Mathematics Commons^™

Large Deviations For Self Intersection Local Times Of Ornstein-Uhlenbeck Processes, Apostolos Gournaris

Doctoral Dissertations

In the area of large deviations, people concern about the asymptotic computation of small probabilities on an exponential scale. The general form of large deviations can be roughly described as: P{Yn ∈ A} ≈ exp{−bnI(A)} (n → ∞), for a random sequence {Yn}, a positive sequence bn with bn → ∞, and a coefficient I(A) ≥ 0. In applications, we often concern about the probability that the random variables take large values, that is we concern about the P{Yn ≥ λ}, where λ > 0. Here, we consider the Ornstein-Uhlenbeck process, study the properties of the local times and self intersection …

Advances In Differentially Methylated Region Detection And Cure Survival Models, Daniel Ahmed Alhassan

Doctoral Dissertations

"This dissertation focuses on two areas of statistics: DNA methylation and survival analysis. The first part of the dissertation pertains to the detection of differentially methylated regions in the human genome. The varying distribution of gaps between succeeding genomic locations, which are represented on the microarray used to quantify methylation, makes it challenging to identify regions that have differential methylation. This emphasizes the need to properly account for the correlation in methylation shared by nearby locations within a specific genomic distance. In this work, a normalized kernel-weighted statistic is proposed to obtain an optimal amount of "information" from neighboring locations …

Recurrent Event Data Analysis With Mismeasured Covariates, Ravinath Alahakoon Mudiyanselage

Doctoral Dissertations

"Consider a study with n units wherein every unit is monitored for the occurrence of an event that can recur with random end of monitoring. At each recurrence, p concomitant variables associated to the event recurrence are recorded with q (q ≤ p) collected with errors. Of interest in this dissertation is the estimation of the regression parameters of event time regression models accounting for the covariates. To circumvent the problem of bias and consistency associated with model's parameter estimation in the presence of measurement errors, we propose inference for corrected estimating functions with well-behaved roots under additive measurement errors …

Efficient High Order Ensemble For Fluid Flow, John Carter

Doctoral Dissertations

"This thesis proposes efficient ensemble-based algorithms for solving the full and reduced Magnetohydrodynamics (MHD) equations. The proposed ensemble methods require solving only one linear system with multiple right-hand sides for different realizations, reducing computational cost and simulation time. Four algorithms utilize a Generalized Positive Auxiliary Variable (GPAV) approach and are demonstrated to be second-order accurate and unconditionally stable with respect to the system energy through comprehensive stability analyses and error tests. Two algorithms make use of Artificial Compressibility (AC) to update pressure and a solenoidal constraint for the magnetic field. Numerical simulations are provided to illustrate theoretical results and demonstrate …

Essays On Conditional Heteroscedastic Time Series Models With Asymmetry, Long Memory, And Structural Changes, K C M R Anjana Bandara Yatawara

Doctoral Dissertations

"The volatility of asset returns is usually time-varying, necessitating the introduction of models with a conditional heteroskedastic variance structure. In this dissertation, several existing formulations, motivated by the Generalized Autoregressive Conditional Heteroskedastic (GARCH) type models, are further generalized to accommodate more dynamic features of asset returns such as asymmetry, long memory, and structural breaks. First, we introduce a hybrid structure that combines short-memory asymmetric Glosten, Jagannathan, and Runkle (GJR) formulation and the long-memory fractionally integrated GARCH (FIGARCH) process for modeling financial volatility. This formulation not only can model volatility clusters and capture asymmetry but also considers the characteristic of long …

Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy

Doctoral Dissertations

Many ecological systems exhibit noisy period-2 oscillations and, when they are spatially extended, they undergo phase transition from synchrony to incoherence in the Ising universality class. Period-2 cycles have two possible phases of oscillations and can be represented as two states in the bistable systems. Understanding the dynamics of ecological systems by representing their oscillations as bistable states and developing dynamical models using the tools from statistical physics to predict their future states is the focus of this thesis. As the ecological oscillators with two-cycle behavior undergo phase transitions in the Ising universality class, many features of synchrony and equilibrium …

Survivor Bond Models For Securitizing Longevity Risk, Priscilla Mansah Codjoe

Doctoral Dissertations

"Longevity risk is the risk that a reference population’s mortality rates deviate from what is projected from prior life tables. This is due to discoveries in biological sciences, improved public health measures, and nutrition, which have dramatically increased life expectancy. Longevity risk raises life insurers’ liability, increasing product costs and reserves. Securitization through longevity derivatives is a way of dealing with this risk.

To enhance the pricing of life contingent products, we present an additive type mortality model in the style of the Lee-Carter. This model incorporates policyholder covariates. By using counting processes and martingale machinery, we obtain close form …

Dynamic Neuromechanical Sets For Locomotion, Aravind Sundararajan

Doctoral Dissertations

Most biological systems employ multiple redundant actuators, which is a complicated problem of controls and analysis. Unless assumptions about how the brain and body work together, and assumptions about how the body prioritizes tasks are applied, it is not possible to find the actuator controls. The purpose of this research is to develop computational tools for the analysis of arbitrary musculoskeletal models that employ redundant actuators. Instead of relying primarily on optimization frameworks and numerical methods or task prioritization schemes used typically in biomechanics to find a singular solution for actuator controls, tools for feasible sets analysis are instead developed …

Bayesian Topological Machine Learning, Christopher A. Oballe

Doctoral Dissertations

Topological data analysis encompasses a broad set of ideas and techniques that address 1) how to rigorously define and summarize the shape of data, and 2) use these constructs for inference. This dissertation addresses the second problem by developing new inferential tools for topological data analysis and applying them to solve real-world data problems. First, a Bayesian framework to approximate probability distributions of persistence diagrams is established. The key insight underpinning this framework is that persistence diagrams may be viewed as Poisson point processes with prior intensities. With this assumption in hand, one may compute posterior intensities by adopting techniques …

Fuzzy Logistic Regression For Detecting Differential Dna Methylation Regions, Tarek M. Bubaker Bennaser

Doctoral Dissertations

“Epigenetics is the study of changes in gene activity or function that are not related to a change in the DNA sequence. DNA methylation is one of the main types of epigenetic modifications, that occur when a methyl chemical group attaches to a cytosine on the DNA sequence. Although the sequence does not change, the addition of a methyl group can change the way genes are expressed and produce different phenotypes. DNA methylation is involved in many biological processes and has important implications in the fields of biomedicine and agriculture.

Statistical methods have been developed to compare DNA methylation at …

Asymptotic Behavior Of The Random Logistic Model And Of Parallel Bayesian Logspline Density Estimators, Konstandinos Kotsiopoulos

Doctoral Dissertations

This dissertation is comprised of two separate projects. The first concerns a Markov chain called the Random Logistic Model. For r in (0,4] and x in [0,1] the logistic map f_r(x) = rx(1 - x) defines, for positive integer t, the dynamical system x_r(t + 1) = f(x_r(t)) on [0,1], where x_r(1) = x. The interplay between this dynamical system and the Markov chain x_r,N(t) defined by perturbing the logistic map by truncated Gaussian noise scaled by N^-1/2, where N -> infinity, is studied. A natural question is …

On Modeling Quantities For Insurer Solvency Against Catastrophe Under Some Markovian Assumptions, Daniel Jefferson Geiger

Doctoral Dissertations

"Insurance companies sometimes face catastrophic losses, yet they must remain solvent enough to meet the legal obligation of covering all claims. Catastrophes can result in large damages to the policyholders, causing the arrival of numerous claims to insurance companies at once. Furthermore, the severity of an event could impact the time until the next occurrence. An insurer needs certain levels of startup capital to meet all claims, and then must have adequate reserves on a continual basis, even more so when catastrophes occur. This work examines two facets of these matters: for an infinite time horizon, we extend and develop …

Statistical Computational Topology And Geometry For Understanding Data, Joshua Lee Mike

Doctoral Dissertations

Here we describe three projects involving data analysis which focus on engaging statistics with the geometry and/or topology of the data.

The first project involves the development and implementation of kernel density estimation for persistence diagrams. These kernel densities consider neighborhoods for every feature in the center diagram and gives to each feature an independent, orthogonal direction. The creation of kernel densities in this realm yields a (previously unavailable) full characterization of the (random) geometry of a dataspace or data distribution.

In the second project, cohomology is used to guide a search for kidney exchange cycles within a kidney paired …

Dependence Structures In Lévy-Type Markov Processes, Eddie Brendan Tu

Doctoral Dissertations

In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and Lévy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of Lévy-type Markov processes include Lévy processes and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with Lévy noise, and subordinated Markov processes. Other examples of Lévy-type Markov processes are time-inhomogeneous Feller evolution systems (FES), which include additive processes. We will provide a tour of various forms of positive dependence, which include association, positive supermodular association (PSA), positive supermodular dependence (PSD), and positive …

Local Holomorphic Extension Of Cauchy Riemann Functions, Brijitta Antony

Doctoral Dissertations

"The purpose of this dissertation is to give an analytic disc approach to the CR extension problem. Analytic discs give a very convenient tool for holomorphic extension of CR functions. The type function is introduced and showed how these type functions have direct application to important questions about CR extension. In this dissertation the CR extension theorem is proved for a rigid hypersurface M in C² given by y = (Re ω)^m(Im ω)ⁿ where m and n are non-negative integers. If the type function is identically zero at the origin, then there is no CR extension. …

Modeling Daily Electricity Load Curve Using Cubic Splines And Functional Principal Components, Abdelmonaem Salem Jornaz

Doctoral Dissertations

"Forecasting electricity load is very important to the electric utilities as well as producers of power because accurate predictions can cut down costs by avoiding power shortages or surpluses. Of specific interest is the 24-hour daily electricity load profile, which provides insight into periods of high demand and periods where the use of electricity is at a minimum. Researchers have proposed many approaches to modeling electricity prices, real-time load, and day-ahead demand, with varying success. In this dissertation three new approaches to modeling and forecasting the 24-hour daily electricity load profiles are presented. The application of the proposed methods is …

Sensitivity Of Mixed Models To Computational Algorithms Of Time Series Data, Gunaime Nevine

Doctoral Dissertations

Statistical analysis is influenced by implementation of the algorithms used to execute the computations associated with various statistical techniques. Over many years; very important criteria for model comparison has been studied and examined, and two algorithms on a single dataset have been performed numerous times. The goal of this research is not comparing two or more models on one dataset, but comparing models with numerical algorithms that have been used to solve them on the same dataset.

In this research, different models have been broadly applied in modeling and their contrasting which are affected by the numerical algorithms in different …

Essays On Unit Root Testing In Time Series, Xiao Zhong

Doctoral Dissertations

"Unit root tests are frequently employed by applied time series analysts to determine if the underlying model that generates an empirical process has a component that can be well-described by a random walk. More specifically, when the time series can be modeled using an autoregressive moving average (ARMA) process, such tests aim to determine if the autoregressive (AR) polynomial has one or more unit roots. The effect of economic shocks do not diminish with time when there is one or more unit roots in the AR polynomial, whereas the contribution of shocks decay geometrically when all the roots are outside …

Small Sample Saddlepoint Confidence Intervals In Epidemiology, Pasan Manuranga Edirisinghe

Doctoral Dissertations

"In section 1, we develop a novel method of confidence interval construction for directly standardized rates. These intervals involve saddlepoint approximations to the intractable distribution of a weighted sum of Poisson random variables and the determination of hypothetical Poisson mean values for each of the age groups. Simulation studies show that, in terms of coverage probability and length, the saddlepoint confidence interval (SP) outperforms four competing confidence intervals obtained from the moment matching (M8), gamma-based (G1,G4) and ABC bootstrap (ABC) methods.

In section 2, we first consider Brillinger's classical model for a vital rate estimate with a random denominator. We …

Small Sample Umpu Equivalence Testing Based On Saddlepoint Approximations, Renren Zhao

Doctoral Dissertations

"In the first section, we consider small sample equivalence tests for exponentiality. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require a much larger sample size, in comparison to classical "difference tests", to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. …

Impacts Of Climate Change On The Evolution Of The Electrical Grid, Melissa Ree Allen

Doctoral Dissertations

Maintaining interdependent infrastructures exposed to a changing climate requires understanding 1) the local impact on power assets; 2) how the infrastructure will evolve as the demand for infrastructure changes location and volume and; 3) what vulnerabilities are introduced by these changing infrastructure topologies. This dissertation attempts to develop a methodology that will a) downscale the climate direct effect on the infrastructure; b) allow population to redistribute in response to increasing extreme events that will increase under climate impacts; and c) project new distributions of electricity demand in the mid-21^st century.

The research was structured in three parts. The first …

Improvements On Segment Based Contours Method For Dna Microarray Image Segmentation, Yang Li

Doctoral Dissertations

DNA microarray is an efficient biotechnology tool for scientists to measure the expression levels of large numbers of genes, simultaneously. To obtain the gene expression, microarray image analysis needs to be conducted. Microarray image segmentation is a fundamental step in the microarray analysis process. Segmentation gives the intensities of each probe spot in the array image, and those intensities are used to calculate the gene expression in subsequent analysis procedures. Therefore, more accurate and efficient microarray image segmentation methods are being pursued all the time.

In this dissertation, we are making efforts to obtain more accurate image segmentation results. We …

Performance Modeling And Optimization Techniques For Heterogeneous Computing, Supada Laosooksathit

Doctoral Dissertations

Since Graphics Processing Units (CPUs) have increasingly gained popularity amoung non-graphic and computational applications, known as General-Purpose computation on GPU (GPGPU), CPUs have been deployed in many clusters, including the world's fastest supercomputer. However, to make the most efficiency from a GPU system, one should consider both performance and reliability of the system.

This dissertation makes four major contributions. First, the two-level checkpoint/restart protocol that aims to reduce the checkpoint and recovery costs with a latency hiding strategy in a system between a CPU (Central Processing Unit) and a GPU is proposed. The experimental results and analysis reveals some benefits, …

New Microarray Image Segmentation Using Segmentation Based Contours Method, Yuan Cheng

Doctoral Dissertations

The goal of the research developed in this dissertation is to develop a more accurate segmentation method for Affymetrix microarray images. The Affymetrix microarray biotechnologies have become increasingly important in the biomedical research field. Affymetrix microarray images are widely used in disease diagnostics and disease control. They are capable of monitoring the expression levels of thousands of genes simultaneously. Hence, scientists can get a deep understanding on genomic regulation, interaction and expression by using such tools.

We also introduce a novel Affymetrix microarray image simulation model and how the Affymetrix microarray image is simulated by using this model. This simulation …

Sieve Bootstrap Based Prediction Intervals And Unit Root Tests For Time Series, Maduka Rupasinghe

Doctoral Dissertations

"The application of the sieve bootstrap procedure, which resamples residuals obtained by fitting a finite autoregressvie (AR) approximation to empirical time series, to obtaining prediction intervals for integrated, long-memory, and seasonal time series as well as constructing a test for seasonal unit roots, is considered. The advantage of this resampling method is that it does not require knowledge about the underlying process generating a given time series and has been shown to work well for ARMA processes. We extend the application of the sieve bootstrap to ARIMA and FARIMA processes. The asymptotic properties of the sieve bootstrap prediction intervals for …

Mathematical And Empirical Modeling Of Chemical Reactions In A Microreactor, Jing Hu

Doctoral Dissertations

This dissertation is concerned with mathematical and empirical modeling to simulate three important chemical reactions (cyclohexene hydrogenation and dehydrogenation, preferential oxidation of carbon monoxide, and the Fischer-Tropsch (F-T) synthesis in a microreaction system.

Empirical modeling and optimization techniques based on experimental design (Central Composite Design (CCD)) and response surface methodology were applied to these three chemical reactions. Regression models were built, and the operating conditions (such as temperature, the ratio of the reactants, and total flow rate) which maximize reactant conversion and product selectivity were determined for each reaction.

A probability model for predicting the probability that a certain species …

Fuzzy Product -Limit Estimators: Soft Computing In The Presence Of Very Small And Highly Censored Data Sets, Kian Lawrence Pokorny

Doctoral Dissertations

When very few data are available and a high proportion of the data is censored, accurate estimates of reliability are problematic. Standard statistical methods require a more complete data set, and with any fewer data, expert knowledge or heuristic methods are required. In the current research a computational system is developed that obtains a survival curve, point estimate, and confidence interval about the point estimate.

The system uses numerical methods to define fuzzy membership functions about each data point that quantify uncertainty due to censoring. The “fuzzy” data are then used to estimate a survival curve, and the mean survival …

Cramer-Rao Bound And Optimal Amplitude Estimator Of Superimposed Sinusoidal Signals With Unknown Frequencies, Shaohui Jia

Doctoral Dissertations

This dissertation addresses optimally estimating the amplitudes of superimposed sinusoidal signals with unknown frequencies. The Cramer-Rao Bound of estimating the amplitudes in white Gaussian noise is given, and the maximum likelihood estimator of the amplitudes in this case is shown to be asymptotically efficient at high signal to noise ratio but finite sample size. Applying the theoretical results to signal resolutions, it is shown that the optimal resolution of multiple signals using a finite sample is given by the maximum likelihood estimator of the amplitudes of signals.