Markov chain Markov field dynamics: models and statistics

Xavier Guyon (1, 2) , Cécile Hardouin (1, 2)

1 SAMOS - Statistique Appliquée et MOdélisation Stochastique
2 MATISSE - UMR 8595 - Modélisation Appliquée, Trajectoires Institutionnelles et Stratégies Socio-Économiques

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 $(E=\Bbb R^+)$, auto-normal dynamics $(E=\Bbb R)$, auto-Poissonian dynamics $(E=\Bbb N)$ and autologistic 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.

Mots clés: Markov Field; Markov Chain Dynamics; Auto-model; Lyapunov Stability Criterion; Martingale Clt Theorem; Model Diagnostic

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Xavier Guyon, Cécile Hardouin. Markov chain Markov field dynamics: models and statistics. Statistics A Journal of Theoretical and Applied Statistics, 2002, 36 (4), pp.339 - 363. ⟨10.1080/02331880213192⟩. ⟨hal-00110370⟩