General and efficient multisplitting of numerical attributes

Often in supervised learning numerical attributes require special treatment and do not fit the learning scheme as well as one could hope. Nevertheless, they are common in practical tasks and, therefore, need to be taken into account. We characterize the well-behavedness of an evaluation function, a property that guarantees the optimal multi-partition of an arbitrary numerical domain to be defined on boundary points. Well-behavedness reduces the number of candidate cut points that need to be examined in multisplitting numerical attributes. Many commonly used attribute evaluation functions possess this property; we demonstrate that the cumulative functions Information Gain and Training Set Error as well as the non-cumulative functions Gain Ratio and Normalized Distance Measure are all well-behaved. We also devise a method of finding optimal multisplits efficiently by examining the minimum number of boundary point combinations that is required to produce partitions which are optimal with respect to a cumulative and well-behaved evaluation function. Our empirical experiments validate the utility of optimal multisplitting: it produces constantly better partitions than alternative approaches do and it only requires comparable time. In top-down induction of decision trees the choice of evaluation function has a more decisive effect on the result than the choice of partitioning strategy; optimizing the value of most common attribute evaluation functions does not raise the accuracy of the produced decision trees. In our tests the construction time using optimal multisplitting was, on the average, twice that required by greedy multisplitting, which in its part required on the average twice the time of binary splitting.