Centroid uncertainty bounds for interval type-2 fuzzy sets: forward and inverse problems

Interval type-2 fuzzy sets (T2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). The centroid of an interval T2 FS, which is an interval T1 FS, provides a measure of the uncertainty in the interval T2 FS. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, associated with the amount of uncertainty in an interval T2 FS. The main purpose of this paper is to demonstrate that our intuition is correct and to quantify the centroid of an interval T2 FS with respect to these geometric properties of its FOU. It is then possible to formulate and solve inverse problems, i.e., going from data to parametric T2 FS models.