Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 30J / a

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- Solution a

Solution

A regression tree chooses a threshold

s

and constant predictions

c_{L}, c_{R}

to minimize the squared-error criterion

min_{s, c_{L}, c_{R}} {\sum_{i : X_{i} \leq s} (Y_{i} - c_{L})^{2} + \sum_{i : X_{i} > s} (Y_{i} - c_{R})^{2}},

where

s

ranges over thresholds producing two nonempty groups. For a fixed split, the least-squares minimizers are the two sample means,

c_{L} = \frac{1}{n _{L}} \sum_{X_{i} \leq s} Y_{i}, c_{R} = \frac{1}{n _{R}} \sum_{X_{i} > s} Y_{i} .

Thus the first split is the candidate threshold with the smallest sum of the two within-node residual sums of squares.

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