HMM with emission process resulting from a special combination of independent Markovian emissions

Abstract

Abstract One of the most used variants of hidden Markov models (HMMs) is the standard case where the time is discrete and the state spaces (hidden and observed spaces) are finite. In this framework, we are interested in HMMs whose emission process results from a combination of independent Markov chains. Principally, we assume that the emission process evolves as follows: given a hidden state realization k at time t, an emission is a realization of a Markov chain Y t k {Y_{t}^{k}} at time t, and for two different hidden states k and k ′ {k^{\prime}} , Y t k {Y_{t}^{k}} and Y t k ′ {Y_{t}^{k^{\prime}}} are assumed independent. Given the hidden process, the considered emission process selects its realizations from independent and homogeneous Markov chains evolving simultaneously. In this paper, we propose to study the three known basic problems of such an HMM variant, by deriving corresponding formulas and algorithms. This allows us to enrich the set of application scenarios of HMMs. Numerical examples are presented to show the applicability of our proposed approach by deriving statistical estimations.