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Minimum Impurity Partitions

David Burshtein, Vincent Della Pietra, Dimitri Kanevsky and Arthur Nadas
The Annals of Statistics
Vol. 20, No. 3 (Sep., 1992), pp. 1637-1646
Stable URL: http://www.jstor.org/stable/2242032
Page Count: 10
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Minimum Impurity Partitions
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Abstract

Let (X, U) be jointly distributed on X × Rn. Let Y = E(U∣ X) and let U be the convex hull of the range of U. Let C: X → C = {1,2,...,k}, k ≥ 1, induce a measurable k way partition {X1,...,Xk} of X. Define the impurity of Xc = C-1(c) to be φ(c, E(U∣ C(X) = c)), where φ: C × U → R1 is a concave function in its second argument. Define the impurity Ψ of the partition as the average impurity of its members: Ψ(C) = Eφ(C(X), E(U∣ C(X)). We show that for any C: X → C there exists a mapping C̃: U → C, such that Ψ(C̃(Y)) ≤ Ψ(C) and such that C̃-1(c) is convex, that is, for each i, j ∈ C, i ≠ j, there exists a separating hyperplane between C̃-1(i) and C̃-1(j). This generalizes some results in statistics and information theory. Suitable choices of U and φ lead to optimal partitions of simple form useful in the construction of classification trees and multidimensional regression trees.

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