Analytical Expressions for Similarity Measure Between Gaussian Interval Type-2 Fuzzy Sets With Uncertain Means

The computation of the Jaccard similarity measure between Gaussian interval type-2 fuzzy sets with uncertain means presents a significant challenge due to the lack of precise formulas and the computationally expensive nature of existing methods. This article addresses this challenge by proposing a novel method that derives analytical formulas for calculating the intersection area between upper and lower membership functions. A comprehensive analysis has identified nine cases for upper membership functions and fifteen cases for lower membership functions, with closed-form equations provided to compute the intersection areas for each identified case. By using these derived equations, the similarity measure can be computed precisely without relying solely on numerical integration. The derived analytical formulas confer the advantage of significantly reducing computational burden and accurately calculating similarity, making the proposed method more suitable for real-world applications. Moreover, it has been demonstrated that calculating Jaccard's similarity measure becomes significantly simpler when interval type-2 fuzzy sets have no uncertainties, implying that these sets are equivalent to type-1 fuzzy sets.