Divergence measure between fuzzy sets

Abstract

In this paper we propose a way of measuring the difference between two fuzzy sets by means of a function which we will call divergence. We define this concept by means of a group of natural axioms and we study in detail the most important classes of such measures, those which have the local property.

Relationships Among Several Fuzzy Measures

“…It should be mentioned here that in some references these functions are called distance measures [3,8,15,26,28,43], in some references these functions are called dissimilarity measures [4,12], but in other references these functions are called divergence measures [1,11,32,33]. The first axiomatic definition for distance measure was proposed by Liu in [43], where four axioms for distance measure were given.…”

“…Based on logarithm operation, Bhandari and Pal [2] defined a divergence measure between fuzzy sets. In [32], Montes et al introduced a new axiomatic definition for divergence measure. Their approach is based on three axioms modeling the minimal requirements for a function that tries to measure the separation or difference between fuzzy sets.…”

“…Thus it is necessary to find some ways to construct more divergence measures and then compare them with other proposals in the literature. Montes et al [32,33] proposed a way of constructing divergence measure. This kind of divergence measure is also a divergence measure in the sense of Liu [43], based on fuzzy union and intersection.…”

“…Let us consider the definition of divergence measure in the sense of Liu [43] and that in the sense of Montes [32].…”

“…The first three axioms in both of the definitions are defined on the basis of the following natural properties [32]:…”