The Fourth Axiom of Similarity Measures

Abstract

In this research, the fourth axiom to improve the well-defined examination of similarity measures is studied, where the measures have a symmetric structure. We first provide a theoretic enhancement of three correlation coefficient similarity measures that were proposed by a source paper. Second, we use the same numerical example proposed by the source paper for pattern recognition problems to illustrate that the weighted correlation coefficient similarity measure is dependent on the set of weights. Finally, we demonstrate that the correlation coefficient similarity measure in the intuitionistic fuzzy set environment can address the issue of practical fault diagnosis when solving the turbine engine problems using similarity measures with symmetric characteristics.

Representations of a Comparison Measure between Two Fuzzy Sets

“…The fourth type IV is the duplicate representations of a comparison measure m(A, B) which appear in type I. For example, for the combination [1]- [5], we obtain that…”

“…which is the same as that of combination [1]- [2]. For simplicity, we adopt the notation [1]-[2] ≡ [1]- [5] to denote the representative equivalence between [1]- [2] and [1]- [5]. There are eight combinations in type IV described as follows:…”

“…A comparison measure calculates the degree of equality or inequality between two compared FSs. Some related definitions such as similarity, similitude, proximity or resemblance were proposed for the equality measures [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], as well as some other dual definitions such as dissimilarity, dissimilitude, divergence or distance for the inequality measures [6,9,13,[18][19][20][21][22][23]. The inequality measures have received much less attention in the literature.…”

“…The degree of comparison measure is an important tool for cluster analysis [8], decision-making [7,14,21,22], e-waste [2,17,20], image processing [10], medical diagnosis [13], pattern recognition [3,4,9,12] and service quality [11,18]. Recently, many papers [5,9,10,12,15,16,18,[20][21][22] have been dedicated to the comparison measures, and research on this area is still carried out in the literature.…”