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added hint to exercise 5

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mhjensen
2018-09-05 16:46:51 +02:00
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doc/src/Projects/2018/Exercises/hw2.do.txt
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@@ -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.