A Fully Interpretable First-Order TSK Fuzzy System and Its Training With Negative Entropic and Rule-Stability-Based Regularization

Abstract

While interpretable antecedent parts of first-order Takagi-Sugeno-Kang (TSK) fuzzy rules can be properly acquired by adopting some clustering methods, this study aims at avoiding the commonly-used yet fully incomprehensive consequent parts and their intractable training, and simultaneously seeking for enhanced generalization performance by determining the weight of each rule. The central idea is to build a mathematically equivalent bridge between a Gaussian mixture model (GMM) and a fully interpretable first-order TSK fuzzy system called FIMG-TSK, with the help of Gaussian-mixture's mean. The resultant FIMG-TSK has a simple expected output expression without summation-to-one defuzzification, which will be helpful in inducing both smaller output variance and a negative entropic and rule-stability-based regularizer for enhancing the generalization performance. After revealing three factors affecting the output stability of FIMG-TSK, the negative entropic and rule-stability-based regularizer is designed through both these factors and the squared entropy to make the output variance of FIMG-TSK as small as possible. Accordingly, a novel training method, whose objective function takes the proposed regularizer as an additional term and hence compromises both accuracy and output stability of FIMG-TSK, is developed to quickly provide an analytical solution to the weight of each rule. The effectiveness of the proposed training method is manifested by the experimental results on ten regression datasets.