Parameter estimation of complex mixed models based on meta-model approach

Abstract

Complex biological processes are usually experimented along time among a collection of individuals. Longitudinal data are then available and the statistical challenge is to better understand the underlying biological mechanisms. The standard statistical approach is mixed-effects model, with regression functions that are now highly-developed to describe precisely the biological processes (solutions of multi-dimensional ordinary differential equations or of partial differential equation). When there is no analytical solution, a classical estimation approach relies on the coupling of a stochastic version of the EM algorithm (SAEM) with a MCMC algorithm. This procedure needs many evaluations of the regression function which is clearly prohibitive when a time-consuming solver is used for computing it. In this work a meta-model relying on a Gaussian process emulator is proposed to replace this regression function. The new source of uncertainty due to this approximation can be incorporated in the model which leads to what is called a mixed meta-model. A control on the distance between the maximum likelihood estimates in this mixed meta-model and the maximum likelihood estimates obtained with the exact mixed model is guaranteed. Eventually, numerical simulations are performed to illustrate the efficiency of this approach.

Population parametrization of costly black box models using iterations between SAEM algorithm and kriging

“…We will use a uniform 20 points basis for the kriging before running SAEM. It is also possible to start from optimized sets of points (Dupuy et al 2015;Pronzato and Muller 2012;Jin et al 2005); however, this leads to bad results, in coherence with the theoretical results established in Barbillon et al (2015) for other models.…”

“…Indeed, the kriging approach (where f is thought as the realization of a Gaussian process Sacks et al 1989;Santner et al 2003;Fang et al 2005) is less sensitive to dimension. Interestingly, kriging to build a fixed grid used by SAEM was later studied in Barbillon et al (2015). They proved the convergence of the SAEM algorithm to the maximum likelihood of an approximate non-linear mixed effect model.…”

“…A remark must be given here: comparison of the global computation cost with a fixed grid approach like in Grenier et al (2014), and Barbillon et al (2015), cannot be done directly on n c since a fixed grid is done once for all SAEM runs, whereas a dynamic metamodel is built at each use of an SAEM algorithm. For instance, if a metamodel is used many times on various data sets, it could be better to use a fixed grid approach than a dynamic grid approach.…”

Meta-model based stochastic simulation of fire barrier cold-side temperature

“…Also, the use of meta-models means additional uncertainty, i.e., one may obtain the converged solution using meta-models but not necessarily the same as the one that corresponds to the actual deterministic model. Studies presenting the use of meta-models provide Introduction the information of likelihood fraction by which the meta-model and actual model prediction differ [49,50,51,52,53,54,55,56]. Nevertheless, such information is not sufficient to explain the discrepancy in terms of output distribution essential for accurate probabilistic prediction.…”