Abstract
This paper considers the problem of interpolation (smoothing) of a partially observable Markov random sequence. For the dynamic observation models, an equation for the interpolation of the posterior probability density is derived. The main goal of this paper is to consider the smoothing problem for the case of unknown distributions of an unobservable component of a random Markov sequence. Successful results were obtained for the strongly stationary Markov processes with mixing and for the conditional density belonging to the exponential family of densities. The resulting method is based on the empirical Bayes approach and kernel nonparametric estimation. The equation for the optimal smoothing estimator is derived in the form independent of unknown distributions of an unobservable process. Such form of the equation allows to use the nonparametric estimates for some conditional functionals in the equation given a set of dependent observations. To compare the nonparametric estimators with optimal mean square smoothing estimators in Kalman scheme, simulation results are given.
Highlights
This paper is devoted to synthesis of interpolation algorithms for an unobservable stationary sequence in conditional Markov scheme
Some efforts are made to construct the sequence of estimators that have the asymptotic behavior close to the behavior of optimal interpolation estimators built under full statistical information on observable and unobservable signal components
The present paper shows that the similar equation is valid for dynamic observation models, in this case the equation is to be supplemented by another recursive equation that reflects the dynamic properties of observations
Summary
This paper is devoted to synthesis of interpolation algorithms for an unobservable stationary sequence in conditional Markov scheme. For the case of full information, the transformation equations for the posterior probability density of unobservable sequences are known. The paper of Khazen (1978) represents the interpolation equation in the form of the normalizing product of filtration posterior probability densities in forward and backward time. This was done only for static observation models. For some conditional probability family of observations it is possible to transform these equations eliminating (explicit) dependence on statistic characteristics unknown a priori. The solution is found on the principles of the empirical Bayes approach and the theory of kernel non-parametric functional estimation (see Vasiliev, Dobrovidov, and Koshkin, 2004)
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