O uso de imagens é crucial na medicina, e seu uso no diagnóstico de doenças é uma das principais ferramentas clínicas da atualidade. Porém, frequentemente necessitam de pós-processamento para serem úteis. Embora ferramentas clássicas sejam utilizadas para esse fim, elas não dão tratamento específico a certas características de imagens fractais, como as provindas de sistemas biológicos. Nesse enfoque, este trabalho objetivou a criação de novas bases de interpolação utilizando a Q-Estatística para verificar se seriam estas seriam adequadas à representação de objetos com características fractais que as bases clássicas. Foram criados dois tipos de splines: uma unidimensional e outra bidimensional, que permitiram um tipo diferente de interpolação, fundamentado na q-Estatística. Os testes demonstraram a potencialidade dessas ferramentas para uso em sinais e imagens médicas, com acentuada redução do erro de interpolação no caso unidimensional (em até 351,876%) e uma redução sutil no caso bidimensional (0,3%). Como resultado adicional, foram criados filtros de imagens e avaliados seus resultados em imagens médicas, que resultaram em melhorias de até 1.340% de ganho efetivo na remoção de ruídos de natureza fractal (marrom). Os resultados sugerem que as q-bases desenvolvidas foram capazes de representar melhor imagens e sinais médicos, bem como é interessante o uso dos filtros desenvolvidos na remoção de diversos tipos de ruído do tipo 1/f^b. / The use of images is crucial in modern medicine, and diagnostic imaging is a major clinical tools used in detecting, monitoring and completion of many treatments. However, often the images need to be post-processed for display to health professionals or automated analysis, searching for signs of abnormalities. Although classical tools are used for that purpose, they do not give special treatment to certain characteristics of fractal images, such as those coming from biological systems. These characteristics are produced, in general, by complex dynamic systems as a result of internal interactions of sub-system components, giving the system a fractal character. In this context, the main objective of this work was to propose interpolation bases using the Q-statistic, creating bases of Q-interpolation, and verify if such bases would be best suited to the representation of objects with fractal characteristics than classical bases, assumed the premise that such a theory model best this kind of phenomenon than classical theory. Based on this hypothesis, we created two types of splines: one-dimensional and one-dimensional, called Q-splines, which allow a different type of interpolation and they can capture behaviors as super-additive or sub-additive among the constituents of a spline. These models have demonstrated numerically the potential use of this type of interpolation for use in signals and medical images, reducing the interpolation error by up to 351.876 % in the one-dimensional case and 0.3 % in two dimensional. As secondary results, were defined two families of image filters, called anisotropic Q-filters and isotropic Q-filters, and their results were evaluated in real medical images. In virtually all analyzes it was possible to obtain the best results from conventional approaches, sometimes with improvements of 1.340 % in some filters, in removing noise fractal nature (brown). The results were more modest for the interpolation of two-dimensional images, however, generally proved exciting and encouraging, clearly showing that these new approaches are not only viable, but also can produce better results compared to classical approaches. Based on these results, we concluded that the Q-bases developed are best able to represent not only signs but medical imaging (1D and 2D) although its use can be improved by the adoption of approaches adapted to the vector representation of information, that favor the use of splines. Similarly, the Q-filters were more suitable for the processing of medical signals when compared to conventional approaches.

In this work was proposed a new distributions, called log-sinh Cauchy, with has bimodal shapes and can be used as alternative to the mixture models. Based in the proposed distribution, the following models were proposed: Regression model based in the GAMLSS framework; models with cure rate based in the mixture and promotion time models; semiparametric models, modeling the parameters using penalized splies; semiparametric models, using the penalized splines to model the non-linear effects present in the cure rate. For all proposed models, the computational codes were implemented in the R software, with is available along of the document as well as some brief introduction on how to use them. / Nesse trabalho foi proposto uma nova distribuição, denominada de exponentiated log-sinh Cauchy, a qual possui densidades bimodais e pode ser utilizada como alternativa aos modelos de mistura. Com base na nova distribuição, foram propostos: modelos de regressão baseados nos modelos GAMLSS; modelos com fração de cura baseados em modelos de mistura e tempo de promoção; modelo semi-paramétrico modelando os parâmetros com splines penalizados; modelo semi-paramétrico com fração de cura utilizando splines para modelar efeitos não lineares na proporção de curados. Para todos os modelos propostos, toda parte computacional foi implementada no software R, sendo disponibilizada ao longo do documento assim como breve descrições de uso.

Non-uniform rational B-splines (NURBS) methodology is presented, on which the isogeometric analysis is based. T-splines are also introduced as a surface design methodology, which are a generalization of NURBS and permit local refinement. Isogeometric analysis using NURBS and T-splines are applied separately to a structural mechanics problem. The results are compared with the closed-form solution. The desirable performance of isogeometric analysis using T-splines on engineering analysis is demonstrated.

Previous healthcare-based studies have reported increasing herpes zoster (HZ) incidence over time; however, this could be an artifact of increased healthcare utilization. This study is a cross-sectional analysis of 15,103 respondents in the 2008 wave of the Health and Retirement Study (HRS) to evaluate changes in HZ incidence from 1940 to 2008. Negative binomial regression is used to model the effect of calendar year, age of onset of HZ, gender and race/ethnicity on HZ incidence. A nonparametric method based on B-spline basis expansion is used to model the effect of calendar year to avoid imposing a predetermined functional form and produce flexible and accurate estimates. This study demonstrates increasing HZ incidence from 1940 to 2008 using self-reported HZ. Although the reason for this increase remains unknown, this study supports the assertion that this trend is real and not an artifact of increasing healthcare utilization for HZ over time.

In clinical experiments, the evolution of a product concentration in tissue over time is often under study. Different products and tissues may be considered. For instance, one could analyse the evolution of drug concentration in plasma over time, by performing successive blood sampling from the subjects participating to the clinical study. One could also observe the evolution of radioactivity uptakes in different regions of the brain during a PET scan (Positron Emission Tomography). The global objective of this thesis is the modelling of such evolutions, which will be called, generically, pharmacokinetic curves (PK curves). Some clinical measures of interest are derived from PK curves. For instance, when analysing the evolution of drug concentration in plasma, PK parameters such as the area under the curve (AUC), the maximal concentration (Cmax) and the time at which it occurs (tmax) are usually reported. In a PET study, one could measure Receptor Occupancy (RO) in some regions of the brain, i.e. the percentage of specific receptors to which the drug is bound. Such clinical measures may be badly estimated if the PK curves are noisy. Our objective is to provide statistical tools to get better estimations of the clinical measures of interest from appropriately smoothed PK curves. Plenty of literature addresses the problem of PK curves fitting using parametric models. It usually relies on a compartmental approach to describe the kinetic of the product under study. The use of parametric models to fit PK curves can lead to problems in some specific cases. Firstly, the estimation procedures rely on algorithms which convergence can be hard to attain with sparse and/or noisy data. Secondly, it may be difficult to choose the adequate underlying compartmental model, especially when a new drug is under study and its kinetic is not well known. The method that we advocate to fit such PK curves is based on Bayesian Penalized splines (P-splines): it provides good results both in terms of PK curves fitting and clinical measures estimations. It avoids the difficult choice of a compartmental model and is more robust than parametric models to a small sample size or a low signal to noise ratio. Working in a Bayesian context provides several advantages: prior information can be injected, models can easily be generalized and extended to hierarchical settings, and uncertainty for associated clinical parameters are straightforwardly derived from credible intervals obtained by MCMC methods. These are major advantages over traditional frequentist approaches.

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Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Neste trabalho apresentamos os modelos lineares parciais generalizados com uma variável explicativa contínua tratada de forma não paramétrica e os modelos lineares parciais aditivos generalizados com no mínimo duas variáveis explicativas contínuas tratadas de tal forma. São utilizados os P-splines para descrever a relação da variável resposta com as variáveis explicativas contínuas. Sendo assim, as funções de verossimilhança penalizadas, as funções escore penalizadas e as matrizes de informação de Fisher penalizadas são desenvolvidas para a obtenção das estimativas de máxima verossimilhança penalizadas por meio da combinação do algoritmo backfitting (Gauss-Seidel) e do processo iterativo escore de Fisher para os dois tipos de modelo. Em seguida, são apresentados procedimentos para a estimação do parâmetro de suavização, bem como dos graus de liberdade efetivos. Por fim, com o objetivo de ilustração, os modelos propostos são ajustados à conjuntos de dados reais. / In this work we present the generalized partial linear models with one continuous explanatory variable treated nonparametrically and the generalized additive partial linear models with at least two continuous explanatory variables treated in such a way. The P-splines are used to describe the relationship among the response and the continuous explanatory variables. Then, the penalized likelihood functions, penalized score functions and penalized Fisher information matrices are derived to obtain the penalized maximum likelihood estimators by the combination of the backfitting (Gauss-Seidel) algorithm and the Fisher escoring iterative method for the two types of model. In addition, we present ways to estimate the smoothing parameter as well as the effective degrees of freedom. Finally, for the purpose of illustration, the proposed models are fitted to real data sets.

Une nouvelle méthode est proposée pour la simulation des plasmas utilisant le modèle cinétique qui couple les équations de Vlasov pour la distribution des particules et de Maxwell pour la contribution des champs électromagnétiques. Cette méthode est semi-Lagrangienne, elle utilise une grille de l'espace des phases et se sert des caractéristiques de l'équation de Vlasov. Ces caractéristiques sont suivies en avançant dans le temps, ce qui permet plusieurs avantages par rapport à la méthode classique. Déjà, cette méthode est explicite, ce qui permet une montée en ordre facilitée qui peut offrir davantage de stabilité, et la possibilité de construire des schémas qui conservent la charge en utilisant ses similitudes avec les méthodes PIC. Ceci est fondamental pour s'assurer que les solutions calculées sont bien physiques.