PhD, University of Cincinnati, 2025, Arts and Sciences: Mathematical Sciences

In this dissertation, we investigate infinite urn schemes and some applications to random matrices and the Chinese restaurant process. The connection between these objects lies in the Kingman representation theorem, which allows us to study the Chinese restaurant process and random permutation matrices using urn scheme techniques. Let (pl)l?N denote a sequence of strictly positive non-increasing real numbers such that ? pl=1. An infinite urn scheme is a classical model in probability where balls are thrown one by one, independently, at urns indexed by N such that each ball lands in urn l with probability pl. If n balls are thrown, we denote the number of urns containing exactly j balls by Dn,j, and we are interested in the limiting distribution of the random variables Sn := n i=1 ai Dn,j for sequences of real numbers (ai)i?N. In Chapter 1, we prove a functional central limit theorem for Sn, where (ai)i?N is satisfying a power law type growth condition. Chapter 2 explains the connection between our results on infinite urn schemes to random permutation matrices. There several theorems are proved for the linear eigenvalue statistics and characteristic polynomials for permutation matrices following the two-parameter Chinese restaurant process. Additionally, in Chapter 3, we present some limit theorems for sums of weighted cycle counts for random permutations following the Ewens' distribution.

Committee: Yizao Wang Ph.D. (Committee Chair); Magda Peligrad Ph.D. (Committee Member); Wlodzimierz Bryc Ph.D. (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

Meta learning is an important sub-field of machine learning in which researchers study ways to train models that can update their parameters well. Many meta-learning papers are presented with strong experimental results to back them up, but oftentimes there is minimal theoretical study shown of how these algorithms perform. These algorithms thus become somewhat of a "black box", and this can hamper the advancement of the field. We discuss various ways in which mathematicians can help to bring clarity to the field of meta-learning, such as in studying algorithm convergence and generalization to unseen data. We show theoretical, as well as experimental, results of studying the DG-SharpMAML, AS-MAML, and AT-GMU algorithms designed to solve optimization problems in machine learning.

Committee: Robert Buckingham Ph.D. (Committee Chair); Justin Zhan Ph.D. (Committee Member); Deniz Bilman Ph.D. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

A cerebral or intracranial aneurysm, commonly referred to as a brain aneurysm, is a potentially life-threatening condition characterized by a bulge or ballooning in a blood vessel in the brain. This condition can affect individuals of any age and, if ruptured, can lead to hemorrhagic stroke, brain damage, and even death. The Brain Aneurysm Foundation reports that an estimated 6.8 million people in the United States have an unruptured brain aneurysm, with approximately 30,000 individuals experiencing a rupture each year. These ruptures have a high fatality rate, with 50% of cases being fatal and 66% of survivors experiencing lasting neurological impairments. Despite the risks, routine screening for brain aneurysms in healthy individuals is uncommon, and aneurysms are often discovered incidentally. Once it is identified, the decision to treat an aneurysm involves significant caution due to the complexity and risks of brain surgery. Hence, a thorough understanding of blood flow in the aneurysm region is essential for developing effective treatment plans. Current research has made strides in understanding aneurysm hemodynamics, mechanical modeling, and treatment devices like stents and coils. While these models offer insight, they often lack flexibility, focusing either on fixed, patient-specific geometries or oversimplified idealized structures that inadequately capture aneurysm dynamics. Recent computational advancements, including AI integration, have improved model accuracy and reduced simulation time, but inconsistencies across studies point to a need for more standardized frameworks. My PhD research addressed these gaps by developing a novel mathematical model that combines anatomical accuracy with flexibility, allowing for varied geometric properties while maintaining structural accuracy. This model aims to predict blood flow dynamics more effectively, supporting personalized aneurysm treatment planning and potentially improving clinical out (open full item for complete abstract)

Committee: Benjamin Vaughan Ph.D. (Committee Chair); Stephan Pelikan Ph.D. (Committee Member); Deniz Bilman Ph.D. (Committee Member) Subjects: Applied Mathematics

PhD, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

The generalized Hastings-McLeod solutions to the inhomogeneous Painleve-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian motions on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings-McLeod solutions as the inhomogeneous parameter approaches infinity using the Deift-Zhou nonlinear steepest-descent method for Riemann-Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically solutions in both regions.

Committee: Robert Buckingham Ph.D. (Committee Chair); Deniz Bilman Ph.D. (Committee Member); Bingyu Zhang Ph.D. (Committee Member); Yizao Wang Ph.D. (Committee Member) Subjects: Materials Science

PhD, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

In this dissertation, we study variational problems related to functions of bounded variation, sets of finite perimeter, and their nonlocal analogs in the metric measure space setting. We begin by considering the least gradient problem, min\{||Du||(Ω): u∊BV(Ω), u=f on ∂Ω}, where Ω⊂X is a bounded domain, f∊L^1(∂Ω) is a prescribed boundary function, and ||Du||(Ω) is the BV-energy of u in Ω. For domains satisfying certain geometric assumptions, we study the sub-class of L^1-boundary data for which solutions to the least gradient problem exist. While solutions are known to exist for continuous boundary data, it is also known that not all L^1-boundary data admit solutions, even in the Euclidean setting. We provide examples in the Euclidean setting which illustrate that the class of solvable L^1-boundary data is nonlocal in a certain sense and does not in general form a vector space. We then prove existence of solutions for various discontinuous boundary data in the metric measure space setting. In the second portion of this dissertation, we introduce and study the following modification of the least gradient problem in the metric measure space setting: min{||Du||(Ω):ψ_1≤ u ≤ψ_2 in Ω, f≤u≤g on ∂Ω}. Here, ψ_1 and ψ_2 are obstacle functions in the domain, and f and g are prescribed L^1-boundary data. We construct solutions to this problem when the obstacle functions and boundary data are continuous. As candidate functions need not be fixed on the boundary, standard compactness arguments fail to yield weak solutions of the form related to the standard least gradient problem. To overcome this issue, we introduce a notion of ε-weak solutions, which minimize BV-energy in slight enlargements of the domain. In the last portion of this dissertation, we adapt a nonlocal variational problem, originally studied by Caffarelli, Roquejoffre, and Savin in ℝ^n, to the metric measure space setting. This minimization problem involves sets of finite fractional perimete (open full item for complete abstract)

Committee: Nageswari Shanmugalingam Ph.D. (Committee Chair); Panu Kalevi Lahti Ph.D M.A B.A. (Committee Member); Stephan Pelikan Ph.D. (Committee Member); Gareth Speight Ph.D. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

This dissertation presents a comprehensive study of a novel enriched Discontinuous Galerkin (xDG) method, designed specifically for solving highly oscillatory differential equations, with a primary focus on its application to High-Intensity Focused Ultrasound (HIFU). HIFU is a medical procedure that employs ultrasound waves, ranging between 0.1 to 20 MHz, to target and ablate abnormal tissues within the body, serving as a motivating application for this research. Central to this study is a comparative analysis between the proposed enriched Symmetric Interior Penalty Galerkin (xSIPG) method and its predecessors, highlighting the advancements in solving crucial differential equations, notably the Helmholtz and Bioheat equations. These equations are pivotal for understanding the propagation of ultrasound waves and their interaction with human tissues. A significant achievement of this research is the optimization of penalty parameters within the xDG framework, which plays an essential role in the accuracy and computational efficiency of the methods. The results indicate xSIPG's significant improvement in modeling efficiency for HIFU simulations, potentially enhancing computational performance by up to three orders of magnitude compared to conventional FEM. Moreover, this study establishes the limitations of previous xDG methods in fully capturing the complexities of the HIFU model, a challenge addressed by the xSIPG approach. This advancement not only highlights the methodological leap facilitated by the xSIPG method but also reinforces the potential of applying xDG techniques to simulate HIFU.

Committee: Benjamin Vaughan Ph.D. (Committee Chair); Deniz Bilman Ph.D. (Committee Member); Sookkyung Lim Ph.D. (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

Much of the current literature about the Heisenberg group H is difficult for those who are in the beginning of their mathematical careers, yet H is endowed with an interesting structure which allows for the generalization of many aspects of analysis in Euclidean space. Such topics include continuity and stronger forms of the same, integral calculus, restrictions and extensions of functions, and Taylor's theorem. The goal of this thesis is to make more accessible a combination of these tenets and others, through examining a Whitney extension theorem in H. We start by building the fundamentals in a more familiar setting, namely in Euclidean 3-space. We then discuss H and its properties, including the notion of horizontality of curves in H. The concept of horizontality provides a natural segue to a version of Whitney's extension theorem for horizontal curves in H; we discuss the necessity and sufficiency of three criteria a curve in Hn must satisfy in order to have a smooth horizontal extension. We conclude by examining two other types of extension theorems, namely Lipschitz maps on metric spaces and continuous maps on normal topological spaces.

Committee: Nageswari Shanmugalingam Ph.D. (Committee Member); Gareth Speight Ph.D. (Committee Chair) Subjects: Mathematics

PhD, University of Cincinnati, 2024, Arts and Sciences: Mathematical Sciences

This dissertation mainly consists of three projects, based on \cite{Shi-2021}, \cite{RoShiZh-2023} and \cite{Shi-2023}. These investigate problems in analysis on metric spaces in various levels of generality. The first topic deals with a generalization of the Gromov-Hausdorff distance to the class of all noncomplete locally compact precompact metric spaces. We introduce a notion of Gromov-Hausdorff distance with boundary, denoted by $\gls{sym-GHBdis}$, to construct a framework of convergence of noncomplete metric spaces. We prove that the class of all isometry classes of noncomplete locally compact precompact metric spaces, denoted by $\gls{sym-GHB}$, with $\gls{sym-GHBdis}$ is a (noncomplete) metric space. We then give a compactness theorem for $\gls{sym-GHBdis}$ using uniform porosity. We present some applications of the compactness theorem to uniform metric spaces and PI spaces. We also present another application to the effective estimate of the Gromov-Hausdorff distance with boundary between uniformized spaces from the pointed Gromov-Hausdorff distance between Gromov hyperbolic spaces. Finally, we investigate some topological properties of $\gls{sym-GHB}$ and $\overline{\gls{sym-GHB}}$. The second topic concerns finding some conditions equivalent to the Gehring-Hayman theorem. The celebrated theorem of Bonk, Heinonen and Koskela states that there is a one-to-one correspondence between quasiisometry classes of proper, geodesic and roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded locally compact uniform spaces. One of the key components was to prove the Gehring-Heyman theorem. We will explore some conditions equivalent to this theorem. As an application, we determine the critical exponents for the uniformized space to be a uniform space in the case of hyperbolic spaces, model spaces $\mathbb{M}^{\kappa}_n$ of sectional curvature $\kappa<0$ and dimension $n \geq 2$ and hyperbolic fillings. The third topic c (open full item for complete abstract)

Committee: Gareth Speight Ph.D. (Committee Chair); Scott Zimmerman Ph.D. (Committee Member); Leonid Slavin Ph.D. (Committee Member); Andrew Lorent Ph.D. (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2023, Arts and Sciences: Mathematical Sciences

In this thesis, we analyze rational solutions built of the Painlevé V equation built from Umemura polynomials with real shape parameter. We first use a Lax pair to encode the solutions to the Painlevé V differential equation. We then construct a Riemann-Hilbert problem and use the nonlinear steepest-descent method to study the regions of the complex plane unoccupied by the zeros and poles as the degree of the polynomials used to build the rational solutions goes to infinity. This thesis will focus specifically on the cubic genus-zero region.

Committee: Deniz Bilman Ph.D. (Committee Member); Robert Buckingham Ph.D. (Committee Chair) Subjects: Mathematics

MS, University of Cincinnati, 2023, Arts and Sciences: Mathematical Sciences

The goal of this work is to perform a preliminary exploration of a potential model for the emergence of the circadian clock in mammalian cells. This will include an overview of the literature regarding the potential mechanisms of this emergence and existing models of the circadian clock. The full identification and validation of a true model for the emergence of the circadian clock is a lengthy process, far outside the scope allowable by the time frame this thesis was written in. It is for this reason that explorations are limited to only a few modifications to existing models, and only the preliminary steps at that. The novel model consists of a core transcriptional/translational feedback loop (TTFL) with a multi-step reaction of phosphorylation steps occurring within the cytoplasm, as well as Michaelis-Menten inspired degradation rates for proteins with enzyme-mediation degradation. The model also includes a stoichiometry-based transcription term, as described in a previous model by Kim and Forger. Parameter values are determined through an iterative search of the parameter space, then undergo stability analysis and Hopf bifurcation point identification. The parameters associated with the proposed mechanisms of emergence, posttranscriptional regulation of Clock mRNA and subcellular localization of PER, are further investigated to determine the suitability of the model. Further modifications to the model are proposed based on the analysis of the parameters of interest and existing literature.

Committee: Sookkyung Lim Ph.D. (Committee Chair); Benjamin Vaughan Ph.D. (Committee Member); Christian Hong Ph.D. (Committee Member) Subjects: Applied Mathematics

MS, University of Cincinnati, 2023, Arts and Sciences: Mathematical Sciences

Partitioning a compact metric space and building an associated graph allows us to show the underlying structure and relationships within the space. In this paper, we aim to explore the fundamental notions of graphs and trees in order to build the necessary understanding for creating partition graphs. We will then show how the ends of the partition graphs created in such a manner are not only homeomorphic to the original compact metric space but also bi-H ¨older to it.

Committee: Nageswari Shanmugalingam Ph.D. (Committee Chair); Pavel Zatitskii Ph.D. (Committee Member); Gareth Speight Ph.D. (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2023, Arts and Sciences: Mathematical Sciences

In this study, we focus on an indirect approach to discuss the solutions of differential equations. Cellular Automata is a well-established method such as other popular numerical methods; Finite Difference, Finite Element, and Finite Volume methods. In addition, there are various types of cellular automata and most of them are categorized based on the transition rule or the local rule. Among them, some authors mentioned stochastic cellular automata as the state-of-the-art method. The results of several types of research on the application of stochastic cellular automata to differential equations are discussed here. One-dimensional heat equation, Ordinary differential equation that describes first-order linear degradation kinetics, Lotka Volterra equations, and Reaction-diffusion systems are identified as differential equations from the existing studies that we can apply cellular automata to discuss solutions.

Committee: Vita Borovyk Ph.D. (Committee Member); Robert Buckingham Ph.D. (Committee Chair) Subjects: Applied Mathematics

MS, University of Cincinnati, 2023, Arts and Sciences: Mathematical Sciences

This thesis concerns traveling wave solutions of linear and nonlinear partial differential equations modeling wave propagation in different physical media. We show nonexistence of so-called solitary traveling wave solutions for a variety of linear models of wave propagation and we discuss the theory of linear waves in order to explain why these linear models do not support solitary waves. Motivated by this, we turn our attention to the famous Korteweg-de Vries equation, a nonlinear partial differential equation modeling propagation of waves with long wavelength in shallow water. We construct solitary wave solutions of the Korteweg-de Vries equation. We also introduce Jacobi elliptic functions and provide a construction of periodic traveling wave solutions of the Korteweg-de Vries equation. Finally, we recover the solitary wave solution from a degeneration of the periodic wave, in a limit where the period tends to infinity.

Committee: Robert Buckingham Ph.D. (Committee Member); Deniz Bilman Ph.D. (Committee Chair) Subjects: Mathematics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

The focus of this thesis is to use deep learning methods for computer model calibration and uncertainty quantification. Computer model calibration is the process of combining information from computer model outputs and observation data to make inference about unknown input parameters of the computer model. The framework for calibration involves an emulation step which faces computational issues when data is high dimensional and a calibration step which faces the inferential issues due to the nonidentifiability between the input parameters and data-model discrepancy. The main aim of this thesis is to address these computational and inferential issues using deep learning methods. This thesis contribute in the field of computer model calibration in the following way: 1) Developing a new inverse model-based computer model calibration framework that utilizes the feature extraction ability of deep neural network to efficiently handle high dimensional data while filtering out the effects of data-model discrepancy. 2) Formulating a computationally efficient generative deep learning model-based emulation method for large spatial data. 3) Siamese neural network and approximate Bayesian computation-based calibration method that can efficiently solve the issue of data-model discrepancy. The proposed methods have been successfully applied to calibrate important climate models such as Weather Research and Forecasting Model (WRF-Hydro) and University of Victoria Earth System Climate Model(UVic ESCM).

Committee: Won Chang Ph.D. (Committee Member); Siva Sivaganesan Ph.D. (Committee Member); Bledar Konomi Ph.D. (Committee Member); Emily Kang Ph.D. (Committee Member) Subjects: Statistics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

Expensive computer models (simulators) are frequently used to simulate the behavior of a complex system in many scientific fields because an explicit experiment is very expensive or dangerous to conduct. Usually, only a limited number of computer runs are available due to limited sources. Therefore, one desires to use the available runs to construct an inexpensive statistical model, an emulator. Then the constructed statistical model can be used as a surrogate for the computer model. Building an emulator for high dimensional outputs with the existing standard method, the Gaussian process model, can be computationally infeasible because it has a cubic computational complexity that scales with the total number of observations. Also, it is common to impose restrictions on the covariance matrix of the Gaussian process model to keep computations tractable. This work constructs a flexible emulator based on a deep neural network (DNN) with feedforward multilayer perceptrons (MLP). High dimensional outputs and limited runs can pose considerable challenges to DNN in learning a complex computer model's behavior. To overcome this challenge, we take advantage of the computer model's spatial structure to engineer features at each spatial location and then make the training of DNN feasible. Also, to improve the predictive performance and avoid overfitting, we adopt a data augmentation technique into our method. Finally, we apply our approach using data from the UVic ESCM model and the PSU3D-ICE model to demonstrate good predictive performance and compare it with an existing state-of-art emulation method.

Committee: Won Chang Ph.D. (Committee Member); Xia Wang Ph.D. (Committee Member); Emily Kang Ph.D. (Committee Member) Subjects: Statistics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

The retrieval algorithms in remote sensing generally involve complex physical forward models that are nonlinear and computationally expensive to evaluate. Statistical emulation provides an alternative with cheap computation and can be used to quantify uncertainty, calibrate model parameters and improve computational efficiency of the retrieval algorithms. Motivated by this, in this thesis, we introduce a framework for building statistical emulators by combining dimension reduction of input and output spaces and Gaussian process modeling. The functional principal component analysis (FPCA) via a conditional expectation method is chosen to reduce the dimension of the output space of the forward model. In addition, the gradient-based kernel dimension reduction (gKDR) method is applied to reduce the dimension of input space when the gradients of the complex forward model are unavailable or computationally prohibitive. A Gaussian process with feasible computation is then constructed at the low-dimensional input and output spaces. Theoretical properties of the resulting statistical emulator are explored, and the proposed method is illustrated by application to NASA's Orbiting Carbon Observatory-2 (OCO-2) data. Though the Gaussian process emulator provides accurate prediction, it loses computational efficiency when the dataset used to train the emulator is large and/or the inputs and outputs of the model are high-dimensional. Dimension reduction in input is often required. In satellite remote sensing, the quantity of interest (QOI) (e.g., the atmospheric state) is inferred from observable radiance spectra. OCO-2's primary QOI is the column averaged dry air mole fraction of CO2 which are key state variables included in the input state vector. Lowering the dimension of input when building emulator can lead to a loss of information and interpretability, and cause negative effects when estimating the atmospheric CO2 in OCO-2 application. To avoid dimension reduction in i (open full item for complete abstract)

Committee: Emily Kang Ph.D. (Committee Member); Seongho Song Ph.D. (Committee Member); Bledar Konomi Ph.D. (Committee Member); Won Chang Ph.D. (Committee Member) Subjects: Statistics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

We provide a self-tallying voting protocol based on the hardness of decision version of the ring learning with errors problem. We present two versions of the protocol, the first one is passively secure against semi-honest adversaries and the other version is actively secure where we additionally employ S'-proofs which are zero-knowledge proofs. We provide proofs of security for two adversarial models, namely a semi-honest one and a malicious one. First, we show that our protocol is secure against semi-honest adversaries by simulating the view given its local inputs and the public messages. We also show that our voting protocol is secure in the sense of privacy of the honest voters' individual votes against a malicious adversary controlling a coalition of dishonest voters. The idea of our security proof follows the real/ideal world-paradigm where we show that our protocol emulates the ideal-world execution of tallying the votes. Hence any attack in the real-world can be translated to an attack in the ideal-world. However, a successful attack in the idealworld is impossible which means any attack in the real-world will not be successful as well.

Committee: Jintai Ding Ph.D. (Committee Member); Seungki Kim Ph.D. (Committee Member); Robert Buckingham Ph.D. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

In this digital age, well tested public-key cryptography is vital for the continuing function of society. An example of one of the uses of cryptography is signature schemes which allow us to digitally sign a document. However, quantum computers utilizing Shor's algorithm threaten the security of all the cryptosystem currently in use. What is needed is post-quantum cryptography: classical cryptographic algorithms able to resist quantum attacks. In 2016, NIST put out a call for proposals for post-quantum cryptosystems for standardization. We are currently in the third round of the “competition,” with many different types of schemes being proposed. In 2017, Ward Beullens et al. submitted the Lifted Unbalanced Oil and Vinegar signature scheme to the NIST competition, which is a modification to the Unbalanced Oil and Vinegar Scheme by Patarin. The main modification is called lifting, which is to take a polynomial over a small finite field and view it as a map over some extension field. LUOV made it into the second round of the competition, but two attacks by Ding et al. showed a flaw in the modifications of LUOV. The first attack was the Subfield Differential Attack (SDA) which prompted a change of parameters by the authors of LUOV. The second was the Nested Subset Differential Attack (NSDA), which broke half of the parameters put forward by the authors of LUOV again. Due to the strengths of these attacks and the possibility stronger ones of a similar nature exist, LUOV did not go into the third round. This dissertation shows that such a stronger attack, which will be called NSDA+, is possible. All three of the attacks SDA, NSDA, and NSDA+ are straightforward but powerful in application against the lifting modification. First in chapter 1, we will discuss what is a public key cryptosystem by looking at the original definition of Diffie and Hellman. Then we will talk of the NIST Post-Quantum Standardizat (open full item for complete abstract)

Committee: Jintai Ding Ph.D. (Committee Member); Seungki Kim Ph.D. (Committee Member); Robert Buckingham Ph.D. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

High-dimensional data, where the number of features or covariates is larger than the number of independent observations, are ubiquitous and are encountered on a regular basis by statistical scientists both in academia and in industry. Due to the modern advancements in data storage and computational power, the high-dimensional data revolution has significantly occupied mainstream statistical research. In this thesis, we undertake the problem of variable selection in high-dimensional generalized linear models, as well as the problem of high-dimensional sparsity selection for covariance matrices in Gaussian graphical models. We first consider a hierarchical generalized linear regression model with the product moment (pMOM) nonlocal prior over coefficients and examine its properties. Under standard regularity assumptions, we establish strong model selection consistency in a high-dimensional setting, where the number of covariates is allowed to increase at a sub-exponential rate with the sample size. The Laplace approximation is implemented for computing the posterior probabilities and the shotgun stochastic search procedure is suggested for exploring the posterior space. The proposed method is validated through simulation studies and illustrated by a real data example on functional activity analysis in fMRI study for predicting Parkinson's disease. Moreover, we consider sparsity selection for the Cholesky factor L of the inverse covariance matrix in high-dimensional Gaussian Directed Acyclic Graph (DAG) models. The sparsity is induced over the space of L via pMOM non-local prior, and the hierarchical hyper-pMOM prior. We also establish model selection consistency for Cholesky factor under more relaxed conditions compared to those in the literature and implement an efficient MCMC algorithm for parallel selecting the sparsity pattern for each column of L. We demonstrate the validity of our theoretical results via numerical simulations, and also use further s (open full item for complete abstract)

Committee: Xuan Cao Ph.D. (Committee Member); Xia Wang Ph.D. (Committee Member); Seongho Song Ph.D. (Committee Member); Lili Ding Ph.D. (Committee Member) Subjects: Statistics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

Joint modeling has been a useful strategy for incorporating latent associations between different types of outcomes simultaneously in the last two decades. This dissertation contributes to the development of a multilevel Bayesian joint model, which is motivated by a longitudinal lung disease study. In Chapter 1, the background of Bayesian methodology for the joint modeling is introduced. Chapters 2 and 3 describe two novel joint models with applications to the multi-center data for cystic fibrosis disease. First, in Chapter 2, a multilevel Bayesian joint model of longitudinal continuous and binary outcomes is proposed. Second, in Chapter 3, a multilevel Bayesian joint model of longitudinal and recurrent outcomes is postulated. Lastly, in Chapter 4, some key takeaways, limitations and future work are discussed.

Committee: Seongho Song Ph.D. (Committee Member); Won Chang Ph.D. (Committee Member); Xia Wang Ph.D. (Committee Member); Rhonda Szczesniak Ph.D. (Committee Member); Hang Joon Kim Ph.D. (Committee Member) Subjects: Statistics