SUMMARYIn this study, we investigated the relationship between similarity measures and entropy for fuzzy sets. First, we developed fuzzy entropy by using the distance measure for fuzzy sets. We pointed out that the distance between the fuzzy set and the corresponding crisp set equals fuzzy entropy. We also found that the sum of the similarity measure and the entropy between the fuzzy set and the corresponding crisp set constitutes the total information in the fuzzy set. Finally, we derived a similarity measure from entropy and showed by a simple example that the maximum similarity measure can be obtained using a minimum entropy formulation.
Fuzzy entropy was designed for non convex fuzzy membership function using well known Hamming distance measure. The proposed fuzzy entropy had the same structure as that of convex fuzzy membership case. Design procedure of fuzzy entropy was proposed by considering fuzzy membership through distance measure, and the obtained results contained more flexibility than the general fuzzy membership function. Furthermore, characteristic analyses for non convex function were also illustrated. Analyses on the mutual information were carried out through the proposed fuzzy entropy and similarity measure, which was also dual structure of fuzzy entropy. By the illustrative example, mutual information was discussed.
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