Divergence and Similarity Characteristics for Two Fuzzy Measures Based on Associated Probabilities

Abstract

Abstract: The article deals with the definitions of the distance, divergence, and similarity characteristics between two finite fuzzy measures, which are generalizations of the same definitions between two finite probability distributions. As is known, a fuzzy measure can be uniquely represented by the so-called its associated probability class (APC). The idea of generalization is that new definitions of distance, divergence, and similarity between fuzzy measures are reduced to the definitions of distance, divergence, and similarity between the APCs of fuzzy measures. These definitions are based on the concept of distance generator. The proof of the correctness of generalizations is provided. Constructed distance, similarity, and divergence relations can be used in such applied problems as: determining the difference between Dempster-Shafer belief structures; Constructions of collaborative filtering similarity relations; non-additive and interactive parameters of machine learning in phase space metrics definition, object clustering, classification and other tasks. In this work, a new concept is used in the fuzzy measure identification problem for a certain multi-attribute decision-making (MADM) environment. For this, a conditional optimization problem with one objective function representing the distance, divergence or similarity index is formulated. Numerical examples are discussed and a comparative analysis of the obtained results is presented.