Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 2, inverse problems

In Part 1 of this two-part paper, we bounded the centroid of a symmetric interval type-2 fuzzy set (T2 FS), and consequently its uncertainty, using geometric properties of its footprint of uncertainty (FOU). We then used these bounds to solve forward problems, i.e., to go from parametric interval T2 FS models to data. The main purpose of the present paper is to formulate and solve inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models, which we call type-2 fuzzistics. Given interval data collected from people about a phrase, and the inherent uncertainties associated with that data, which can be described statistically using the first- and second-order statistics about the end-point data, we establish parametric FOUs such that their uncertainty bounds are directly connected to statistical uncertainty bounds. These results should find applicability in computing with words.