Study on Entropy and Similarity Measure for Fuzzy Set

Abstract

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.

Mutual Information Analysis with Similarity Measure

“…There are many fuzzy entropies satisfying Definition 2.1, following entropies are satisfying four axioms of Definition 2.1, and the proofs are found in our previous results [5,6].…”

“…Applying similarity measure, there must be needed comparing data sets. Conventional similarity measure has been designed based on the fuzzy number and distance measure [4][5][6]. Similarity measure with fuzzy number can be found in references [4].…”

Analysis of a Similarity Measure for Non-Overlapped Data

“…Among the research on the fuzzy set theory, the analysis of data uncertainties has been carried out by numerous researchers through fuzzy data sets [6][7][8]. Of those researches based on fuzzy data sets, the similarity measure design problem-i.e., the design of a measure evaluating the degree of similarity between two data sets [8][9][10][11][12][13][14][15]-has been attracting an increasing amount of attention from the research community due to the increasing number of applications, including data mining, pattern recognition, and clustering [16,17]. The design of the similarity measure based on fuzzy numbers is convenient, but it can only use triangular or trapezoidal fuzzy membership functions [11,12].…”

“…The design of the similarity measure based on fuzzy numbers is convenient, but it can only use triangular or trapezoidal fuzzy membership functions [11,12]. If we design a similarity measure based on a distance measure, we can use a general fuzzy membership function without any limit on its shape [13][14][15]. Note that most conventional similarity measures-whether based on fuzzy number or distance measure-can be applied to overlapped data sets only [8,11,13,14,[18][19][20][21].…”

“…For each data sequence, its data distribution is represented by various structures. We designed similarity measures for data that have been shown as overlapped data in previous works [13][14][15]. In the following, we explain several types of data with examples:…”