Thesis (Ph.D.)--University of California, San Francisco, 2005. / Includes bibliographical references. Also available online.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 271-274).

Thesis (Ph.D.)--Baylor University, 2007. / Includes bibliographical references (p. 225-231).

Thesis (Ph.D.)--Baylor University, 2007. / Includes bibliographical references (p. 96-98).

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Electrical and Computer Engineering." Includes bibliographical references (p. 149-154).

Thesis (Ph. D.)--Washington State University, August 2006. / Includes bibliographical references (p. 67-68).

In recent years, high-dimensional data sets are widely available in many scientific areas, such as gene expression study, finance and others. Estimating the covariance matrix is a significant issue in such high-dimensional data analysis. This thesis focuses on high-dimensional covariance matrix estimation and its application. First, this thesis focuses on the covariance matrix estimation. In Chapter 2, a new optimal shrinkage estimation of the covariance matrices is proposed. This method is motivated by the quadratic discriminant analysis where many covariance matrices need to be estimated simultaneously. We shrink the sample covariance matrix towards the pooled sample covariance matrix through a shrinkage parameter. Some properties of the optimal shrinkage parameter are investigated and we also provide how to estimate the optimal shrinkage parameter. Simulation studies and real data analysis are also conducted. In Chapter 4, we estimate the determinant of the covariance matrix using some recent proposals for estimating high-dimensional covariance matrix. Specifically, a total of nine covariance matrix estimation methods will be considered for comparison. Through extensive simulation studies, we explore and summarize some interesting comparison results among all compared methods. A few practical guidelines are also made on the sample size, the dimension, and the correlation of the data set for estimating the determinant of high-dimensional covariance matrix. Finally, from a perspective of the loss function, the comparison study in this chapter also serves as a proxy to assess the performance of the covariance matrix estimation. Second, this thesis focuses on the application of high-dimensional covariance matrix estimation. In Chapter 3, we consider to estimate the high-dimensional covariance matrix based on the diagonal matrix of the sample covariance matrix and apply it to the Hotelling’s tests. In this chapter, we propose a shrinkage-based diagonal Hotelling’s test for both one-sample and two-sample cases. We also propose several different ways to derive the approximate null distribution under different scenarios of p and n for our proposed shrinkage-based test. Simulation studies show that the proposed method performs comparably to existing competitors when n is moderate or large, and it is better when n is small. In addition, we analyze four gene expression data sets and they demonstrate the advantage of our proposed shrinkage-based diagonal Hotelling’s test. Apart from the covariance matrix estimation, we also develop a new classification method for a specific type of high-dimensional data, RNA-sequencing data. In Chapter 5, we propose a negative binomial linear discriminant analysis for RNA-Seq data. By Bayes’ rule, we construct the classifier by fitting a negative binomial model, and propose some plug-in rules to estimate the unknown parameters in the classifier. The relationship between the negative binomial classifier and the Poisson classifier is explored, with a numerical investigation of the impact of dispersion on the discriminant score. Simulation results show the superiority of our proposed method. We also analyze four real RNA-Seq data sets to demonstrate the advantage of our method in real-world applications. Keywords: Covariance matrix, Discriminant analysis, High-dimensional data, Hotelling’s test, Log determinant, RNA-sequencing data.

Cachaça é a terceira das bebidas alcoólicas fermento-destilado mais consumida no mundo, com uma produção de aproximadamente 2,5 litros de bilhões por ano. Tradicionalmente, a cachaça é produzida a partir da destilação do mosto fermentado da cana-de-açúcar. Porém, ela pode sofrer a contaminação por HPAs (Hidrocarbonetos Policíclicos Aromáticos) quando a cana-de-açúcar utilizada na sua produção é queimada antes da sua colheita. Objetivando a distinção entre as amostras de cachaça produzidas a partir da cana-de-açúcar colhida após a queima ou não dos canaviais, foram analisados 15 HPAs em 26 amostras de cachaça produzidas com cana-de-açúcar queimada e 105 amostras de cachaça produzidas com cana-de-açúcar não queimada. As amostras de cachaça foram previamente concentradas por extração em fase sólida (SPE) e analisadas por cromatografia liquida de alta eficiência (HPLC), acoplada a um detector de fluorescência. Este método apresentou uma boa separação cromatográfica para a análise dos seguintes HPAs estudados: naftaleno, acenafteno, fluoreno, fenantreno, antraceno, fluoranteno, pireno, benzo(a)antraceno, criseno, benzo(b)fluoranteno, benzo(k)fluoranteno, benzo(a)pireno, dibenzo(a,h)antraceno, benzo(g,h,i)perileno e indeno(1,2,3-c,d)pireno. O método cromatográfico apresentou bons limites de detecção (5,68E-04 a 45,4 µg L-1), boa porcentagem de extração e recuperação (81,5 ± 9 % a 113 ± 5 %) e uma boa reprodutibilidade (> 90,0 %). As amostras produzidas a partir de cana-de-açúcar queimada apresentaram teores médios totais dos HPAs de 21,1 µg L-1, enquanto que as amostras produzidas com cana-de-açúcar não queimada apresentaram teores médios totais dos HPAs de 1,91 µg L-1. Os dados analíticos foram tratados por meio de análise multivariada (PCA, FA, PLS, DA, LDA, QDA e CDA), possibilitando uma boa distinção entre as amostras produzidas com cana queimada e não queimada. Os resultados dos modelos estatísticos revelaram uma elevada probabilidade (85,0 %) na diferenciação entre estes dois grupos de cachaça / Cachaça is the third most consumed fermented alcoholic beverage in the world, with a production of nearly 2.5 billion liters a year. Traditionally, cachaça is made from the distillation of the fermented sugar cane juice. However, it may be contaminated by PAHs (Polycyclic Aromatic Hydrocarbons) when the sugar cane used for its production is burned before harvesting. Aiming at distinguishing spirits made from burned and non-burned sugar cane, 15 PAHs in 26 cachaças samples obtained from non-burned and 105 cachaça samples obtained from burned sugar cane have been analyzed. The cachaça samples were previously concentrated by solid phase extraction (SPE) and analyzed by high performance liquid chromatography (HPLC), coupled with a fluorescence detector. This method presented good chromatographic separation for the analysis of the following PAHs: naphthalene, acenaphthene, fluorene, phenanthrene, anthracene, fluoranthene, pyrene, benz(a)anthracene, chrysene, benz(b)fluoranthene, benz(k)fluoranthene, benz(a)pyrene, dibenz(a,h)anthracene, benz(g,h,i)perylene, and indeno(1,2,3- c,d)pyrene. The chromatographic method showed good detection limits (5.68E-04 to 45.4 ? g L-1), good index for the clean up percentage and recovery (from 81.5 ± 9% to 113 ± 5%) and a good reproducibility (> 90.0%). The samples obtained from burned sugar cane presented 21.1 ? g L-1 total PAHs average contents while the ones obtained from non-burned sugar cane showed PAHs average contents of 1.91 ?g L-1. Such analytical data were treated by means of the multivariate statistical methods (PCA, FA, PLS, LDA, QDA and CDA), thus allowing good distinction among the samples obtained from burned and non-burned sugar cane. Results of the Abstract IQSC-USP statistical models presented high probability (85.0%) for the distinction between these two groups of spirits.

Thesis (Ph.D.)--McMaster University (Canada), 1991. / Source: Dissertation Abstracts International, Volume: 53-11, Section: B, page: 5750. Supervisor: N. Balakrishnan.

Thesis (Ph.D.)--University of Wollongong, 2008. / Typescript. Includes bibliographical references: leaf 243-250.