Dependence measures are frequently applied in neuroimagining studies as a tool for analysis and classification of fMRI data. The aim of this thesis is to evaluate an algorithm for its use in classifying fMRI data using dependence measures. The focus is on evaluating the algorithm under a few changes, for example without adding voxel-based tests in voxel selection, for future use in classification. Additionally, the thesis aims to compare the performance of two dependence measures, the RV coefficient and its modified version. The classification performance of the algorithm is evaluated on a simulated fMRI data as well as resting-state and task-based fMRI data sets. On simulated fMRI data the algorithm yields an estimated accuracy of 81.41 percent versus 75.00 percent for the classifier using the RV coefficient and the modified RV coefficient, respectively. However, when evaluated on real fMRI data the estimated accuracy is close to, or even lower, than 50 percent. This indicates that the classification performance is not far from what would be expected from a classifier picked at random. It is expected that implementing additional tests to select a subset of voxels, to use in the classification step of the algorithm, may prove helpful. Further, some differences in classification performance of the RV coefficients are found. Based on the observed differences it is not possible to conclude that one measure can be preferred over the other.

Diferente das medidas de associação global (coeficiente de correlação linear de Pearson, de Spearman, tau de Kendall, por exemplo), as medidas de dependência local descrevem o comportamento da dependência localmente em diferentes regiões. Nesta tese, as medidas de dependência local para variáveis aleatórias propostas por Bairamov et al. (2003), Bjerve e Doksum (1993) e Sibuya (1960), são estudadas sob o enfoque de processos estocásticos estacionários bivariados e univariados, neste caso, estudando o comportamento da dependência local ao longo das defasagens da série temporal. Para as duas primeiras medidas, discutimos as suas propriedades, e estudamos os seus estimadores, além da consistência dos mesmos. Para a medida de Sibuya, além de discutir suas propriedades, propomos três estimadores para variáveis aleatórias e dois para séries temporais, verificando a consistência dos mesmos. O comportamento das três medidas locais e dos seus estimadores foram avaliados através de simulações e aplicações a dados reais (neste caso, fizemos uma comparação destas com cópula e densidade cópula). / Unlike global association measures (Pearson´s linear correlation coefficient, Spearman´s rho, Kendall´s tau, for example), local dependence measures describe the behaviour of dependence locally in different regions. In this thesis, the local dependence measures for random variables proposed by Bairamov et al. (2003), Bjerve and Doksum (1993) and Sibuya (1960), are studied in the context of bivariate and univariate stationary stochastic processes, in this case, evaluating the performance of local dependence along time lags. We discussed the properties and studied the estimators and consistence of the first two measures. As for the Sibuya measure, in addition to discussing its properties, we propose three estimators for random variables and two for time series while checking their consistence. The behaviour of the three local measures and their respective estimators was evaluated by simulations and application to real data (in this case, a comparison was drawn with copula and copula density).

Diferente das medidas de associação global (coeficiente de correlação linear de Pearson, de Spearman, tau de Kendall, por exemplo), as medidas de dependência local descrevem o comportamento da dependência localmente em diferentes regiões. Nesta tese, as medidas de dependência local para variáveis aleatórias propostas por Bairamov et al. (2003), Bjerve e Doksum (1993) e Sibuya (1960), são estudadas sob o enfoque de processos estocásticos estacionários bivariados e univariados, neste caso, estudando o comportamento da dependência local ao longo das defasagens da série temporal. Para as duas primeiras medidas, discutimos as suas propriedades, e estudamos os seus estimadores, além da consistência dos mesmos. Para a medida de Sibuya, além de discutir suas propriedades, propomos três estimadores para variáveis aleatórias e dois para séries temporais, verificando a consistência dos mesmos. O comportamento das três medidas locais e dos seus estimadores foram avaliados através de simulações e aplicações a dados reais (neste caso, fizemos uma comparação destas com cópula e densidade cópula). / Unlike global association measures (Pearson´s linear correlation coefficient, Spearman´s rho, Kendall´s tau, for example), local dependence measures describe the behaviour of dependence locally in different regions. In this thesis, the local dependence measures for random variables proposed by Bairamov et al. (2003), Bjerve and Doksum (1993) and Sibuya (1960), are studied in the context of bivariate and univariate stationary stochastic processes, in this case, evaluating the performance of local dependence along time lags. We discussed the properties and studied the estimators and consistence of the first two measures. As for the Sibuya measure, in addition to discussing its properties, we propose three estimators for random variables and two for time series while checking their consistence. The behaviour of the three local measures and their respective estimators was evaluated by simulations and application to real data (in this case, a comparison was drawn with copula and copula density).

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The determination of the sufficient amount of economic capital and its allocation to the business lines is the key issue for insurance companies. In this thesis we introduce two methods of aggregating economic capital. One is based on linear correlation and the second deals with copulas. A multitude of allocation principles have been proposed in the literature. We choose those which are the most used in practice and compare advantages and disadvantages of their application. The last chapter is devoted to the numerical examples of capital aggregation and allocation principles. 1

La modélisation probabiliste des événements climatiques et environnementaux doit prendre en compte leur nature spatiale. Cette thèse porte sur l’étude de mesures de risque pour des processus spatiaux. Dans une première partie, nous introduisons des mesures de risque à même de prendre en compte la structure de dépendance des processus spatiaux sous-jacents pour traiter de données environnementales. Une deuxième partie est consacrée à l’estimation des paramètres de processus de type max-mélange. La première partie de la thèse est dédiée aux mesures de risque. Nous étendons les travaux réalisés dans [44] d’une part à des processus gaussiens, d’autre part à d’autres processus max-stables et à des processus max-mélange, d’autres structures de dépendance sont ainsi considérées. Les mesures de risque considérées sont basées sur la moyenne L(A,D) de pertes ou de dommages D sur une région d’intérêt A. Nous considérons alors l’espérance et la variance de ces dommages normalisés. Dans un premier temps, nous nous intéressons aux propriétés axiomatiques des mesures de risque, à leur calcul et à leur comportement asymptotique (lorsque la taille de la région A tend vers l’infini). Nous calculons les mesures de risque dans différents cas. Pour un processus gaussien, X, on considère la fonction d’excès : D+ X,u = (X−u)+ où u est un seuil fixé. Pour des processus max-stables et max-mélange X, on considère la fonction puissance : DνX = Xν. Dans certains cas, des formules semi-explicites pour les mesures de risque correspondantes sont données. Une étude sur simulations permet de tester le comportement des mesures de risque par rapport aux nombreux paramètres en jeu et aux différentes formes de noyau de corrélation. Nous évaluons aussi la performance calculatoire des différentes méthodes proposées. Celle-ci est satisfaisante. Enfin, nous avons utilisé une étude précédente sur des données de pollution dans le Piémont italien, celle-ci peuvent être considérées comme gaussiennes. Nous étudions la mesure de risque associée au seuil légal de pollution donnée par la directive européenne 2008/50/EC. Dans une deuxième partie, nous proposons une procédure d’estimation des paramètres d’un processus max-mélange, alternative à la méthode d’estimation par maximum de vraisemblance composite. Cette méthode plus classique d’estimation par maximum de vraisemblance composite est surtout performante pour estimer les paramètres de la partie max-stable du mélange (et moins performante pour estimer les paramètres de la partie asymptotiquement indépendante). Nous proposons une méthode de moindres carrés basée sur le F-madogramme : minimisation de l’écart quadratique entre le F-madogramme théorique et le F-madogramme empirique. Cette méthode est évaluée par simulation et comparée à la méthode par maximum de vraisemblance composite. Les simulations indiquent que la méthode par moindres carrés du F-madogramme est plus performante pour estimer les paramètres de la partie asymptotiquement indépendante / When dealing with environmental or climatic changes, a natural spatial dependence aspect appears. This thesis is dedicated to the study of risk measures in this spatial context. In the first part (Chapters 3 and 4), we study risk measures, which include the natural spatial dependence structure in order to assess the risks due to extreme environmental events and in the last part (Chapter 5), we propose estimation procedures for underlying processes, such as isotropic and stationary max-mixture processes. In the first part dedicated to risk measures, we extended the work in [44] in order to obtain spatial risk measures for various spatial processes and different dependence structures. We based these risk measures on the mean losses over a region A of interest. Risk measures are then defined as the expectation E[L(A,D)] and variance Var(L(A,D)) of the normalized loss. In the study of these measures, we focused on the axiomatic properties of asymptotic behavior (as the size of the region interest goes to infinity) and on computational aspects. We calculated two risk measures: risk measure for the gaussian process based on the damage function called access damage D+ X,u and risk measure for extreme processes based on the power damage function DνX . In simulation study and for each risk measure provided, we emphasized the theoretical results of asymptotic behavior by various parameters of a model and different Kernels for the correlation function. We also evaluated the performance of these risk measures. The results were encouraging. Finally, we implemented the risk measure corresponding to gaussian on the real data of pollution in Piemonte, Italy. We assessed the risks associated with this pollution when an excess of it was over the legal level determined by the European directive 2008/50/EC. With respect to estimation, we proposed a semi-parametric estimation procedure in order to estimate the parameters of a max-mixture model and also of a max-stable model ( inverse max-stable model) as an alternative to composite likelihood. A good estimation by the proposed estimator required the dependence measure to detect all dependence structures in the model, especially when dealing with the max-mixture model. We overcame this challenge by using the F-madogram. The semi-parametric estimation was then based on a quasi least square method, by minimizing the square difference between the theoretical F-madogram and an empirical one. We evaluated the performance of this estimator through a simulation study. It was shown that on a mean, the estimation is performed well, although in some cases, it encountered some difficulties