FUZZY FUNCTION APPROXIMATION WITH ELLIPSOIDAL RULES

A fuzzy rule can have the shape of an ellipsoid in the input-output st ate space of a system, Then an additive fuzzy system approximates a fu nction by covering its graph with ellipsoidal rule patches, It average s rule patches that overlap, The best fuzzy rules cover the extrema or bumps in the function, Neural or statistical clustering systems can a pproximate the unknown fuzzy rules from training data, Neural systems can then both tune these rules and add rules to improve the function a pproximation. We use a hybrid neural system that combines unsupervised and supervised learning to find and tune the rules in the form of ell ipsoids. Unsupervised competitive learning finds the first-order and s econd-order statistics of clusters in the training data. The covarianc e matrix of each cluster gives an ellipsoid centered at the vector or centroid of the data cluster, The supervised neural system learns with gradient descent. It locally minimizes the mean-squared error of the fuzzy function approximation, In the hybrid system unsupervised learni ng initializes the gradient descent. The hybrid system tends to give a more accurate function approximation than does the lone unsupervised or supervised system, We found a closed-form model for the optimal rul es when only the centroids of the ellipsoids change. We used numerical techniques to find the optimal rules in the general case.