Markov Chain Markov Field dynamics: Models and statistics

Abstract

This study deals with time dynamics of Markov fields defined on a finite set of sites with state space <$>E<$>, focussing on Markov Chain Markov Field (MCMF) evolution. Such a model is characterized by two families of potentials: the instantaneous interaction potentials, and the time delay potentials. Four models are specified: auto-exponential dynamics (<$>E = {\\of R}^+<$>), auto-normal dynamics (<$>E = {\\of R}<$>), auto-Poissonian dynamics (<$>E = {\\of N}<$>) and auto-logistic dynamics ( E qualitative and finite). Sufficient conditions ensuring ergodicity and strong law of large numbers are given by using a Lyapunov criterion of stability, and the conditional pseudo-likelihood statistics are summarized. We discuss the identification procedure of the two Markovian graphs and look for validation tests using martingale central limit theorems. An application to meteorological data illustrates such a modelling.