Stable Nonparametric Signal Filtration in Nonlinear Models

Abstract

A stationary two-component hidden Markov process (X _n , S _n ) _{n ⩾1} is considered where the first component is observable and the second one is non-observable. The problem of filtering a random signal (S _n ) _{n ⩾1} from the mixture with a noise by observations X ₁ ⁿ = X ₁, ⋯ , X _n is solved under a nonparametric uncertainty regarding the distribution of the desired signal. This means that a probabilistic parametric model of the useful signal (S _n ) is assumed to be completely unknown. In these assumptions, it is impossible generally to build an optimal Bayesian estimator of S _n explicitly. However, for a more restricted class of static observation models, in which the conditional density f(x _n  | s _n ) belongs to the exponential family, the Bayesian estimator satisfies the optimal filtering equation which depends on probabilistic characteristics of the observable process (X _n ) only. These unknown characteristics can be restored from the observations X ₁ ⁿ by using stable nonparametric estimation procedures adapted to dependent data. Up to this work, the author investigated the case where the domain of the desirable signal S _n is the whole real axis. To solve the problem in this case the approach of nonparametric kernel estimation with symmetric kernel functions has been used. In this paper we consider the case where S _n  ∈ 0, ∞). This assumption leads to nonlinear models of observation and non-Gaussian noise which in turn requires more complex mathematical constructions including non-symmetric kernel functions. The nonlinear multiplicative observation model with non-Gaussian noise is considered in detail, and the nonparametric estimator of an unknown gain coefficient is constructed. The choice of smoothing and regularization parameters plays the crucial role to build stable nonparametric procedures. The optimal choice of these parameters leading to an automatic algorithm of nonparametric filtering is proposed.