idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k β | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 β | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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6,401 | When to use fixed effects vs using cluster SEs? | These answers are fine, but the most recent and best answer is provided by Abadie et al. (2019) "When Should You Adjust Standard Errors for Clustering?" With fixed effects, a main reason to cluster is you have heterogeneity in treatment effects across the clusters. There are other reasons, for example if the clusters ... | When to use fixed effects vs using cluster SEs? | These answers are fine, but the most recent and best answer is provided by Abadie et al. (2019) "When Should You Adjust Standard Errors for Clustering?" With fixed effects, a main reason to cluster i | When to use fixed effects vs using cluster SEs?
These answers are fine, but the most recent and best answer is provided by Abadie et al. (2019) "When Should You Adjust Standard Errors for Clustering?" With fixed effects, a main reason to cluster is you have heterogeneity in treatment effects across the clusters. There... | When to use fixed effects vs using cluster SEs?
These answers are fine, but the most recent and best answer is provided by Abadie et al. (2019) "When Should You Adjust Standard Errors for Clustering?" With fixed effects, a main reason to cluster i |
6,402 | When to use fixed effects vs using cluster SEs? | @Kishore Gawande referenced the NBER working paper by Alberto Abadie, Susan Athey, Guido W. Imbens, and Jeffrey Wooldridge but I think it would be useful to repeat the key conclusions here as (per my reading) they do not necessarily align with every aspect of the the most accepted answers here.
First, clustered standar... | When to use fixed effects vs using cluster SEs? | @Kishore Gawande referenced the NBER working paper by Alberto Abadie, Susan Athey, Guido W. Imbens, and Jeffrey Wooldridge but I think it would be useful to repeat the key conclusions here as (per my | When to use fixed effects vs using cluster SEs?
@Kishore Gawande referenced the NBER working paper by Alberto Abadie, Susan Athey, Guido W. Imbens, and Jeffrey Wooldridge but I think it would be useful to repeat the key conclusions here as (per my reading) they do not necessarily align with every aspect of the the most... | When to use fixed effects vs using cluster SEs?
@Kishore Gawande referenced the NBER working paper by Alberto Abadie, Susan Athey, Guido W. Imbens, and Jeffrey Wooldridge but I think it would be useful to repeat the key conclusions here as (per my |
6,403 | How to test hypothesis of no group differences? | I think you are asking about testing for equivalence. Essentially you need to decide how large a difference is acceptable for you to still conclude that the two groups are effectively equivalent. That decision defines the 95% (or other) confidence interval limits, and sample size calculations are made on this basis.
T... | How to test hypothesis of no group differences? | I think you are asking about testing for equivalence. Essentially you need to decide how large a difference is acceptable for you to still conclude that the two groups are effectively equivalent. That | How to test hypothesis of no group differences?
I think you are asking about testing for equivalence. Essentially you need to decide how large a difference is acceptable for you to still conclude that the two groups are effectively equivalent. That decision defines the 95% (or other) confidence interval limits, and sam... | How to test hypothesis of no group differences?
I think you are asking about testing for equivalence. Essentially you need to decide how large a difference is acceptable for you to still conclude that the two groups are effectively equivalent. That |
6,404 | How to test hypothesis of no group differences? | Besides the already mentioned possibility of some kind of equivalence test, of which most of them, to the best of my knowledge, are mostly routed in the good old frequentist tradition, there is the possibility of conducting tests which really provide a quantification of evidence in favor of a null-hyptheses, namely bay... | How to test hypothesis of no group differences? | Besides the already mentioned possibility of some kind of equivalence test, of which most of them, to the best of my knowledge, are mostly routed in the good old frequentist tradition, there is the po | How to test hypothesis of no group differences?
Besides the already mentioned possibility of some kind of equivalence test, of which most of them, to the best of my knowledge, are mostly routed in the good old frequentist tradition, there is the possibility of conducting tests which really provide a quantification of e... | How to test hypothesis of no group differences?
Besides the already mentioned possibility of some kind of equivalence test, of which most of them, to the best of my knowledge, are mostly routed in the good old frequentist tradition, there is the po |
6,405 | How to test hypothesis of no group differences? | Following Thylacoleo's answer, I did a little research.
The equivalence package in R has the tost() function.
See Robinson and Frose (2004) "Model validation using equivalence tests" for more info. | How to test hypothesis of no group differences? | Following Thylacoleo's answer, I did a little research.
The equivalence package in R has the tost() function.
See Robinson and Frose (2004) "Model validation using equivalence tests" for more info. | How to test hypothesis of no group differences?
Following Thylacoleo's answer, I did a little research.
The equivalence package in R has the tost() function.
See Robinson and Frose (2004) "Model validation using equivalence tests" for more info. | How to test hypothesis of no group differences?
Following Thylacoleo's answer, I did a little research.
The equivalence package in R has the tost() function.
See Robinson and Frose (2004) "Model validation using equivalence tests" for more info. |
6,406 | How to test hypothesis of no group differences? | There are a few papers I know of that could be helpful to you:
Tryon, W. W. (2001). Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: An integrated alternative method of conducting null hypothesis statistical tests. Psychological Methods, 6, 371-386. (FREE PDF)
A... | How to test hypothesis of no group differences? | There are a few papers I know of that could be helpful to you:
Tryon, W. W. (2001). Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: An integra | How to test hypothesis of no group differences?
There are a few papers I know of that could be helpful to you:
Tryon, W. W. (2001). Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: An integrated alternative method of conducting null hypothesis statistical tests. ... | How to test hypothesis of no group differences?
There are a few papers I know of that could be helpful to you:
Tryon, W. W. (2001). Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: An integra |
6,407 | How to test hypothesis of no group differences? | I have recently thought about an alternative way of "equivalence testing" based on a distance between the two distributions rather than between their means.
There are some methods providing confidence intervals for the overlap of two Gaussian distributions:
The overlap $O(P_1,P_2)$ of (between?) two distributions $P_... | How to test hypothesis of no group differences? | I have recently thought about an alternative way of "equivalence testing" based on a distance between the two distributions rather than between their means.
There are some methods providing confidenc | How to test hypothesis of no group differences?
I have recently thought about an alternative way of "equivalence testing" based on a distance between the two distributions rather than between their means.
There are some methods providing confidence intervals for the overlap of two Gaussian distributions:
The overlap ... | How to test hypothesis of no group differences?
I have recently thought about an alternative way of "equivalence testing" based on a distance between the two distributions rather than between their means.
There are some methods providing confidenc |
6,408 | How to test hypothesis of no group differences? | In the medical sciences, it is preferable to use a confidence interval approach as opposed to two one-sided tests (tost). I also recommend graphing the point estimates, CIs, and a priori-determined equivalence margins to make things very clear.
Your question would likely be addressed by such an approach.
The CONSORT gu... | How to test hypothesis of no group differences? | In the medical sciences, it is preferable to use a confidence interval approach as opposed to two one-sided tests (tost). I also recommend graphing the point estimates, CIs, and a priori-determined eq | How to test hypothesis of no group differences?
In the medical sciences, it is preferable to use a confidence interval approach as opposed to two one-sided tests (tost). I also recommend graphing the point estimates, CIs, and a priori-determined equivalence margins to make things very clear.
Your question would likely ... | How to test hypothesis of no group differences?
In the medical sciences, it is preferable to use a confidence interval approach as opposed to two one-sided tests (tost). I also recommend graphing the point estimates, CIs, and a priori-determined eq |
6,409 | How to test hypothesis of no group differences? | Yes. This is equivalence testing. Basically you reverse the null and alternative hypothesis and base the sample size on the power to show that the difference of the means is within the window of equivalence. Blackwelder called it "Proving the null hypothesis." This is commonly done in pharmaceutical clinical trials ... | How to test hypothesis of no group differences? | Yes. This is equivalence testing. Basically you reverse the null and alternative hypothesis and base the sample size on the power to show that the difference of the means is within the window of equ | How to test hypothesis of no group differences?
Yes. This is equivalence testing. Basically you reverse the null and alternative hypothesis and base the sample size on the power to show that the difference of the means is within the window of equivalence. Blackwelder called it "Proving the null hypothesis." This is ... | How to test hypothesis of no group differences?
Yes. This is equivalence testing. Basically you reverse the null and alternative hypothesis and base the sample size on the power to show that the difference of the means is within the window of equ |
6,410 | How to test hypothesis of no group differences? | Bootstrap differences (e.g. the difference between the means) between the 2 sample groups and check for statistical significance. A more detailed description of this approach, albeit in a different context, can be found here http://www.automated-trading-system.com/a-different-application-of-the-bootstrap/ | How to test hypothesis of no group differences? | Bootstrap differences (e.g. the difference between the means) between the 2 sample groups and check for statistical significance. A more detailed description of this approach, albeit in a different co | How to test hypothesis of no group differences?
Bootstrap differences (e.g. the difference between the means) between the 2 sample groups and check for statistical significance. A more detailed description of this approach, albeit in a different context, can be found here http://www.automated-trading-system.com/a-diffe... | How to test hypothesis of no group differences?
Bootstrap differences (e.g. the difference between the means) between the 2 sample groups and check for statistical significance. A more detailed description of this approach, albeit in a different co |
6,411 | What test can I use to compare slopes from two or more regression models? | To answer these questions with R code, use the following:
1. How can I test the difference between slopes?
Answer: Examine the ANOVA p-value from the interaction of Petal.Width by Species, then compare the slopes using lsmeans::lstrends, as follows.
library(lsmeans)
m.interaction <- lm(Sepal.Length ~ Petal.Width*Spec... | What test can I use to compare slopes from two or more regression models? | To answer these questions with R code, use the following:
1. How can I test the difference between slopes?
Answer: Examine the ANOVA p-value from the interaction of Petal.Width by Species, then comp | What test can I use to compare slopes from two or more regression models?
To answer these questions with R code, use the following:
1. How can I test the difference between slopes?
Answer: Examine the ANOVA p-value from the interaction of Petal.Width by Species, then compare the slopes using lsmeans::lstrends, as fol... | What test can I use to compare slopes from two or more regression models?
To answer these questions with R code, use the following:
1. How can I test the difference between slopes?
Answer: Examine the ANOVA p-value from the interaction of Petal.Width by Species, then comp |
6,412 | What test can I use to compare slopes from two or more regression models? | How can I test the difference between slopes?
Include a dummy for species, let it interact with $P_i$, and see if this dummy is significant. Let $L_i$ be the sepal length and $P_i$ be the pedal width and $S_1, S_2, S_3$ be the dummy variables for the three species. The compare the model
$$ E(L_i) = \beta_0 + \beta_1 P... | What test can I use to compare slopes from two or more regression models? | How can I test the difference between slopes?
Include a dummy for species, let it interact with $P_i$, and see if this dummy is significant. Let $L_i$ be the sepal length and $P_i$ be the pedal width | What test can I use to compare slopes from two or more regression models?
How can I test the difference between slopes?
Include a dummy for species, let it interact with $P_i$, and see if this dummy is significant. Let $L_i$ be the sepal length and $P_i$ be the pedal width and $S_1, S_2, S_3$ be the dummy variables for... | What test can I use to compare slopes from two or more regression models?
How can I test the difference between slopes?
Include a dummy for species, let it interact with $P_i$, and see if this dummy is significant. Let $L_i$ be the sepal length and $P_i$ be the pedal width |
6,413 | What test can I use to compare slopes from two or more regression models? | I agree with the previous suggestion. You should fit a multiple regression model with a dummy variable for each data set. This will allow you to test whether the intercepts differ. If you also want to know if the slopes differ, then you need to also include interactions between the dummies and the variable in questi... | What test can I use to compare slopes from two or more regression models? | I agree with the previous suggestion. You should fit a multiple regression model with a dummy variable for each data set. This will allow you to test whether the intercepts differ. If you also want | What test can I use to compare slopes from two or more regression models?
I agree with the previous suggestion. You should fit a multiple regression model with a dummy variable for each data set. This will allow you to test whether the intercepts differ. If you also want to know if the slopes differ, then you need t... | What test can I use to compare slopes from two or more regression models?
I agree with the previous suggestion. You should fit a multiple regression model with a dummy variable for each data set. This will allow you to test whether the intercepts differ. If you also want |
6,414 | Yolo Loss function explanation | Explanation of the different terms :
The 3 $\lambda$ constants are just constants to take into account more one aspect of the loss function. In the article $\lambda_{coord}$ is the highest in order to have the more importance in the first term
The prediction of YOLO is a $S*S*(B*5+C)$ vector : $B$ bbox predictions fo... | Yolo Loss function explanation | Explanation of the different terms :
The 3 $\lambda$ constants are just constants to take into account more one aspect of the loss function. In the article $\lambda_{coord}$ is the highest in order | Yolo Loss function explanation
Explanation of the different terms :
The 3 $\lambda$ constants are just constants to take into account more one aspect of the loss function. In the article $\lambda_{coord}$ is the highest in order to have the more importance in the first term
The prediction of YOLO is a $S*S*(B*5+C)$ v... | Yolo Loss function explanation
Explanation of the different terms :
The 3 $\lambda$ constants are just constants to take into account more one aspect of the loss function. In the article $\lambda_{coord}$ is the highest in order |
6,415 | Yolo Loss function explanation | \begin{align}
&\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt{w_i}-\sqrt{\hat{w}_i})^2 +(\sqrt{h_i}-\sqrt{\hat{h}_i})^2 ]\\
&+ \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}(C_i... | Yolo Loss function explanation | \begin{align}
&\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt | Yolo Loss function explanation
\begin{align}
&\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt{w_i}-\sqrt{\hat{w}_i})^2 +(\sqrt{h_i}-\sqrt{\hat{h}_i})^2 ]\\
&+ \sum_{i=0}^{S^2}\sum_{j... | Yolo Loss function explanation
\begin{align}
&\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt |
6,416 | Yolo Loss function explanation | The loss formula you wrote is of the original YOLO paper loss, not the v2, or v3 loss.
There are some major differences between versions. I suggest reading the papers, or checking the code implementations. Papers: v2, v3.
Some major differences I noticed:
Class probability is calculated per bounding box (hence output ... | Yolo Loss function explanation | The loss formula you wrote is of the original YOLO paper loss, not the v2, or v3 loss.
There are some major differences between versions. I suggest reading the papers, or checking the code implementat | Yolo Loss function explanation
The loss formula you wrote is of the original YOLO paper loss, not the v2, or v3 loss.
There are some major differences between versions. I suggest reading the papers, or checking the code implementations. Papers: v2, v3.
Some major differences I noticed:
Class probability is calculated ... | Yolo Loss function explanation
The loss formula you wrote is of the original YOLO paper loss, not the v2, or v3 loss.
There are some major differences between versions. I suggest reading the papers, or checking the code implementat |
6,417 | Yolo Loss function explanation | Your loss function is for YOLO v1 and not YOLO v2. I was also confused with the difference in the two loss functions and seems like many people are:
https://groups.google.com/forum/#!topic/darknet/TJ4dN9R4iJk
YOLOv2 paper explains the difference in architecture from YOLOv1 as follows:
We remove the fully connected la... | Yolo Loss function explanation | Your loss function is for YOLO v1 and not YOLO v2. I was also confused with the difference in the two loss functions and seems like many people are:
https://groups.google.com/forum/#!topic/darknet/TJ | Yolo Loss function explanation
Your loss function is for YOLO v1 and not YOLO v2. I was also confused with the difference in the two loss functions and seems like many people are:
https://groups.google.com/forum/#!topic/darknet/TJ4dN9R4iJk
YOLOv2 paper explains the difference in architecture from YOLOv1 as follows:
W... | Yolo Loss function explanation
Your loss function is for YOLO v1 and not YOLO v2. I was also confused with the difference in the two loss functions and seems like many people are:
https://groups.google.com/forum/#!topic/darknet/TJ |
6,418 | Yolo Loss function explanation | Here is my Study Note
Loss function: sum-squared error
a. Reason: Easy to optimize
b. Problem: (1) Does not perfectly align with our goal of maximize average precision. (2) In every image, many grid cells do not contain any object. This pushes the confidence scores of those cells towards 0, often overpowering the grad... | Yolo Loss function explanation | Here is my Study Note
Loss function: sum-squared error
a. Reason: Easy to optimize
b. Problem: (1) Does not perfectly align with our goal of maximize average precision. (2) In every image, many grid | Yolo Loss function explanation
Here is my Study Note
Loss function: sum-squared error
a. Reason: Easy to optimize
b. Problem: (1) Does not perfectly align with our goal of maximize average precision. (2) In every image, many grid cells do not contain any object. This pushes the confidence scores of those cells towards... | Yolo Loss function explanation
Here is my Study Note
Loss function: sum-squared error
a. Reason: Easy to optimize
b. Problem: (1) Does not perfectly align with our goal of maximize average precision. (2) In every image, many grid |
6,419 | Interpretation of p-value in hypothesis testing | (Technically, the P-value is the probability of observing data at least as extreme as that actually observed, given the null hypothesis.)
Q1. A decision to reject the null hypothesis on the basis of a small P-value typically depends on 'Fisher's disjunction': Either a rare event has happened or the null hypothesis is f... | Interpretation of p-value in hypothesis testing | (Technically, the P-value is the probability of observing data at least as extreme as that actually observed, given the null hypothesis.)
Q1. A decision to reject the null hypothesis on the basis of a | Interpretation of p-value in hypothesis testing
(Technically, the P-value is the probability of observing data at least as extreme as that actually observed, given the null hypothesis.)
Q1. A decision to reject the null hypothesis on the basis of a small P-value typically depends on 'Fisher's disjunction': Either a rar... | Interpretation of p-value in hypothesis testing
(Technically, the P-value is the probability of observing data at least as extreme as that actually observed, given the null hypothesis.)
Q1. A decision to reject the null hypothesis on the basis of a |
6,420 | Interpretation of p-value in hypothesis testing | +1 to @MichaelLew, who has provided you with a good answer. Perhaps I can still contribute by providing a way of thinking about Q2. Consider the following situation:
The null hypothesis is true. (Note that if the null hypothesis is not true, no type I errors are possible, and it's not clear what meaning the $p$-v... | Interpretation of p-value in hypothesis testing | +1 to @MichaelLew, who has provided you with a good answer. Perhaps I can still contribute by providing a way of thinking about Q2. Consider the following situation:
The null hypothesis is true. | Interpretation of p-value in hypothesis testing
+1 to @MichaelLew, who has provided you with a good answer. Perhaps I can still contribute by providing a way of thinking about Q2. Consider the following situation:
The null hypothesis is true. (Note that if the null hypothesis is not true, no type I errors are pos... | Interpretation of p-value in hypothesis testing
+1 to @MichaelLew, who has provided you with a good answer. Perhaps I can still contribute by providing a way of thinking about Q2. Consider the following situation:
The null hypothesis is true. |
6,421 | Interpretation of p-value in hypothesis testing | I'd like to make a comment related to "the insignificance of null hypothesis significance testing" but which does not answer the question of the OP.
In my opinion, the main problem is not the misinterpretation of the $p$-value. Many practitioners often test for a "significant difference" for instance, and they wrongly ... | Interpretation of p-value in hypothesis testing | I'd like to make a comment related to "the insignificance of null hypothesis significance testing" but which does not answer the question of the OP.
In my opinion, the main problem is not the misinter | Interpretation of p-value in hypothesis testing
I'd like to make a comment related to "the insignificance of null hypothesis significance testing" but which does not answer the question of the OP.
In my opinion, the main problem is not the misinterpretation of the $p$-value. Many practitioners often test for a "signifi... | Interpretation of p-value in hypothesis testing
I'd like to make a comment related to "the insignificance of null hypothesis significance testing" but which does not answer the question of the OP.
In my opinion, the main problem is not the misinter |
6,422 | How to find a good fit for semi-sinusoidal model in R? | It can be done with linear regression -
You just need both a $\sin$ and a $\cos$ term at each frequency.
The reason why you can use a $\sin$ and $\cos$ term in a linear regression to handle seasonality with any amplitude and phase is because of the following trigonometric identity:
A 'general' sine wave with amplitude... | How to find a good fit for semi-sinusoidal model in R? | It can be done with linear regression -
You just need both a $\sin$ and a $\cos$ term at each frequency.
The reason why you can use a $\sin$ and $\cos$ term in a linear regression to handle seasonali | How to find a good fit for semi-sinusoidal model in R?
It can be done with linear regression -
You just need both a $\sin$ and a $\cos$ term at each frequency.
The reason why you can use a $\sin$ and $\cos$ term in a linear regression to handle seasonality with any amplitude and phase is because of the following trigo... | How to find a good fit for semi-sinusoidal model in R?
It can be done with linear regression -
You just need both a $\sin$ and a $\cos$ term at each frequency.
The reason why you can use a $\sin$ and $\cos$ term in a linear regression to handle seasonali |
6,423 | How to find a good fit for semi-sinusoidal model in R? | The temperature you provide in your question repeats exactly every year. I suspect this aren't really measured temperatures over four years. In your example, you wouldn't need a model, because the temperatures just repeat exactly. But otherwise you could use the nls function to fit a sine curve:
ToY <- c(1/12,2/12,3/12... | How to find a good fit for semi-sinusoidal model in R? | The temperature you provide in your question repeats exactly every year. I suspect this aren't really measured temperatures over four years. In your example, you wouldn't need a model, because the tem | How to find a good fit for semi-sinusoidal model in R?
The temperature you provide in your question repeats exactly every year. I suspect this aren't really measured temperatures over four years. In your example, you wouldn't need a model, because the temperatures just repeat exactly. But otherwise you could use the nl... | How to find a good fit for semi-sinusoidal model in R?
The temperature you provide in your question repeats exactly every year. I suspect this aren't really measured temperatures over four years. In your example, you wouldn't need a model, because the tem |
6,424 | How does Factor Analysis explain the covariance while PCA explains the variance? | The distinction between Principal component analysis and Factor analysis is discussed in numerous textbooks and articles on multivariate techniques. You may find the full thread, and a newer one, and odd answers, on this site, too.
I'm not going to make it detailed. I've already given a concise answer and a longer one ... | How does Factor Analysis explain the covariance while PCA explains the variance? | The distinction between Principal component analysis and Factor analysis is discussed in numerous textbooks and articles on multivariate techniques. You may find the full thread, and a newer one, and | How does Factor Analysis explain the covariance while PCA explains the variance?
The distinction between Principal component analysis and Factor analysis is discussed in numerous textbooks and articles on multivariate techniques. You may find the full thread, and a newer one, and odd answers, on this site, too.
I'm not... | How does Factor Analysis explain the covariance while PCA explains the variance?
The distinction between Principal component analysis and Factor analysis is discussed in numerous textbooks and articles on multivariate techniques. You may find the full thread, and a newer one, and |
6,425 | How does Factor Analysis explain the covariance while PCA explains the variance? | "Explaining covariance" vs. explaining variance
Bishop actually means a very simple thing. Under the factor analysis model (eq. 12.64) $$p(\mathbf x|\mathbf z) = \mathcal N(\mathbf x | \mathbf W \mathbf z + \boldsymbol \mu, \boldsymbol \Psi)$$ the covariance matrix of $\mathbf x$ is going to be (eq. 12.65) $$\mathbf C... | How does Factor Analysis explain the covariance while PCA explains the variance? | "Explaining covariance" vs. explaining variance
Bishop actually means a very simple thing. Under the factor analysis model (eq. 12.64) $$p(\mathbf x|\mathbf z) = \mathcal N(\mathbf x | \mathbf W \math | How does Factor Analysis explain the covariance while PCA explains the variance?
"Explaining covariance" vs. explaining variance
Bishop actually means a very simple thing. Under the factor analysis model (eq. 12.64) $$p(\mathbf x|\mathbf z) = \mathcal N(\mathbf x | \mathbf W \mathbf z + \boldsymbol \mu, \boldsymbol \Ps... | How does Factor Analysis explain the covariance while PCA explains the variance?
"Explaining covariance" vs. explaining variance
Bishop actually means a very simple thing. Under the factor analysis model (eq. 12.64) $$p(\mathbf x|\mathbf z) = \mathcal N(\mathbf x | \mathbf W \math |
6,426 | Variance of product of dependent variables | Well, using the familiar identity you pointed out,
$$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$
Using the analogous formula for covariance,
$$ E(X^{2}Y^{2}) = {\rm cov}(X^{2}, Y^{2}) + E(X^2)E(Y^2) $$
and
$$ E(XY)^{2} = [ {\rm cov}(X,Y) + E(X)E(Y) ]^{2} $$
which implies that, in general, ${\rm var}(XY)$ can be... | Variance of product of dependent variables | Well, using the familiar identity you pointed out,
$$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$
Using the analogous formula for covariance,
$$ E(X^{2}Y^{2}) = {\rm cov}(X^{2}, Y^{2}) + E(X^2)E(Y | Variance of product of dependent variables
Well, using the familiar identity you pointed out,
$$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$
Using the analogous formula for covariance,
$$ E(X^{2}Y^{2}) = {\rm cov}(X^{2}, Y^{2}) + E(X^2)E(Y^2) $$
and
$$ E(XY)^{2} = [ {\rm cov}(X,Y) + E(X)E(Y) ]^{2} $$
which impli... | Variance of product of dependent variables
Well, using the familiar identity you pointed out,
$$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$
Using the analogous formula for covariance,
$$ E(X^{2}Y^{2}) = {\rm cov}(X^{2}, Y^{2}) + E(X^2)E(Y |
6,427 | Variance of product of dependent variables | This is an addendum to @Macro's very nice answer which lays out
exactly what needs to known in order to determine the variance of
the product of two correlated random variables. Since
\begin{align}
\operatorname{var}(XY) &= E\left[(XY)^2\right] - \left(E[XY]\right)^2
\tag{1}\\
&= E[(XY)^2] - \left(\operatorname{cov}(... | Variance of product of dependent variables | This is an addendum to @Macro's very nice answer which lays out
exactly what needs to known in order to determine the variance of
the product of two correlated random variables. Since
\begin{align}
| Variance of product of dependent variables
This is an addendum to @Macro's very nice answer which lays out
exactly what needs to known in order to determine the variance of
the product of two correlated random variables. Since
\begin{align}
\operatorname{var}(XY) &= E\left[(XY)^2\right] - \left(E[XY]\right)^2
\tag{1}... | Variance of product of dependent variables
This is an addendum to @Macro's very nice answer which lays out
exactly what needs to known in order to determine the variance of
the product of two correlated random variables. Since
\begin{align}
|
6,428 | Bootstrapping vs Bayesian Bootstrapping conceptually? | The (frequentist) bootstrap takes the data as a reasonable approximation to the unknown population distribution. Therefore, the sampling distribution of a statistic (a function of the data) can be approximated by repeatedly resampling the observations with replacement and computing the statistic for each sample.
Let $... | Bootstrapping vs Bayesian Bootstrapping conceptually? | The (frequentist) bootstrap takes the data as a reasonable approximation to the unknown population distribution. Therefore, the sampling distribution of a statistic (a function of the data) can be app | Bootstrapping vs Bayesian Bootstrapping conceptually?
The (frequentist) bootstrap takes the data as a reasonable approximation to the unknown population distribution. Therefore, the sampling distribution of a statistic (a function of the data) can be approximated by repeatedly resampling the observations with replaceme... | Bootstrapping vs Bayesian Bootstrapping conceptually?
The (frequentist) bootstrap takes the data as a reasonable approximation to the unknown population distribution. Therefore, the sampling distribution of a statistic (a function of the data) can be app |
6,429 | Regression coefficients that flip sign after including other predictors | Multicollinearity is the usual suspect as JoFrhwld mentioned. Basically, if your variables are positively correlated, then the coefficients will be negatively correlated, which can lead to a wrong sign on one of the coefficients.
One check would be to perform a principal components regression or ridge regression. Thi... | Regression coefficients that flip sign after including other predictors | Multicollinearity is the usual suspect as JoFrhwld mentioned. Basically, if your variables are positively correlated, then the coefficients will be negatively correlated, which can lead to a wrong si | Regression coefficients that flip sign after including other predictors
Multicollinearity is the usual suspect as JoFrhwld mentioned. Basically, if your variables are positively correlated, then the coefficients will be negatively correlated, which can lead to a wrong sign on one of the coefficients.
One check would b... | Regression coefficients that flip sign after including other predictors
Multicollinearity is the usual suspect as JoFrhwld mentioned. Basically, if your variables are positively correlated, then the coefficients will be negatively correlated, which can lead to a wrong si |
6,430 | Regression coefficients that flip sign after including other predictors | I believe effects like these are frequently caused by collinearity (see this question). I think the book on multilevel modeling by Gelman and Hill talks about it. The problem is that IV1 is correlated with one or more of the other predictors, and when they are all included in the model, their estimation becomes erratic... | Regression coefficients that flip sign after including other predictors | I believe effects like these are frequently caused by collinearity (see this question). I think the book on multilevel modeling by Gelman and Hill talks about it. The problem is that IV1 is correlated | Regression coefficients that flip sign after including other predictors
I believe effects like these are frequently caused by collinearity (see this question). I think the book on multilevel modeling by Gelman and Hill talks about it. The problem is that IV1 is correlated with one or more of the other predictors, and w... | Regression coefficients that flip sign after including other predictors
I believe effects like these are frequently caused by collinearity (see this question). I think the book on multilevel modeling by Gelman and Hill talks about it. The problem is that IV1 is correlated |
6,431 | Regression coefficients that flip sign after including other predictors | See Simpson's Paradox. In short the main effect observed can reverse when an interaction is added to a model. At the linked page most of the examples are categorical but there is a figure at the top of the page one could imagine continuously. For example, if you have a categorical and continuous predictor then the cont... | Regression coefficients that flip sign after including other predictors | See Simpson's Paradox. In short the main effect observed can reverse when an interaction is added to a model. At the linked page most of the examples are categorical but there is a figure at the top o | Regression coefficients that flip sign after including other predictors
See Simpson's Paradox. In short the main effect observed can reverse when an interaction is added to a model. At the linked page most of the examples are categorical but there is a figure at the top of the page one could imagine continuously. For e... | Regression coefficients that flip sign after including other predictors
See Simpson's Paradox. In short the main effect observed can reverse when an interaction is added to a model. At the linked page most of the examples are categorical but there is a figure at the top o |
6,432 | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of normality? | You can't really even compare the two since the Kolmogorov-Smirnov is for a completely specified distribution (so if you're testing normality, you must specify the mean and variance; they can't be estimated from the data*), while the Shapiro-Wilk is for normality, with unspecified mean and variance.
* you also can't st... | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of | You can't really even compare the two since the Kolmogorov-Smirnov is for a completely specified distribution (so if you're testing normality, you must specify the mean and variance; they can't be est | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of normality?
You can't really even compare the two since the Kolmogorov-Smirnov is for a completely specified distribution (so if you're testing normality, you must specify the mean and variance; they can't be estimated f... | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of
You can't really even compare the two since the Kolmogorov-Smirnov is for a completely specified distribution (so if you're testing normality, you must specify the mean and variance; they can't be est |
6,433 | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of normality? | Briefly stated, the Shapiro-Wilk test is a specific test for normality, whereas the method used by Kolmogorov-Smirnov test is more general, but less powerful (meaning it correctly rejects the null hypothesis of normality less often). Both statistics take normality as the null and establishes a test statistic based on t... | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of | Briefly stated, the Shapiro-Wilk test is a specific test for normality, whereas the method used by Kolmogorov-Smirnov test is more general, but less powerful (meaning it correctly rejects the null hyp | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of normality?
Briefly stated, the Shapiro-Wilk test is a specific test for normality, whereas the method used by Kolmogorov-Smirnov test is more general, but less powerful (meaning it correctly rejects the null hypothesis ... | What is the difference between the ShapiroβWilk test of normality and the KolmogorovβSmirnov test of
Briefly stated, the Shapiro-Wilk test is a specific test for normality, whereas the method used by Kolmogorov-Smirnov test is more general, but less powerful (meaning it correctly rejects the null hyp |
6,434 | How to draw valid conclusions from "big data"? | Your fears are well founded and perceptive. Yahoo and probably several other companies are doing randomized experiments on users and doing it well. But observational data are frought with difficulties. It is a common misperception that problems diminish as the sample size increases. This is true for variance, but b... | How to draw valid conclusions from "big data"? | Your fears are well founded and perceptive. Yahoo and probably several other companies are doing randomized experiments on users and doing it well. But observational data are frought with difficulti | How to draw valid conclusions from "big data"?
Your fears are well founded and perceptive. Yahoo and probably several other companies are doing randomized experiments on users and doing it well. But observational data are frought with difficulties. It is a common misperception that problems diminish as the sample si... | How to draw valid conclusions from "big data"?
Your fears are well founded and perceptive. Yahoo and probably several other companies are doing randomized experiments on users and doing it well. But observational data are frought with difficulti |
6,435 | How to draw valid conclusions from "big data"? | There are a number of techniques in experimental design and analysis that can help you reduce your bias, but this again always boils down to the same thing: One has to know what one is doing. Big data analysis has the same problem as any other data analysis; it suffers from a lack of hypotheses.
A clear example is mult... | How to draw valid conclusions from "big data"? | There are a number of techniques in experimental design and analysis that can help you reduce your bias, but this again always boils down to the same thing: One has to know what one is doing. Big data | How to draw valid conclusions from "big data"?
There are a number of techniques in experimental design and analysis that can help you reduce your bias, but this again always boils down to the same thing: One has to know what one is doing. Big data analysis has the same problem as any other data analysis; it suffers fro... | How to draw valid conclusions from "big data"?
There are a number of techniques in experimental design and analysis that can help you reduce your bias, but this again always boils down to the same thing: One has to know what one is doing. Big data |
6,436 | Why is the exponential family so important in statistics? | Excellent questions.
Regarding A:
A sufficient statistic is nothing more than a distillation of the information that is contained in the sample with respect to a given model. As you would expect, if you have a sample $x_i \sim N(\mu,\sigma^2)$ for $i \in \{1, \ldots, N\}$ and each independent, it is clear that so long ... | Why is the exponential family so important in statistics? | Excellent questions.
Regarding A:
A sufficient statistic is nothing more than a distillation of the information that is contained in the sample with respect to a given model. As you would expect, if y | Why is the exponential family so important in statistics?
Excellent questions.
Regarding A:
A sufficient statistic is nothing more than a distillation of the information that is contained in the sample with respect to a given model. As you would expect, if you have a sample $x_i \sim N(\mu,\sigma^2)$ for $i \in \{1, \l... | Why is the exponential family so important in statistics?
Excellent questions.
Regarding A:
A sufficient statistic is nothing more than a distillation of the information that is contained in the sample with respect to a given model. As you would expect, if y |
6,437 | Why is the exponential family so important in statistics? | Since I do not see the result mentioned on that thread so far, let me mention an often neglected if significant issue, namely that, concerning question A, exponential families are closely linked with the notion of sufficiency due to the Pitman-Koopman-Darmois lemma:
Suppose $X_n$ , $n = 1 , 2 , 3 , \dots$ are independ... | Why is the exponential family so important in statistics? | Since I do not see the result mentioned on that thread so far, let me mention an often neglected if significant issue, namely that, concerning question A, exponential families are closely linked with | Why is the exponential family so important in statistics?
Since I do not see the result mentioned on that thread so far, let me mention an often neglected if significant issue, namely that, concerning question A, exponential families are closely linked with the notion of sufficiency due to the Pitman-Koopman-Darmois le... | Why is the exponential family so important in statistics?
Since I do not see the result mentioned on that thread so far, let me mention an often neglected if significant issue, namely that, concerning question A, exponential families are closely linked with |
6,438 | Why is the exponential family so important in statistics? | Great questions. There are lots of ways to answer these. John Madden does an excellent job, but I'm going to crib a little bit from Ben's answer here regarding sufficient statistics.
The loss function for a Gaussian linear model (as Ben notes) is
$$ \ell_{\mathbf{y}, \mathbf{x}}(\boldsymbol{\beta}, \sigma)=-n \ln \sig... | Why is the exponential family so important in statistics? | Great questions. There are lots of ways to answer these. John Madden does an excellent job, but I'm going to crib a little bit from Ben's answer here regarding sufficient statistics.
The loss functio | Why is the exponential family so important in statistics?
Great questions. There are lots of ways to answer these. John Madden does an excellent job, but I'm going to crib a little bit from Ben's answer here regarding sufficient statistics.
The loss function for a Gaussian linear model (as Ben notes) is
$$ \ell_{\math... | Why is the exponential family so important in statistics?
Great questions. There are lots of ways to answer these. John Madden does an excellent job, but I'm going to crib a little bit from Ben's answer here regarding sufficient statistics.
The loss functio |
6,439 | Why is the exponential family so important in statistics? | None of the properties that OP mentions were important when the most popular exponential family distributions were discovered or were put in use. That is not to say that the properties are irrelevant or not important. These are all interesting and useful features, but these are not the reasons for popularity of the dis... | Why is the exponential family so important in statistics? | None of the properties that OP mentions were important when the most popular exponential family distributions were discovered or were put in use. That is not to say that the properties are irrelevant | Why is the exponential family so important in statistics?
None of the properties that OP mentions were important when the most popular exponential family distributions were discovered or were put in use. That is not to say that the properties are irrelevant or not important. These are all interesting and useful feature... | Why is the exponential family so important in statistics?
None of the properties that OP mentions were important when the most popular exponential family distributions were discovered or were put in use. That is not to say that the properties are irrelevant |
6,440 | Why is the exponential family so important in statistics? | Regarding a point you raise in A: the sufficient statistic for the parameters $\beta$ of a logistic regression - that's a good question. There is no analytical solution to Logistic regression, only numerical. So, calling the final solution a function of the data $T(X,y)$ is a bit of a stretch to the definition of a fun... | Why is the exponential family so important in statistics? | Regarding a point you raise in A: the sufficient statistic for the parameters $\beta$ of a logistic regression - that's a good question. There is no analytical solution to Logistic regression, only nu | Why is the exponential family so important in statistics?
Regarding a point you raise in A: the sufficient statistic for the parameters $\beta$ of a logistic regression - that's a good question. There is no analytical solution to Logistic regression, only numerical. So, calling the final solution a function of the data... | Why is the exponential family so important in statistics?
Regarding a point you raise in A: the sufficient statistic for the parameters $\beta$ of a logistic regression - that's a good question. There is no analytical solution to Logistic regression, only nu |
6,441 | Step-by-step example of reverse-mode automatic differentiation | Let's say we have expression $z = x_1x_2 + \sin(x_1)$ and want to find derivatives $\frac{dz}{dx_1}$ and $\frac{dz}{dx_2}$. Reverse-mode AD splits this task into 2 parts, namely, forward and reverse passes.
Forward pass
First, we decompose our complex expression into a set of primitive ones, i.e. expressions consistin... | Step-by-step example of reverse-mode automatic differentiation | Let's say we have expression $z = x_1x_2 + \sin(x_1)$ and want to find derivatives $\frac{dz}{dx_1}$ and $\frac{dz}{dx_2}$. Reverse-mode AD splits this task into 2 parts, namely, forward and reverse p | Step-by-step example of reverse-mode automatic differentiation
Let's say we have expression $z = x_1x_2 + \sin(x_1)$ and want to find derivatives $\frac{dz}{dx_1}$ and $\frac{dz}{dx_2}$. Reverse-mode AD splits this task into 2 parts, namely, forward and reverse passes.
Forward pass
First, we decompose our complex expr... | Step-by-step example of reverse-mode automatic differentiation
Let's say we have expression $z = x_1x_2 + \sin(x_1)$ and want to find derivatives $\frac{dz}{dx_1}$ and $\frac{dz}{dx_2}$. Reverse-mode AD splits this task into 2 parts, namely, forward and reverse p |
6,442 | Why does minimizing the MAE lead to forecasting the median and not the mean? | It's useful to take a step back and forget about the forecasting aspect for a minute. Let's consider just any distribution $F$ and assume we wish to summarize it using a single number.
You learn very early in your statistics classes that using the expectation of $F$ as a single number summary will minimize the expected... | Why does minimizing the MAE lead to forecasting the median and not the mean? | It's useful to take a step back and forget about the forecasting aspect for a minute. Let's consider just any distribution $F$ and assume we wish to summarize it using a single number.
You learn very | Why does minimizing the MAE lead to forecasting the median and not the mean?
It's useful to take a step back and forget about the forecasting aspect for a minute. Let's consider just any distribution $F$ and assume we wish to summarize it using a single number.
You learn very early in your statistics classes that using... | Why does minimizing the MAE lead to forecasting the median and not the mean?
It's useful to take a step back and forget about the forecasting aspect for a minute. Let's consider just any distribution $F$ and assume we wish to summarize it using a single number.
You learn very |
6,443 | Why does minimizing the MAE lead to forecasting the median and not the mean? | Stephan answer gives you an intuitive explanation of why the minimizing the absolute average error give you the median. Now to answer which of the MSE, MAE or MAPE to use:
The MAE is robust, meaning it is less sensitive to outliers. Imagine a series with an error a million time greater that what it should. On the MSE, ... | Why does minimizing the MAE lead to forecasting the median and not the mean? | Stephan answer gives you an intuitive explanation of why the minimizing the absolute average error give you the median. Now to answer which of the MSE, MAE or MAPE to use:
The MAE is robust, meaning i | Why does minimizing the MAE lead to forecasting the median and not the mean?
Stephan answer gives you an intuitive explanation of why the minimizing the absolute average error give you the median. Now to answer which of the MSE, MAE or MAPE to use:
The MAE is robust, meaning it is less sensitive to outliers. Imagine a ... | Why does minimizing the MAE lead to forecasting the median and not the mean?
Stephan answer gives you an intuitive explanation of why the minimizing the absolute average error give you the median. Now to answer which of the MSE, MAE or MAPE to use:
The MAE is robust, meaning i |
6,444 | Why does minimizing the MAE lead to forecasting the median and not the mean? | All the aforementioned explanations are great, just suggesting a shorter one.
Assuming you would use some value which isn't the median to minimize MAE then there are A examples above the value and B examples below it such that w.l.o.g $A>B$. Then by increasing the value by $\epsilon>0$ the error reduces by $\epsilon$ f... | Why does minimizing the MAE lead to forecasting the median and not the mean? | All the aforementioned explanations are great, just suggesting a shorter one.
Assuming you would use some value which isn't the median to minimize MAE then there are A examples above the value and B e | Why does minimizing the MAE lead to forecasting the median and not the mean?
All the aforementioned explanations are great, just suggesting a shorter one.
Assuming you would use some value which isn't the median to minimize MAE then there are A examples above the value and B examples below it such that w.l.o.g $A>B$. T... | Why does minimizing the MAE lead to forecasting the median and not the mean?
All the aforementioned explanations are great, just suggesting a shorter one.
Assuming you would use some value which isn't the median to minimize MAE then there are A examples above the value and B e |
6,445 | Dropping one of the columns when using one-hot encoding | This depends on the models (and maybe even software) you want to use. With linear regression, or generalized linear models estimated by maximum likelihood (or least squares) (in R this means using functions lm or glm), you need to leave out one column. Otherwise you will get a message about some columns "left out beca... | Dropping one of the columns when using one-hot encoding | This depends on the models (and maybe even software) you want to use. With linear regression, or generalized linear models estimated by maximum likelihood (or least squares) (in R this means using fun | Dropping one of the columns when using one-hot encoding
This depends on the models (and maybe even software) you want to use. With linear regression, or generalized linear models estimated by maximum likelihood (or least squares) (in R this means using functions lm or glm), you need to leave out one column. Otherwise ... | Dropping one of the columns when using one-hot encoding
This depends on the models (and maybe even software) you want to use. With linear regression, or generalized linear models estimated by maximum likelihood (or least squares) (in R this means using fun |
6,446 | Dropping one of the columns when using one-hot encoding | In chapter 5 of this book Feature engineering for machine learning has an example can illustrate kjetil's answer.
City Rent
0 SF 3999
1 SF 4000
2 SF 4001
3 NYC 3499
4 NYC 3500
5 NYC 3501
6 Seattle 2499
7 Seattle 2500
8 Seattle 2501
One-hot encoding:
San Francisco Β Β Β Β Β 1Β Β Β Β 0Β Β Β Β 0
New York Β Β Β Β Β Β Β Β Β Β Β Β Β 0Β Β Β Β 1Β ... | Dropping one of the columns when using one-hot encoding | In chapter 5 of this book Feature engineering for machine learning has an example can illustrate kjetil's answer.
City Rent
0 SF 3999
1 SF 4000
2 SF 4001
3 NYC 3499
4 NYC 3500
5 NYC 3501
6 Seattle 249 | Dropping one of the columns when using one-hot encoding
In chapter 5 of this book Feature engineering for machine learning has an example can illustrate kjetil's answer.
City Rent
0 SF 3999
1 SF 4000
2 SF 4001
3 NYC 3499
4 NYC 3500
5 NYC 3501
6 Seattle 2499
7 Seattle 2500
8 Seattle 2501
One-hot encoding:
San Francisco ... | Dropping one of the columns when using one-hot encoding
In chapter 5 of this book Feature engineering for machine learning has an example can illustrate kjetil's answer.
City Rent
0 SF 3999
1 SF 4000
2 SF 4001
3 NYC 3499
4 NYC 3500
5 NYC 3501
6 Seattle 249 |
6,447 | Understanding shape and calculation of confidence bands in linear regression | The standard error of the regression line at point $X$ (i.e. $s_{\widehat{Y}_{X}}$) is hand calculated (Yech!) using:
$s_{\widehat{Y}_{X}} = s_{Y|X}\sqrt{\frac{1}{n}+\frac{\left(X-\overline{X}\right)^{2}}{\sum_{i=1}^{n}{\left(X_{i}-\overline{X}\right)^{2}}}}$,
where the standard error of the estimate (i.e. $s_{Y|X}$) i... | Understanding shape and calculation of confidence bands in linear regression | The standard error of the regression line at point $X$ (i.e. $s_{\widehat{Y}_{X}}$) is hand calculated (Yech!) using:
$s_{\widehat{Y}_{X}} = s_{Y|X}\sqrt{\frac{1}{n}+\frac{\left(X-\overline{X}\right)^ | Understanding shape and calculation of confidence bands in linear regression
The standard error of the regression line at point $X$ (i.e. $s_{\widehat{Y}_{X}}$) is hand calculated (Yech!) using:
$s_{\widehat{Y}_{X}} = s_{Y|X}\sqrt{\frac{1}{n}+\frac{\left(X-\overline{X}\right)^{2}}{\sum_{i=1}^{n}{\left(X_{i}-\overline{X... | Understanding shape and calculation of confidence bands in linear regression
The standard error of the regression line at point $X$ (i.e. $s_{\widehat{Y}_{X}}$) is hand calculated (Yech!) using:
$s_{\widehat{Y}_{X}} = s_{Y|X}\sqrt{\frac{1}{n}+\frac{\left(X-\overline{X}\right)^ |
6,448 | Understanding shape and calculation of confidence bands in linear regression | Nice question. It's important to understand these concepts and they're not straightforward.
The 95% confidence bands you see around the regression line are generated by the 95% confidence intervals that the true value for $\bar y$ falls within that range for each individual x. So take a vertical slice, say at x = 50. T... | Understanding shape and calculation of confidence bands in linear regression | Nice question. It's important to understand these concepts and they're not straightforward.
The 95% confidence bands you see around the regression line are generated by the 95% confidence intervals th | Understanding shape and calculation of confidence bands in linear regression
Nice question. It's important to understand these concepts and they're not straightforward.
The 95% confidence bands you see around the regression line are generated by the 95% confidence intervals that the true value for $\bar y$ falls within... | Understanding shape and calculation of confidence bands in linear regression
Nice question. It's important to understand these concepts and they're not straightforward.
The 95% confidence bands you see around the regression line are generated by the 95% confidence intervals th |
6,449 | Classification/evaluation metrics for highly imbalanced data | Yes, your assumptions about Kappa seem about right. Kappa as single, scalar metrics is mostly and advantage over other single, scalar metrics like accuracy, which will not reflect prediction performance of smaller classes (shadowed by performance of any much bigger class). Kappa solves this problem more elegantly, as y... | Classification/evaluation metrics for highly imbalanced data | Yes, your assumptions about Kappa seem about right. Kappa as single, scalar metrics is mostly and advantage over other single, scalar metrics like accuracy, which will not reflect prediction performan | Classification/evaluation metrics for highly imbalanced data
Yes, your assumptions about Kappa seem about right. Kappa as single, scalar metrics is mostly and advantage over other single, scalar metrics like accuracy, which will not reflect prediction performance of smaller classes (shadowed by performance of any much ... | Classification/evaluation metrics for highly imbalanced data
Yes, your assumptions about Kappa seem about right. Kappa as single, scalar metrics is mostly and advantage over other single, scalar metrics like accuracy, which will not reflect prediction performan |
6,450 | Classification/evaluation metrics for highly imbalanced data | Besides the AUC and Kohonen's kappa already discussed in the other answers, I'd also like to add a few metrics I've found useful for imbalanced data. They are both related to precision and recall. Because by averaging these you get a metric weighing $TP$s and both types of errors ($FP$ and $FN$):
F1 score, which is th... | Classification/evaluation metrics for highly imbalanced data | Besides the AUC and Kohonen's kappa already discussed in the other answers, I'd also like to add a few metrics I've found useful for imbalanced data. They are both related to precision and recall. Bec | Classification/evaluation metrics for highly imbalanced data
Besides the AUC and Kohonen's kappa already discussed in the other answers, I'd also like to add a few metrics I've found useful for imbalanced data. They are both related to precision and recall. Because by averaging these you get a metric weighing $TP$s and... | Classification/evaluation metrics for highly imbalanced data
Besides the AUC and Kohonen's kappa already discussed in the other answers, I'd also like to add a few metrics I've found useful for imbalanced data. They are both related to precision and recall. Bec |
6,451 | Classification/evaluation metrics for highly imbalanced data | For imbalanced datasets, the Average Precision metric is sometimes a better alternative to the AUROC. The AP score is the area under the precision-recall curve.
Here's a discussion with some code (Python)
Here's a paper.
Also see Peter Flach's Precision-Recall-Gain curves, along with a discussion about the shortcoming... | Classification/evaluation metrics for highly imbalanced data | For imbalanced datasets, the Average Precision metric is sometimes a better alternative to the AUROC. The AP score is the area under the precision-recall curve.
Here's a discussion with some code (Py | Classification/evaluation metrics for highly imbalanced data
For imbalanced datasets, the Average Precision metric is sometimes a better alternative to the AUROC. The AP score is the area under the precision-recall curve.
Here's a discussion with some code (Python)
Here's a paper.
Also see Peter Flach's Precision-Reca... | Classification/evaluation metrics for highly imbalanced data
For imbalanced datasets, the Average Precision metric is sometimes a better alternative to the AUROC. The AP score is the area under the precision-recall curve.
Here's a discussion with some code (Py |
6,452 | What does the anova() command do with a lmer model object? | Use the Source, Luke. We can peek inside the ANOVA function by doing getAnywhere(anova.Mermod). The first part of that function is for comparing two different models. The anova on the fixed effects comes in the big else block in the second half:
dc <- getME(object, "devcomp")
X <- getME(object, "X")
as... | What does the anova() command do with a lmer model object? | Use the Source, Luke. We can peek inside the ANOVA function by doing getAnywhere(anova.Mermod). The first part of that function is for comparing two different models. The anova on the fixed effects co | What does the anova() command do with a lmer model object?
Use the Source, Luke. We can peek inside the ANOVA function by doing getAnywhere(anova.Mermod). The first part of that function is for comparing two different models. The anova on the fixed effects comes in the big else block in the second half:
dc <- getME(ob... | What does the anova() command do with a lmer model object?
Use the Source, Luke. We can peek inside the ANOVA function by doing getAnywhere(anova.Mermod). The first part of that function is for comparing two different models. The anova on the fixed effects co |
6,453 | What is compound symmetry in plain english? | Compound symmetry is essentially the "exchangeable" correlation structure, except with a specific decomposition for the total variance. For example, if you have mixed model for the subject $i$ in cluster $j$ response, $Y_{ij}$, with only a random intercept by cluster
$$ Y_{ij} = \alpha + \gamma_{j} + \varepsilon_{ij} ... | What is compound symmetry in plain english? | Compound symmetry is essentially the "exchangeable" correlation structure, except with a specific decomposition for the total variance. For example, if you have mixed model for the subject $i$ in clus | What is compound symmetry in plain english?
Compound symmetry is essentially the "exchangeable" correlation structure, except with a specific decomposition for the total variance. For example, if you have mixed model for the subject $i$ in cluster $j$ response, $Y_{ij}$, with only a random intercept by cluster
$$ Y_{i... | What is compound symmetry in plain english?
Compound symmetry is essentially the "exchangeable" correlation structure, except with a specific decomposition for the total variance. For example, if you have mixed model for the subject $i$ in clus |
6,454 | What is compound symmetry in plain english? | For me, the best answer I've seen about compound symmetry is from David Howell (here):
In other words, the correlation between trial 1 and trial 2 is equal to the correlation between trial 1 and trial 4 or trial 3 and trial 4, etc. But a more direct way to think about compound symmetry is to say that it requires tha... | What is compound symmetry in plain english? | For me, the best answer I've seen about compound symmetry is from David Howell (here):
In other words, the correlation between trial 1 and trial 2 is equal to the correlation between trial 1 and tr | What is compound symmetry in plain english?
For me, the best answer I've seen about compound symmetry is from David Howell (here):
In other words, the correlation between trial 1 and trial 2 is equal to the correlation between trial 1 and trial 4 or trial 3 and trial 4, etc. But a more direct way to think about comp... | What is compound symmetry in plain english?
For me, the best answer I've seen about compound symmetry is from David Howell (here):
In other words, the correlation between trial 1 and trial 2 is equal to the correlation between trial 1 and tr |
6,455 | What is compound symmetry in plain english? | Compound Symmetry just means that all the variances are equal and all the covariances are equal. So the same variance and covariance are used for all subjects. If you think this applies to the factors in your ANOVA model, compound symmetry is a good covariance structure to use because of its simple structure. | What is compound symmetry in plain english? | Compound Symmetry just means that all the variances are equal and all the covariances are equal. So the same variance and covariance are used for all subjects. If you think this applies to the factors | What is compound symmetry in plain english?
Compound Symmetry just means that all the variances are equal and all the covariances are equal. So the same variance and covariance are used for all subjects. If you think this applies to the factors in your ANOVA model, compound symmetry is a good covariance structure to us... | What is compound symmetry in plain english?
Compound Symmetry just means that all the variances are equal and all the covariances are equal. So the same variance and covariance are used for all subjects. If you think this applies to the factors |
6,456 | Maximum likelihood estimators for a truncated distribution | Consider any location-scale family determined by a "standard" distribution $F$,
$$\Omega_F = \left\{F_{(\mu, \sigma)}: x \to F\left(\frac{x-\mu}{\sigma}\right) \mid \sigma \gt 0\right\}.$$
Assuming $F$ differentiable, basic rules of differentiation show its probability elements are $\frac{1}{\sigma}f\left((x-\mu)/\sigm... | Maximum likelihood estimators for a truncated distribution | Consider any location-scale family determined by a "standard" distribution $F$,
$$\Omega_F = \left\{F_{(\mu, \sigma)}: x \to F\left(\frac{x-\mu}{\sigma}\right) \mid \sigma \gt 0\right\}.$$
Assuming $F | Maximum likelihood estimators for a truncated distribution
Consider any location-scale family determined by a "standard" distribution $F$,
$$\Omega_F = \left\{F_{(\mu, \sigma)}: x \to F\left(\frac{x-\mu}{\sigma}\right) \mid \sigma \gt 0\right\}.$$
Assuming $F$ differentiable, basic rules of differentiation show its pro... | Maximum likelihood estimators for a truncated distribution
Consider any location-scale family determined by a "standard" distribution $F$,
$$\Omega_F = \left\{F_{(\mu, \sigma)}: x \to F\left(\frac{x-\mu}{\sigma}\right) \mid \sigma \gt 0\right\}.$$
Assuming $F |
6,457 | Clojure versus R: advantages and disadvantages for data analysis | Let me start by saying that I love both languages: you can't go wrong with either, and they are certainly better than something like C++ or Java for doing data analysis.
For basic data analysis I would suggest R (especially with plyr). IMO, R is a little easier to learn than Clojure, although this isn't completely obv... | Clojure versus R: advantages and disadvantages for data analysis | Let me start by saying that I love both languages: you can't go wrong with either, and they are certainly better than something like C++ or Java for doing data analysis.
For basic data analysis I woul | Clojure versus R: advantages and disadvantages for data analysis
Let me start by saying that I love both languages: you can't go wrong with either, and they are certainly better than something like C++ or Java for doing data analysis.
For basic data analysis I would suggest R (especially with plyr). IMO, R is a little... | Clojure versus R: advantages and disadvantages for data analysis
Let me start by saying that I love both languages: you can't go wrong with either, and they are certainly better than something like C++ or Java for doing data analysis.
For basic data analysis I woul |
6,458 | Clojure versus R: advantages and disadvantages for data analysis | I have been a heavy R user for the past 6-7 years. As a language, it has several design limitations. Yet, for work in econometrics and in data analysis, I still wholeheartedly recommend it. It has a large number of packages that would be relevant to you for econometrics, time series, consumer choice modeling etc. and o... | Clojure versus R: advantages and disadvantages for data analysis | I have been a heavy R user for the past 6-7 years. As a language, it has several design limitations. Yet, for work in econometrics and in data analysis, I still wholeheartedly recommend it. It has a l | Clojure versus R: advantages and disadvantages for data analysis
I have been a heavy R user for the past 6-7 years. As a language, it has several design limitations. Yet, for work in econometrics and in data analysis, I still wholeheartedly recommend it. It has a large number of packages that would be relevant to you f... | Clojure versus R: advantages and disadvantages for data analysis
I have been a heavy R user for the past 6-7 years. As a language, it has several design limitations. Yet, for work in econometrics and in data analysis, I still wholeheartedly recommend it. It has a l |
6,459 | Clojure versus R: advantages and disadvantages for data analysis | Update (August 2014): as @gappy comments below, as of R version 3.0.0 the limits are higher and means R is capable of handling larger datasets.
Here's a data point: R has a "big data ceiling", useful to know if you plan on working with huge data sets.
I'm unsure whether the same limitations apply to Clojure/Incanter, w... | Clojure versus R: advantages and disadvantages for data analysis | Update (August 2014): as @gappy comments below, as of R version 3.0.0 the limits are higher and means R is capable of handling larger datasets.
Here's a data point: R has a "big data ceiling", useful | Clojure versus R: advantages and disadvantages for data analysis
Update (August 2014): as @gappy comments below, as of R version 3.0.0 the limits are higher and means R is capable of handling larger datasets.
Here's a data point: R has a "big data ceiling", useful to know if you plan on working with huge data sets.
I'm... | Clojure versus R: advantages and disadvantages for data analysis
Update (August 2014): as @gappy comments below, as of R version 3.0.0 the limits are higher and means R is capable of handling larger datasets.
Here's a data point: R has a "big data ceiling", useful |
6,460 | Do null and alternative hypotheses have to be exhaustive or not? | On principle, there is no reason for hypotheses to be exhaustive. If the test is about a parameter $\theta$ with $H_0$ being the restriction $\theta\in\Theta_0$, the alternative $H_a$ can be of any form $\theta\in\Theta_a$ as long as $$\Theta_0\cap\Theta_a=\emptyset.$$
An example as to why exhaustivity does not make m... | Do null and alternative hypotheses have to be exhaustive or not? | On principle, there is no reason for hypotheses to be exhaustive. If the test is about a parameter $\theta$ with $H_0$ being the restriction $\theta\in\Theta_0$, the alternative $H_a$ can be of any fo | Do null and alternative hypotheses have to be exhaustive or not?
On principle, there is no reason for hypotheses to be exhaustive. If the test is about a parameter $\theta$ with $H_0$ being the restriction $\theta\in\Theta_0$, the alternative $H_a$ can be of any form $\theta\in\Theta_a$ as long as $$\Theta_0\cap\Theta_... | Do null and alternative hypotheses have to be exhaustive or not?
On principle, there is no reason for hypotheses to be exhaustive. If the test is about a parameter $\theta$ with $H_0$ being the restriction $\theta\in\Theta_0$, the alternative $H_a$ can be of any fo |
6,461 | Do null and alternative hypotheses have to be exhaustive or not? | The main reason you see the requirement that hypotheses be exhaustive is the problem of what happens if the true parameter value is in the region which is not covered by either the null or alternative hypothesis. Then, testing at the $\alpha %$ level of confidence becomes meaningless, or, perhaps worse, your test will... | Do null and alternative hypotheses have to be exhaustive or not? | The main reason you see the requirement that hypotheses be exhaustive is the problem of what happens if the true parameter value is in the region which is not covered by either the null or alternative | Do null and alternative hypotheses have to be exhaustive or not?
The main reason you see the requirement that hypotheses be exhaustive is the problem of what happens if the true parameter value is in the region which is not covered by either the null or alternative hypothesis. Then, testing at the $\alpha %$ level of ... | Do null and alternative hypotheses have to be exhaustive or not?
The main reason you see the requirement that hypotheses be exhaustive is the problem of what happens if the true parameter value is in the region which is not covered by either the null or alternative |
6,462 | Do null and alternative hypotheses have to be exhaustive or not? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
The alternative does not need to be exhaustive neither... | Do null and alternative hypotheses have to be exhaustive or not? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| Do null and alternative hypotheses have to be exhaustive or not?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
... | Do null and alternative hypotheses have to be exhaustive or not?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
6,463 | Good games for learning statistical thinking? | How about the game show "Deal or No Deal". Though not emphasized the banker is checking the probability given the number of remaining suitcases what the probability is that the 1,000,000 dollar prize is in your suitcase. Based on the odds he makes an offer that favors him. Whenever I watch this I am hoping the cont... | Good games for learning statistical thinking? | How about the game show "Deal or No Deal". Though not emphasized the banker is checking the probability given the number of remaining suitcases what the probability is that the 1,000,000 dollar priz | Good games for learning statistical thinking?
How about the game show "Deal or No Deal". Though not emphasized the banker is checking the probability given the number of remaining suitcases what the probability is that the 1,000,000 dollar prize is in your suitcase. Based on the odds he makes an offer that favors hi... | Good games for learning statistical thinking?
How about the game show "Deal or No Deal". Though not emphasized the banker is checking the probability given the number of remaining suitcases what the probability is that the 1,000,000 dollar priz |
6,464 | Good games for learning statistical thinking? | Fantasy sports incentivize players to think with statistical intuition. For example, every week in fantasy football you must choose which players to start based on, e.g.:
that player's career stats,
the trend in the time series of that player's stats,
the team he is going up against,
weather,
injuries,
and many more... | Good games for learning statistical thinking? | Fantasy sports incentivize players to think with statistical intuition. For example, every week in fantasy football you must choose which players to start based on, e.g.:
that player's career stats, | Good games for learning statistical thinking?
Fantasy sports incentivize players to think with statistical intuition. For example, every week in fantasy football you must choose which players to start based on, e.g.:
that player's career stats,
the trend in the time series of that player's stats,
the team he is going... | Good games for learning statistical thinking?
Fantasy sports incentivize players to think with statistical intuition. For example, every week in fantasy football you must choose which players to start based on, e.g.:
that player's career stats, |
6,465 | Good games for learning statistical thinking? | Poker is a good game for learning probabilistic thinking. The game has been dominated at the highest level in recent years by "math nerds" who have spent a lot of time studying the odds and spend very little time trying to read their opponents. Check out the Time article World Series of Poker: Attack of the Math Brat... | Good games for learning statistical thinking? | Poker is a good game for learning probabilistic thinking. The game has been dominated at the highest level in recent years by "math nerds" who have spent a lot of time studying the odds and spend ver | Good games for learning statistical thinking?
Poker is a good game for learning probabilistic thinking. The game has been dominated at the highest level in recent years by "math nerds" who have spent a lot of time studying the odds and spend very little time trying to read their opponents. Check out the Time article ... | Good games for learning statistical thinking?
Poker is a good game for learning probabilistic thinking. The game has been dominated at the highest level in recent years by "math nerds" who have spent a lot of time studying the odds and spend ver |
6,466 | Good games for learning statistical thinking? | Democracy by Positech is mostly a web model of all the statistical and political interactions that get news coverage. It includes polling and bias as a game mechanic but pretty much nothing else requested in the opening post. However having to conduct those studies to reveal interactions would gel very well with the ga... | Good games for learning statistical thinking? | Democracy by Positech is mostly a web model of all the statistical and political interactions that get news coverage. It includes polling and bias as a game mechanic but pretty much nothing else reque | Good games for learning statistical thinking?
Democracy by Positech is mostly a web model of all the statistical and political interactions that get news coverage. It includes polling and bias as a game mechanic but pretty much nothing else requested in the opening post. However having to conduct those studies to revea... | Good games for learning statistical thinking?
Democracy by Positech is mostly a web model of all the statistical and political interactions that get news coverage. It includes polling and bias as a game mechanic but pretty much nothing else reque |
6,467 | Good games for learning statistical thinking? | I have made some statistics games for teaching. They deal with confidence intervals, hypothesis tests (Neyman-Pearson) and significance tests (Fisherian).
They are excellent tools for engaging students in lectures/workshops and a subset of the students spend quite a lot of time playing them.
I have been going to make a... | Good games for learning statistical thinking? | I have made some statistics games for teaching. They deal with confidence intervals, hypothesis tests (Neyman-Pearson) and significance tests (Fisherian).
They are excellent tools for engaging student | Good games for learning statistical thinking?
I have made some statistics games for teaching. They deal with confidence intervals, hypothesis tests (Neyman-Pearson) and significance tests (Fisherian).
They are excellent tools for engaging students in lectures/workshops and a subset of the students spend quite a lot of ... | Good games for learning statistical thinking?
I have made some statistics games for teaching. They deal with confidence intervals, hypothesis tests (Neyman-Pearson) and significance tests (Fisherian).
They are excellent tools for engaging student |
6,468 | ANOVA on binomial data | No to ANOVA, which assumes a normally distributed outcome variable (among other things). There are "old school" transformations to consider, but I would prefer logistic regression (equivalent to a chi square when there is only one independent variable, as in your case). The advantage of using logistic regression over a... | ANOVA on binomial data | No to ANOVA, which assumes a normally distributed outcome variable (among other things). There are "old school" transformations to consider, but I would prefer logistic regression (equivalent to a chi | ANOVA on binomial data
No to ANOVA, which assumes a normally distributed outcome variable (among other things). There are "old school" transformations to consider, but I would prefer logistic regression (equivalent to a chi square when there is only one independent variable, as in your case). The advantage of using log... | ANOVA on binomial data
No to ANOVA, which assumes a normally distributed outcome variable (among other things). There are "old school" transformations to consider, but I would prefer logistic regression (equivalent to a chi |
6,469 | ANOVA on binomial data | Maybe some consider it old-fashioned, but if you only want to test the null hypothesis of all groups having equal success probability, then you can define
$X_k$ as number of successes in group $k$, $n_k$ as number of trials in group $k$, the estimated probability in group $k$ will be $\hat{p}_k=X_k/n_k$, and then use t... | ANOVA on binomial data | Maybe some consider it old-fashioned, but if you only want to test the null hypothesis of all groups having equal success probability, then you can define
$X_k$ as number of successes in group $k$, $n | ANOVA on binomial data
Maybe some consider it old-fashioned, but if you only want to test the null hypothesis of all groups having equal success probability, then you can define
$X_k$ as number of successes in group $k$, $n_k$ as number of trials in group $k$, the estimated probability in group $k$ will be $\hat{p}_k=X... | ANOVA on binomial data
Maybe some consider it old-fashioned, but if you only want to test the null hypothesis of all groups having equal success probability, then you can define
$X_k$ as number of successes in group $k$, $n |
6,470 | ANOVA on binomial data | I would like to differ from what you think about Chi-Sq test. It is applicable even if the data is not binomial. It's based on the asymptotic normality of mle (in most of the cases).
I would do a logistic regression like this:
$$\log \frac {\hat{\pi}} {1-\hat{\pi}} = \beta_0 + \beta_1 \times D_1 + \beta_2 \times D_2$$... | ANOVA on binomial data | I would like to differ from what you think about Chi-Sq test. It is applicable even if the data is not binomial. It's based on the asymptotic normality of mle (in most of the cases).
I would do a log | ANOVA on binomial data
I would like to differ from what you think about Chi-Sq test. It is applicable even if the data is not binomial. It's based on the asymptotic normality of mle (in most of the cases).
I would do a logistic regression like this:
$$\log \frac {\hat{\pi}} {1-\hat{\pi}} = \beta_0 + \beta_1 \times D_1... | ANOVA on binomial data
I would like to differ from what you think about Chi-Sq test. It is applicable even if the data is not binomial. It's based on the asymptotic normality of mle (in most of the cases).
I would do a log |
6,471 | ANOVA on binomial data | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
I think you are right that ANOVA should not be used to... | ANOVA on binomial data | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| ANOVA on binomial data
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
I think you are right that ANOV... | ANOVA on binomial data
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
6,472 | How to present results of a Lasso using glmnet? | My understanding is that you can't necessarily say much about which variables are "important" or have "real" effects based on whether their coefficients are nonzero. To give an extreme example, if you have two predictors that are perfectly collinear, the lasso will pick one of them essentially at random to get the ful... | How to present results of a Lasso using glmnet? | My understanding is that you can't necessarily say much about which variables are "important" or have "real" effects based on whether their coefficients are nonzero. To give an extreme example, if yo | How to present results of a Lasso using glmnet?
My understanding is that you can't necessarily say much about which variables are "important" or have "real" effects based on whether their coefficients are nonzero. To give an extreme example, if you have two predictors that are perfectly collinear, the lasso will pick ... | How to present results of a Lasso using glmnet?
My understanding is that you can't necessarily say much about which variables are "important" or have "real" effects based on whether their coefficients are nonzero. To give an extreme example, if yo |
6,473 | How to present results of a Lasso using glmnet? | I just wanted to point out that there is recent work trying to develop a test statistic specifically for the LASSO, which takes into account the feature selection being performed:
A significance test for the lasso.
Richard Lockhart, Jonathan Taylor, Ryan J. Tibshirani, Robert Tibshirani.
http://arxiv.org/abs/1301.716... | How to present results of a Lasso using glmnet? | I just wanted to point out that there is recent work trying to develop a test statistic specifically for the LASSO, which takes into account the feature selection being performed:
A significance test | How to present results of a Lasso using glmnet?
I just wanted to point out that there is recent work trying to develop a test statistic specifically for the LASSO, which takes into account the feature selection being performed:
A significance test for the lasso.
Richard Lockhart, Jonathan Taylor, Ryan J. Tibshirani, R... | How to present results of a Lasso using glmnet?
I just wanted to point out that there is recent work trying to develop a test statistic specifically for the LASSO, which takes into account the feature selection being performed:
A significance test |
6,474 | How to present results of a Lasso using glmnet? | Regarding inference for LASSO or elastic net models have a look at CRAN packages selectiveInference and hdi, they do exactly that whilst taking into account the variable selection step! | How to present results of a Lasso using glmnet? | Regarding inference for LASSO or elastic net models have a look at CRAN packages selectiveInference and hdi, they do exactly that whilst taking into account the variable selection step! | How to present results of a Lasso using glmnet?
Regarding inference for LASSO or elastic net models have a look at CRAN packages selectiveInference and hdi, they do exactly that whilst taking into account the variable selection step! | How to present results of a Lasso using glmnet?
Regarding inference for LASSO or elastic net models have a look at CRAN packages selectiveInference and hdi, they do exactly that whilst taking into account the variable selection step! |
6,475 | When should one use Coordinate descent vs. gradient descent? | I think it usually is a matter of how simple/easy it is to work out the gradient of the smooth part of the function and/or the proximal operator of the penalty.
Sometimes, it is a lot more simple to find an exact solution of the problem in the case with one single variable (or a block or variables), than it is to work ... | When should one use Coordinate descent vs. gradient descent? | I think it usually is a matter of how simple/easy it is to work out the gradient of the smooth part of the function and/or the proximal operator of the penalty.
Sometimes, it is a lot more simple to f | When should one use Coordinate descent vs. gradient descent?
I think it usually is a matter of how simple/easy it is to work out the gradient of the smooth part of the function and/or the proximal operator of the penalty.
Sometimes, it is a lot more simple to find an exact solution of the problem in the case with one s... | When should one use Coordinate descent vs. gradient descent?
I think it usually is a matter of how simple/easy it is to work out the gradient of the smooth part of the function and/or the proximal operator of the penalty.
Sometimes, it is a lot more simple to f |
6,476 | When should one use Coordinate descent vs. gradient descent? | Coordinate descent updates one parameter at a time, while gradient descent attempts to update all parameters at once.
It's hard to specify exactly when one algorithm will do better than the other. For example, I was very shocked to learn that coordinate descent was state of the art for LASSO. And I was not the only on... | When should one use Coordinate descent vs. gradient descent? | Coordinate descent updates one parameter at a time, while gradient descent attempts to update all parameters at once.
It's hard to specify exactly when one algorithm will do better than the other. Fo | When should one use Coordinate descent vs. gradient descent?
Coordinate descent updates one parameter at a time, while gradient descent attempts to update all parameters at once.
It's hard to specify exactly when one algorithm will do better than the other. For example, I was very shocked to learn that coordinate desc... | When should one use Coordinate descent vs. gradient descent?
Coordinate descent updates one parameter at a time, while gradient descent attempts to update all parameters at once.
It's hard to specify exactly when one algorithm will do better than the other. Fo |
6,477 | When should one use Coordinate descent vs. gradient descent? | I realise that this is an old question and has some very good answers. I would like to share some practical personal experience.
When working with generative machine learning techniques, you are usually working with some sort of probabilities. An example may be the mixture probabilities of the $k$ components in a mixtu... | When should one use Coordinate descent vs. gradient descent? | I realise that this is an old question and has some very good answers. I would like to share some practical personal experience.
When working with generative machine learning techniques, you are usual | When should one use Coordinate descent vs. gradient descent?
I realise that this is an old question and has some very good answers. I would like to share some practical personal experience.
When working with generative machine learning techniques, you are usually working with some sort of probabilities. An example may ... | When should one use Coordinate descent vs. gradient descent?
I realise that this is an old question and has some very good answers. I would like to share some practical personal experience.
When working with generative machine learning techniques, you are usual |
6,478 | (Why) do overfitted models tend to have large coefficients? | In the regularisation context a "large" coefficient means that the estimate's magnitude is larger than it would have been, if a fixed model specification had been used. It's the impact of obtaining not just the estimates, but also the model specification, from the data.
Consider what a procedure like stepwise regressio... | (Why) do overfitted models tend to have large coefficients? | In the regularisation context a "large" coefficient means that the estimate's magnitude is larger than it would have been, if a fixed model specification had been used. It's the impact of obtaining no | (Why) do overfitted models tend to have large coefficients?
In the regularisation context a "large" coefficient means that the estimate's magnitude is larger than it would have been, if a fixed model specification had been used. It's the impact of obtaining not just the estimates, but also the model specification, from... | (Why) do overfitted models tend to have large coefficients?
In the regularisation context a "large" coefficient means that the estimate's magnitude is larger than it would have been, if a fixed model specification had been used. It's the impact of obtaining no |
6,479 | (Why) do overfitted models tend to have large coefficients? | One very simple answer without looking into your details: When you are overfitting, the parameter estimators tend to get large variances, and with large variances large values are just what you should expect! | (Why) do overfitted models tend to have large coefficients? | One very simple answer without looking into your details: When you are overfitting, the parameter estimators tend to get large variances, and with large variances large values are just what you shoul | (Why) do overfitted models tend to have large coefficients?
One very simple answer without looking into your details: When you are overfitting, the parameter estimators tend to get large variances, and with large variances large values are just what you should expect! | (Why) do overfitted models tend to have large coefficients?
One very simple answer without looking into your details: When you are overfitting, the parameter estimators tend to get large variances, and with large variances large values are just what you shoul |
6,480 | (Why) do overfitted models tend to have large coefficients? | David. I think the problem with your example is you haven't normalised your data ( ie X^10>> X.
So david is right that it shrinks larger coefficients more ( so you can end up with lots of small coefficients, whilst L1 regularisation might give you one large and the rest zero)
so basically it is encapsulating that smal... | (Why) do overfitted models tend to have large coefficients? | David. I think the problem with your example is you haven't normalised your data ( ie X^10>> X.
So david is right that it shrinks larger coefficients more ( so you can end up with lots of small coeffi | (Why) do overfitted models tend to have large coefficients?
David. I think the problem with your example is you haven't normalised your data ( ie X^10>> X.
So david is right that it shrinks larger coefficients more ( so you can end up with lots of small coefficients, whilst L1 regularisation might give you one large an... | (Why) do overfitted models tend to have large coefficients?
David. I think the problem with your example is you haven't normalised your data ( ie X^10>> X.
So david is right that it shrinks larger coefficients more ( so you can end up with lots of small coeffi |
6,481 | (Why) do overfitted models tend to have large coefficients? | This image is from my note of Andrew Ng's DL course, pls let me know if you have question | (Why) do overfitted models tend to have large coefficients? | This image is from my note of Andrew Ng's DL course, pls let me know if you have question | (Why) do overfitted models tend to have large coefficients?
This image is from my note of Andrew Ng's DL course, pls let me know if you have question | (Why) do overfitted models tend to have large coefficients?
This image is from my note of Andrew Ng's DL course, pls let me know if you have question |
6,482 | How to split dataset for time-series prediction? | This link from Rob Hyndman's blog has some info that may be useful: http://robjhyndman.com/hyndsight/crossvalidation/
In my experience, splitting data into chronological sets (year 1, year 2, etc) and checking for parameter stability over time is very useful in building something that's robust. Furthermore, if your dat... | How to split dataset for time-series prediction? | This link from Rob Hyndman's blog has some info that may be useful: http://robjhyndman.com/hyndsight/crossvalidation/
In my experience, splitting data into chronological sets (year 1, year 2, etc) and | How to split dataset for time-series prediction?
This link from Rob Hyndman's blog has some info that may be useful: http://robjhyndman.com/hyndsight/crossvalidation/
In my experience, splitting data into chronological sets (year 1, year 2, etc) and checking for parameter stability over time is very useful in building ... | How to split dataset for time-series prediction?
This link from Rob Hyndman's blog has some info that may be useful: http://robjhyndman.com/hyndsight/crossvalidation/
In my experience, splitting data into chronological sets (year 1, year 2, etc) and |
6,483 | How to split dataset for time-series prediction? | 1) Technically speaking, you don't need to test out of sample if you use AIC and similar criteria because they help avoid overfitting.
3) I don't see how you can do the standard CV because it implies training a time series model with some missing values. Instead, try using a rolling window for training and predict the ... | How to split dataset for time-series prediction? | 1) Technically speaking, you don't need to test out of sample if you use AIC and similar criteria because they help avoid overfitting.
3) I don't see how you can do the standard CV because it implies | How to split dataset for time-series prediction?
1) Technically speaking, you don't need to test out of sample if you use AIC and similar criteria because they help avoid overfitting.
3) I don't see how you can do the standard CV because it implies training a time series model with some missing values. Instead, try usi... | How to split dataset for time-series prediction?
1) Technically speaking, you don't need to test out of sample if you use AIC and similar criteria because they help avoid overfitting.
3) I don't see how you can do the standard CV because it implies |
6,484 | How to split dataset for time-series prediction? | In your case you don't have a lot of options. You only have one bakery, it seems. So, to run an out-of-sample test your only option is the time separation, i.e. the training sample would from the beginning to some recent point in time, and the holdout would from that point to today.
If your model is not time series, th... | How to split dataset for time-series prediction? | In your case you don't have a lot of options. You only have one bakery, it seems. So, to run an out-of-sample test your only option is the time separation, i.e. the training sample would from the begi | How to split dataset for time-series prediction?
In your case you don't have a lot of options. You only have one bakery, it seems. So, to run an out-of-sample test your only option is the time separation, i.e. the training sample would from the beginning to some recent point in time, and the holdout would from that poi... | How to split dataset for time-series prediction?
In your case you don't have a lot of options. You only have one bakery, it seems. So, to run an out-of-sample test your only option is the time separation, i.e. the training sample would from the begi |
6,485 | How to split dataset for time-series prediction? | I often approach problems from a Bayesian perspective. In this case, I'd consider using overimputation as a strategy. This means setting up a likelihood for your data, but omit some of your outcomes. Treat those values as missing, and model those missing outcomes using their corresponding covariates. Then rotate throug... | How to split dataset for time-series prediction? | I often approach problems from a Bayesian perspective. In this case, I'd consider using overimputation as a strategy. This means setting up a likelihood for your data, but omit some of your outcomes. | How to split dataset for time-series prediction?
I often approach problems from a Bayesian perspective. In this case, I'd consider using overimputation as a strategy. This means setting up a likelihood for your data, but omit some of your outcomes. Treat those values as missing, and model those missing outcomes using t... | How to split dataset for time-series prediction?
I often approach problems from a Bayesian perspective. In this case, I'd consider using overimputation as a strategy. This means setting up a likelihood for your data, but omit some of your outcomes. |
6,486 | How to split dataset for time-series prediction? | Disclaimer: The method described here is original research, not based on a thorough reading of the litterature. It is my best attempt at improvising a K-fold CV method for a multivariate timeseries analysis with relatively short input window lengths (assuming no/low dependence over longer time spans), where there was a... | How to split dataset for time-series prediction? | Disclaimer: The method described here is original research, not based on a thorough reading of the litterature. It is my best attempt at improvising a K-fold CV method for a multivariate timeseries an | How to split dataset for time-series prediction?
Disclaimer: The method described here is original research, not based on a thorough reading of the litterature. It is my best attempt at improvising a K-fold CV method for a multivariate timeseries analysis with relatively short input window lengths (assuming no/low depe... | How to split dataset for time-series prediction?
Disclaimer: The method described here is original research, not based on a thorough reading of the litterature. It is my best attempt at improvising a K-fold CV method for a multivariate timeseries an |
6,487 | How do bottleneck architectures work in neural networks? | The bottleneck architecture is used in very deep networks due to computational considerations.
To answer your questions:
56x56 feature maps are not represented in the above image. This block is taken from a ResNet with input size 224x224. 56x56 is the downsampled version of the input at some intermediate layer.
64-d r... | How do bottleneck architectures work in neural networks? | The bottleneck architecture is used in very deep networks due to computational considerations.
To answer your questions:
56x56 feature maps are not represented in the above image. This block is taken | How do bottleneck architectures work in neural networks?
The bottleneck architecture is used in very deep networks due to computational considerations.
To answer your questions:
56x56 feature maps are not represented in the above image. This block is taken from a ResNet with input size 224x224. 56x56 is the downsample... | How do bottleneck architectures work in neural networks?
The bottleneck architecture is used in very deep networks due to computational considerations.
To answer your questions:
56x56 feature maps are not represented in the above image. This block is taken |
6,488 | How do bottleneck architectures work in neural networks? | As far as I understand, illustration on the right shows that input to this block already has 256 features. So we are deep into some ResNet architecture and already created 256 features (we lost some w x h due to conv 3x3 before but gained features instead).
Still, calculating 256 channels (features) can take too much t... | How do bottleneck architectures work in neural networks? | As far as I understand, illustration on the right shows that input to this block already has 256 features. So we are deep into some ResNet architecture and already created 256 features (we lost some w | How do bottleneck architectures work in neural networks?
As far as I understand, illustration on the right shows that input to this block already has 256 features. So we are deep into some ResNet architecture and already created 256 features (we lost some w x h due to conv 3x3 before but gained features instead).
Still... | How do bottleneck architectures work in neural networks?
As far as I understand, illustration on the right shows that input to this block already has 256 features. So we are deep into some ResNet architecture and already created 256 features (we lost some w |
6,489 | How do bottleneck architectures work in neural networks? | I just want to answer your questions:
What stride length is used and at what layers?
Stride is not relevant in this discussion/figure. The "layers" are represented as rectangles and are 2d convolutional layers.
How are the 56x56 feature maps represented in the diagram above?
Feature maps are the result of a conv. layer... | How do bottleneck architectures work in neural networks? | I just want to answer your questions:
What stride length is used and at what layers?
Stride is not relevant in this discussion/figure. The "layers" are represented as rectangles and are 2d convolution | How do bottleneck architectures work in neural networks?
I just want to answer your questions:
What stride length is used and at what layers?
Stride is not relevant in this discussion/figure. The "layers" are represented as rectangles and are 2d convolutional layers.
How are the 56x56 feature maps represented in the di... | How do bottleneck architectures work in neural networks?
I just want to answer your questions:
What stride length is used and at what layers?
Stride is not relevant in this discussion/figure. The "layers" are represented as rectangles and are 2d convolution |
6,490 | When is the bootstrap estimate of bias valid? | I think your formula is wrong. The last $t$ should have a star rather than a hat:
\begin{equation}
\mathrm{bias}_t \approx \frac{1}{N}\sum_i \tilde{t}_i- t^*
\end{equation}
You want to use the actual statistic evaluated on the empirical distribution (this is often easy, since the original sample is a finite set), rathe... | When is the bootstrap estimate of bias valid? | I think your formula is wrong. The last $t$ should have a star rather than a hat:
\begin{equation}
\mathrm{bias}_t \approx \frac{1}{N}\sum_i \tilde{t}_i- t^*
\end{equation}
You want to use the actual | When is the bootstrap estimate of bias valid?
I think your formula is wrong. The last $t$ should have a star rather than a hat:
\begin{equation}
\mathrm{bias}_t \approx \frac{1}{N}\sum_i \tilde{t}_i- t^*
\end{equation}
You want to use the actual statistic evaluated on the empirical distribution (this is often easy, sin... | When is the bootstrap estimate of bias valid?
I think your formula is wrong. The last $t$ should have a star rather than a hat:
\begin{equation}
\mathrm{bias}_t \approx \frac{1}{N}\sum_i \tilde{t}_i- t^*
\end{equation}
You want to use the actual |
6,491 | When is the bootstrap estimate of bias valid? | The problem you describe is a problem of interpretation, not one of validity. The bootstrap bias estimate for your constant estimator isn't invalid, it is in fact perfect.
The bootstrap estimate of bias is between an estimator $\hat\theta = s(x)$ and a parameter $\theta = t(F),$ where $F$ is some unknown distribution ... | When is the bootstrap estimate of bias valid? | The problem you describe is a problem of interpretation, not one of validity. The bootstrap bias estimate for your constant estimator isn't invalid, it is in fact perfect.
The bootstrap estimate of b | When is the bootstrap estimate of bias valid?
The problem you describe is a problem of interpretation, not one of validity. The bootstrap bias estimate for your constant estimator isn't invalid, it is in fact perfect.
The bootstrap estimate of bias is between an estimator $\hat\theta = s(x)$ and a parameter $\theta = ... | When is the bootstrap estimate of bias valid?
The problem you describe is a problem of interpretation, not one of validity. The bootstrap bias estimate for your constant estimator isn't invalid, it is in fact perfect.
The bootstrap estimate of b |
6,492 | When is the bootstrap estimate of bias valid? | You make one mistake and maybe that is the reason it is confusing. You say:
if my estimator simply returns a constant that is independent of the
observations, the above estimate of bias is clearly invalid
Bootstrap is not about how much your method is biased, but how much your results obtained by some function, giv... | When is the bootstrap estimate of bias valid? | You make one mistake and maybe that is the reason it is confusing. You say:
if my estimator simply returns a constant that is independent of the
observations, the above estimate of bias is clearly | When is the bootstrap estimate of bias valid?
You make one mistake and maybe that is the reason it is confusing. You say:
if my estimator simply returns a constant that is independent of the
observations, the above estimate of bias is clearly invalid
Bootstrap is not about how much your method is biased, but how mu... | When is the bootstrap estimate of bias valid?
You make one mistake and maybe that is the reason it is confusing. You say:
if my estimator simply returns a constant that is independent of the
observations, the above estimate of bias is clearly |
6,493 | When is the bootstrap estimate of bias valid? | I find it useful to think about the bootstrap procedures in terms of the functionals of the distributions they operate on -- I gave an example in this answer to a different bootstrap question.
The estimate you gave is what it is -- an estimate. Nobody says it does not suffer from problems that statistical estimates may... | When is the bootstrap estimate of bias valid? | I find it useful to think about the bootstrap procedures in terms of the functionals of the distributions they operate on -- I gave an example in this answer to a different bootstrap question.
The est | When is the bootstrap estimate of bias valid?
I find it useful to think about the bootstrap procedures in terms of the functionals of the distributions they operate on -- I gave an example in this answer to a different bootstrap question.
The estimate you gave is what it is -- an estimate. Nobody says it does not suffe... | When is the bootstrap estimate of bias valid?
I find it useful to think about the bootstrap procedures in terms of the functionals of the distributions they operate on -- I gave an example in this answer to a different bootstrap question.
The est |
6,494 | Quantile regression: Which standard errors? | Did you go through the paper Koenker and Hallock(2000): Quantile Regression: An Introduction (econ.uiuc.edu/~roger/research/intro/rq.pdf)? Bootstrap is preferable because it makes no assumption about the distribution of response (p. 47, Quantile regressions, Hao and Naiman, 2007). Also, note that the "...assumptions fo... | Quantile regression: Which standard errors? | Did you go through the paper Koenker and Hallock(2000): Quantile Regression: An Introduction (econ.uiuc.edu/~roger/research/intro/rq.pdf)? Bootstrap is preferable because it makes no assumption about | Quantile regression: Which standard errors?
Did you go through the paper Koenker and Hallock(2000): Quantile Regression: An Introduction (econ.uiuc.edu/~roger/research/intro/rq.pdf)? Bootstrap is preferable because it makes no assumption about the distribution of response (p. 47, Quantile regressions, Hao and Naiman, 2... | Quantile regression: Which standard errors?
Did you go through the paper Koenker and Hallock(2000): Quantile Regression: An Introduction (econ.uiuc.edu/~roger/research/intro/rq.pdf)? Bootstrap is preferable because it makes no assumption about |
6,495 | Are there statistical lessons from the "Bible Code" episode | Edit This is one of those questions whose answer may elicit negative reactions irrespective of whether or not the answer is correct. There have been several deleted answers so far, and it is doubtful that any answer will satisfy a majority of readers. Frankly, I do not care, but also any answer has to deal with the con... | Are there statistical lessons from the "Bible Code" episode | Edit This is one of those questions whose answer may elicit negative reactions irrespective of whether or not the answer is correct. There have been several deleted answers so far, and it is doubtful | Are there statistical lessons from the "Bible Code" episode
Edit This is one of those questions whose answer may elicit negative reactions irrespective of whether or not the answer is correct. There have been several deleted answers so far, and it is doubtful that any answer will satisfy a majority of readers. Frankly,... | Are there statistical lessons from the "Bible Code" episode
Edit This is one of those questions whose answer may elicit negative reactions irrespective of whether or not the answer is correct. There have been several deleted answers so far, and it is doubtful |
6,496 | Maximum Mean Discrepancy (distance distribution) | It might help to give slightly more of an overview of MMD.$\DeclareMathOperator{\E}{\mathbb E}\newcommand{\R}{\mathbb R}\newcommand{\X}{\mathcal X}\newcommand{\h}{\mathcal H}\DeclareMathOperator{\MMD}{MMD}$
In general, MMD is defined by the idea of representing distances between distributions as distances between mean ... | Maximum Mean Discrepancy (distance distribution) | It might help to give slightly more of an overview of MMD.$\DeclareMathOperator{\E}{\mathbb E}\newcommand{\R}{\mathbb R}\newcommand{\X}{\mathcal X}\newcommand{\h}{\mathcal H}\DeclareMathOperator{\MMD} | Maximum Mean Discrepancy (distance distribution)
It might help to give slightly more of an overview of MMD.$\DeclareMathOperator{\E}{\mathbb E}\newcommand{\R}{\mathbb R}\newcommand{\X}{\mathcal X}\newcommand{\h}{\mathcal H}\DeclareMathOperator{\MMD}{MMD}$
In general, MMD is defined by the idea of representing distances... | Maximum Mean Discrepancy (distance distribution)
It might help to give slightly more of an overview of MMD.$\DeclareMathOperator{\E}{\mathbb E}\newcommand{\R}{\mathbb R}\newcommand{\X}{\mathcal X}\newcommand{\h}{\mathcal H}\DeclareMathOperator{\MMD} |
6,497 | Maximum Mean Discrepancy (distance distribution) | Here is how I interpretted MMD. Two distributions are similar if their moments are similar. By applying a kernel, I can transform the variable such that all moments (first, second, third etc.) are computed. In the latent space I can compute the difference between the moments and average it. This gives a measure of the ... | Maximum Mean Discrepancy (distance distribution) | Here is how I interpretted MMD. Two distributions are similar if their moments are similar. By applying a kernel, I can transform the variable such that all moments (first, second, third etc.) are com | Maximum Mean Discrepancy (distance distribution)
Here is how I interpretted MMD. Two distributions are similar if their moments are similar. By applying a kernel, I can transform the variable such that all moments (first, second, third etc.) are computed. In the latent space I can compute the difference between the mom... | Maximum Mean Discrepancy (distance distribution)
Here is how I interpretted MMD. Two distributions are similar if their moments are similar. By applying a kernel, I can transform the variable such that all moments (first, second, third etc.) are com |
6,498 | Maximum Mean Discrepancy (distance distribution) | What we know today as Maximum Mean Discrepancy is actually derived from the following Integral Probability Metric [A]:
If p and q are two distributions and $\mathcal{F}$ is a class of real
valued bounded bounded measurable functions, then the metric is
defined as, $$D(p, q, \mathcal{F}) = \sup_{f \in \mathcal{F}} \lef... | Maximum Mean Discrepancy (distance distribution) | What we know today as Maximum Mean Discrepancy is actually derived from the following Integral Probability Metric [A]:
If p and q are two distributions and $\mathcal{F}$ is a class of real
valued bou | Maximum Mean Discrepancy (distance distribution)
What we know today as Maximum Mean Discrepancy is actually derived from the following Integral Probability Metric [A]:
If p and q are two distributions and $\mathcal{F}$ is a class of real
valued bounded bounded measurable functions, then the metric is
defined as, $$D(p... | Maximum Mean Discrepancy (distance distribution)
What we know today as Maximum Mean Discrepancy is actually derived from the following Integral Probability Metric [A]:
If p and q are two distributions and $\mathcal{F}$ is a class of real
valued bou |
6,499 | Maximum Mean Discrepancy (distance distribution) | For the Gaussian kernel $K({\mathbf x}, {\mathbf y})=e^{-||{\mathbf x}-{\mathbf y}||^2/4\sigma^2}$ on ${\mathbb R}^n$, the MMD satisfies:
${\rm MMD}(P,Q) \propto \sup\limits_{f\in L_2({\mathbb R}^n), ||f||_{L_2}\leq 1} {\mathbb E}_{X\sim P, \epsilon\sim N(0, \sigma^2I_n)} f(X+\epsilon)-{\mathbb E}_{Y\sim Q, \epsilon'\s... | Maximum Mean Discrepancy (distance distribution) | For the Gaussian kernel $K({\mathbf x}, {\mathbf y})=e^{-||{\mathbf x}-{\mathbf y}||^2/4\sigma^2}$ on ${\mathbb R}^n$, the MMD satisfies:
${\rm MMD}(P,Q) \propto \sup\limits_{f\in L_2({\mathbb R}^n), | Maximum Mean Discrepancy (distance distribution)
For the Gaussian kernel $K({\mathbf x}, {\mathbf y})=e^{-||{\mathbf x}-{\mathbf y}||^2/4\sigma^2}$ on ${\mathbb R}^n$, the MMD satisfies:
${\rm MMD}(P,Q) \propto \sup\limits_{f\in L_2({\mathbb R}^n), ||f||_{L_2}\leq 1} {\mathbb E}_{X\sim P, \epsilon\sim N(0, \sigma^2I_n)... | Maximum Mean Discrepancy (distance distribution)
For the Gaussian kernel $K({\mathbf x}, {\mathbf y})=e^{-||{\mathbf x}-{\mathbf y}||^2/4\sigma^2}$ on ${\mathbb R}^n$, the MMD satisfies:
${\rm MMD}(P,Q) \propto \sup\limits_{f\in L_2({\mathbb R}^n), |
6,500 | Random walk on the edges of a cube | I suggest modeling the problem as a Markov chain where each state represents the distance between the spider and the ant. In this case we have 4 possible states $S_i$ as the distances $i$ can be $\{0,1,2,3\}$.
When the spider is at the opposite corner of the cube, it is at a distance of 3 steps from the ant. It is in s... | Random walk on the edges of a cube | I suggest modeling the problem as a Markov chain where each state represents the distance between the spider and the ant. In this case we have 4 possible states $S_i$ as the distances $i$ can be $\{0, | Random walk on the edges of a cube
I suggest modeling the problem as a Markov chain where each state represents the distance between the spider and the ant. In this case we have 4 possible states $S_i$ as the distances $i$ can be $\{0,1,2,3\}$.
When the spider is at the opposite corner of the cube, it is at a distance ... | Random walk on the edges of a cube
I suggest modeling the problem as a Markov chain where each state represents the distance between the spider and the ant. In this case we have 4 possible states $S_i$ as the distances $i$ can be $\{0, |
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