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Predicting rock permeability from well logs in uncored wells is an important task in reservoir characterization. Due to the high costs of coring and laboratory analysis, typically cores are acquired in only a few wells. Since most wells are logged, the common practice is to estimate permeability from logs using correlation equations developed from limited core data. Most commonly, permeability is estimated from various well logs using statistical regression. For sandstones, often the logs of permeability can be correlated with porosity, but in carbonates the porosity permeability relationship tends to be much more complex and erratic. For this reason permeability prediction is a critical aspect of reservoir characterization in complex reservoirs such as carbonate reservoirs. In order to improve the permeability estimation in these reservoirs, several statistical regression techniques have already been tested in previous work to correlate permeability with different well logs. It has been shown that statistical regression for data correlation is quite promising in predicting complex reservoirs. But using all the possible well logs to predict permeability is not appropriate because the possibility of spurious correlation increases if you use more well logs. In statistics, variable selection is used to remove unnecessary independent variables and give a better prediction. So we apply variable selection to the permeability prediction procedures in order to further improve permeability estimation. We present three approaches to further improve reservoir permeability prediction based on well logs via data correlation and variable selection in this research. The first is a combination of stepwise algorithm with ACE technique. The second approach is the application of tree regression and cross-validation. The third is multivariate adaptive regression splines. Three methods are tested and compared at two complex carbonate reservoirs in west Texas: Salt Creek Field Unit (SCFU) and North Robertson Unit (NRU). The result of SCFU shows that permeability prediction is improved by applying variable selection to non-parametric regression ACE while tree regression is unable to predict permeability because it can not preserve the continuity of permeability. In NRU, none of these three methods can predict permeability accurately. This is due to the high complexity of NRU reservoir and measurement accuracy. In this reservoir, high permeability is discrete from low permeability, which makes prediction even more difficult. Permeability predictions based on well logs in complex carbonate reservoirs can be further improved by selecting appropriate well logs for data correlation. In comparing the relative predictive performance of the three regression methods, the stepwise with ACE method appears to outperform the other two methods.

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Recent advances in technology for brain imaging and high-throughput genotyping have motivated studies examining the influence of genetic variation on brain structure. In this setting, high-dimensional regression for multi-SNP association analysis is challenging as the brain imaging phenotypes are multivariate and there is a desire to incorporate a biological group structure among SNPs based on their belonging genes. Wang et al. (Bioinformatics, 2012) have recently developed an approach for simultaneous estimation and SNP selection based on penalized regression with regularization based on a novel group l_{2,1}-norm penalty, which encourages sparsity at the gene level. A problem with the proposed approach is that it only provides a point estimate. We solve this problem by developing a corresponding Bayesian formulation based on a three-level hierarchical model that allows for full posterior inference using Gibbs sampling. For the selection of tuning parameters, we consider techniques based on: (i) a fully Bayes approach with hyperpriors, (ii) empirical Bayes with implementation based on a Monte Carlo EM algorithm, and (iii) cross-validation (CV). When the number of SNPs is greater than the number of observations we find that both the fully Bayes and empirical Bayes approaches overestimate the tuning parameters, leading to overshrinkage of regression coefficients. To understand this problem we derive an approximation to the marginal likelihood and investigate its shape under different settings. Our investigation sheds some light on the problem and suggests the use of cross-validation or its approximation with WAIC (Watanabe, 2010) when the number of SNPs is relatively large. Properties of our Gibbs-WAIC approach are investigated using a simulation study and we apply the methodology to a large dataset collected as part of the Alzheimer's Disease Neuroimaging Initiative. / Graduate

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Increasing use of rock materials like shale in building, roofing, embankment filling, brick manufacturing, and in other civil structure application makes it an important rock to consider in construction engineering. Knowledge of thermal and physical properties of shale as building material is required to predict the rock's strength and permanence against weathering. Inconsistent heat capacity of anisotropic rock can result in differential heat flow. This tendency can expand the building materials leading to reduction in strength and initiate disintegration. Authors have studied various thermo-physical properties of anisotropic shale from Tennessee, which is commonly used as building stones and bricks. Experiment was designed to measure the basic thermal property, 'heat capacity' of shale. Series of laboratory tests including durability, strength, specific gravity, moisture content, and porosity were conducted to determine the physical and mechanical behavior of the samples. Results indicated that properties like porosity, strength and heat capacity varied significantly within samples, where as specific gravity and moisture content yielded steady values. Multivariate regression analysis was performed to evaluate possible correlations among the tested properties. Strong positive relationship was evident between heat capacity, and porosity. Heat capacity and Unconfined Compressive Strength of shale were inversely related. This study emphasized that physical and thermal properties of shale are directly linked with strength and durability of the rock mass.

Master of Science / Department of Statistics / Kun Chen / The singular value decomposition (SVD) is a commonly used matrix factorization technique in statistics, and it is very e ective in revealing many low-dimensional structures in a noisy data matrix or a coe cient matrix of a statistical model. In particular, it is often desirable to obtain a sparse SVD, i.e., only a few singular values are nonzero and their corresponding left and right singular vectors are also sparse. However, in several existing methods for sparse SVD estimation, the exact orthogonality among the singular vectors are often sacri ced due to the di culty in incorporating the non-convex orthogonality constraint in sparse estimation. Imposing orthogonality in addition to sparsity, albeit di cult, can be critical in restricting and guiding the search of the sparsity pattern and facilitating model interpretation. Combining the ideas of penalized regression and Bregman iterative methods, we propose two methods that strive to achieve the dual goal of sparse and orthogonal SVD estimation, in the general framework of high dimensional multivariate regression. We set up simulation studies to demonstrate the e cacy of the proposed methods.

Recursive binary partitioning is a popular tool for regression analysis. Two fundamental problems of exhaustive search procedures usually applied to fit such models have been known for a long time: Overfitting and a selection bias towards covariates with many possible splits or missing values. While pruning procedures are able to solve the overfitting problem, the variable selection bias still seriously effects the interpretability of tree-structured regression models. For some special cases unbiased procedures have been suggested, however lacking a common theoretical foundation. We propose a unified framework for recursive partitioning which embeds tree-structured regression models into a well defined theory of conditional inference procedures. Stopping criteria based on multiple test procedures are implemented and it is shown that the predictive performance of the resulting trees is as good as the performance of established exhaustive search procedures. It turns out that the partitions and therefore the models induced by both approaches are structurally different, indicating the need for an unbiased variable selection. The methodology presented here is applicable to all kinds of regression problems, including nominal, ordinal, numeric, censored as well as multivariate response variables and arbitrary measurement scales of the covariates. Data from studies on animal abundance, glaucoma classification, node positive breast cancer and mammography experience are re-analyzed. / Series: Research Report Series / Department of Statistics and Mathematics

Conjuntos de dados com respostas multivariadas aparecem frequentemente em pesquisas em que os dados são provenientes de questionários. Exemplos mais comuns são pesquisas de opinião, mais especificamente, pesquisas de marketing em que a preferência do consumidor em potencial é avaliado: pelo produto, marca, preço, praça, promoção e etc. Um tipo pesquisa de opinião que ganha grande destaque no Brasil de dois em dois anos são as pesquisas eleitorais de intenção de votos. Nós introduzimos nesta tese uma classe de modelos de regressão multivariados com funções de ligação assimétricas para o ajuste de conjuntos de dados com respostas multivariadas binárias. As funções de ligação consideradas são bastante flexíveis e robustas, contemplando funções de ligação simétricas como casos particulares. Devido a complexidade do modelo, nós discutimos a sua identificabilidade. A abordagem Bayesiana foi considerada e alguns algoritmos de Monte Carlo via Cadeia de Markov (MCMC) foram desenvolvidos. Nós descrevemos algumas ferramentas de seleção de modelos, os quais incluem o Critério de Informação da Deviance (DIC), a Pseudo-Verossimilhança Marginal e o Pseudo-Fator de Bayes. Adicionalmente, um estudo de simulação foi desenvolvido com dois objetivos; i) verificar a qualidade dos algoritmos desenvolvidos e ii) verificar a importância da escolha da função de ligação . No final da tese uma aplicação em um conjunto de dados real é considerada com o objetivo de ilustrar as metodologias e técnicas apresentadas. / Data sets with multivariate responses often appear in surveys where the data came from questionnaires. Opinion poll, sometimes simply referred to as a poll, are common examples of studies in which the responses are multivariate. One type poll that gain great prominence in Brazil in election years, is the survey of vote intent. However, despite the higher visibility of prognostic studies of election, opnion polls is a tool widely used to detect trends and positions of different social segments on various topics, be they political, social or governmental. We introduce in this work a class of multivariate regression models with asymmetric link functions to fit data sets with multivariate binary responses. The link functions here considered are quite flexible and robust, contemplating symmetrical link functions as special cases. Due to the complexity of the model, we discuss its identifiability. The Bayesian approach was considered and some Monte Carlo Markov Chain (MCMC) algorithms have been developed. Simulation studies have been developed with two objectives: i) verify the quality of the algorithms developed and ii) to verify the importance of choosing the link function. At the end of this work an application in a real data set is considered in order to illustrate the methodologies and techniques presented.

Conjuntos de dados com respostas multivariadas aparecem frequentemente em pesquisas em que os dados são provenientes de questionários. Exemplos mais comuns são pesquisas de opinião, mais especificamente, pesquisas de marketing em que a preferência do consumidor em potencial é avaliado: pelo produto, marca, preço, praça, promoção e etc. Um tipo pesquisa de opinião que ganha grande destaque no Brasil de dois em dois anos são as pesquisas eleitorais de intenção de votos. Nós introduzimos nesta tese uma classe de modelos de regressão multivariados com funções de ligação assimétricas para o ajuste de conjuntos de dados com respostas multivariadas binárias. As funções de ligação consideradas são bastante flexíveis e robustas, contemplando funções de ligação simétricas como casos particulares. Devido a complexidade do modelo, nós discutimos a sua identificabilidade. A abordagem Bayesiana foi considerada e alguns algoritmos de Monte Carlo via Cadeia de Markov (MCMC) foram desenvolvidos. Nós descrevemos algumas ferramentas de seleção de modelos, os quais incluem o Critério de Informação da Deviance (DIC), a Pseudo-Verossimilhança Marginal e o Pseudo-Fator de Bayes. Adicionalmente, um estudo de simulação foi desenvolvido com dois objetivos; i) verificar a qualidade dos algoritmos desenvolvidos e ii) verificar a importância da escolha da função de ligação . No final da tese uma aplicação em um conjunto de dados real é considerada com o objetivo de ilustrar as metodologias e técnicas apresentadas. / Data sets with multivariate responses often appear in surveys where the data came from questionnaires. Opinion poll, sometimes simply referred to as a poll, are common examples of studies in which the responses are multivariate. One type poll that gain great prominence in Brazil in election years, is the survey of vote intent. However, despite the higher visibility of prognostic studies of election, opnion polls is a tool widely used to detect trends and positions of different social segments on various topics, be they political, social or governmental. We introduce in this work a class of multivariate regression models with asymmetric link functions to fit data sets with multivariate binary responses. The link functions here considered are quite flexible and robust, contemplating symmetrical link functions as special cases. Due to the complexity of the model, we discuss its identifiability. The Bayesian approach was considered and some Monte Carlo Markov Chain (MCMC) algorithms have been developed. Simulation studies have been developed with two objectives: i) verify the quality of the algorithms developed and ii) to verify the importance of choosing the link function. At the end of this work an application in a real data set is considered in order to illustrate the methodologies and techniques presented.