Controlling Membership Spread in Type-2 Fuzzy Clustering

Abstract

Fuzzy C-means (FCM) has been a prominent clustering algorithm for a long time. It was extended to a type-2 framework by the Linguistic Fuzzy C-means (LFCM) that operates on vectors of fuzzy numbers utilizing the extension principle, the decomposition theorem, and interval analyses. The purpose of this paper is to study the effects of the iterative type-2 fuzzy clustering algorithm. The LFCM incorporates uncertainty through type-2 fuzzy sets, but it is prone to membership spread, i.e., the uncertainty in a membership function can become too large or broad during the iterative alternating optimization procedure. We devise three dampening approaches to mitigate the problem. Vertical cut dampening, linear dampening, and reflection dampening are defined along with the experiments conducted on a synthetic dataset named the butterfly dataset. We also illustrate the updated memberships (fuzzy numbers) and the resulting cluster prototypes (fuzzy vectors) from visual standpoints. Applying any of these dampening approaches will result in thinner membership functions and helps us control the uncertainty not to grow rapidly, and in fact, aid in convergence.