Dissertation Defense Announcements

In this dissertation, we propose a broad class of so-called Cox-Aalen transformation models that incorporate both multiplicative and additive covariate effects on the baseline hazard function through a transformation framework. The proposed model offers a high degree of flexibility and versatility, encompassing the Cox-Aalen model and transformation models as special cases. For right-censored data, we propose an estimating equation approach and devise an Expectation-Solving (ES) algorithm that involves fast and robust calculations. The resulting estimator is shown to be consistent and asymptotically normal via empirical process techniques. Finally, we assess the performance of the proposed procedures by conducting simulation studies and applying them in two randomized, placebo-controlled HIV prevention efficacy trials.

We also consider the regression analysis of the Cox-Aalen transformation models with partly interval-censored data, which comprise exact and interval-censored observations. We construct a set of estimating equations and implement an ES algorithm that ensures stability and fast convergence. Under regularity assumptions, we demonstrate that the estimators obtained are consistent and asymptotically normal, and we propose using weighted bootstrapping techniques to estimate their variance consistently. To evaluate the proposed methods, we perform thorough simulation experiments and apply them to analyze data from a randomized HIV/AIDS trial.

The increasing prevalence of roads and vehicle traffic, most particularly in urban areas, has a corresponding impact on road mortality, especially for avian species that make use of foraging opportunities along roadside verges. In many cases, raptors, or birds of prey, are vulnerable to vehicle collisions because they forage along roads. The purpose of my research was to conducted a comprehensive investigation into the traffic, habitat and road verge factors that influence collision risk for both nocturnal and diurnal raptors. In addition, I examined the impact that species and individual traits have on the location of vehicle collisions involving birds of prey. I expected to find a notable difference in collision vulnerability between nocturnal and diurnal species. I also expected that road verge vegetation would play a significant role in vehicle collision risk for birds of prey.
Although I did not observe a significant difference in collision risk for raptors based on time of activity, I did find that prey cover in the form of complex vegetation along road verges was an important predictor of collision risk. Dense brush, shrubs or tall grass provide habitat for prey items such as small birds and mammals, which in turn attracts foraging raptors to roadsides, thus increasing the risk of being struck by a passing vehicle.
My analysis of species and individual traits showed that body size and reproductive output were the most important predictors of collision risk. Larger species and those with smaller clutch sizes were most likely to be hit by cars, regardless of road and road verge conditions or habitat characteristics.

Studying urban dynamics is essential given the ever-increasing changes in urban areas with all its ensuing consequences, whether negative or positive. It is of paramount importance to take into account the temporal dimension of urban dynamics when studying its patterns and processes. Nevertheless, the majority of studies overlook this consideration and take cross-sectional research approaches. Moreover, a large body of literature in urban dynamics is dedicated to the explanatory analysis and causal inference only, neglecting the importance of predictive analysis. Addressing these two main gaps, this research explores urban dynamics through both causal inference and predictive modeling using longitudinal research designs. Urban dynamics are studied from two aspects in this work; transportation/land-use interactions, and economic growth. In the first article, the impact of built environment on commuting duration is assessed in 2000 and 2015 in Mecklenburg County, NC using spatial panel data models. Results show that the built environment has a statistically significant impact on commuting duration. However, it is important to note that the practical magnitude of the impact is small. In the second and third articles, the business performance of businesses are forecasted for non-business services and business services respectively in Mecklenburg County, NC, using recurrent neural networks long short-term memory deep learning method. After building and training the sequential model, its predictive performance is assessed using out-of-sample evaluation.

This dissertation is focused on developing efficient numerical methods and theoretical analysis for solving various inverse problems that arise in the fields of mathematics, physics, engineering, and beyond. We propose in this dissertation a unified framework with two stages to solve severely ill-posed and highly nonlinear inverse problems. In the first stage, we derive a system of partial differential equations by introducing a new variable and truncating the Fourier series of the solution to the governing equation. In the second stage, we solve the system derived in the first stage using the quasi-reversibility method, the Carleman contraction mapping method, and the convexification method. The obtained solutions of this stage directly yield the desired solutions to the inverse problems. An important contribution of the dissertation is that we will rigorously and numerically prove the efficiency of this framework, including its global convergence to the true solution. The analytic proofs are based on some Carleman estimates, and the numerical proofs are provided by successfully testing our methods with highly noisy simulated data and experimental data provided by US Army Research Laboratory engineers.