Simulation and Parametric Inference of a Mixed Effects Model with Stochastic Differential Equations Using the Fokker-Planck Equation Solution

This chapter is concerned with estimation method for multidimensional and nonlinear dynamical models including stochastic differential equations containing random effects (random parameters). This type of model has proved useful for describing continuous random processes, for distinguishing intra- and interindividual variability as well as for accounting for uncertainty in the dynamic model itself. Pharmacokinetic/pharmacodynamic modeling often involves repeated measurements on a series of experimental units, and random effects are incorporated into the model to simulate the individual behavior in the entire population. Unfortunately, the estimation of this kind of models could involve some difficulties, because in most cases, the transition density of the diffusion process given the random effects is not available. In this work, we focus on the approximation of the transition density of a such process in a closed form in order to obtain parameter estimates in this kind of model, using the Fokker-Planck equation and the Risken approximation. In addition, the chapter discusses a simulation study using Markov Chain Monte Carlo simulation, to provide results of the proposed methodology and to illustrate an application of mixed effects models with SDEs in the epidemiology using the minimal model describing glucose-insulin kinetics

Bayesian Estimation of Multivariate Autoregressive Hidden Markov Model with Application to Breast Cancer Biomarker Modeling

In this work, a first-order autoregressive hidden Markov model (AR(1)HMM) is proposed. It is one of the suitable models to characterize a marker of breast cancer disease progression essentially the progression that follows from a reaction to a treatment or caused by natural developments. The model supposes we have observations that increase or decrease with relation to a hidden phenomenon. We would like to discover if the information about those observations can let us learn about the progression of the phenomenon and permit us to evaluate the transition between its states (supposed discrete here). The hidden states governed by the Markovian process would be the disease stages, and the marker observations would be the depending observations. The parameters of the autoregressive model are selected at the first level according to a Markov process, and at the second level, the next observation is generated from a standard autoregressive model of first order (unlike other models considering the successive observations are independents). A Markov Chain Monte Carlo (MCMC) method is used for the parameter estimation, where we develop the posterior density for each parameter and we use a joint estimation of the hidden states or block update of the states

Non parametric estimation for fractional diffusion processes with random effects

El Omari M, El Maroufy H, Fuchs C. Non parametric estimation for fractional diffusion processes with random effects. STATISTICS. 2019;53(4):753-769.We propose a nonparametric estimation for a class of fractional stochastic differential equations (FSDE) with random effects. We precisely consider general linear fractional stochastic differential equations with drift depending on random effects and non-random diffusion. We build ordinary kernel estimators and histogram estimators and study their risk (), when . Asymptotic results are evaluated as both and N tend to infinity

An MCMC computational approach for a continuous time state-dependent regime switching diffusion process

Hibbah EH, El Maroufy H, Fuchs C, Ziad T. An MCMC computational approach for a continuous time state-dependent regime switching diffusion process. JOURNAL OF APPLIED STATISTICS. 2019;47:1354-1374.State-dependent regime switching diffusion processes or hybrid switching diffusion (HSD) processes are hard to simulate with classical methods which leads us to adopt a Markov chain Monte Carlo (MCMC) Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm

An MCMC computational approach for a continuous time state-dependent regime switching diffusion process

State-dependent regime switching diffusion processes or hybrid switching diffusion (HSD) processes are hard to simulate with classical methods which leads us to adopt a Markov chain Monte Carlo (MCMC) Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm

Inférence bayésienne pour modèle épidémique stochastique SIR non linéaire

Inference for epidemic parameters can be challenging, in part due to data that are intrinsically stochastic and tend to be observed by means of discrete-time sampling, which are limited in their completeness. The problem is particularly acute when the likelihood of the data is computationally intractable. Consequently standard statistical techniques can become too complicated to implement effectively. In this work, we develop a powerful method for bayesian paradigm for SIR stochastic epidemic models via data-augmented Markov Chain Monte Carlo. The latter samples missing values as well as the model parameters, where the missing values and parameters are treated as random variables. These routines are based on the approximation of the discrete-time epidemic by diffusion process. We illustrate our techniques using simulated epidemics.L'inférence pour les paramètres d'un modèle épidémique peut être difficile, en partie en raison de données qui sont intrinsèquement stochastiques et ont tendance à être observés par échantillonnage à temps discret, qui sont limités dans leur intégralité. Le problème est particulièrement aigu lorsque la vraisemblance des données est intraitable. Par conséquent les techniques statistiques standards peuvent devenir trop compliquées à mettre en œuvre efficacement. Dans ce travail, nous développons une méthode puissante pour le paradigme bayésien pour les modèles stochastiques épidémiques de type SIR par données augmentées à partir d'un algorithme chaînes de Markov Monte Carlo (MCMC). Ce dernier échantillonne les données manquantes ainsi que les paramètres du modèle, où les données manquantes et les paramètres sont traités comme des variables aléatoires. Ces routines sont basées sur l'approximation du modèle d'épidémie en temps discret par un processus de diffusion. Nous illustrons nos techniques par des épidémies simulées