Partitioning Nominal Attributes in Decision Trees

To find the optimal branching of a nominal attribute at a node in an L-ary decision tree, one is often forced to search over all possible L-ary partitions for the one that yields the minimum impurity measure. For binary trees (L = 2) when there are just two classes a short-cut search is possible that is linear in n, the number of distinct values of the attribute. For the general case in which the number of classes, k, may be greater than two, Burshtein et al. have shown that the optimal partition satisfies a condition that involves the existence of2L hyperplanes in the class probability space. We derive a property of the optimal partition for concave impurity measures (including in particular the Gini and entropy impurity measures) in terms of the existence ofL vectors in the dual of the class probability space, which implies the earlier condition.Unfortunately, these insights still do not offer a practical search method when n and k are large, even for binary trees. We therefore present a new heuristic search algorithm to find a good partition. It is based on ordering the attribute's values according to their principal component scores in the class probability space, and is linear in n. We demonstrate the effectiveness of the new method through Monte Carlo simulation experiments and compare its performance against other heuristic methods.