Classification and Regression Trees by Breiman Leo & Friedman Jerome & Stone Charles J. & Olshen R.A

Author:Breiman, Leo & Friedman, Jerome & Stone, Charles J. & Olshen, R.A. [Breiman, Leo]
Language: eng
Format: epub
Publisher: Apress - A
Published: 1983-12-31T14:00:00+00:00

APPENDIX

The Linear Combination Search Algorithm

Suppose that at a given node t, a goodness of split criterion Φ(s, t) is defined. Hereafter, when we say “best split,” we mean with respect to the Φ(s, t) criterion.

At the node there are N = N(t) measurement vectors x1, ..., xN with M1 ordered coordinates. Let the ordered coordinates of the nth case be xm,n, m = 1, ..., M1. At the first step of the algorithm, the N values xm,n, n = 1, ..., N, are centered at their median and divided by their interquartile range for each m, m = 1, ..., M1. Only these normalized values are used in the algorithm, and, to avoid a plethora of notation, we denote them also by x m,n.

The algorithm cycles through the variables x1, ..., x , at each step doing a search for an improved linear combination split. At the beginning of the Lth cycle, let the current linear combination split be v < c, where v = βmxm.

For fixed γ, consider the problem of finding the best split of the form

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