A nonlinear integral defined on partition and its application to decision trees

Nonlinear integrals play an important role in information fusion. So far, all existing nonlinear integrals of a function with respect to a set function are defined on a subset of a space. In many of the problems with information fusion, such as decision tree generation in inductive learning, we often need to deal with the function defined on a partition of the space. Motivated by minimizing the classification information entropy of a partition while generating decision trees, this paper proposes a nonlinear integral of a function with respect to a nonnegative set function on a partition, and provides the conclusion that the sum of the weighted entropy of the union of several subsets is not less than the sum of the weighted entropy of a single subset. It is shown that selecting the entropy of a single attribute is better than selecting the entropy of the union of several attributes in generating rules by decision trees.