The autoregressive hidden Markov chain (AR-HMC) is widely used for various problems, including signal and image processing, econometrics, recognition, health sciences and etc. The AR-HMC enhances the HMC architecture by introducing a direct stochastic dependence between observations. In this paper, we develop a method of nonlinear filtering of a hidden (unobservable) Markov chain with two states (for simplicity) in AR-HMC model. This Markov chain controls coefficients of AR(p) model. The problem is to estimate the states of Markov chain by statistically dependent observations drawn from AR(p) model in the case of an unknown probability transition matrix and prior probabilities of the Markov chain, and completely unknown conditional distribution describing one of the two AR models. Another conditional distribution is proposed to be known. In this statement, the problem cannot be parameterized and consequently solved by the well-known EM-method [1]. Therefore, the proposed solution is based on the kernel nonparametric approach adapted to dependent observations [2]. We show, via simulations, that with an increase in sample size the performance of the proposed algorithm approaches the performance of the optimal (Bayes) nonlinear filtering developed for the first time by [3]. This method can be classified as an unsupervised estimation technique but, in contrast to EM-method, under nonparametric uncertainty.
[1] Baum L.E., Petrie T., Soules G. and Weiss N. (1970). A maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains. The Annals of Mathematical Statistics, 41, No. 1, 164–171.
[2] Alexander Dobrovidov, Gennady Koshkin and Vyacheslav Vasiliev (2012). Non-parametric state space models. Kendrick Press. USA.
[3] Stratonovich R.L. (1960). Conditional Markov Processes. Theory of Probability and its Applications, 5, No. 2, 156–178.