Kalman filtering for triplet Markov chains: applications and extensions

An important problem in signal processing consists in estimating an unobservable process x = {x_n}_nεIN from an observed process y = {y_n}_nεIN. In linear Gaussian hidden Markov chains (LGHMC), the classical recursive solution is given by the Kalman filter. In this paper, we consider linear Gaussian triplet Markov chains (LGTMC) by assuming that the triplet (x, r, y) (in which r = {r_n}_n∈N is some additional process) is Markovian and Gaussian. We first show that this model encompasses and generalizes the classical linear stochastic dynamical models with autoregressive process and/or measurement noise. We next propose (for the regular and for the perfect-measurement cases) restoration Kalman-like algorithms for general LGTMC.