On the Relationship Between Similarity Measures and Thresholds of Statistical Significance in the Context of Comparing Fuzzy Sets

Comparing fuzzy sets by computing their similarity is common, with a large set of measures of similarity available. However, while commonplace in the computational intelligence community, the application and results of similarity measures are less common in the wider scientific context, where statistical approaches are the standard for comparing distributions. This is challenging, as it means that developments around similarity measures arising from the fuzzy community are inaccessible to the wider scientific community and that the fuzzy community fails to take advantage of a strong statistical understanding, which may be applicable to comparing (fuzzy membership) functions. In this paper, we commence a body of work on systematically relating the outputs of similarity measures to the notion of statistically significant difference; that is, how (dis)similar do two fuzzy sets need to be for them to be statistically different? We explain that in this context, it is useful to initially focus on dis-similarity, rather than similarity, as the former aligns directly with the widely used concept of statistical difference. We propose two methods of applying statistical tests to the outputs of fuzzy dissimilarity measures to determine significant difference. We show how the proposed work provides deeper insight into the behavior and possible interpretation of degrees of dis-similarity and, consequently, similarity, and how the interpretation differs with respect to context (e.g., the complexity of the fuzzy sets).