A Novel Generic Hebbian Ordering-Based Fuzzy Rule Base Reduction Approach to Mamdani Neuro-Fuzzy System

There are two important issues in neuro-fuzzy modeling: (1) interpretability--the ability to describe the behavior of the system in an interpretable way--and (2) accuracy--the ability to approximate the outcome of the system accurately. As these two objectives usually exert contradictory requirements on the neuro-fuzzy model, certain compromise has to be undertaken. This letter proposes a novel rule reduction algorithm, namely, Hebb rule reduction, and an iterative tuning process to balance interpretability and accuracy. The Hebb rule reduction algorithm uses Hebbian ordering, which represents the degree of coverage of the samples by the rule, as an importance measure of each rule to merge the membership functions and hence reduces the number of the rules. Similar membership functions (MFs) are merged by a specified similarity measure in an order of Hebbian importance, and the resultant equivalent rules are deleted from the rule base. The rule with a higher Hebbian importance will be retained among a set of rules. The MFs are tuned through the least mean square (LMS) algorithm to reduce the modeling error. The tuning of the MFs and the reduction of the rules proceed iteratively to achieve a balance between interpretability and accuracy. Three published data sets by Nakanishi (Nakanishi, Turksen, & Sugeno, 1993), the Pat synthetic data set (Pal, Mitra, & Mitra, 2003), and the traffic flow density prediction data set are used as benchmarks to demonstrate the effectiveness of the proposed method. Good interpretability, as well as high modeling accuracy, are derivable simultaneously and are suitably benchmarked against other well-established neuro-fuzzy models.