Fuzzy Integral Defined On Partition Of Set And ID3 Algorithm

Fuzzy integral is generalization of the classical (Lebesgue) integral, the measure in fuzzy integral is non-additive.The decision tree algorithm is an algorithm of inductive learning, ID3 algorithm is one of the most representative decision tree algorithms, ID3 uses the entropy as the criterion to select the expanded attributes in generating decision tree recursively. The process of the generating decision tree is that of parting the given set. Considering the partition of the set effect on the size of the decision tree, Motivated by ID3 algorithm generating decision trees, in this paper, we propose a fuzzy integral of a function with respect to a non-negative set function on a partition, presents its basic properties, and gives its computing in special case. Using the new integral in decision tree, the paper 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. So, this paper provides a mathematically theoretic basis for the ID3 algorithm.