Study on Entropy and Similarity Measures for Fuzzy Sets.pdf

IEICE TRANS. INF. SYST., VOL.E92–D, NO.9 SEPTEMBER 2009 1783 LETTER Study on Entropy and Similarity Measure for Fuzzy Set † †† ††a) Sang-Hyuk LEE , Member, Keun Ho RYU , Nonmember, and Gyoyong SOHN , Member SUMMARY In this study, we investigated the relationship between derivation of similarity measures. However, derived similar- similarity measures and entropy for fuzzy sets. First, we developed fuzzy ity measures are restricted to triangular or trapezoidal mem- entropy by using the distance measure for fuzzy sets. We pointed out that bership functions [11], [12]. In contrast, similarity measures 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 based on the distance measure are applicable to general the entropy between the fuzzy set and the corresponding crisp set consti- fuzzy membership functions, including nonconvex fuzzy tutes the total information in the fuzzy set. Finally, we derived a similarity membership functions [13], [14]. measure from entropy and showed by a simple example that the maximum The correlation between entropy and similarity for similarity measure can be obtained using a minimum entropy formulation. fuzzy sets has been presented from various viewpoints [15]. key words: similarity measure, distance measure, fuzzy entropy Liu also proposed a r