Section: New Results

Inference in mixed-effects diffusion models

Participants : Maud Delattre, Marc Lavielle.

The structure of mixed effects models allows a suitable consideration of the whole variability characterizing such data, which is usually split into some intra-individual variability - i.e., the variability occurring within the dynamics of each individual - and some between-subjects variability. In a mixed-effects model, the same structural model is used for describing each individual sequence of observations, but the parameters of this model vary randomly among the individuals, which allows a correct account of the differences between subjects. In a mixed-effects diffusion model, the description of each individual series of observations is based on stochastic differential equations (SDEs). Diffusion is known to be a relevant tool for describing random variability in dynamical systems, and is widely used in applications in many domains.

Although many methods are available for the inference in classical fixed-effects diffusion models, there is still a need for a general, fast and easy to implement method for the inference in mixed-effects diffusion models. Indeed, except in very specific classes of mixed-effects diffusion models, the likelihood of the observations does not have any closed-form expression, making maximum likelihood estimation of the model parameters an intricate issue. The difficulty is twofold for computing the observed likelihood since it involves the transition densities of the underlying individual diffusion processes and integrals over the unobserved individual parameters that can rarely be computed in a closed form. Specific versions of the SAEM algorithm have already been proposed for estimating the population parameters in mixed-effects diffusion models (using for instance an Euler-Maruyama approximation of the individual processes or some particle Markov Chain Monte-Carlo methods). In these two versions of SAEM however, simulation of both the random individual parameters and the individual latent processes is required at simulation step, which is computationally cumbersome.

We have proposed a new inference methodology for mixed-effects diffusion models which consists in coupling the SAEM algorithm with the extended Kalman filter for estimating the population parameters. The relevant article has been submitted in 2012. In this new version of the SAEM algorithm, we only need to simulate the individual parameters at each iteration. We also provide tools for estimating the individual parameters and the individual diffusion trajectories.