Separable Approximation Property of Hierarchical Fuzzy Systems

This paper discusses the capabilities of standard hierarchical fuzzy systems to approximate continuous functions with natural hierarchical structure. The separable approximation property of hierarchical fuzzy systems is proved, that is, the construction of a hierarchical fuzzy system with required approximation accuracy can be achieved by the separate construction of each sub-system with required approximation accuracy. This property provides a simple method to construct hierarchical fuzzy systems for function approximation. Based on the separable approximation property, it is further proved the structure approximation property of hierarchical fuzzy systems

I. Introduction

Since the pioneering work of Backley [1], Kosko [11], and Wang [18], the universal approximation property of fuzzy systems has attracted much attention [2], [5], [13], [20]–[23]. However, almost all these fuzzy approximation schemes, based on standard fuzzy systems, suffer the curse of dimensionality, which can be viewed from the following three perspectives: 1)

Rule dimensionality: The total number of rules in the fuzzy rule base increases exponentially with the number of input variables;

2)

Parameter dimensionality: The total number of parameters in the mathematical models of fuzzy systems increases exponentially with the number of input variables.

3)

Data or information dimensionality: The number of data or knowledge sets required to identify fuzzy systems increases exponentially with the number of input variables.