Doctor of Philosophy / Department of Statistics / Juan Du / It is common to assume the spatial or spatio-temporal data are realizations of underlying random elds or stochastic processes. E ective approaches to modelling of the underlying autocorrelation structure of the same random eld and the association among multiple processes are of great demand in many areas including atmospheric sciences, meteorology and agriculture. To this end, this dissertation studies methods and application of the spatial modeling of large-scale dependence structure and spatio-temporal regression modelling. First, variogram and variogram matrix functions play important roles in modeling dependence structure among processes at di erent locations in spatial statistics. With more and more data collected on a global scale in environmental science, geophysics, and related elds, we focus on the characterizations of the variogram models on spheres of all dimensions for both stationary and intrinsic stationary, univariate and multivariate random elds. Some e cient approaches are proposed to construct a variety of variograms including simple polynomial structures. In particular, the series representation and spherical behavior of intrinsic stationary random elds are explored in both theoretical and simulation study. The applications of the proposed model and related theoretical results are demonstrated using simulation and real data analysis. Second, knowledge of the influential factors on the number of days suitable for fieldwork (DSFW) has important implications on timing of agricultural eld operations, machinery decision, and risk management. To assess how some global climate phenomena such as El Nino Southern Oscillation (ENSO) a ects DSFW and capture their complex associations in space and time, we propose various spatio-temporal dynamic models under hierarchical Bayesian framework. The Integrated Nested Laplace Approximation (INLA) is used and adapted to reduce the computational burden experienced when a large number of geo-locations and time points is considered in the data set. A comparison study between dynamics models with INLA viewing spatial domain as discrete and continuous is conducted and their pros and cons are evaluated based on multiple criteria. Finally a model with time- varying coefficients is shown to reflect the dynamic nature of the impact and lagged effect of ENSO on DSFW in US with spatio-temporal correlations accounted.

Traumatic injuries lead to articular cartilage lesion formation and result in the development of osteoarthritis. Recent research suggests that the early stage of mechanical injuries involve cell death (apoptosis and necrosis) and inflammation. In this thesis, we focus on building mathematical models to investigate the biological mechanism involving chondrocyte death and inflammatory responses in the process of cartilage degeneration. Chapter 1 describes the structure of articular cartilage, the process of carti- lage degeneration, and reviews of existing mathematical models. Chapter 2 presents a delay-diffusion-reaction model of cartilage lesion formation under cyclic loading. Computational methods were used to simulate the impact of varying loading stresses and erythropoietin levels. The model is parameterized with experimental results, and is therefore clinically relevant. Due to numerical limitations using delay differential equations, a new model is presented using tools for population dynamics. Chapter 3 presents an age and space-structured model of articular cartilage lesion formation un- der a single blunt impact. Age structure is introduced to represent the time delay in cytokine synthesis and cell transition. Numerical simulations produce similar tempo- ral and spatial patterns to our experimental data. In chapter 4, we extend our model under the cyclic loading setting. Chapter 5 builds a spatio-temporal model adapted from the former models, and investigates the distribution of model parameters using experimental data and statistical methods. Chapter 6 concludes.

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L’objectif général de cette thèse est de caractériser la dynamique des transferts des bactéries fécales à l’aide d’une modélisation spatio-temporelle, à l’échelle du bassin versant (BV) dans une région agricole et à l’échelle événementielle. Ce projet vise à mieux comprendre l'influence des processus hydrologiques, les facteurs environnementaux et temporels impliqués dans l’explication des épisodes de contamination microbienne des eaux récréatives. Premièrement, un modèle bayésien hiérarchique a été développé pour quantifier et cartographier les niveaux de probabilité des eaux à être contaminées par des effluents agricoles, sur la base des données spectrales et des variables géomorphologiques. Par cette méthode, nous avons pu calculer les relations pondérées entre les concentrations d’Escherichia coli et la distribution de l’ensemble des paramètres agro-pédo-climatiques qui régissent sa propagation. Les résultats ont montré que le modèle bayésien développé peut être utilisé en mode prédictif de la contamination microbienne des eaux récréatives. Ce modèle avec un taux de succès de 71 % a mis en évidence le rôle significatif joué par la pluie qui est la cause principale du transport des polluants. Deuxièmement, le modèle bayésien a fait l’objet d'une analyse de sensibilité liée aux paramètres spatiaux, en utilisant les indices de Sobol. Cette démarche a permis (i) la quantification des incertitudes sur les variables pédologiques, d’occupation du sol et de la distance et (2) la propagation de ces incertitudes dans le modèle probabiliste c'est-à-dire le calcul de l’erreur induite dans la sortie par les incertitudes des entrées spatiales. Enfin, une analyse de sensibilité des simulations aux différentes sources d’incertitude a été effectuée pour évaluer la contribution de chaque facteur sur l’incertitude globale en prenant en compte leurs interactions. Il apparaît que sur l’ensemble des scénarios, l’incertitude de la contamination microbienne dépend directement de la variabilité des sols argileux. Les indices de premier ordre de l’analyse de Sobol ont montré que parmi les facteurs les plus susceptibles d’influer la contamination microbienne, la superficie des zones agricoles est le premier facteur important dans l'évaluation du taux de coliformes. C’est donc sur ce paramètre que l’attention devra se porter dans le contexte de prévision d'une contamination microbienne. Ensuite, la deuxième variable la plus importante est la zone urbaine avec des parts de sensibilité d’environ 30 %. Par ailleurs, les estimations des indices totaux sont meilleures que celles des indices de premier ordre, ce qui signifie que l’impact des interactions paramétriques est nettement significatif pour la modélisation de la contamination microbienne Enfin, troisièmement, nous proposons de mettre en œuvre une modélisation de la variabilité temporelle de la contamination microbiologique du bassin versant du lac Massawippi, à partir du modèle AVSWAT. Il s'agit d'une modélisation couplant les composantes temporelles et spatiales qui caractérisent la dynamique des coliformes. La synthèse des principaux résultats démontrent que les concentrations de coliformes dans différents sous-bassins versants se révèlent influencées par l’intensité de pluie. La recherche a également permis de conclure que les meilleures performances en calage sont obtenues au niveau de l'optimisation multi-objective. Les résultats de ces travaux ouvrent des perspectives encourageantes sur le plan opérationnel en fournissant une compréhension globale de la dynamique de la contamination microbienne des eaux de surface. / Abstract : The aim of this study was to predict water faecal contamination from a bayesian probabilistic model, on a watershed scale in a farming area and on a factual scale. This project aims to better understand the influence of hydrological, environmental and temporal factors involved in the explanation of microbial contamination episodes of recreational waters. First, a bayesian probabilistic model: Weight of Evidence was developed to identify and map the probability of water levels to be contaminated by agricultural effluents, on the basis of spectrals data and geomorphologic variables. By this method, we were able to calculate weighted relationships between concentrations of Escherichia coli and distribution of key agronomic, pedologic and climatic parameters that influence the spread of these microorganisms. The results showed that the Bayesian model that was developed can be used as a prediction of microbial contamination of recreational waters. This model, with a success rate of 71%, highlighted the significant role played by the rain, which is the main cause of pollution transport. Secondly, the Bayesian probabilistic model has been the subject of a sensitivity analysis related to spatial parameters, using Sobol indications. This allowed (1) quantification of uncertainties on soil variables, land use and distance and (2) the spread of these uncertainties in the probabilistic model that is to say, the calculation of induced error in the output by the uncertainties of spatial inputs. Lastly, simulation sensitivity analysis to the various sources of uncertainty was performed to assess the contribution of each factor on the overall uncertainty taking into account their interactions. It appears that of all the scenarios, the uncertainty of the microbial contamination is directly dependent on the variability of clay soils. Sobol prime indications analysis showed that among the most likely to influence the microbial factors, the area of farmland is the first important factor in assessing the coliforms. Importance must be given on this parameter in the context of preparation for microbial contamination. Then, the second most important variable is the urban area with sensitivity shares of approximately 30%. Furthermore, estimates of the total indications are better than those of the first order, which means that the impact of parametric interaction is clearly significant for the modeling of microbial contamination. Thirdly, we propose to implement a temporal variability model of microbiological contamination on the watershed of Lake Massawippi, based on the AVSWAT model. This is a model that couples the temporal and spatial components that characterize the dynamics of coliforms. The synthesis of the main results shows that concentrations of Escherichia coli in different sub-watersheds are influenced by rain intensity. Research also concluded that best performance is obtained by multi-objective optimization. The results of these studies show the prospective of operationally providing a comprehensive understanding of the dynamics of microbial contamination of surface water.

Neste trabalho utilizamos um modelo espaço-temporal proposto em Rojas (2004) para medir a interação social da criminalidade na região metropolitana de São Paulo. Realizamos simulações de Monte Carlo para testar a capacidade de estimação do modelo em diferentes cenários. Observamos que a estimação melhora com o aumento de observações ao longo do tempo. Já os resultados empíricos indicam que a região metropolitana de São Paulo é um hot spot no estado, pois é encontrado um maior grau de interação social no índice de homicídio em relação aos índices de roubo e furto. / In this paper we employ a spatio-temporal model proposed in Rojas (2004) to evaluate the social interaction in crime in São Paulo metropolitan area. We carry out Monte Carlo simulations to test the model estimation capability in different scenarios. We notice that the estimation gets better as the number of observations in time raises. The results point out that São Paulo metropolitan area is a hot spot in the state since we found out a greater social interaction for the homicide index, compared to robbery and thievery.

Neste trabalho utilizamos um modelo espaço-temporal proposto em Rojas (2004) para medir a interação social da criminalidade na região metropolitana de São Paulo. Realizamos simulações de Monte Carlo para testar a capacidade de estimação do modelo em diferentes cenários. Observamos que a estimação melhora com o aumento de observações ao longo do tempo. Já os resultados empíricos indicam que a região metropolitana de São Paulo é um hot spot no estado, pois é encontrado um maior grau de interação social no índice de homicídio em relação aos índices de roubo e furto. / In this paper we employ a spatio-temporal model proposed in Rojas (2004) to evaluate the social interaction in crime in São Paulo metropolitan area. We carry out Monte Carlo simulations to test the model estimation capability in different scenarios. We notice that the estimation gets better as the number of observations in time raises. The results point out that São Paulo metropolitan area is a hot spot in the state since we found out a greater social interaction for the homicide index, compared to robbery and thievery.

Neste trabalho utilizamos um modelo espaço-temporal proposto em Rojas (2004) para medir a interação social da criminalidade na região metropolitana de São Paulo. Realizamos simulações de Monte Carlo para testar a capacidade de estimação do modelo em diferentes cenários. Observamos que a estimação melhora com o aumento de observações ao longo do tempo. Já os resultados empíricos indicam que a região metropolitana de São Paulo é um hot spot no estado, pois é encontrado um maior grau de interação social no índice de homicídio em relação aos índices de roubo e furto. / In this paper we employ a spatio-temporal model proposed in Rojas (2004) to evaluate the social interaction in crime in São Paulo metropolitan area. We carry out Monte Carlo simulations to test the model estimation capability in different scenarios. We notice that the estimation gets better as the number of observations in time raises. The results point out that São Paulo metropolitan area is a hot spot in the state since we found out a greater social interaction for the homicide index, compared to robbery and thievery.

This biomedical engineering thesis explores the opportunities and challenges of 2D+t contrast echocardiography for left ventricle functional analysis, both clinically and within a computer vision atlas-based deformable template model framework. A database was created for the experiments in this thesis, with 21 studies of contrast Dobutamine Stress Echo, in all 4 principal planes. The database includes clinical variables, human expert hand-traced myocardial contours and visual scoring. First the problem is studied from a clinical perspective. Quantification of endocardial global and local function using standard measures shows expected values and agreement with human expert visual scoring, but the results are less reliable for myocardial thickening. Next, the problem of segmenting the endocardium with a computer is posed in a standard landmark and atlas-based deformable template model framework. The underlying assumption is that these models can emulate human experts in terms of integrating previous knowledge about the anatomy and physiology with three sources of information from the image: texture, geometry and kinetics. Probabilistic atlases of contrast echocardiography are computed, while noting from histograms at selected anatomical locations that modelling texture with just mean intensity values may be too naive. Intensity analysis together with the clinical results above suggest that lack of external boundary definition may preclude this imaging technique for appropriate measuring of myocardial thickening, while endocardial boundary definition is appropriate for evaluation of wall motion. Geometry is presented in a Principal Component Analysis (PCA) context, highlighting issues about Gaussianity, the correlation and covariance matrices with respect to physiology, and analysing different measures of dimensionality. A popular extension of deformable models ---Active Appearance Models (AAMs)--- is then studied in depth. Contrary to common wisdom, it is contended that using a PCA texture space instead of a fixed atlas is detrimental to segmentation, and that PCA models are not convenient for texture modelling. To integrate kinetics, a novel spatio-temporal model of cardiac contours is proposed. The new explicit model does not require frame interpolation, and it is compared to previous implicit models in terms of approximation error when the shape vector changes from frame to frame or remains constant throughout the cardiac cycle. Finally, the 2D+t atlas-based deformable model segmentation problem is formulated and solved with a gradient descent approach. Experiments using the similarity transformation suggest that segmentation of the whole cardiac volume outperforms segmentation of individual frames. A relatively new approach ---the inverse compositional algorithm--- is shown to decrease running times of the classic Lucas-Kanade algorithm by a factor of 20 to 25, to values that are within real-time processing reach.