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.
Highlights
Since Zadeh [1] developed fuzzy sets and Atanassov [2] constructed intuitionistic fuzzy sets (IFSs), numerous studies have examined fuzzy sets and IFSs to determine their theoretic evolution and devise applications to practical problems
Based on our proven Lemma 1 and Lemma 2, we verify that the first similarity measure proposed by Zhang et al [4] satisfies the fourth axiom (A4)
For three FSs A, B and C satisfying A ⊆ B ⊆ C we prove that SIFS (A, C) ≤ SIFS (A, B)
Summary
Since Zadeh [1] developed fuzzy sets and Atanassov [2] constructed intuitionistic fuzzy sets (IFSs), numerous studies have examined fuzzy sets and IFSs to determine their theoretic evolution and devise applications to practical problems. A series of papers—Deng et al [5], Tang et al [6], Lan et al [7], Yang et al [8], Deng [9], Chang et al [10], Jung et al [11], and Deng et al [12]—make revisions to existing proofs Motivated by these articles, Zhang et al [4]. The first goal of this paper is to provide a revision to enhance the proof of Zhang et al [4] on their similarity measures for the fourth axiom of Li and Cheng [14]. We demonstrate that the second similarity measure proposed by Zhang et al [4] addressed a practical pattern recognition problem of fault diagnosis for the turbine engine.
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