A Self-Evolving Fuzzy System Which Learns Dynamic Threshold Parameter by Itself

Abstract

This paper proposes an online learning algorithm for data streams, namely self-evolving fuzzy system (SEFS). Unlike the fixed control parameters commonly used in evolving fuzzy systems, the SEFS uses online training errors, which measure the quality of an identified model in presenting the dynamics of the data stream, to set a dynamic threshold automatically for rule generation. This self-tuning parameter, which controls the speed and coverage for fuzzy rule generation, helps the SEFS properly deal with the underfitting/overfitting problems relying on two facts: 1) Large training errors present an underfitted model, which is too coarse to represent the highly complicated and rapidly dynamic (e.g., highly nonlinear, nonstationary) behavior of the data segment. Then, finer rules need to be added; and 2) tiny training errors reflect an overfitted model, which can ideally represent any slight dynamic behavior of the data stream. In this case, coarse rule base should be used. Besides, an $L^{2}$ -distance-based geometric similarity measure is proposed in the rule merging phase. With this similarity measure, the SEFS computes the similarity between Gaussian membership functions accurately without making an approximation of the Gaussian membership function beforehand. In addition, a weighted recursive least-squares algorithm with a variable forgetting factor, which minimizes the mean square of the noise-free posterior error signal, is applied to learn the consequent parameters. Several benchmark examples across both artificial and real-life datasets verify that the SEFS has the ability to give better performance compared with many state-of-the-art algorithms.