A Novel Rule-Centered Fuzzy Model Induced by Epanechnikov Mixtures

This work explores how mixture model based on Epanechnikov kernel functions, termed here as Epanechnikov mixture model (EMM), can be translated into a rule-centered generalized fuzzy model (RCGFM) with multidimensional Epanechnikov membership functions that take into consideration of the correlation among components of sample data and therefore result in a more effective partition of the input space. Thus, the new fuzzy model can be interpreted probabilistically, i.e., the conditional means of an EMM corresponds to the defuzzified output of a RCGFM and the coefficients represented by the means and/or variances in probability theory in the consequent polynomials of fuzzy rules can be exactly interpreted as Taylor series coefficients. The differential theory is combined with the fuzzy theory. Furthermore, the proposed model yields a smaller number of fuzzy rules and inputs, because the compactly supported polynomial Epanechnikov membership functions minimize the asymptotic mean integrated squared error with the optimal kernel efficiency. As demonstrated by the experimental results in several benchmarking datasets, the EMM linked RCGFM as the fuzzy model EMM-RCGFM is superior to other fuzzy models in modeling accuracy and robustness, besides generating a small set of probabilistically interpretable fuzzy rules. This work represents a very first attempt to exhibit the applicability of Epanechnikov kernel functions as fuzzy membership functions.