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Approximation accuracy of some neuro-fuzzy approaches

Many methods have been proposed in the literature for designing fuzzy systems from input-output data (the so-called neuro-fuzzy methods), but very little was done to analyze the performance of the methods from a rigorous mathematical point of view. In this paper, we establish approximation bounds for two of these methods - the table lookup scheme proposed by Wang et al. (1992) and the clustering method studied by Wang (1993, 1997). We derive detailed formulas of the error bounds between the nonlinear function to be approximated and the fuzzy systems designed using the methods based on input-output data. These error bounds show explicitly how the parameters in the two methods influence their approximation capability. We also propose modified versions for the two methods such that the designed fuzzy systems are well-defined over the whole input domain.