Similarity Measures for Closed General Type-2 Fuzzy Sets: Overview, Comparisons, and a Geometric Approach
Dongrui Wu, Jerry M. Mendel · IEEE Transactions on Fuzzy Systems · 2018
The similarity between two fuzzy sets (FSs) is an important concept in fuzzy logic. As the research interest on general type-2 (GT2) FSs has increased recently, many similarity measures for them have also been proposed. This paper gives a comprehensive overview of existing similarity measures for GT2 FSs, points out their limitations, and, by using an intuitive geometric explanation, proposes a Jaccard similarity measure for GT2 FSs that is an extension of the popular Jaccard similarity measure for type-1 and interval type-2 FSs. The fundamental difference between the proposed Jaccard similarity measure for GT2 FSs and all existing similarity measures is that the Jaccard similarity measure considers the overall geometries of two GT2 FSs and does not depend on a specific representation of the GT2 FSs, whereas all existing similarity measures for GT2 FSs depend either on the vertical slice representation or the α-plane representation. We show that the Jaccard similarity measure for GT2 FSs satisfies four properties of a similarity measure and demonstrate its reasonableness using two examples.