Monotone Data Samples Do Not Always Generate Monotone Fuzzy If-Then Rules

Abstract

The Wang–Mendel (WM) method is one of the earliest methods to learn fuzzy If-Then rules from data. In this article, the WM method is used to generate fuzzy If-Then rules for a zero-order Takagi–Sugeno–Kang (TSK) fuzzy inference system (FIS) from a set of multi-attribute monotone data. Convex and normal trapezoid fuzzy sets are used as fuzzy membership functions. Besides that, a strong fuzzy partition strategy is used. Our empirical analysis shows that a set of multi-attribute monotone data may lead to non-monotone fuzzy If-Then rules. The same observation can be made, empirically, using adaptive neuro-fuzzy inference system (ANFIS), a well-known and popular FIS model with neural learning capability. This finding is important for the modeling of a monotone FIS model, because it shows that even with a “clean” data set pertaining to a monotone system, the generated fuzzy If-Then rules may need to be preprocessed, before being used for FIS modeling. In short, it is imperative to develop methods for preprocessing non-monotone fuzzy rules from data, e.g., monotone fuzzy rules relabeling, or removing non-monotone fuzzy rules, is important (and is potentially necessary) during the course of developing data-driven FIS models.