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 ct ranges through the interval (0, 2] and gives the Gaussian bell curve when 0t = 2 and gives the Cauchy bell curve when ~t = 1. The impulsiveness grows as ~ 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. 1. Filtering impulsive noise Impulsive noise is not Gaussian. The bell-curve density o f impulsive noise has thicker tails than a Gaussian bell curve has. The thicker tails give rise to more frequent bursts o f noise. The Cauchy density f ( x ) = 1 / n ( l + x 2) has this property and so do all alpha-stable probability densities [17] for index parameter ~ in 0 < ~t < 2. The Cauchy density is the special case when ct = 1. The thicker polynomial tails give an infinite variance and give infinite higher-order moments. The lower-order fractional moments are still finite. The Gaussian density is the special case o f an alpha-stable density when ct = 2. It is unique in this family because it has exponential tails and has finite variance and higher-order moments. Fig. 1 shows how alpha-stable noise grows more impulsive as the index ~t falls from 2 to 1. The Gaussian bell curve is by far the most common bell curve in science and engineering. It fits some data well and depends on just the first and second moments o f a process. This gives simple closed-form solutions to many problems. Limiting sums o f independent random variables converge to a standard Gaussian variable if the variables have finite variance. The key term e -x2 is in effect invariant under linear transforms * Corresponding author. 0165-0114/96/$09.50 (~) 1996 Elsevier Science B.V. All rights reserved SSD! 0165-0 114(95)001 23 -9 16 H.M. Kim, R KoskolFuzzy Sets and Systems 77 (1996) 15 33