Using the singular value decomposition, show that the variance of the direction vector -\( \hat{z}_i=\hat{X}\hat{v}_i \) is equal to (equation (3.49) of Hastie et al.) +\( \hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1 \) is equal to (equation (3.49) of Hastie et al.) $$ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, $$ -where \( d_i \) are the singular values of the matrix \( \hat{X} \). Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where \( d_i \) are the singular values of the matrix \( \hat{X} \). In Hastie et al, the matrix elements of \( X \) are centered. The consequence is that the mean values of for example \( \hat{u}_i \) are zero. + +
+Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise.
diff --git a/doc/Projects/2018/hw2/html/hw2.html b/doc/Projects/2018/hw2/html/hw2.html index 66482871e..102ba1e39 100644 --- a/doc/Projects/2018/hw2/html/hw2.html +++ b/doc/Projects/2018/hw2/html/hw2.html @@ -136,12 +136,15 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge
Using the singular value decomposition, show that the variance of the direction vector -\( \hat{z}_i=\hat{X}\hat{v}_i \) is equal to (equation (3.49) of Hastie et al.) +\( \hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1 \) is equal to (equation (3.49) of Hastie et al.) $$ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, $$ -where \( d_i \) are the singular values of the matrix \( \hat{X} \). Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where \( d_i \) are the singular values of the matrix \( \hat{X} \). In Hastie et al, the matrix elements of \( X \) are centered. The consequence is that the mean values of for example \( \hat{u}_i \) are zero. + +
+Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. diff --git a/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz b/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz index eddf88e9a..dda671634 100644 Binary files a/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz and b/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz differ diff --git a/doc/Projects/2018/hw2/pdf/hw2.p.tex b/doc/Projects/2018/hw2/pdf/hw2.p.tex index 7ba297184..101bd04f8 100644 --- a/doc/Projects/2018/hw2/pdf/hw2.p.tex +++ b/doc/Projects/2018/hw2/pdf/hw2.p.tex @@ -211,11 +211,13 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and \subsection{Exercise 5} Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie \emph{et al.}) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie \emph{et al.}) \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie \emph{et al}, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. % ------------------- end of main content --------------- diff --git a/doc/Projects/2018/hw2/pdf/hw2.pdf b/doc/Projects/2018/hw2/pdf/hw2.pdf index fd9334f70..7fbb0fe01 100644 Binary files a/doc/Projects/2018/hw2/pdf/hw2.pdf and b/doc/Projects/2018/hw2/pdf/hw2.pdf differ diff --git a/doc/Projects/2018/hw2/pdf/hw2.tex b/doc/Projects/2018/hw2/pdf/hw2.tex index bb1c897c9..8670ad27a 100644 --- a/doc/Projects/2018/hw2/pdf/hw2.tex +++ b/doc/Projects/2018/hw2/pdf/hw2.tex @@ -181,11 +181,13 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and \subsection*{Exercise 5} Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie \emph{et al.}) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie \emph{et al.}) \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie \emph{et al}, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. % ------------------- end of main content --------------- diff --git a/doc/src/Projects/2018/Exercises/hw2.do.txt b/doc/src/Projects/2018/Exercises/hw2.do.txt index 2b6e8071f..987cef3ec 100644 --- a/doc/src/Projects/2018/Exercises/hw2.do.txt +++ b/doc/src/Projects/2018/Exercises/hw2.do.txt @@ -54,10 +54,12 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and ===== Exercise 5 ===== Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie *et al.*) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie *et al.*) !bt \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] !et -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie *et al*, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise.