Enhanced Interval Approach for Encoding Words Into Interval Type-2 Fuzzy Sets and Its Convergence Analysis

Construction of interval type-2 fuzzy set models is the first step in the perceptual computer, which is an implementation of computing with words. The interval approach (IA) has, so far, been the only systematic method to construct such models from data intervals that are collected from a survey. However, as pointed out in this paper, it has some limitations, and its performance can be further improved. This paper proposes an enhanced interval approach (EIA) and demonstrates its performance on data that are collected from a web survey. The data part of the EIA has more strict and reasonable tests than the IA, and the fuzzy set part of the EIA has an improved procedure to compute the lower membership function. We also perform a convergence analysis to answer two important questions: 1) Does the output interval type-2 fuzzy set from the EIA converge to a stable model as increasingly more data intervals are collected, and 2) if it converges, then how many data intervals are needed before the resulting interval type-2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals? We show that the EIA converges in a mean-square sense, and generally, 30 data intervals seem to be a good compromise between cost and accuracy.