Automatic nonparametric signal filtration

Abstract

Recent results in nonparametric bandwidth selection allow us to create data- based algorithms of automatic nonparametric signal Їltration. Such algorithms are based on the optimal Їltering equation and its nonparametric counterpart from the theory of nonparametric signal processing [1, 2]. This approach was developed for the case when state equation and probability distribution of unob- servable useful signal are unknown, but the observation equation and perturba- tion distribution are known completely. Term "automatic Їltration" means that the output data of the observation equation is only used to derive a nonparamet- ric signal Їltration equation. The estimation equation contains a term that is a non-parametric estimator of logarithmic derivative of density, which depends on bandwidths for probability and its derivative estimates. Using the results of [3, 4] for bandwidth selection by Smoothed Cross-Validation method, we give an automatic Їltration method. To obtain a stable non-parametric estimator of log- arithmic density derivative some regularization procedure is used that is named piecewise smooth approximation [5]. Modeling was carried out to compare the behavior of nonparametric estimates with the optimal Kalman ones.