update book

This commit is contained in:
Morten Hjorth-Jensen
2024-09-02 16:29:05 +02:00
parent 548e735c20
commit 41f98033a3
8 changed files with 91 additions and 171 deletions
+2 -11
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@@ -59,24 +59,15 @@ that $\vert\vert \bm{\beta}\vert\vert_2^2\le t$, where $t$ is
a finite number larger than zero. We will not implement that here.
=== a) Expression for Ridge regression ===
Show that the optimal parameters
!bt
\[
\hat{\bm{\beta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y},
\]
!et
with $\bm{I}$ being a $p\times p$ identity matrix.
with $\bm{I}$ being a $p\times p$ identity matrix,
with the constraint that
!bt
\[
\sum_{i=0}^{p-1} \beta_i^2 \leq t,
\]
!et
with $t$ a finite positive number. In the optimization, we will not require that the latter is satisfied.
The ordinary least squares result is
!bt