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 shape emerges as the best. The sine function often does well and has tractable learning, but its undulating side-lobes may have no linguistic meaning. This suggests that function-approximation accuracy may sometimes have to outweigh linguistic or philosophical interpretations. We divide the if-part sets into two large classes. The first consists of n-dimensional joint sets that factor into n scalar sets. These sets ignore the correlations among input vector components. Fuzzy systems suffer in general from exponential rule explosion in high dimensions when they blindly approximate functions. The factorable fuzzy sets themselves also suffer from a curse of dimensionality: they tend to become binary spikes in high dimension. The second class 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 sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning and inscrutable if-then rules. We prove that some of these sets still suffer from spikiness