Extending Similarity Measures of Interval Type-2 Fuzzy Sets to General Type-2 Fuzzy Sets

TL;DR

This paper presents a novel method to extend existing similarity measures from interval type-2 fuzzy sets to general type-2 fuzzy sets using zSlices representation, preserving key properties.

Contribution

It introduces a general approach for extending interval type-2 similarity measures to general type-2 fuzzy sets with property preservation.

Findings

Extended similarity measures demonstrate consistent properties.

Comparative analysis shows effectiveness of the proposed extension.

Examples illustrate the applicability of the method.

Abstract

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.