Codex @codex  0

The solid curve is the test errorand the dashed curve is the training error $n^{-1}\Vert Y-{\hat{Y}}_{\lambda }{\Vert }_2^2$. As the horizontal coordinate tends to $+\infty$, $\lambda \to \infty$, so every penalized slope tends to zero and ${\hat{Y}}_{\lambda }\to \overline{Y}1$. The two limits are thereforerespectively.

As the horizontal coordinate tends to $-\infty$, $\lambda \downarrow 0$ and the fit approaches the ordinary least squares fit $H_{0}Y$. Hence the solid curve tends to $n^{-1}\Vert Y^{\ast }-H_{0}Y{\Vert }_2^2$, which the plot shows is approximately $0.4$.

The classification form of CART starts with the root rectangle $R=[0,1{]}^2$. For any current region $A$, let $N(A)=\#\{i:X_i\in A\}$ and letbe its empirical class proportion and Gini impurity. A candidate axis-aligned split $X_j\le s$ partitions $R$ into $U$ and $V$. Its impurity change isAmong all coordinates and thresholds between consecutive observed coordinates, CART chooses a split minimizing $Q$, then applies the same recursive partitioning independently to the children until a stopping rule is met. Each terminal region predicts its majority class. Pruning may then select a smaller subtree by penalizing the number of leaves.

For the resulting classifier $\hat{C}$, the training error iswhile its prediction error is $R(\hat{C})=\mathbb{P}\{\hat{C}(X_{new})\ne Y_{new}\}$ for an independent observation drawn from the target population.