In this thesis, a dynamic spatial model for the spread of Chronic Wasting Disease in Colorado mule deer is derived from a system of differential equations that captures the qualitative spatial and temporal behaviour of the disease. These differential equations are incorporated into an empirical Bayesian hierarchical model through the unusual step of deterministic autoregressive updates. Spatial effects in the model are described directly in the differential equations rather than through the use of correlations in the data. The use of deterministic updates is a simplification that reduces the number of parameters that must be estimated, yet still provides a flexible model that gives reasonable predictions for the disease. The posterior distribution generated by the data model hierarchy possesses characteristics that are atypical for many Markov chain Monte Carlo simulation techniques. To address these difficulties, a new MCMC technique is developed that has qualities similar to recently introduced tempered Langevin type algorithms. The methodology is used to fit the CWD model, and posterior parameter estimates are then used to obtain predictions about Chronic Wasting Disease.

I have developed time-varying coefficient models for recurrent event data to evaluate the temporal profiles for recurrence rate and covariate effects. There are three major parts in this dissertation. The first two parts propose a mixed Poisson process model with gamma frailties for single type recurrent events. The third part proposes a Bayesian joint model based on multivariate log-normal frailties for multi-type recurrent events. In the first part, I propose an approach based on penalized B-splines to obtain smooth estimation for both time-varying coefficients and the log baseline intensity. An EM algorithm is developed for parameter estimation. One issue with this approach is that the estimating procedure is conditional on smoothing parameters, which have to be selected by cross-validation or optimizing certain performance criterion. The procedure can be computationally demanding with a large number of time-varying coefficients. To achieve objective estimation of smoothing parameters, I propose a mixed-model representation approach for penalized splines. Spline coefficients are treated as random effects and smoothing parameters are to be estimated as variance components. An EM algorithm embedded with penalized quasi-likelihood approximation is developed to estimate the model parameters. The third part proposes a Bayesian joint model with time-varying coefficients for multi-type recurrent events. Bayesian penalized splines are used to estimate time-varying coefficients and the log baseline intensity. One challenge in Bayesian penalized splines is that the smoothness of a spline fit is considerably sensitive to the subjective choice of hyperparameters. I establish a procedure to objectively determine the hyperparameters through a robust prior specification. A Markov chain Monte Carlo procedure based on Metropolis-adjusted Langevin algorithms is developed to sample from the high-dimensional distribution of spline coefficients. The procedure includes a joint sampling scheme to achieve better convergence and mixing properties. Simulation studies in the second and third part have confirmed satisfactory model performance in estimating time-varying coefficients under different curvature and event rate conditions. The models in the second and third part were applied to data from a commercial truck driver naturalistic driving study. The application results reveal that drivers with 7-hours-or-less sleep prior to a shift have a significantly higher intensity after 8 hours of on-duty driving and that their intensity remains higher after taking a break. In addition, the results also show drivers' self-selection on sleep time, total driving hours in a shift, and breaks. These applications provide crucial insight into the impact of sleep time on driving performance for commercial truck drivers and highlights the on-road safety implications of insufficient sleep and breaks while driving. This dissertation provides flexible and robust tools to evaluate the temporal profile of intensity for recurrent events. / PHD / The overall objective of this dissertation is to develop models to evaluate the time-varying profiles for event occurrences and the time-varying effects of risk factors upon event occurrences. There are three major parts in this dissertation. The first two parts are designed for single event type. They are based on approaches such that the whole model is conditional on a certain kind of tuning parameter. The value of this tuning parameter has to be pre-specified by users and is influential to the model results. Instead of pre-specifying the value, I develop an approach to achieve an objective estimate for the optimal value of tuning parameter and obtain model results simultaneously. The third part proposes a model for multi-type events. One challenge is that the model results are considerably sensitive to the subjective choice of hyperparameters. I establish a procedure to objectively determine the hyperparameters. Simulation studies have confirmed satisfactory model performance in estimating the temporal profiles for both event occurrences and effects of risk factors. The models were applied to data from a commercial truck driver naturalistic driving study. The results reveal that drivers with 7-hours-or-less sleep prior to a shift have a significantly higher intensity after 8 hours of on-duty driving and that their driving risk remains higher after taking a break. In addition, the results also show drivers’ self-selection on sleep time, total driving hours in a shift, and breaks. These applications provide crucial insight into the impact of sleep time on driving performance for commercial truck drivers and highlights the on-road safety implications of insufficient sleep and breaks while driving. This dissertation provides flexible and robust tools to evaluate the temporal profile of both event occurrences and effects of risk factors.

Cette thèse porte sur le problème de l'échantillonnage en grande dimension et est basée sur l'algorithme de Langevin non ajusté (ULA).Dans une première partie, nous proposons deux extensions d'ULA et fournissons des garanties de convergence précises pour ces algorithmes. ULA n'est pas applicable lorsque la distribution cible est à support compact; grâce à une régularisation de Moreau Yosida, il est néanmoins possible d'échantillonner à partir d'une distribution suffisamment proche de la distribution cible. ULA diverge lorsque les queues de la distribution cible sont trop fines; en renormalisant correctement le gradient, cette difficulté peut être surmontée.Dans une deuxième partie, nous donnons deux applications d'ULA. Nous fournissons un algorithme pour estimer les constantes de normalisation de densités log concaves à partir d'une suite de distributions dont la variance augmente graduellement. En comparant ULA avec la diffusion de Langevin, nous développons une nouvelle méthode de variables de contrôle basée sur la variance asymptotique de la diffusion de Langevin.Dans une troisième partie, nous analysons Stochastic Gradient Langevin Dynamics (SGLD), qui diffère de ULA seulement dans l'estimation stochastique du gradient. Nous montrons que SGLD, appliqué avec des paramètres habituels, peut être très éloigné de la distribution cible. Cependant, avec une technique appropriée de réduction de variance, son coût calcul peut être bien inférieur à celui d'ULA pour une précision similaire. / This thesis focuses on the problem of sampling in high dimension and is based on the unadjusted Langevin algorithm (ULA).In a first part, we suggest two extensions of ULA and provide precise convergence guarantees for these algorithms. ULA is not feasible when the target distribution is compactly supported; thanks to a Moreau Yosida regularization, it is nevertheless possible to sample from a probability distribution close enough to the distribution of interest. ULA diverges when the tails of the target distribution are too thin; by taming appropriately the gradient, this difficulty can be overcome.In a second part, we give two applications of ULA. We provide an algorithm to estimate normalizing constants of log concave densities based on a sequence of distributions with increasing variance. By comparison of ULA with the Langevin diffusion, we develop a new control variates methodology based on the asymptotic variance of the Langevin diffusion.In a third part, we analyze Stochastic Gradient Langevin Dynamics (SGLD), which differs from ULA only in the stochastic estimation of the gradient. We show that SGLD, applied with usual parameters, may be very far from the target distribution. However, with an appropriate variance reduction technique, its computational cost can be much lower than ULA for the same accuracy.