MAP state sequence estimation for jump Markov linear systems via the expectation-maximization algorithm

In a jump Markov linear system the state matrix, observation matrix and the noise covariance matrices evolve according to the realization of a finite state Markov chain. Given a realization of the observation process, the aim is to estimate the state of the Markov chain assuming known model parameters. In this paper, we present three expectation maximization (EM) algorithms for state estimation to obtain maximum a posteriori state sequence estimates (MAPSE). Our first EM algorithm yields the MAPSE for the entire sequence of the finite state Markov chain. The second EM algorithm yields the MAPSE of the (continuous) state of the jump linear system. Our third EM algorithm computes the joint MAPSE of the finite and continuous states. The three EM algorithms, optimally combine a hidden Markov model estimator and a Kalman smoother in three different ways to compute the desired MAPSEs.