Learning for Hierarchical Fuzzy Systems Based on the Gradient-Descent Method

Standard fuzzy systems suffer the "curse of dimensionality" which has become the bottleneck when applying fuzzy systems to solve complex and high dimensional application problems. This curse of dimensionality results in a larger number of fuzzy rules which reduces the transparency of fuzzy systems. Furthermore too many rules also reduce the generalization capability of fuzzy systems. Hierarchical fuzzy systems have emerged as an effective alternative to overcome this curse of dimensionality and have attracted much attention. However, research on learning methods for hierarchical fuzzy systems and applications is rare. In this paper, we propose a scheme to construct general hierarchical fuzzy systems based on the gradient-descent method. To show the advantages of the proposed method (in terms of accuracy, transparency, generalization capability and fewer rules), this method is applied to a function approximation problem and the result is compared with those obtained by standard (flat) fuzzy systems.

I. Introduction

Standard fuzzy systems have been applied in many fields, such as approximation [1], [2], [3], control [4], [5], classification [6] and clustering [7]. Indeed, fuzzy systems are universal approximators which can approximate arbitrary continuous functions to any accuracy [1], [2], [3], [4], [8]. The success of these applications is due to the flexibility and expressive ease of fuzzy systems, and the related theoretical results [1], [2], [4], [8] for fuzzy logic obtained during the last four decades.