Continuous time hidden Markov Model with bidimensional observed process

Abstract

A hidden Markov process (HMP) is a bivariate process (state process and observed process) de¯ned such that both of the joint process and the marginal state process are markovian. It's characterized by a special structure of its transition kernel which allows to deduce its statistical properties from the similar properties of the underlying state process. The use of HMP as a model, commonly referred to as hidden Markov model (HMM), is frequently restricted to the study of one or more unobserved cate- gorical variables for which only indirected measurements from a unidimensional space are available, but here we allow relaxation of this restriction. We will present a class of HMM that consists of a background process in continu- ous time with a bidimensional observed process since the indirected measurements can be related to more than one variable. The analysis will be illustrated by an example of hidden Markov models with binormal observed process. A likelihood ratio test is taken to compare the continuous time hidden HMM with bidimensional observed pro- cess versus the continuous time HMM with unidimensional observed process. The test provides the usefulness of the ¯rst model instead of the second one. Estimation of quantities of interest is performed using the Gibbs sampler algorithm within Metropolis accept-reject step, which is a stochastic algorithm belonging to the families of Monte Carlo Markov Chain methods.