Complexity reduction in fuzzy inference systems

Abstract

Despite ever-growing processor speed, application of fuzzy technology is still hindered by rule explosion, the phenomenon in which an increase in the number of antecedents results in the exponential growth of the number of fuzzy if-then rules. As a result most contemporary fuzzy inference systems are simply not scalable. That is, one cannot easily extend existing inference systems (incrementally or otherwise) to a large functional multi-input, multi-output (MIMO) system, since the addition of new antecedents quickly renders the system intractable. While much work has addressed rule explosion, most complexity reduction techniques entail post-design offline processing. These techniques restrict fuzzy systems to static processes, as rules cannot be adapted in real time in the reduced complexity format. Clearly, for fuzzy systems to be truly scalable while retaining the ability to function dynamically, complexity reduction cannot take the form of a post-design fix. Instead, the underlying structure of contemporary fuzzy systems must evolve.In this work, we explore an alternative architecture for fuzzy inferencing that often reduces exponential rule explosion to a more manageable linear increase in the number of fuzzy if-then rules. Two types of problems are discussed in due course. First, we examine those problems that satisfy a certain constraint---additive seperability. These problems incur a minimum amount of complexity under the novel architecture. Secondly, we discuss those problems that do not satisfy this constraint and how one may obtain a maximum amount of complexity reduction for this class of tougher problems. In order to analyze desired input/output manifolds we develop a novel transform that assists in the design of minimum complexity systems. The transform is derived from the multidimensional Fourier transform and its relationship with the Radon transform is also discussed. In this work, a chapter is dedicated strictly to the properties and computation of this transform. We prove how one may utilize this transform to derive an approximation to a desired input/output manifold that is both implementable in the novel complexity saving architecture and has a minimum mean-square error.