This thesis explores one kind of equation used to model the physics behind one beam with two ends fixed. Initially, Woinowsky Krieger sets a nonlinear partial differential equation (PDE) model by attaching one nonlinear term to the classic linear beam equation. From Zdzislaw Brezezniak, Bohdan Maslowski, Jan Seidler, they demonstrate this model mixed with one Brownian motion term describing random fluctuation. After stochastic modifications, this model becomes more accurate to the behaviors of beam vibrations in reality, and theoretically, the solution has better properties. In this thesis, the model includes more complex noises which cover the condition of random uncontinuous disturbance in the language of Poisson random measure. The major breakthrough of this work is the proof of existence and uniqueness of solutions to this stochastic beam equation and solves the flaws of previous work on proof.

Spatial statistics methods are widely used since several areas of knowledge such as environmental sciences, geology, agronomy, among others, involve the understanding of the spatial distribution of processes from spatially referenced data. With the advancement of Geographic Information Systems and the Global Positioning Systems this use has been extended. Many methods used in spatial statistics are computationally demanding, and therefore, the development of more computationally efficient methods has received a lot of attention in recent years. One such important development is the introduction of the integrated nested Laplace approximation method which is able to carry out Bayesian analysis in a more efficient way. The use of this method for geostatistical data is commonly done considering the stochastic partial differential equation approach that requires the creation of a mesh overlying the study area. This is the first and an important step since all results will depend on the choice of this mesh. As there is no formal and close way to specify the mesh, we investigate possible guidelines on how a suitable mesh is chosen for a specific problem. Through simulations studies, we tried to create guidelines for the construction of the mesh for random, regular and cluster data set and we aplly this guidelines in real data set. / Métodos de estatística espacial são amplamente utilizados, uma vez que várias áreas do conhecimento, como ciências ambientais, geologia, agronomia, entre outros, envolvem a compreensão da distribuição espacial de processos a partir de dados referenciados espacialmente. Com o avanço dos Sistemas de Informação Geográfica e dos Sistemas de Posicionamento Global, esse uso foi ampliado. Muitos métodos utilizados na estatística espacial são computacionalmente exigentes e, portanto, o desenvolvimento de métodos mais eficientes recebeu muita atenção nos últimos anos. Um desenvolvimento importante foi a introdução do método de aproximação de Laplace aninhado integrado, capaz de realizar análises Bayesianas de forma mais eficiente. O uso deste método para dados geoestatísticos é comumente feito considerando a abordagem de equações diferenciais parciais estocásticas que requer a criação de uma malha que cobre a área de estudo. Este é o primeiro e um importante passo, pois todos os resultados dependerão da escolha desta malha. Como não existe uma maneira formal e direta de especificar a malha, investigamos possíveis diretrizes sobre como uma malha adequada é escolhida para um problema específico. Através de estudos de simulações, tentamos criar diretrizes para a construção da malha para conjunto de dados aleatórios, regulares e de cluster e aplicamos essas diretrizes em conjunto de dados reais.

Spatial statistics methods are widely used since several areas of knowledge such as environmental sciences, geology, agronomy, among others, involve the understanding of the spatial distribution of processes from spatially referenced data. With the advancement of Geographic Information Systems and the Global Positioning Systems this use has been extended. Many methods used in spatial statistics are computationally demanding, and therefore, the development of more computationally efficient methods has received a lot of attention in recent years. One such important development is the introduction of the integrated nested Laplace approximation method which is able to carry out Bayesian analysis in a more efficient way. The use of this method for geostatistical data is commonly done considering the stochastic partial differential equation approach that requires the creation of a mesh overlying the study area. This is the first and an important step since all results will depend on the choice of this mesh. As there is no formal and close way to specify the mesh, we investigate possible guidelines on how a suitable mesh is chosen for a specific problem. Through simulations studies, we tried to create guidelines for the construction of the mesh for random, regular and cluster data set and we aplly this guidelines in real data set. / Métodos de estatística espacial são amplamente utilizados, uma vez que várias áreas do conhecimento, como ciências ambientais, geologia, agronomia, entre outros, envolvem a compreensão da distribuição espacial de processos a partir de dados referenciados espacialmente. Com o avanço dos Sistemas de Informação Geográfica e dos Sistemas de Posicionamento Global, esse uso foi ampliado. Muitos métodos utilizados na estatística espacial são computacionalmente exigentes e, portanto, o desenvolvimento de métodos mais eficientes recebeu muita atenção nos últimos anos. Um desenvolvimento importante foi a introdução do método de aproximação de Laplace aninhado integrado, capaz de realizar análises Bayesianas de forma mais eficiente. O uso deste método para dados geoestatísticos é comumente feito considerando a abordagem de equações diferenciais parciais estocásticas que requer a criação de uma malha que cobre a área de estudo. Este é o primeiro e um importante passo, pois todos os resultados dependerão da escolha desta malha. Como não existe uma maneira formal e direta de especificar a malha, investigamos possíveis diretrizes sobre como uma malha adequada é escolhida para um problema específico. Através de estudos de simulações, tentamos criar diretrizes para a construção da malha para conjunto de dados aleatórios, regulares e de cluster e aplicamos essas diretrizes em conjunto de dados reais.

No description available.

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present. However, recent environmental data applications suggest that asymptotic independence is equally important. Therefore, development of flexible subasymptotic models is in pressing need. This dissertation consists of four major contributions to subasymptotic modeling of multivariate and spatial extremes. Firstly, the dissertation proposes a new spatial copula model for extremes based on the multivariate generalized hyperbolic distribution. The extremal dependence of this distribution is revisited and a corrected theoretical description is provided. Secondly, the dissertation thoroughly investigates the extremal dependence of stochastic processes driven by exponential-tailed Lévy noise. It shows that the discrete approximation models, which are linear transformations of a random vector with independent components, bridge asymptotic independence and asymptotic dependence in a novel way, whilst the exact stochastic processes exhibit only asymptotic independence. Thirdly, the dissertation explores two different notions of optimal prediction for extremes, and compares the classical linear kriging predictor and the conditional mean predictor for certain non-Gaussian models. Finally, the dissertation proposes a multivariate skew-elliptical link model for correlated highly-imbalanced (extreme) binary responses, and shows that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior.

Cette thèse porte sur l’estimation spatiale de la teneur superficielle en carbone organiquedu sol ou teneur en SOC (pour ’Soil Organic Carbon content’), à l’échelle d’une petite région agricolefrancilienne. La variabilité de la teneur en SOC a été identifiée comme étant l’une des principales sourcesd’incertitude de la prédiction des stocks de SOC, dont l’accroissement favorise la fertilité des sols etl’atténuation des émissions de gaz à effet de serre. Nous utilisons des données provenant de sourceshétérogènes décrites selon différentes résolutions spatiales (prélèvements de sol, carte pédologique, imagessatellitaires multispectrales, etc) dans le but de produire d’une part une information spatiale exhaustive,et d’autre part des estimations précises de la teneur en SOC sur la région d’étude ainsi qu’une uneévaluation des incertitudes associées. Plusieurs modèles originaux, dont certains tiennent compte duchangement du support, sont construits et plusieurs approches et méthodes de prédiction sont considérées.Parmi elles, on retrouve des méthodes bayésiennes récentes et performantes permettant non seulementd’inférer des modèles sophistiqués intégrant conjointement des données de résolution spatiale différentemais aussi de traiter des données en grande dimension. Afin d’optimiser la qualité de la prédictiondes modélisations multi-sources, nous proposons également une approche efficace et rapide permettantd’accroître l’influence d’un type de données importantes mais sous-représentées dans l’ensemble de toutesles données initialement intégrées. / In this thesis, we are interested in the spatial estimation of the topsoil organic carbon(SOC) content over a small agricultural area located West of Paris. The variability of the SOC contenthas been identified as one of the main sources of prediction uncertainty of SOC stocks, whose increasepromotes soil fertility and mitigates greenhouse gas emissions. We use data issued from heterogeneoussources defined at different spatial resolutions (soil samples, soil map, multispectral satellite images, etc)with the aim of providing on the one hand an exhaustive spatial information, and on the other accurateestimates of the SOC content in the study region and an assessment of the related uncertainties. Severaloriginal models, some of which incorporate the change of support, are built and several approaches andprediction methods are considered. These include recent and powerful Bayesian methods enabling notonly the inference of sophisticated models integrating jointly data of different spatial resolutions butalso the exploitation of large data sets. In order to optimize the quality of prediction of the multi-sourcedata modellings, we also propose an efficient and fast approach : it allows to increase the influence of animportant but under-represented type of data, in the set of all initially integrated data.

No description available.

Dans cette thèse nous nous intéresserons aux équations aux dérivées partielles stochastiques (EDPS) d'un point de vue aussi bien théorique que numérique. Ces équations peuvent être vues comme une généralisation du concept d'équations aux dérivées partielles (EDP) déterministes, équations donnant des modèles dans de nombreux domaines tel que la physique, la biologie ou encore l'économie. L'aspect stochastique apparaît avec la volonté de prendre en compte des données que l'on ne connaît pas de façon déterministe et dont nous avons uniquement des informations statistiques. Ces données peuvent être aussi bien un coefficient de l'équation qu'un terme de force, on qualifie alors ces données de "bruits". De par leurs complexités, il est courant de ne pas avoir de solution formelle pour certaines EDPS, la résolution numérique est alors l'unique moyen d'obtenir certaines statistiques de la solution inconnues formellement. La discrétisation de cette source d'information représentée par les termes de bruit pose le problème de leur troncature. L'information contenue dans ces termes de bruits est infini, ainsi tout comme il est impossible de représenter numériquement, sauf cas particulier, de façon exacte une fonction sur l'intervalle $[0,1]$, il est impossible de stocker de façon exacte ces termes de bruits, se pose alors la question du traitement numérique de ces termes de bruits. Une des méthodes consiste à simuler le bruit afin d'obtenir une famille de trajectoires du bruit et résoudre pour chacune de ces trajectoires l'équation associée afin de pouvoir faire des statistiques sur l'ensemble des solutions obtenues, cette méthode correspond à la méthode de Monte-Carlo. Elle offre l'avantage d'être relativement simple à mettre en œuvre mais se pose alors des problèmes de lenteur de convergence dûs au coût unitaire des intégrations numériques de chaque trajectoire qui dépendent en général de la méthode déterministe utilisée, de la dimension du problème et de la variance des moments que l'on souhaite estimer. Une deuxième philosophie est la décomposition du bruit sur une base polynomiale adaptée à une mesure de référence (ici la mesure de Wiener). C'est la méthode principalement étudiée dans cette thèse. Nous décrirons comment à l'aide d'une décomposition dite en chaos il est possible d'obtenir des statistiques de solutions d'EDPS, mais également comment on peut se servir d'une telle décomposition afin de réduire la variance dans une méthode de Monte Carlo / In this thesis, we will be interested by the numerical approximation of SPDEs. Such equations can be seen as generalization of deterministic PDEs whose coefficients have been perturbed in order to take into account incertainties. Usually those incertainties are only known through their statistical properties. This kind of data could be included into the coefficients of the PDE or can be modelized through an infinite dimensional diffusion term in the second member. The main purpose of the numerical investigations concerning SPDEs is the estimation of the joint probability distribution of its solution, and practically the estimation of some moments or some event's probabilities. The discretization of the noise's information in the small scales implies a large number of additionnal parameters and yields, in general, problems. The first and most popular method used usually is the Monte Carlo method. It relies upon the simulation of a large number of trajectories of the noise followed by the numerical integration of the associated SPDE's solution. Its main advantage is its simplicity and its capacity to be parallelized. Nevertheless, its main drawback is the rather slow convergence due to the unit cost of numerical integration of each trajectory which depend on the deterministic method used, the problem's dimension. Also the convergence can be slowed down because of the large variance of the statistical moments we want to estimate. A second approach consists in the chaos expansion of the coefficients based on a reference measure (Wiener's mesure e.g.). It will be the main purpose of this thesis. We will describe how such an expansion can be made possible in the SPDEs' framework, through the examples of the KdV and Burgers stochastic equations, in order to obtain statistical moments of the solutions but also in order to reduce wariance within a Monte Carlo method

Esta tesis está basada en la modelización jerárquica espacial desde la perspectiva Bayesiana para el estudio de enfermedades en cultivos agrícolas. La necesidad de controlar la variabilidad espacial presente en la mayoría de los datos observados en Agricultura, exige la búsqueda de nuevas alternativas de modelización capaces de recoger adecuadamente la estructura de interrelaciones entre los individuos estudiados. En este sentido, el objetivo general de la tesis es el aporte de herramientas de modelización generales en el ámbito del análisis espacial, que permitan estudiar la presencia de enfermedades en cultivos agrícolas y describan la distribución de los patrones de contagio cuando se tiene poca información y no se tienen covariables explicativas.En general, las modelizaciones propuestas reconocen la existencia de correlación espacial a pequeña escala. Al ilustrar la metodología con datos reales, se reconoce la importancia de la variabilidad espacial y es gracias a ella que puede llegar a comprenderse la dinámica de contagio y el patrón de movilidad de los agentes causantes de la enfermedad en el contexto agrícola. Por lo tanto, un proceso espacial combinado con modelos jerárquicos y vistos desde el paradigma Bayesiano, permite la construcción de herramientas útiles en estudios epidemiológicos en cualquier contexto, y permiten estudiar la incidencia y extensión de fenómenos asociados a un proceso espacial. / Monsalve Graterol, NC. (2013). Modelos jerárquicos bayesianos espaciales en epidemiología agrícola [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19161 / Palancia

This thesis represents a small contribution to our understanding of metastable patterns in various stochastic models from physics and biology. By a \emph{metastable pattern}, we mean a pattern that appears to persist in a regular fashion on some timescale, but disappears or undergoes an irregular change on a longer timescale. Metastable patterns frequently result from stochastic perturbations of patterns that are stable without perturbation. In this thesis, we study stochastic perturbations of stable spatiotemporal patterns in several classes of PDE and integral equations. In particular, we address two major questions: \begin{enumerate}[Q1.] \item When perturbed by noise, for how long does a pattern that is stable without noise persist? \item How does the stochastic perturbation affect the average behaviour of a pattern on the timescale where it appears to persist? \end{enumerate} To address these questions, we pursue two lines of inquiry: the first based on the theory of \emph{quasi-ergodic measures}, and the second based on \emph{phase decomposition techniques}. In our first line of inquiry we present novel, rigorous connections between metastability of general infinite dimensional stochastic evolution systems and the spectral properties of their sub-Markov generators using the theory of quasi-ergodic measures. To do so, we develop a novel $L^p$-approach to the study of quasi-ergodic measures. We are then able to draw conclusions about the metastability of travelling waves and other patterns in a class of stochastic reaction-diffusion equations. For instance, we obtain a rigorous definition of the \emph{quasi-asymptotic speed}~of a travelling wave in a stochastic PDE. We moreover find that stochastic perturbations of amplitude $\sigma>0$ cause the quasi-asymptotic speed of certain travelling waves to deviate from the deterministic wave speed by a constant that is approximately proportional to $\sigma^2$. In our second line of inquiry, the dynamics of our (infinite dimensional) stochastic evolution system are projected onto a finite dimensional manifold that captures some property of a metastable pattern. While most previous studies using phase reduction techniques have used the \emph{variational phase}, we take an approach based on the \emph{isochronal phase}, inspired by classical work on finite dimensional oscillatory systems. When the pattern in question is a travelling wave, the isochronal phase captures the position of the wave at a given point in time. By exploiting the regularity properties of the isochronal phase, we are able to prove several novel results about the metastable behaviour of the reduced dynamics in the small noise regime in a very large class of stochastic evolution systems. These results allow us to moreover compute the noise-induced changes in the speed of stochastically perturbed travelling waves and other patterns. The results we obtain using this approach are numerically precise, and may be applied to a very general class of stochastic evolution systems.