Fuzzy prediction and filtering in impulsive noise

Abstract

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 α ranges through the interval (0, 2] and gives the Gaussian bell curve when α = 2 and gives the Cauchy bell curve when α = 1. The impulsiveness grows as α 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.