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Thesis (M.S.)--Michigan State University. Comparative Medicine and Integrative Biology, 2006. / Title from PDF t.p. (viewed on Nov. 20, 2008) Includes bibliographical references (p. 94-102). Also issued in print.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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.

Amongst the most threatened ecosystems on Earth, mangrove forests are also one of the more difficult to work in due to their growth in mud and open water coastal zones and their dense tangled stems, branches and prop roots. Consequently, there has been an impetus to employ remotely sensed imagery as a means for rapid inventory of these coastal wetlands. To date, the majority of mangrove maps derived from satellite imagery utilize a simple mangrove classification scheme which does not distinguish mangrove species and may not be useful for conservation and management purposes. Although more elaborate satellite based mangrove classification schemes are being developed, given their enhanced complexity they deserve additional justification for end users. The purpose of this study was to statistically examine the appropriateness of one such classification scheme based on an inventory of field data. In January of 2007 and May of 2008, 61 field sample plots were selected in a stratified random fashion based on a previous classification of a degraded mangrove forest of the Isla La Palma (Sinaloa, Mexico) using Landsat TM5 data. Unlike other previous Landsat TM based classifications of this region, which simply identified the mangrove forests as one class, the mangroves were classified (i. e. mapped) according to four conditions; healthy tall, healthy dwarf, poor condition, and dead mangroves. Within each sample plot, all mangroves of diameter of breast height (dbh) greater than 2.5 cm were identified and their height, condition and dbh recorded. An estimated Leaf Area Index (LAI) value also was obtained for each sample and the shortest distance from the center of each sample plot to open flowing water was determined using a geographic information system (GIS) overlay procedure. These data were then used to calculate mean values for the four classes as well as to determine stem densities, basal areas, and the Shannon-Wiener diversity index. In order to assess the appropriateness of this mangrove classification scheme a discriminant analysis approach was then applied to these field data. The results indicate this forest has undergone severe degradation, with decreasing mean tree heights, mean dbh and species diversity. In regards to the discriminant analysis procedure, further classification of these field plots and cross-validation based on these significant variables provided high classification accuracy thus validating the appropriateness of the satellite based image classification scheme. Moreover, the discriminant analysis indicated that the estimated LAI, mean height, and mean dbh are significant in the separation of the classification of mangrove forest condition along these field sample plots.

The spectral separability of southern pines is a perplexing issue due to limited variance of spectral reflectance in species with similar morphological characteristics. Understory vegetation reflectance may exacerbate the ability to accurately identify various overstory tree species, specifically those of longleaf and loblolly pines in the southeastern US. In this study, identification of target level overstory crowns with varying degrees of understory vegetation cover based on fire return frequency was used to assess the role of understory reflectance on target crown species discernment. Seasonal variations of understory vegetation in late dormant and late growing seasons were compared for disparities in potential reflectance contribution from understory vegetation. Overall, the impact of understory vegetation was considered negligible in the spectral separability of longleaf and loblolly pines based on discriminant analysis results. Classification of WorldView-2 relative spectral profiles resulted in overall accuracies of 92% for dormant season and 96% for growing season imagery.

The objective of this study is to establish whether companies that utilise their resources more efficiently present specific characteristics in their financial profile, and whether on the basis of these characteristics a classification model can be constructed that includes, alongside resource utilisation measures, predictors related to other financial dimensions calculated from published information. The- research proceeds by examining the factors influencing companies' performance, and the reliabilty of published accounts. Discriminant analysis is chosen as the most appropriate technique of analysis. Its applications in the field of financial analysis are discussed -and an examination of the discriminant analysis technique is undertaken. For reasons of comparability and access to a large quantity of information, the analytical part of the study is based on data extracted from a computer readable tape provided by Extel Statistical Services Ltd. It starts by describing the financial variables to be used later on in the study, and proposing a classification framework that would be of assistance in identifying the financial dimensions of importance in relation to the problem under investigation. A discriminant model that correctly classifies 85 per cent of the companies is then constructed. It includes, besides measures of resources utilisation, measures of financial levarage, working capital management, cash position and stability of past performance. The-part of the analysis on the identification of potential well performing companies indicates that, although specific characteristics can be noticed up to five year before, it is only possible to construct a classification model with sufficient accuracy one year before a high level of performance is actually reached. Finally, an index of financial performance based on normal approximations of the z-score distributions from the model used to identify well performing companies is suggested and an assessment of the structural change experienced by companies rising from a less well to a well performaing status is presented. / Algerian Ministere de l'Hydraulique