Thesis (M. Phil.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 81-89). Also available in print.

Thesis (Ph. D.)--University of Florida, 1981. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 108-109).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 124-131).

The focus of this thesis is on the design and selection of systolic architectures for ASIC implementation of the real-time digital signal processing task of Modi- fied Covariance spectral estimation. When used with pulsed Doppler ultrasound blood flow detectors, the Modified Covariance spectral estimator offers increased sensitivity in the detection of arterial disease over conventional Fourier transform based methods. The systolic model of computation is considered because through pipelining and parallel processing high levels of concurrency can be achieved to attain the nec- essary throughput for real-time operation. Systolic arrays of simple processing units are also well suited for implementation on VLSI. The versatility of the de- sign of systolic arrays using the rigorous data dependence graph methodology is demonstrated throughout the thesis by application to all sections of the spectral estimator design at both word and bit levels. Systolic array design for the model order 4 Modified Covariance spectral estima- tor, known to offer accurate estimation of blood flow mean velocity and d1stur- bance at an acceptable computational burden, is initially discussed. A variety of problem size dependent systolic arrays for real-time implementation of the fixed model order spectral estimator are designed using data dependence graph mapping methods. Optimal designs are chosen by comparison of hardware, com- munication and control costs, as well as efficiency, timing, data flow and accuracy considerations. A cost/benefit analysis, based on results from structural simula- tion of the arrays, allows the most suitable word-lengths to be chosen. Problem size independent systolic arrays are then discussed as means of coping with the huge increases in computational burden for a Modified Covariance spec- tral estimator which is programmable up to high model orders. This type of array can be used to reduce the number of PEs and increase efficiency when compared to the problem size dependent arrays and the research culminates in the proposal of a novel spiral systolic array for Cholesky decomposition.

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.

A Wald type test with the wrong dispersion matrix is used when the dispersion matrix is not a consistent estimator of the asymptotic covariance matrixof the test statistic. One class of such tests occurs when there are k groups and it is assumed that the population covariance matrices from the k groups are equal, but the common covariance matrix assumption does not hold. The pooled t test, one way AVOVA F test, and one way MANOVA F test are examples of this class. Two bootstrap confidence regions are modified to obtain large sample Wald type tests with the wrong dispersion matrix.

This thesis develops two semiparametric methods for censored survival data when the covariates involved are time-dependent. Respectively in the two parts of this thesis, we introduce an interquantile regression model and a censored quantile regression model that account for the commonly observed time-dependent covariates in survival analysis. The proposed quantile-based techniques offer a greater model flexibility comparing to the Cox proportional hazards model and the accelerated failure time model. The first half of this thesis introduces a censored interquantile regression model with time-dependent covariates. Conventionally, censored quantile regression stipulates a specific, pointwise conditional quantile of the survival time given covariates. Despite its model flexibility and straightforward interpretation, the pointwise formulation oftentimes yields rather unstable estimates across neighbouring quantile levels with large variances. In view of this phenomenon, we propose a new class of censored interquantile regression models with time-dependent covariates that can capture the relationship between the failure time and the covariate processes of a target population that falls within a specific quantile bracket. The pooling of information within a homogeneous neighbourhood facilitates more efficient estimates hence more consistent conclusion on statistical significances of the variables concerned. This new formulation can also be regarded as a generalization of the accelerated failure time model for survival data in the sense that it relaxes the assumption of global homogeneity for the error at all quantile levels. By introducing a class of weighted rank-based estimation procedure, our framework allows a quantile-based inference on the covariate effect with a less restrictive set of assumptions. Numerical studies demonstrate that the proposed estimator outperforms existing alternatives under various settings in terms of smaller empirical bias and standard deviation. A perturbation-based resampling method is also developed to reconcile the asymptotic distribution of the parameter estimates. Finally, consistency and weak convergence of the proposed estimator are established via empirical process theory. In the second half of this thesis, we propose a class of censored quantile regression models for right censored failure time data with time-dependent covariates that only requires a standard conditionally independent censorship. Upon a quantile based transformation, a system of functional estimating equations for the quantile parameters is derived based on the martingale construction. While time-dependent covariates naturally arise in time to event analysis, the few existing literature requires either an independent censoring mechanism or a fully observed covariate process even after the event has occured. The proposed formulation extends the existing censored quantile regression model so that only the covariate history up to the observed event time is required as in the Cox proportional hazards model for time-dependent covariates. A recursive algorithm is developed to evaluate the estimator numerically. Asymptotic properties including uniform consistency and weak convergence of the proposed estimator as a process of the quantile level is established. Monte Carlo simulations and numerical studies on the clinical trial data of the AIDS Clinical Trials Group is presented to illustrate the numerical performance of the proposed estimator.

No / In this work we present the angle of arrival estimation techniques and their comparison at different values of SNR using a 5 element UCA. The techniques that have been considered include phase interferometry, Multiple Signal Classification and covariance. The results show that for very low values of SNR the performance of the covariance matrix based algorithm is the best but for slightly higher values of SNR, MUSIC algorithm outperforms covariance.

Estimation of covariance matrix and its inverse is of great importance in multivariate statistics with broad applications such as dimension reduction, portfolio optimization, linear discriminant analysis and gene expression analysis. However, accurate estimation of covariance or inverse covariance matrices is challenging due to the positive definiteness constraint and large number of parameters, especially in the high-dimensional cases. In this thesis, I develop several approaches for estimating large covariance and inverse covariance matrices with different applications. In Chapter 2, I consider an estimation of time-varying covariance matrices in the analysis of multivariate financial data. An order-invariant Cholesky-log-GARCH model is developed for estimating the time-varying covariance matrices based on the modified Cholesky decomposition. This decomposition provides a statistically interpretable parametrization of the covariance matrix. The key idea of the proposed model is to consider an ensemble estimation of covariance matrix based on the multiple permutations of variables. Chapter 3 investigates the sparse estimation of inverse covariance matrix for the highdimensional data. This problem has attracted wide attention, since zero entries in the inverse covariance matrix imply the conditional independence among variables. I propose an orderinvariant sparse estimator based on the modified Cholesky decomposition. The proposed estimator is obtained by assembling a set of estimates from the multiple permutations of variables. Hard thresholding is imposed on the ensemble Cholesky factor to encourage the sparsity in the estimated inverse covariance matrix. The proposed method is able to catch the correct sparse structure of the inverse covariance matrix. Chapter 4 focuses on the sparse estimation of large covariance matrix. Traditional estimation approach is known to perform poorly in the high dimensions. I propose a positive-definite estimator for the covariance matrix using the modified Cholesky decomposition. Such a decomposition provides a exibility to obtain a set of covariance matrix estimates. The proposed method considers an ensemble estimator as the center" of these available estimates with respect to Frobenius norm. The proposed estimator is not only guaranteed to be positive definite, but also able to catch the underlying sparse structure of the true matrix. / Ph. D.

A single compressor station site along a gathering line network was monitored for fugitive methane emissions to quantify long-term emissions in Appalachia Virginia. Continuous monitoring was conducted from January 2021 to April 2021. The compressor station undergoing monitoring operated two CAT3516 Tale and one CAT3516 B engines operating at 80% of max output flow. Data presented on methane emissions during this period was gathered with an eddy covariance monitoring station. This station was equipped with an LI-7700 methane analyzer, LI-7500A - CO_2/H_2 O analyzer as well as a sonic anemometer these sensors could be observed remotely through cellular connection. The data is represented in flux output ((µmol)/(s m^2 )) as well as kg CO_2 equivalence of methane outlined by the EPA greenhouse gas inventory. The average daily emissions for this compressor station are estimated to be 136 kg CO_2 equivalent emissions. This study shows that the site during the observational period the compressor station emitted on average are estimated to be 5.43 kg of CH_4 per day. / Master of Science / There has been an increased interest in quantifying and recording methane (CH_4) emissions among all sectors. A main focus of interest among methane is to understand fugitive gasses and emissions resulting from the natural gas sector. Leaks along pipelines are most likely occurring at connection points between components. This study aimed to continuously monitor a pipeline compressor station in Appalachia Virginia. Compressor stations are just one component of the pipeline network as well as the natural gas production and delivery chain attributed with CH_4 emissions. To monitor methane emissions at the site a stationary eddy covariance monitoring station was installed that was equipped with an open path methane analyzer, open path CO_2 and H_2 O analyzer, and a sonic anemometer. The data gathered was used to calculate the flux of methane which is the amount of methane being generated or absorbed by the area of interest. The goal of this study was to continuously monitor methane emissions of a natural gas compressor station. Data presented in this study was collected from January 2021 to April 2021. Data was presented in the flux output ((µmol)/(s m^2 )) as well as kg CO_2 equivalence of methane outlined by the EPA greenhouse gas inventory.