Physical Sciences and Mathematics Commons^™

Model Checks For Two-Sample Location-Scale, Atefeh Javidialsaadi

Dissertations

Two-sample location-scale refers to a model that permits a pair of standardized random variables to have a common distribution. This means that if X₁ and X₂ are two random variables with means µ₁ and µ₂ and standard deviations ?₁ and ?₂, then (X₁ - µ₁)/?₁ and (X₂ - µ₂)/?₂ have some common unspecified standard or base distribution F₀. Function-based hypothesis testing for these models refers to formal tests that would help determine whether or not two samples may have come from some location-scale …

Dependent Censoring In Survival Analysis, Zhongcheng Lin

Dissertations

This dissertation mainly consists of two parts. In the first part, some properties of bivariate Archimedean Copulas formed by two time-to-event random variables are discussed under the setting of left censoring, where these two variables are subject to one left-censored independent variable respectively. Some distributional results for their joint cdf under different censoring patterns are presented. Those results are expected to be useful in both model fitting and checking procedures for Archimedean copula models with bivariate left-censored data. As an application of the theoretical results that are obtained, a moment estimator of the dependence parameter in Archimedean copula models is …

On Non-Linear Network Embedding Methods, Huong Yen Le

Dissertations

As a linear method, spectral clustering is the only network embedding algorithm that offers both a provably fast computation and an advanced theoretical understanding. The accuracy of spectral clustering depends on the Cheeger ratio defined as the ratio between the graph conductance and the 2nd smallest eigenvalue of its normalizedLaplacian. In several graph families whose Cheeger ratio reaches its upper bound of Theta(n), the approximation power of spectral clustering is proven to perform poorly. Moreover, recent non-linear network embedding methods have surpassed spectral clustering by state-of-the-art performance with little to no theoretical understanding to back them.

The dissertation includes work …

Modeling Dewetting, Demixing, And Thermal Effects In Nanoscale Metal Films, Ryan Howard Allaire

Dissertations

Thin film dynamics, particularly on the nanoscale, is a topic of extensive interest. The process by which thin liquids evolve is far from trivial and can lead to dewetting and drop formation. Understanding this process involves not only resolving the fluid mechanical aspects of the problem, but also requires the coupling of other physical processes, including liquid-solid interactions, thermal transport, and dependence of material parameters on temperature and material composition. The focus of this dissertation is on the mathematical modeling and simulation of nanoscale liquid metal films, which are deposited on thermally conductive substrates, liquefied by laser heating, and subsequently …

Modeling And Design Optimization For Membrane Filters, Yixuan Sun

Dissertations

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for: (1) pleated membrane …

Asymmetric Multivariate Archimedean Copula Models And Semi-Competing Risks Data Analysis, Ziyan Guo

Dissertations

Many multivariate models have been proposed and developed to model high dimensional data when the dimension of a data set is greater than 2 (d ≥ 3). The existing multivariate models often force the “exchangeable” structure for part or the whole model, are not very flexible which tends to be of limited use in practice. There is a demand for developing and studying multivariate models with any pre-specified bivariate margins.

Suppose there exists such a class of flexible models with any pre-specified bivariate margins. Given a multivariate data, what is the distribution function and how to easily estimate the parameters …

Stationary Probability Distributions Of Stochastic Gradient Descent And The Success And Failure Of The Diffusion Approximation, William Joseph Mccann

Theses

In this thesis, Stochastic Gradient Descent (SGD), an optimization method originally popular due to its computational efficiency, is analyzed using Markov chain methods. We compute both numerically, and in some cases analytically, the stationary probability distributions (invariant measures) for the SGD Markov operator over all step sizes or learning rates. The stationary probability distributions provide insight into how the long-time behavior of SGD samples the objective function minimum.

A key focus of this thesis is to provide a systematic study in one dimension comparing the exact SGD stationary distributions to the Fokker-Planck diffusion approximation equations —which are commonly used in …

Mechanisms Of Oscillations And Polyglot Entrainment In Neuronal And Circadian Models, Emel Khan

Dissertations

Entrainment is a type of synchronization in which the period of an endogenous oscillator matches the period of an external forcing signal and a stable phase relationship is maintained between them. Entrainment patterns are described in terms of the number of input oscillations (N) that are phase-locked to a number of output oscillations (M), referred to as N:M patterns. Arnold tongue diagrams are used to depict the regions of N:M entrainment patterns in the input period-amplitude parameter space. Although the entrainment of self-sustained oscillators by periodic forcing are well investigated is a well-studied problem, entrainment of damped oscillators has been …