This dissertation addresses three important open questions in the context of generating random vectors having discrete support. The first question relates to the "NORmal To Anything" (NORTA) procedure, which is easily the most widely used amongst methods for general random vector generation. While NORTA enjoys such popularity, there remain issues surrounding its efficient and correct implementation particularly when generating random vectors having denumerable support. These complications stem primarily from having to safely compute (on a digital computer) certain infinite summations that are inherent to the NORTA procedure. This dissertation addresses the summation issue within NORTA through the construction of easily computable truncation rules that can be applied for a range of discrete random vector generation contexts. The second question tackled in this dissertation relates to developing a customized algorithm for generating multivariate Poisson random vectors. The algorithm developed (TREx) is uniformly fast—about hundred to thousand times faster than NORTA—and presents opportunities for straightforward extensions to the case of negative binomial marginal distributions. The third and arguably most important question addressed in the dissertation is that of exact nonparametric random vector generation on finite spaces. Specifically, it is wellknown that NORTA does not guarantee exact generation in dimensions higher than two. This represents an important gap in the random vector generation literature, especially in view of contexts that stipulate strict adherence to the dependency structure of the requested random vectors. This dissertation fully addresses this gap through the development of Maximum Entropy methods. The methods are exact, very efficient, and work on any finite discrete space with stipulated nonparametric marginal distributions. All code developed as part of the dissertation was written in MATLAB, and is publicly accessible through the Web site https://filebox.vt.edu/users/pasupath/pasupath.htm. / Ph. D.

In this thesis we introduce the spherical, central, angular, halfspace and regression symmetry of random vectors and their measures. Firstly we deal with their mutual relations and equivalent expressions. We also study the uniqueness of the center of individual symmetries and other interesting properties. Then we define the halfspace, projection, spatial and regression multidimensional median and show their properties. Finally we look at the relationships between these medians and symmetric distributions. 1

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Cluster randomized trials (CRT) are comparative studies designed to evaluate interventions where the unit of analysis and randomization is the cluster but the unit of observation is individuals within clusters. Typically such designs involve a limited number of clusters and thus the variation between clusters is left uncontrolled. Experimental designs and analysis strategies that minimize this variance are required. In this work we focus on the CRT with pre-post intervention measures. By incorporating the baseline measure into the analysis, we can effectively reduce the variance of the treatment effect. Well known methods such as adjustment for baseline as a covariate and analysis of differences of pre and post measures are two ways to accomplish this. An alternate way of incorporating baseline measures in the data analysis is to order the clusters on baseline means and pairmatch the two clusters with the smallest means, pair-match the next two, and so on. Our results show that matching on baseline helps to control the between cluster variation when there is a high correlation between the pre-post measures. Six cases of designs and analysis are evaluated by comparing the variance of the treatment effect and the power of related hypothesis tests. We observed that - given our assumptions - the adjusted analysis for baseline as a covariate without pair-matching is the best choice in terms of variance. Future work may reveal that other matching schemes that reflect the natural clustering of experimental units could reduce the variance and increase the power over the standard methods.

The scatter halfspace depth is a quite recently established concept which extends the idea of the location halfspace depth for positive definite matrices. It provides an interest- ing insight into the problem of suitability quantification of a matrix for the description of the covariance structure of the multivariate distribution. The thesis focuses on the investigation of theoretical properties of the depth for both general and more specific probability distributions which can be used for data analysis. It turns out that the es- timators of scatter parameters based on the empirical scatter depth are quite effective even under relatively weak assumptions. These estimators are useful especially for dealing with a sample containing outliers or contaminating observations. 1

This thesis contains theory of probability and statistical sets. Solved and unsolved problems of probability, random variable and distributions random variable, random vector, statistical sets, regression and correlation analysis. Unsolved problems contains solutions.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Hypothyroidism, in both its clinical and subclinical manifestations, is a highly prevalent endocrinopathy among pregnant women. Given the high fetal dependency on maternal thyroid hormones (TH), a shortage of these hormones can be related to long-term effects on organisms, as TH have ubiquitous actions. Particularly, effects of gestational hypothyroidism on kidneys could cause an elevation of blood pressure later in life. An indicator of fetal programming can be the somatic growth of an organism, which, in rats, is hard to be modelled due to persistent growth post-puberty. With this purpose, use of body weight in analysis of variance (ANOVA), taking animal age as repeated measures factor (RMF), can cause nonsphericity in this statistical model. A goal was to evaluate, on adult offspring, the influence of maternal TH deficiency on renal function and on somatic growth, as well the relationship between renal function and systolic arterial pressure (SAP). Additionally, we sought an adequate method of analysis for body weight curves of rats and investigated consequences of using ANOVA as usual for this purpose. Female Wistar rats were divided in three main experimental groups: control euthyroid; hypothyroid by addition of methimazole (0.02%) in drinking water; and euthyroid by hormonal replacement, receiving both methimazole (0.02%) and levothyroxine (T4, 100 μg/L) in drinking water. Treatment was given between gestational days (GD) 9 and 21. On offspring, either nephrectomy or sham-surgery was done at post-coitus day (PCD) 26. Blood pressure was evaluated at around postnatal day (PND) 90, through a tail-plethysmograph. Blood and urine were collected at around PND 110 to evaluate creatinine clearance. To evaluate somatic growth, animals were weighed once to thrice weekly after weaning. Analysis of weight data was done by fitting a modified tetraparametric sigmoid curve to weight data and reducing its parameters by principal component analysis, whereby two principal components were able to retain 92% of original variance of parameters. Experimental gestational hypothyroidism (EGH) had no effect on SAP (p* > 0,200), except through its interaction with estrous cycle. There was a third order interaction concerning estrous phase, nephrectomy and T4 replacement, where the luteal SAP drop was intensified (p* = 0,039). Body weight at PCD 26 did not show any correlation with SAP (p* = 0,716). There was no correlation between SAP and creatinine clearance (p* = 0,803). EGH had no significant effect on creatinine clearance nor on weight curve principal components (p* > 0,200). Thus, given current approach, it was not possible to detect any EGH effect on SAP, neither on renal function, nor on somatic growth. Analysis of body weight curves allowed a precise definition of the beginning of adulthood of animals regarding their somatic growth and elaboration of a simplified model of somatic growth of rats. Also, it was possible to establish an empirical maximum for the coefficient of sphericity of studies which use repeated measures ANOVA to model continuous variables which have been discretized and used as RMF; this can be useful for both experimental planning and previous results reevaluation. / O hipotireoidismo, em suas formas clínica e subclínica, é endocrinopatia de prevalência significativa em mulheres gestantes. Dada a dependência fetal de hormônios tireoideanos (HT) maternos, a sua carência pode estar relacionada a efeitos duradouros no organismo, tendo-se em vista a ação ubíqua dos HT. Em particular, efeitos do hipotireoidismo gestacional no desenvolvimento renal têm o potencial de levar a uma elevação da pressão arterial (PA) na vida adulta. Um indicador da programação fetal pode ser o crescimento somático do organismo, que, em ratos, é difícil de ser modelado devido ao crescimento persistente pós-puberal. Nesse sentido, o uso do peso corporal em análises de variância (ANOVA), tomando-se como fator de medidas repetidas (FMR) a idade do animal, é capaz de violar a premissa de esfericidade do modelo estatístico. Um dos objetivos do estudo foi avaliar, na vida adulta da prole de ratas, a influência da carência dos HT maternos na função renal e no crescimento somático, bem como a relação da função renal com a PA sistólica (PAS). Além disso, buscou-se caracterizar um método adequado de análise de curvas de peso corporal de ratos e investigar as consequências do uso de ANOVA em sua forma tradicional para esse propósito. Ratas Wistar prenhas foram divididas em três grupos experimentais principais: eutireoideanas controle; hipotireoideanas por adição de metimazol (0,02%) na água de beber; e eutireoideanas por reposição hormonal, recebendo tanto metimazol (0,02%) quanto levotiroxina (T4, 100 μg/L) na água de beber. Os tratamentos foram realizados do 9º ao 21º dia de gestação. Com as proles das ratas, foram feitas: cirurgias de nefrectomia ou pseudocirurgia equivalente no 26º dia pós-cópula (DPC); medidas de PAS com cerca de 90 dias pós-natal (DPN) através de um pletismógrafo de cauda; e, coletas de sangue e urina para avaliação do clearance de creatinina com cerca de 110 DPN. Para a avaliação do crescimento somático, os animais foram pesados entre 1 e 3 vezes por semana após o desmame. A análise de curvas de peso corporal foi feita por ajuste de sigmoide de quatro parâmetros, reduzidos a dois componentes principais, mantendo-se 92% da variância original dos parâmetros. O hipotireoidismo gestacional experimental (HGE) não teve efeito significativo sobre a PAS (p* > 0,200), exceto em sua interação com o ciclo estral: houve interação do ciclo estral com a reposição de T4 da mãe e nefrectomia, em que a queda de PAS da fase lútea foi acentuada (p* = 0,039). O peso corporal no 26º DPC não apresentou correlação com a PAS (p* = 0,716). Também não houve correlação entre a PAS e o clearance de creatinina (p* = 0,803). O HGE não teve efeito significativo sobre o clearance de creatinina nem sobre os componentes principais das curvas de peso corporal (p* > 0,200). Portanto, não foi possível verificar, com a abordagem utilizada, qualquer efeito do HGE sobre a PA da prole adulta, nem sobre a função renal e tampouco sobre o crescimento somático. A análise do peso corporal permitiu uma definição precisa do início da vida adulta dos animais quanto ao seu crescimento somático e a elaboração de um modelo conceitual simplificado do crescimento somático de ratos. Além disso, foi possível estabelecer um máximo empírico para o coeficiente de esfericidade de estudos que usam ANOVA de medidas repetidas para modelar variáveis contínuas que tenham sido discretizadas e usadas como FMR; isso pode ser útil tanto no planejamento experimental quanto na reavaliação de achados anteriores. / São Cristóvão, SE

In recent years, Polarimetric Synthetic Aperture Radar (PolSAR) has been one of the most important instruments for earth observation, and is increasingly used in various remote sensing applications. Statistical modelling and scattering analysis are two main ways for PolSAR data interpretation, and have been intensively investigated in the past two decades. Moreover, spatial analysis was applied in the analysis of PolSAR data and found to be beneficial to achieve more accurate interpretation results. This thesis focuses on extracting typical spatial information, i.e., edges and regions by exploring the statistical characteristics of PolSAR data. The existing spatial analysing methods are mainly based on the complex Wishart distribution, which well characterizes the inherent statistical features in homogeneous areas. However, the non-Gaussian models can give better representation of the PolSAR statistics, and therefore have the potential to improve the performance of spatial analysis, especially in heterogeneous areas. In addition, the traditional fixed-shape windows cannot accurately estimate the distribution parameter in some complicated areas, leading to the loss of the refined spatial details. Furthermore, many of the existing methods are not spatially adaptive so that the obtained results are promising in some areas whereas unsatisfactory in other areas. Therefore, this thesis is dedicated to extracting spatial information by applying the non-Gaussian statistical models and spatially adaptive strategies. The specific objectives of the thesis include: (1) to develop reliable edge detection method, (2) to develop spatially adaptive superpixel generation method, and (3) to investigate a new framework of region-based segmentation. Automatic edge detection plays a fundamental role in spatial analysis, whereas the performance of classical PolSAR edge detection methods is limited by the fixed-shape windows. Paper 1 investigates an enhanced edge detection method using the proposed directional span-driven adaptive (DSDA) window. The DSDA window has variable sizes and flexible shapes, and can overcome the limitation of fixed-shape windows by adaptively selecting homogeneous samples. The spherically invariant random vector (SIRV) product model is adopted to characterize the PolSAR data, and a span ratio is combined with the SIRV distance to highlight the dissimilarity measure. The experimental results demonstrated that the proposed method can detect not only the obvious edges, but also the tiny and inconspicuous edges in heterogeneous areas. Edge detection and region segmentation are two important aspects of spatial analysis. As to the region segmentation, paper 2 presents an adaptive PolSAR superpixel generation method based on the simple linear iterative clustering (SLIC) framework. In the k-means clustering procedure, multiple cues including polarimetric, spatial, and texture information are considered to measure the distance. Since the constant weighting factor which balances the spectral similarity and spatial proximity may cause over- or under-superpixel segmentation in different areas, the proposed method sets the factor adaptively based on the homogeneity analysis. Then, in heterogeneous areas, the spectral similarity is more significant than the spatial constraint, generating superpixels which better preserved local details and refined structures. Paper 3 investigates another PolSAR superpixel generation method, which is achieved from the global optimization aspect, using the entropy rate method. The distance between neighbouring pixels is calculated based on their corresponding DSDA regions. In addition, the SIRV distance and the Wishart distance are combined together. Therefore, the proposed method makes good use of the entropy rate framework, and also incorporates the merits of the SIRV distance and the Wishart distance. The superpixels are generated in a homogeneity-adaptive manner, resulting in smooth representation of the land covers in homogeneous areas, and well preserved details in heterogeneous areas. / <p>QC 20171123</p>

The thesis presents methods and criteria for creation and optimization of design of computer experiments. Using the core of a program Freet the optimized designs were created by combination of these methods and criteria. Then, the suitability of the designs for statistical analysis of the tasks vith input random variables was assessed by comparison of the obtained results of six selected functions and the exact (analytically obtained) solutions. Basic theory, definitions of the evaluated functions, description of the setting of optimization and the discussion of the obtained results, including recommendations related to identified weaknesses of certain designs, are presented. The thesis also contains a description of an application that was created to display the results.