On computing the similarity of trapezoidal fuzzy sets using an Automated Area Method

Abstract

This paper provides a new Automated Area Method (AAM) for computing the Jaccard similarity of non-normal trapezoidal fuzzy sets (with triangles and rectangles as special cases), needed, e.g. in XAI for rule-based fuzzy systems, Computing With Words using the Perceptual Computer, Perceptual reasoning, etc. The most intensive part of the Jaccard similarity computation is computing the area under the minimum of two fuzzy sets. The AAM does this very efficiently by initially using figures for the 40 possible cases of the minimum of two non-normal trapezoidal type-1 FSs, and by then providing area formulas for each of them using simple geometry. Flowcharted algorithms are then provided for automatically determining which of the 40 cases one is in, which do not need any figures, and only need the coordinates of the vertices of the two non-normal trapezoidal type-1 FSs. Examples are provided that illustrate these new results, both for type-1 and interval type-2 FSs. The AAM can also be used to compute other similarity measures and to compute the subsethood between fuzzy sets.