Analyzing the behavior of a CC-integral in a Fuzzy Rule-Based Classification System

Abstract

In a recent paper, it was introduced the concept of Choquet-like Copula-based integral (CC-integral for short). This kind of function extends the standard Choquet integral and generalizes it by copula functions. These functions were applied in the Fuzzy Reasoning Method (FRM) of a Fuzzy Rule-Based Classification System (FRBCS), presenting an example where the CC-integral based on the minimum t-norm had different behaviors according to the values being aggregated. Therefore, the resulting FRM is theoretically more flexible than those associated with classical aggregation functions like the maximum. In this work, we present a methodology to study the flexibility of the aggregation function used in the FRM. Specifically, we conduct an analysis of 3 different methods to aggregate values in the FRM, namely, the CC-integral based on the minimum t-norm, the standard Choquet integral and the maximum (classical FRM of the winning rule — WR). We prove that the CC-integral behaves in different ways according to the values to be aggregated, whereas the Choquet integral offers an averaging behavior and the WR presents an strict behavior, since it considers only the rule having the maximum compatibility with the example.