Approximation of Fuzzy Sets by Interval Type-2 Trapezoidal Fuzzy Sets

In this paper, we propose a gradient-based method to approximate a fuzzy set through a trapezoidal fuzzy set (TFS). By adding some constraints in the formulated optimization problem, the major characteristics of the fuzzy set such as the core, the major part of the support, and the shape of the membership function could be preserved; also the form of the optimized result as a TFS is guaranteed. We regard the optimized TFS as the "skeleton" (blueprint) of the original fuzzy set. Based on this skeleton, we further extend the TFS to a higher type, that is, an interval type-2 TFS (IT2 TFS), so that more information about the original fuzzy set could be captured but the number of the parameters used to describe the original fuzzy set is still maintained low (nine parameters are required for an IT2 TFS). The principle of justifiable granularity is used to ensure that the formed type-2 information granule exhibits a sound interpretation. Both synthetic fuzzy sets and those constructed by the fuzzy C-means algorithm applied to the publicly available data have been used to demonstrate the usefulness of the proposed approximation methods.