Fuzzy function learning with covariance ellipsoids

It is shown how first- and second-order statistics can estimate fuzzy rules and sets from input-output data. The fuzzy system approximates the function by covering its graph with fuzzy patches in the input-output state space. The neural quantizer system uses unsupervised competitive learning to estimate the local centroids and covariances of pattern classes. The covariance matrix of each random quantization vector defines an ellipsoid around the centroid of the pattern class. The ellipsoids define fuzzy patches or rules that cover the graph of the function. Regions of sparse data give rise to large ellipsoids or less certain rules. The approximation error falls as the number of patches grows. Ellipsoidal covariance learning estimates the control surface for a car velocity controller