Nonparametric PU Learning of State Estimation in Markov Switching Model

In this contribution, we develop methods of nonlinear filtering and prediction of an unobservable Markov chain which controls the states of observable stochastic process. This process is a mixture of two subsidiary stochastic processes, the switching of which is controlled by the Markov chain. Each of this subsidiary processes is described by conditional distribution density (cdd). The feature of the problem is that cdd’s and transition probability matrix of the Markov chain are unknown, but a training sample (positive labeled) from one of the two subsidiary processes and training sample (unlabeled) from the mixture process are available. Construction of process binary classifier using positive and unlabeled samples in machine learning is called PU learning. To solve this problem for stochastic processes, nonparametric kernel estimators based on weakly dependent observations are applied. We examine the novel method performance on simulated data and compare it with the same performance of the optimal Bayesian solution with known cdd’s and the transition matrix of the Markov chain. The modeling shows close results for the optimal task and the PU learning problem even in the case of a strong overlapping of the conditional densities of subsidiary processes.