Asymptotically Optimal Nonparametric Signal Interpolation

Abstract

The problem of interpolation (smoothing) of a partially observable Markov random sequence is considered. For the dynamic observation models, an equation in the interpolation posterior probability density is derived. This equation has a certain form of the normalized product of the posterior probability densities in forward and backward times and differs from its counterpart for static observation models [3, 1] in an additional equation. The aim of this paper is to consider the problem of smoothing for the case of unknown distributions of the unobservable component of the random Markov sequence. For the strongly stationary Markov processes with mixing and for the conditional density of observation model belonging to the exponent family success was reached. A resultant method is based on the empirical Bayes approach and the kernel non-parametric estimation [5]. The equation of the nonlinear optimal smoothing estimate is derived in a form independent of the unknown distributions of an unobservable process. Such form of equation allows one to use the non-parametric estimates of some conditional statistics given any set of dependent observations. Modeling was carried out to compare the nonparametric estimates with optimal mean-square smoothing estimates in Kalman scheme