The shape of fuzzy sets in adaptive function approximation

The shape of if-part fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate if-part sets and derive supervised learning laws that tune them. Then we test how well the resulting adaptive fuzzy systems approximate a battery of test functions. No one set shape emerges as the best shape. The sinc function often does well and has a tractable learning law. But its undulating sidelobes may have no linguistic meaning. This suggests that the engineering goal of function-approximation accuracy may sometimes have to outweigh the linguistic or philosophical interpretations of fuzzy sets that have accompanied their use in expert systems. We divide the if-part sets into two large classes. The first class consists of n-dimensional joint sets that factor into n scalar sets as found in almost all published fuzzy systems. These sets ignore the correlations among vector components of input vectors. Fuzzy systems that use factorable if-part sets suffer in general from exponential rule explosion in high dimensions when they blindly approximate functions without knowledge of the functions. The factorable fuzzy sets themselves also suffer from what we call the second curse of dimensionality: The fuzzy sets tend to become binary spikes in high dimension. The second class of if-part sets consists of the more general but less common n-dimensional joint sets that do not factor into n scalar fuzzy sets. We present a method for constructing such unfactorable joint sets from scalar distance measures. Fuzzy systems that use unfactorable if-part sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning laws and inscrutable if-then rules. We prove that some of these unfactorable joint sets still suffer the second curse of dimensionality of spikiness. The search for the best if-part sets in fuzzy function approximation has just begun.