Fuzzy prediction and filtering in impulsive noise

Additive fuzzy systems can filter impulsive noise from signals. Alpha-stable statistics model the impulsiveness as a parametrized family of probability density functions or unit-area bell curves. The bell-curve parameter alpha ranges through the interval (0,2] and gives the Gaussian bell curve when alpha = 2 and gives the Cauchy bell curve when alpha = 1. The impulsiveness grows as alpha falls and the bell curves have thicker tails. Only the Gaussian statistics have finite variances or finite higher moments. An additive fuzzy system can learn ellipsoidal fuzzy rule patches from a new pseudo-covariation matrix or measure of alpha-stable covariation. Mahalanobis distance gives a joint set function for the learned if-part fuzzy sets of the if-then rules, The joint set function preserves input correlations that factored set functions ignore, Competitive learning tunes the local means and pseudo-covariations of the alpha-stable statistics and thus tunes the fuzzy rules. Then the covariation rules can both predict nonlinear signals in impulsive noise and filter the impulsive noise in time-series data, The fuzzy system filtered such noise better than did a benchmark radial basis neural network.