Using PMCMC in EM algorithm for stochastic mixed models: theoretical and practical issues

Donnet, Sophie; Samson, Adeline

Journal de la société française de statistique, Tome 155 (2014) no. 1, pp. 49-72.

Des données longitudinales mesurées chez plusieurs individus d’un processus biologique sont classiquement analysées par modeles mixtes. Récemment, des version stochastiques de ces modeles ont été proposées pour tenir compte de la variabilité dans le temps. Meme si la vraisemblance de ces modeles mixtes stochastiques n’est pas explicite, de nombreuses méthodes d’estimation ont été proposées dans le cas où le processus stochastique est une chaine de Markov discrète à espace d’état fini. Quand le processus stochastique caché est un processus à temps continu sur un espace d’état non fini, il existe peu de références. Nous nous interessons à ce cadre. Nous proposons de combiner un algorithme MCMC particulaire à un algorithme SAEM pour calculer l’estimateur du maximum de vraisemblance du modele. Les propriétés théoriques et numériques de l’algorithme sont discutées. A partir de deux exemples, un processus d’Ornstein-Uhlenbeck et un processus non homogène en temps et à volatilité stochastique, nous illustrons la convergence de l’algorithme.

Biological processes measured repeatedly among a series of individuals are standardly analyzed by mixed models. Recently, stochastic processes have been introduced to model the variability along time for each subject. Although the likelihood of these stochastic mixed models is intractable, various estimation methods have been proposed when the latent stochastic process is a discrete time finite state Markov chain. This is not the case when the hidden stochastic process is a continuous time process with non finite state space. This paper focuses on mixed models defined by parametric Stochastic Differential Equations (SDEs). We propose to use Particle MCMC algorithm for the maximum likelihood estimation of mixed SDE models, by combining it with SAEM algorithm. Theoretical and numerical convergence properties are discussed. Two simulated examples, an Ornstein-Uhlenbeck process and a time-inhomogeneous SDE with stochastic volatility, illustrate this estimator convergence, including the volatility parameter which is known to be hard to estimate.

Keywords: Mixed models, Stochastic Differential Equations, SAEM algorithm, Particle Filter, PMCMC
Mot clés : modèles mixtes, equations différentielles stochastiques, algorithme SAEM, filtre particulaire, PMCMC

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