package, she named this method w |
51,977 | Discrete probability distribution with two 'tails' | There's an infinite number of such distributions -- one need merely a way to get a probability for each value of $i$ and one has just such a distribution. It's a simple matter to spend a leisurely weekend afternoon or an evening inventing a hundred of the things.
However, few are explicitly named in the literature.
On... | Discrete probability distribution with two 'tails' | There's an infinite number of such distributions -- one need merely a way to get a probability for each value of $i$ and one has just such a distribution. It's a simple matter to spend a leisurely wee | Discrete probability distribution with two 'tails'
There's an infinite number of such distributions -- one need merely a way to get a probability for each value of $i$ and one has just such a distribution. It's a simple matter to spend a leisurely weekend afternoon or an evening inventing a hundred of the things.
Howe... | Discrete probability distribution with two 'tails'
There's an infinite number of such distributions -- one need merely a way to get a probability for each value of $i$ and one has just such a distribution. It's a simple matter to spend a leisurely wee |
51,978 | Discrete probability distribution with two 'tails' | $$X \sim Y\times \mbox{Poisson}(\lambda)$$
where $Y$ follows a Rademacher distribution (i.e. $Y$ is $1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$). | Discrete probability distribution with two 'tails' | $$X \sim Y\times \mbox{Poisson}(\lambda)$$
where $Y$ follows a Rademacher distribution (i.e. $Y$ is $1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$). | Discrete probability distribution with two 'tails'
$$X \sim Y\times \mbox{Poisson}(\lambda)$$
where $Y$ follows a Rademacher distribution (i.e. $Y$ is $1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$). | Discrete probability distribution with two 'tails'
$$X \sim Y\times \mbox{Poisson}(\lambda)$$
where $Y$ follows a Rademacher distribution (i.e. $Y$ is $1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$). |
51,979 | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$? | If it's only continuous but not differentiable at the mode, you can't. Consider the Laplace distribution. | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$? | If it's only continuous but not differentiable at the mode, you can't. Consider the Laplace distribution. | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$?
If it's only continuous but not differentiable at the mode, you can't. Consider the Laplace distribution. | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$?
If it's only continuous but not differentiable at the mode, you can't. Consider the Laplace distribution. |
51,980 | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$? | Find the maximum value. The Laplace distribution is defined piece-wise, and not smooth at the mode, which is at the piece-wise common point (A.K.A., corner point), and such a common, junctional corner point in a piece-wise defined function has no general continuity guarantee of either the separate piecewise defined fun... | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$? | Find the maximum value. The Laplace distribution is defined piece-wise, and not smooth at the mode, which is at the piece-wise common point (A.K.A., corner point), and such a common, junctional corner | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$?
Find the maximum value. The Laplace distribution is defined piece-wise, and not smooth at the mode, which is at the piece-wise common point (A.K.A., corner point), and such a common, junctional corner point in a piece-wi... | If $f(x)$ is a unimodal probability density function, how can I show that its mode is at $f'(x)=0$?
Find the maximum value. The Laplace distribution is defined piece-wise, and not smooth at the mode, which is at the piece-wise common point (A.K.A., corner point), and such a common, junctional corner |
51,981 | auto.arima: why forecast converges to mean after some periods? | In your ARIMA specification, the middle number in both the first and the second bracket is zero. That means, there is no [simple] differencing and no seasonal differencing. Thus your series appears to be mean-stationary (rather than integrated or seasonally integrated). (As far as I know, the AR and MA parameters yield... | auto.arima: why forecast converges to mean after some periods? | In your ARIMA specification, the middle number in both the first and the second bracket is zero. That means, there is no [simple] differencing and no seasonal differencing. Thus your series appears to | auto.arima: why forecast converges to mean after some periods?
In your ARIMA specification, the middle number in both the first and the second bracket is zero. That means, there is no [simple] differencing and no seasonal differencing. Thus your series appears to be mean-stationary (rather than integrated or seasonally... | auto.arima: why forecast converges to mean after some periods?
In your ARIMA specification, the middle number in both the first and the second bracket is zero. That means, there is no [simple] differencing and no seasonal differencing. Thus your series appears to |
51,982 | auto.arima: why forecast converges to mean after some periods? | As a complement to the first part of the answer given by @RichardHardy, it could be checked analytically that the forecasts converge to a constant as the forecasting horizon goes to infinity.
For simplicity, let's take an ARMA(1,1) model defined as follows:
$$
y_t = \delta + \phi y_{t-1} + \epsilon_t + \theta \epsilon_... | auto.arima: why forecast converges to mean after some periods? | As a complement to the first part of the answer given by @RichardHardy, it could be checked analytically that the forecasts converge to a constant as the forecasting horizon goes to infinity.
For simp | auto.arima: why forecast converges to mean after some periods?
As a complement to the first part of the answer given by @RichardHardy, it could be checked analytically that the forecasts converge to a constant as the forecasting horizon goes to infinity.
For simplicity, let's take an ARMA(1,1) model defined as follows:... | auto.arima: why forecast converges to mean after some periods?
As a complement to the first part of the answer given by @RichardHardy, it could be checked analytically that the forecasts converge to a constant as the forecasting horizon goes to infinity.
For simp |
51,983 | Interpreting regression with transformed variables | I agree strongly with the suggestion of COOLSerdash and Srikanth Guhan that Poisson regression is much more natural for your problem. So the idea of transforming the response is a much weaker solution than using an appropriate count model. Poisson regression is well documented in many good texts and in this forum. In e... | Interpreting regression with transformed variables | I agree strongly with the suggestion of COOLSerdash and Srikanth Guhan that Poisson regression is much more natural for your problem. So the idea of transforming the response is a much weaker solution | Interpreting regression with transformed variables
I agree strongly with the suggestion of COOLSerdash and Srikanth Guhan that Poisson regression is much more natural for your problem. So the idea of transforming the response is a much weaker solution than using an appropriate count model. Poisson regression is well do... | Interpreting regression with transformed variables
I agree strongly with the suggestion of COOLSerdash and Srikanth Guhan that Poisson regression is much more natural for your problem. So the idea of transforming the response is a much weaker solution |
51,984 | Derivative of softmax and squared error | Yes, your formula is correct. The formula in the draft chapter was for the sigmoid not for the softmax. We will fix it. Thanks for pointing it out.
-- Yoshua Bengio | Derivative of softmax and squared error | Yes, your formula is correct. The formula in the draft chapter was for the sigmoid not for the softmax. We will fix it. Thanks for pointing it out.
-- Yoshua Bengio | Derivative of softmax and squared error
Yes, your formula is correct. The formula in the draft chapter was for the sigmoid not for the softmax. We will fix it. Thanks for pointing it out.
-- Yoshua Bengio | Derivative of softmax and squared error
Yes, your formula is correct. The formula in the draft chapter was for the sigmoid not for the softmax. We will fix it. Thanks for pointing it out.
-- Yoshua Bengio |
51,985 | Derivative of softmax and squared error | The formula you quote from the book
$$ \frac{d p_i}{d a_j} = \sum_i p_i (1 - p_i) $$
cannot be correct because it has no dependence on $j$. Also, the relationship
$$\sum_k p_k = 1$$
implies that
$$ \sum_k \frac{d p_k}{d a_j} = 0 $$
and this doesn't hold for the book's proposed formula.
I think your formula is correct.... | Derivative of softmax and squared error | The formula you quote from the book
$$ \frac{d p_i}{d a_j} = \sum_i p_i (1 - p_i) $$
cannot be correct because it has no dependence on $j$. Also, the relationship
$$\sum_k p_k = 1$$
implies that
$$ \ | Derivative of softmax and squared error
The formula you quote from the book
$$ \frac{d p_i}{d a_j} = \sum_i p_i (1 - p_i) $$
cannot be correct because it has no dependence on $j$. Also, the relationship
$$\sum_k p_k = 1$$
implies that
$$ \sum_k \frac{d p_k}{d a_j} = 0 $$
and this doesn't hold for the book's proposed f... | Derivative of softmax and squared error
The formula you quote from the book
$$ \frac{d p_i}{d a_j} = \sum_i p_i (1 - p_i) $$
cannot be correct because it has no dependence on $j$. Also, the relationship
$$\sum_k p_k = 1$$
implies that
$$ \ |
51,986 | Do the residual plot and QQ plot look normal? | Pretty obviously not normal. A step function is not a straight line. However, you also seem to be checking (unconditional) normality of the response, which is not assumed to be normal in a mixed model (you'd have some mixture of normals, depending on the fixed effects)
You clearly have discrete data. So your response's... | Do the residual plot and QQ plot look normal? | Pretty obviously not normal. A step function is not a straight line. However, you also seem to be checking (unconditional) normality of the response, which is not assumed to be normal in a mixed model | Do the residual plot and QQ plot look normal?
Pretty obviously not normal. A step function is not a straight line. However, you also seem to be checking (unconditional) normality of the response, which is not assumed to be normal in a mixed model (you'd have some mixture of normals, depending on the fixed effects)
You ... | Do the residual plot and QQ plot look normal?
Pretty obviously not normal. A step function is not a straight line. However, you also seem to be checking (unconditional) normality of the response, which is not assumed to be normal in a mixed model |
51,987 | Do the residual plot and QQ plot look normal? | Likert data simply cannot be normal. Although in some cases it is safe enough to treat it as normal, it isn't actually ever normal and treating it as such is potentially dangerous.
In addition to the points @Glen_b has made, your residual plot doesn't look good. The residuals should be symmetrical (vertically) arou... | Do the residual plot and QQ plot look normal? | Likert data simply cannot be normal. Although in some cases it is safe enough to treat it as normal, it isn't actually ever normal and treating it as such is potentially dangerous.
In addition to t | Do the residual plot and QQ plot look normal?
Likert data simply cannot be normal. Although in some cases it is safe enough to treat it as normal, it isn't actually ever normal and treating it as such is potentially dangerous.
In addition to the points @Glen_b has made, your residual plot doesn't look good. The res... | Do the residual plot and QQ plot look normal?
Likert data simply cannot be normal. Although in some cases it is safe enough to treat it as normal, it isn't actually ever normal and treating it as such is potentially dangerous.
In addition to t |
51,988 | What are distribution assumptions in Ridge and Lasso regression models? | Your question is not entirely clear. This answer assumes that by "distribution of the features" you mean the conditional distributions of the response across explanatory variables.
Under the General Linear Model regression estimates are obtained by minimising the Residual Sum of Squares
$RSS = \sum\limits_{i=1}^{n} \Bi... | What are distribution assumptions in Ridge and Lasso regression models? | Your question is not entirely clear. This answer assumes that by "distribution of the features" you mean the conditional distributions of the response across explanatory variables.
Under the General L | What are distribution assumptions in Ridge and Lasso regression models?
Your question is not entirely clear. This answer assumes that by "distribution of the features" you mean the conditional distributions of the response across explanatory variables.
Under the General Linear Model regression estimates are obtained by... | What are distribution assumptions in Ridge and Lasso regression models?
Your question is not entirely clear. This answer assumes that by "distribution of the features" you mean the conditional distributions of the response across explanatory variables.
Under the General L |
51,989 | What are distribution assumptions in Ridge and Lasso regression models? | From a Bayesian standpoint, the assumptions are simply in the priors on the coefficients. Ridge regression is equivalent to using a Gaussian prior, whereas LASSO is equivalent to using a Laplace prior. As @whuber said, these models don't make assumptions on the distribution of the explanatory variables. | What are distribution assumptions in Ridge and Lasso regression models? | From a Bayesian standpoint, the assumptions are simply in the priors on the coefficients. Ridge regression is equivalent to using a Gaussian prior, whereas LASSO is equivalent to using a Laplace prior | What are distribution assumptions in Ridge and Lasso regression models?
From a Bayesian standpoint, the assumptions are simply in the priors on the coefficients. Ridge regression is equivalent to using a Gaussian prior, whereas LASSO is equivalent to using a Laplace prior. As @whuber said, these models don't make assum... | What are distribution assumptions in Ridge and Lasso regression models?
From a Bayesian standpoint, the assumptions are simply in the priors on the coefficients. Ridge regression is equivalent to using a Gaussian prior, whereas LASSO is equivalent to using a Laplace prior |
51,990 | Does high number of values outside of 95% Confidence Interval imply non-normality? | It doesn't mean that there isn't a problem, but you are comparing apples with oranges. The confidence interval is for the mean -- not the population. With a huge amount of data, the confidence interval for the mean will be very narrow because you can estimate the mean very accurately -- but almost all the data values w... | Does high number of values outside of 95% Confidence Interval imply non-normality? | It doesn't mean that there isn't a problem, but you are comparing apples with oranges. The confidence interval is for the mean -- not the population. With a huge amount of data, the confidence interva | Does high number of values outside of 95% Confidence Interval imply non-normality?
It doesn't mean that there isn't a problem, but you are comparing apples with oranges. The confidence interval is for the mean -- not the population. With a huge amount of data, the confidence interval for the mean will be very narrow be... | Does high number of values outside of 95% Confidence Interval imply non-normality?
It doesn't mean that there isn't a problem, but you are comparing apples with oranges. The confidence interval is for the mean -- not the population. With a huge amount of data, the confidence interva |
51,991 | Does high number of values outside of 95% Confidence Interval imply non-normality? | Here is a picture that shows what @rvl said. With reasonably large sample sizes, a tiny fraction of the values are within the 95% confidence interval of the mean. Source
. | Does high number of values outside of 95% Confidence Interval imply non-normality? | Here is a picture that shows what @rvl said. With reasonably large sample sizes, a tiny fraction of the values are within the 95% confidence interval of the mean. Source
. | Does high number of values outside of 95% Confidence Interval imply non-normality?
Here is a picture that shows what @rvl said. With reasonably large sample sizes, a tiny fraction of the values are within the 95% confidence interval of the mean. Source
. | Does high number of values outside of 95% Confidence Interval imply non-normality?
Here is a picture that shows what @rvl said. With reasonably large sample sizes, a tiny fraction of the values are within the 95% confidence interval of the mean. Source
. |
51,992 | Any other non-parametric alternative to Kruskal-Wallis? | The generalization of the Kruskal-Wallis test is the proportional odds ordinal logistic model. Such a model can provide the multiple degree of freedom overall test as you get with K-W but also can provide general contrasts (on the log odds ratio scale) including pairwise comparisons. | Any other non-parametric alternative to Kruskal-Wallis? | The generalization of the Kruskal-Wallis test is the proportional odds ordinal logistic model. Such a model can provide the multiple degree of freedom overall test as you get with K-W but also can pr | Any other non-parametric alternative to Kruskal-Wallis?
The generalization of the Kruskal-Wallis test is the proportional odds ordinal logistic model. Such a model can provide the multiple degree of freedom overall test as you get with K-W but also can provide general contrasts (on the log odds ratio scale) including ... | Any other non-parametric alternative to Kruskal-Wallis?
The generalization of the Kruskal-Wallis test is the proportional odds ordinal logistic model. Such a model can provide the multiple degree of freedom overall test as you get with K-W but also can pr |
51,993 | Any other non-parametric alternative to Kruskal-Wallis? | You can still run the Kruskal-Wallis, all you need to do is run subsequent pair-wise tests comparing each group to the other groups.
After running a Kruskal-Wallis test and determining that there is a significant difference, you could run additional post hoc tests, for example a Dunn's test, to compare each individual ... | Any other non-parametric alternative to Kruskal-Wallis? | You can still run the Kruskal-Wallis, all you need to do is run subsequent pair-wise tests comparing each group to the other groups.
After running a Kruskal-Wallis test and determining that there is a | Any other non-parametric alternative to Kruskal-Wallis?
You can still run the Kruskal-Wallis, all you need to do is run subsequent pair-wise tests comparing each group to the other groups.
After running a Kruskal-Wallis test and determining that there is a significant difference, you could run additional post hoc tests... | Any other non-parametric alternative to Kruskal-Wallis?
You can still run the Kruskal-Wallis, all you need to do is run subsequent pair-wise tests comparing each group to the other groups.
After running a Kruskal-Wallis test and determining that there is a |
51,994 | Revert minmax normalization to original value | Using the same formula as you used to standardize from 0 to 1, now use true min and max to standardize to the true range, most commonly: Xi = (Xi - Xmin)/(Xmax-Xmin) | Revert minmax normalization to original value | Using the same formula as you used to standardize from 0 to 1, now use true min and max to standardize to the true range, most commonly: Xi = (Xi - Xmin)/(Xmax-Xmin) | Revert minmax normalization to original value
Using the same formula as you used to standardize from 0 to 1, now use true min and max to standardize to the true range, most commonly: Xi = (Xi - Xmin)/(Xmax-Xmin) | Revert minmax normalization to original value
Using the same formula as you used to standardize from 0 to 1, now use true min and max to standardize to the true range, most commonly: Xi = (Xi - Xmin)/(Xmax-Xmin) |
51,995 | Revert minmax normalization to original value | Because your output is in [0, 1], I guess you used some output functions for classification, such as sigmoid. However, your loss function is not for classification. Thus, I suggest either using classification loss (such as sigmoid cross entropy) with [0, 1] output values, or using regression loss (such as squared error... | Revert minmax normalization to original value | Because your output is in [0, 1], I guess you used some output functions for classification, such as sigmoid. However, your loss function is not for classification. Thus, I suggest either using classi | Revert minmax normalization to original value
Because your output is in [0, 1], I guess you used some output functions for classification, such as sigmoid. However, your loss function is not for classification. Thus, I suggest either using classification loss (such as sigmoid cross entropy) with [0, 1] output values, o... | Revert minmax normalization to original value
Because your output is in [0, 1], I guess you used some output functions for classification, such as sigmoid. However, your loss function is not for classification. Thus, I suggest either using classi |
51,996 | If P(A)=0, is A a null event? | First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As the first interpretation makes the question a tautology (of course if the definition of null event is that it's probabi... | If P(A)=0, is A a null event? | First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As | If P(A)=0, is A a null event?
First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As the first interpretation makes the question a tautology (of course if the definition of n... | If P(A)=0, is A a null event?
First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As |
51,997 | If P(A)=0, is A a null event? | A probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a probability measure on $\Sigma$.
One of the properties of $\Sigma$ is that it contains the empty set $\emptyset$, and t... | If P(A)=0, is A a null event? | A probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a | If P(A)=0, is A a null event?
A probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a probability measure on $\Sigma$.
One of the properties of $\Sigma$ is that it contains th... | If P(A)=0, is A a null event?
A probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a |
51,998 | If P(A)=0, is A a null event? | The idea of a Null event is used to emulate the idea of a failed experiment.
Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.
First you have the probability that you did, in fact flip a coin. This has a probability of 1 technically speaking.
The second is the Null event (usua... | If P(A)=0, is A a null event? | The idea of a Null event is used to emulate the idea of a failed experiment.
Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.
First you have the probability | If P(A)=0, is A a null event?
The idea of a Null event is used to emulate the idea of a failed experiment.
Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.
First you have the probability that you did, in fact flip a coin. This has a probability of 1 technically speaking.
The ... | If P(A)=0, is A a null event?
The idea of a Null event is used to emulate the idea of a failed experiment.
Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.
First you have the probability |
51,999 | How to calculate the scale parameter of a Cauchy random variable | Note that the mean of the Cauchy distribution doesn't exist (so we can't assume it to be 0). However, I assume you mean the center of symmetry of the Cauchy (which is both the mode and the median and various other measures of location).
Let's call the center $\mu$ and the scale parameter $\sigma\,$:
$$f(x; \mu,\sigma)... | How to calculate the scale parameter of a Cauchy random variable | Note that the mean of the Cauchy distribution doesn't exist (so we can't assume it to be 0). However, I assume you mean the center of symmetry of the Cauchy (which is both the mode and the median and | How to calculate the scale parameter of a Cauchy random variable
Note that the mean of the Cauchy distribution doesn't exist (so we can't assume it to be 0). However, I assume you mean the center of symmetry of the Cauchy (which is both the mode and the median and various other measures of location).
Let's call the ce... | How to calculate the scale parameter of a Cauchy random variable
Note that the mean of the Cauchy distribution doesn't exist (so we can't assume it to be 0). However, I assume you mean the center of symmetry of the Cauchy (which is both the mode and the median and |
52,000 | Nonlinear Statistics? | I will list some of the models/methods so that you can more easily locate information on them (whether here, on Wikipedia and other such sources, via google, in the titles of books and articles and so on) on your own. This is not a complete list but covers a number of the more common approaches. I'll also mention some ... | Nonlinear Statistics? | I will list some of the models/methods so that you can more easily locate information on them (whether here, on Wikipedia and other such sources, via google, in the titles of books and articles and so | Nonlinear Statistics?
I will list some of the models/methods so that you can more easily locate information on them (whether here, on Wikipedia and other such sources, via google, in the titles of books and articles and so on) on your own. This is not a complete list but covers a number of the more common approaches. I... | Nonlinear Statistics?
I will list some of the models/methods so that you can more easily locate information on them (whether here, on Wikipedia and other such sources, via google, in the titles of books and articles and so |
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