Fuzzy function approximation with ellipsoidal rules

A fuzzy rule can have the shape of an ellipsoid in the input-output state spare of a system. Then an additive fuzzy system approximates a function by covering its graph with ellipsoidal rule patches. It averages rule patches that overlap. The best fuzzy rules cover the extrema or bumps in the function. Neural or statistical clustering systems can approximate the unknown fuzzy rules from training data. Neural systems can then both tune these rules and add rules to improve the function approximation. We use a hybrid neural system that combines unsupervised and supervised learning to find and tune the rules in the form of ellipsoids. Unsupervised competitive learning finds the first-order and second-order statistics of clusters in the training data. The covariance 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 learning 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 rules when only the centroids of the ellipsoids change. We used numerical techniques to find the optimal rules in the general case.