Solution

Start with the root region

R^{p}

. For any current terminal region

R

, coordinate

k

, and threshold

s

, form

R_{-} = {x \in R : x_{k} \leq s}, R_{+} = {x \in R : x_{k} > s},

(198)

discarding splits that leave an empty child. For each child use its training-response mean

γ_{\pm} = \frac{\sum _{i} Y _{i}^{'} 1 _{R_{\pm}} ( X _{i}^{'} )}{\sum _{i} 1 _{R_{\pm}} ( X _{i}^{'} )} .

(199)

Compare candidate splits by the resulting residual sum of squares

\sum_{X_{i}^{'} \in R_{-}} (Y_{i}^{'} - γ_{-})^{2} + \sum_{X_{i}^{'} \in R_{+}} (Y_{i}^{'} - γ_{+})^{2},

(200)

together with the unchanged residual sums in the other leaves. Choose the leaf, coordinate, and threshold minimizing the total. Replace that leaf by its two children and repeat until the stopping rule or the prescribed

J

leaves is reached. The resulting terminal regions are

R_{1}, \dots, R_{J}

, and

γ_{j} = \frac{\sum _{i} Y _{i}^{'} 1 _{R_{j}} ( X _{i}^{'} )}{\sum _{i} 1 _{R_{j}} ( X _{i}^{'} )} .

(201)

This is the standard regression tree construction.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 31K / a / Solution

 Ancestors (11)