Numéro spécial : données longitudinales quantitatives, événementielles, incomplètement observées
Using PMCMC in EM algorithm for stochastic mixed models: theoretical and practical issues
[Utilisation de PMCMC dans l’algorithme EM pour des modèles mixtes stochastiques : enjeux théoriques et pratiques]
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.

Mots clés : modèles mixtes, equations différentielles stochastiques, algorithme SAEM, filtre particulaire, PMCMC
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Donnet, Sophie; Samson, Adeline. Using PMCMC in EM algorithm for stochastic mixed models: theoretical and practical issues. Journal de la société française de statistique, Tome 155 (2014) no. 1, pp. 49-72. http://www.numdam.org/item/JSFS_2014__155_1_49_0/

[1] Andrieu, C.; Doucet, A.; Holenstein, R. Particle Markov chain Monte Carlo methods, J. R. Statist. Soc. B, Volume 72(3) (2010) no. 72, pp. 1-33

[2] Allassonnière, Stéphanie; Kuhn, Estelle; Trouvé, Alain Construction of Bayesian deformable models via stochastic approximation algorithm: A convergence study., Bernoulli, Volume 641-678 (2010), pp. 605-908 | Zbl 1220.62101

[3] Altman, R. Mixed Hidden Markov Models: An Extension of the Hidden Markov Model to the Longitudinal Data Setting, J. Amer. Statist. Assoc, Volume 102 (407) (2007), pp. 201-210 | Zbl 1284.62803

[4] Andrieu, Christophe; Roberts, Gareth O. The pseudo-marginal approach for efficient Monte Carlo computations, Ann. Statist., Volume 37 (2009) no. 2, pp. 697-725 | Zbl 1185.60083

[5] Benveniste, Albert; Métivier, Michel; Priouret, Pierre Adaptive algorithms and stochastic approximations, Applications of Mathematics (New York), 22, Springer-Verlag, Berlin, 1990, xii+365 pages | Zbl 0752.93073

[6] Chopin, N.; Singh, S. S. On the particle Gibbs sampler, ArXiv e-prints (2013) | arXiv:1304.1887

[7] Donnet, S.; Foulley, J.L.; Samson, A. Bayesian analysis of growth curves using mixed models defined by stochastic differential equations, Biometrics, Volume 66 (2010), pp. 733-741 | Zbl 1203.62187

[8] Davidian, M.; Giltinan, D.M. Nonlinear models to repeated measurement data, Chapman and Hall, 1995

[9] Delattre, M.; Genon-Catalot, V.; Samson, A. Maximum likelihood estimation for stochastic differential equations with random effects., Scand. J. Statistics, Volume 40(2) (2013), pp. 322-343 | Zbl 1328.62148

[10] Delattre, M.; Lavielle, M. Maximum Likelihood Estimation in Discrete Mixed Hidden Markov Models using the SAEM algorithm, Comput. Statist. Data Anal., Volume 56 (2012) no. 6, pp. 2073-2085 | Zbl 1243.62111

[11] Delattre, M.; Lavielle, M. Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models, Stat. Interface (2013) | Zbl 1326.93121

[12] Delyon, B.; Lavielle, M.; Moulines, E. Convergence of a stochastic approximation version of the EM algorithm, Ann. Statist., Volume 27 (1999), pp. 94-128 | Zbl 0932.62094

[13] Dempster, A.P.; Laird, N.M.; Rubin, D.B. Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. B, Volume 39 (1977), pp. 1-38 | Zbl 0364.62022

[14] Donnet, S.; Samson, A. Parametric inference for mixed models defined by stochastic differential equations, ESAIM P&S, Volume 12 (2008), pp. 196-218 | Zbl 1182.62164

[15] Kitawaga, G. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, J. Comp. Graph. Statist., Volume 5(1) (1996) no. 5, pp. 1-25

[16] Kuhn, E.; Lavielle, M. Maximum likelihood estimation in nonlinear mixed effects models, Comput. Statist. Data Anal., Volume 49 (2005), pp. 1020-1038 | Zbl 1429.62279

[17] Klim, S.; Mortensen, S.B.; Kristensen, N.R.; Overgaard, R.V.; Madsen, H. Population stochastic modelling (PSM) - An R package for mixed-effects models based on stochastic differential equations, Comput Meth Prog Bio, Volume 94 (2009), pp. 279-289

[18] Maruotti, A. Mixed Hidden Markov Models for Longitudinal Data: An Overview, International Statistical Review, Volume 79 (3) (2011), pp. 427-454 | Zbl 1238.62094

[19] Overgaard, R.V.; Jonsson, N.; Tornøe, C.W.; Madsen, H. Non-Linear Mixed-Effects Models with Stochastic Differential Equations: Implementation of an Estimation Algorithm., J Pharmacokinet. Pharmacodyn., Volume 32 (2005) no. 1, pp. 85-107

[20] Oksendal, B. Stochastic differential equations: an introduction with applications, Springer-Verlag, Berlin-Heidelberg, 2007

[21] Pinheiro, J.C.; Bates, D.M. Mixed-effect models in S and Splus, Springer-Verlag, 2000 | Zbl 0953.62065

[22] Picchini, U.; Ditlevsen, S.; De Gaetano, A.; Lansky, P. Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural computation, Volume 20(11) (2008), pp. 2696-2714 | Zbl 1169.68580

[23] Picchini, U.; De Gaetano, A.; Ditlevsen, S. Stochastic Differential Mixed-Effects Models, Scand. J. Statist., Volume 37 (2010), pp. 67-90 | Zbl 1224.62041

[24] Pitt, Michael K.; dos Santos Silva, Ralph; Giordani, Paolo; Kohn, Robert On some properties of Markov chain Monte Carlo simulation methods based on the particle filter, J Econometrics, Volume 171 (2012) no. 2, pp. 134 -151 | Zbl 1443.62499

[25] Pitt, M.K.; Shephard, N. Auxiliary variable based particle filters, Sequential Monte Carlo methods in practice (Stat. Eng. Inf. Sci.), Springer, New York, 2001, pp. 273-293 | MR 1847796 | Zbl 1056.93590

[26] Tierney, Luke Markov chains for exploring posterior distributions, Ann. Statist., Volume 22 (1994) no. 4, pp. 1701-1762 | Zbl 0829.62080

[27] Tornøe, C.W.; Overgaard, R.V.; Agersø, H.; Nielsen, H.A.; Madsen, H.; Jonsson, E.N. Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res., Volume 22 (2005) no. 8, p. 1247-58

[28] West, Mike Approximating posterior distributions by mixtures, J. R. Statist. Soc. B, Volume 55 (1993) no. 2, pp. 409-422 | Zbl 0800.62221

[29] Wolfinger, R. Laplace’s approximation for nonlinear mixed models, Biometrika, Volume 80 (1993) no. 4, pp. 791-795 | Zbl 0800.62351