Mathematics Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2793
Browse
475 results
Search Results
Item Markov multi-state models for survival analysis with recurrent events(2019) Zhang, Tianhui; Yang, Grace; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Markov models are a major class within the system of multi-state models for the analysis of lifetime or event-time data. Applications abound, including the estimation of lifetime of ultra-cold neutrons, the bias correction of the apparent magnitude distribution of the stars in a certain area of the sky, and the survival analysis of clinical trials. This thesis addresses some of the problems arising in the analysis of right-censored lifetime data. Clinical trials are used as examples to investigate these problems. A Markov model that takes a patient's disease development into account for the analysis of right-censored data was first constructed by Fix and Neyman (1951). The Fix-Neyman (F-N) model is a homogeneous Markov process with two transient and two absorbing states that describes a patient's status over a period of time during a cancer clinical trial. This thesis extends the F-N model by assuming the transition rates (hazard rates) to be both state and time dependent. Recurrent transitions between relapse and recovery are allowed in the extended model. By relaxing the condition of time-independent hazard rates, the extension increases the applicability of the Markov models. The extended models are used to compute the model survival functions, cumulative hazard functions that take into consideration of right censored observations as it has been done in the celebrated Kaplan-Meier estimator. Using the Fix-Neyman procedure and the Kolmogorov forward equations, closed-form solutions are obtained for certain irreversible 4-state extended models while numerical solutions are obtained for the model with recurrent events. The 4-state model is motivated by an Aplastic Anemia data set. The computational method works for general irreversible and reversible models with no restriction on the number of states. Simulations of right-censored Markov processes are performed by using a sequence of competing risks models. Simulated data are used for checking the performance of nonparametric estimators for various sample sizes. In addition, applying Aalen's (1978) results, estimators are shown have asymptotic normal distributions. A brief review of some of the literature relevant to this thesis is provided. References are readily available from a vast literature on the survival analysis including many text books. General Markov process models for survival analysis are described, e.g., in Andersen, Borgan, Gill and Keiding (1993).Item Regression Analysis of Recurrent Events with Measurement Errors(2019) Ren, Yixin; Smith, Paul J; He, Xin; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Recurrent event data and panel count data are often encountered in longitudinal follow-up studies. The main difference between the two types of data is the observation process. Continuous observations will result in recurrent event data; and discrete observations will lead to panel count data. In statistical literature, regression analysis of the two types of data have been well studied; and a typical assumption of those studies is that all covariates are accurately recorded. However, in many applications, it is common to have measurement errors in some of the covariates. For example, in a clinical trial, a medical index might have been measured multiple times. Then dealing with the differences among those measurements is an essential topic for statisticians. For recurrent event data, we present a class of semiparametric regression models that allow correlations between censoring time and recurrent event process via frailty. An estimating equation based approach is developed to account for the presence of measurement errors in some of the covariates. Both large and finite sample properties of the proposed estimators are established. An example from the study of gamma interferon in chronic granulomatous disease is provided. For panel count data, we consider two situations in which the observation process is independent or dependent of covariates. Estimating equations are developed for the estimation of the regression parameters for both cases. Simulation studies indicate that the proposed inference procedures perform well for practical situations. An example of bladder cancer study is used to demonstrate the value of the proposed method.Item Modeling Imatinib-Treated Chronic Myelogenous Leukemia and the Immune System(2019) Peters, Cara Disa; Levy, Doron; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chronic myelogenous leukemia can be considered as a chronic condition thanks to the development of tyrosine kinase inhibitors in the early 2000s. Most CML patients are able to manage the disease, but unending treatment can affect quality of life. The focus of much clinical research has thus transitioned to treatment cessation, where many clinical trials have demonstrated that treatment free remission is possible. While there are a lot of existing questions surrounding the criteria for cessation candidates, much evidence indicates the immune system plays a significant role. Mathematical modeling provides a complementary component to clinical research. Existing models well-describe the dynamics of CML in the first phase of treatment where most patients experience a biphasic decline in the BCR-ABL ratio. The Clapp model is one of the first to incorporate the immune system and capture the often-seen oscillations in the BCR-ABL ratio that occur later in therapy. However, these models are far from capable of being used in a predictive manner and do not fully capture the dynamics surrounding treatment cessation. Based on clinical research demonstrating the importance of immune response, we hypothesize that a mathematical model of CML should include a more detailed description of the immune system. We therefore present a new model that is an extension of the Clapp model. The model is then fit to patient data and determined to be a good qualitative description of CML dynamics. With this model it can be shown that treatment free remission is possible. However, the model introduces new parameters that must be correctly identified in order for it to have predictive power. We next consider the parameter identification problem. Since the dynamics of CML can be considered in two phases, the biphasic decline of and oscillations in the BCR-ABL ratio, we hypothesize that parameter values may differ over the course of treatment and look to identify which parameters are most variable by refitting the model to different windows of data. It is determined that parameters associated with immune response and regulation are most difficult to identify and could be key to selecting good treatment cessation candidates. To increase the predictive power of our model, we consider data assimilation techniques which are successfully used in weather forecasting. The extended Kalman filter is used to assimilate CML patient data. Although we determine that the EKF is not the ideal technique for our model, it is shown that data assimilation methods in general hold promising value to the search for a predictive model of CML. In order to have the most success, new techniques should be considered, data should be collected more frequently, and immune assay data should be made available.Item Developments in Lagrangian Data Assimilation and Coupled Data Assimilation to Support Earth System Model Initialization(2019) Sun, Luyu; Carton, James A.; Penny, Stephen G.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The air-sea interface is one of the most physically active interfaces of the Earth's environments and significantly impacts the dynamics in both the atmosphere and ocean. In this doctoral dissertation, developments are made to two types of Data Assimilation (DA) applied to this interface: Lagrangian Data Assimilation (LaDA) and Coupled Data Assimilation (CDA). LaDA is a DA method that specifically assimilates position information measured from Lagrangian instruments such as Argo floats and surface drifters. To make a better use of this Lagrangian information, an augmented-state LaDA method is proposed using Local Ensemble Transform Kalman Filter (LETKF), which is intended to update the ocean state (T/S/U/V) at both the surface and at depth by directly assimilating the drifter locations. The algorithm is first tested using "identical twin" Observing System Simulation Experiments (OSSEs) in a simple double gyre configuration with the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model version 4.1 (MOM4p1). Results from these experiments show that with a proper choice of localization radius, the estimation of the state is improved not only at the surface, but throughout the upper 1000m. The impact of localization radius and model error in estimating accuracy of both fluid and drifter states are investigated. Next, the algorithm is applied to a realistic eddy-resolving model of the Gulf of Mexico (GoM) using Modular Ocean Model version 6 (MOM6) numerics, which is related to the 1/4-degree resolution configuration in transition to operational use at NOAA/NCEP. Atmospheric forcing is first used to produce the nature run simulation with forcing ensembles created using the spread provided by the 20 Century Reanalysis version 3 (20CRv3). In order to assist the examination on the proposed LaDA algorithm, an updated online drifter module adapted to MOM6 is developed, which resolves software issues present in the older MOM4p1 and MOM5 versions of MOM. In addition, new attributions are added, such as: the output of the intermediate trajectories and the interpolated variables: temperature, salinity, and velocity. The twin experiments with the GoM also show that the proposed algorithm provides positive impacts on estimating the ocean state variables when assimilating the drifter position together with surface temperature and salinity. Lastly, an investigation of CDA explores the influence of different couplings on improving the simultaneous estimation of atmosphere and ocean state variables. Synchronization theory of the drive-response system is applied together with the determination of Lyapunov Exponents (LEs) as an indication of the error convergence within the system. A demonstration is presented using the Ensemble Transform Kalman Filter on the simplified Modular Arbitrary-Order Ocean-Atmosphere Model, a three-layer truncated quasi-geostrophic model. Results show that strongly coupled data assimilation is robust in producing more accurate state estimates and forecasts than traditional approaches of data assimilation.Item Topics in Stochastic Optimization(2019) Sun, Guowei N/A; Fu, Michael C; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis, we work with three topics in stochastic optimization: ranking and selection (R&S), multi-armed bandits (MAB) and stochastic kriging (SK). For R&S, we first consider the problem of making inferences about all candidates based on samples drawn from one. Then we study the problem of designing efficient allocation algorithms for problems where the selection objective is more complex than the simple expectation of a random output. In MAB, we use the autoregressive process to capture possible temporal correlations in the unknown reward processes and study the effect of such correlations on the regret bounds of various bandit algorithms. Lastly, for SK, we design a procedure for dynamic experimental design for establishing a good global fit by efficiently allocating simulation budgets in the design space. The first two Chapters of the thesis work with variations of the R&S problem. In Chapter 1, we consider the problem of choosing the best design alternative under a small simulation budget, where making inferences about all alternatives from a single observation could enhance the probability of correct selection. We propose a new selection rule exploiting the relative similarity between pairs of alternatives and show its improvement on selection performance, evaluated by the Probability of Correct Selection, compared to selection based on collected sample averages. We illustrate the effectiveness by applying our selection index on simulated R\&S problems using two well-known budget allocation policies. In Chapter 2, we present two sequential allocation frameworks for selecting from a set of competing alternatives when the decision maker cares about more than just the simple expected rewards. The frameworks are built on general parametric reward distributions and assume the objective of selection, which we refer to as utility, can be expressed as a function of the governing reward distributional parameters. The first algorithm, which we call utility-based OCBA (UOCBA), uses the Delta-technique to find the asymptotic distribution of a utility estimator to establish the asymptotically optimal allocation by solving the corresponding constrained optimization problem. The second, which we refer to as utility-based value of information (UVoI) approach, is a variation of the Bayesian value of information (VoI) techniques for efficient learning of the utility. We establish the asymptotic optimality of both allocation policies and illustrate the performance of the two algorithms through numerical experiments. Chapter 3 considers the restless bandit problem where the rewards on the arms are stochastic processes with strong temporal correlations that can be characterized by the well-known stationary autoregressive-moving-average time series models. We argue that despite the statistical stationarity of the reward processes, a linear improvement in cumulative reward can be obtained by exploiting the temporal correlation, compared to policies that work under the independent reward assumption. We introduce the notion of temporal exploration-exploitation trade-off, where a policy has to balance between learning more recent information to track the evolution of all reward processes and utilizing currently available predictions to gain better immediate reward. We prove a regret lower bound characterized by the bandit problem complexity and correlation strength along the time index and propose policies that achieve a matching upper bound. Lastly, Chapter 4 proposes a fully sequential experimental design procedure for the stochastic kriging (SK) methodology of fitting unknown response surfaces from simulation experiments. The procedure first estimates the current SK model performance by jackknifing the existing data points. Then, an additional SK model is fitted on the jackknife error estimates to capture the landscape of the current SK model performance. Methodologies for balancing exploration and exploitation trade-off in Bayesian optimization are employed to select the next simulation point. Compared to existing experimental design procedures relying on the posterior uncertainty estimates from the fitted SK model for evaluating model performance, our method is robust to the SK model specifications. We design a dynamic allocation algorithm, which we call kriging-based dynamic stochastic kriging (KDSK), and illustrate its performance through two numerical experiments.Item Moduli Spaces of Sheaves on Hirzebruch Orbifolds(2019) Wang, Weikun; Gholampour, Amin; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $\mu$-stable torsion free sheaves of rank $1$ and $2$ on $\mathcal{H}_{r}^{ab}$. Finally, we show that if $\mathcal{X}$ is the total space of the canonical bundle over a Hirzebruch orbifold, then we can obtain generating functions of Donaldson-Thomas invariants.Item Locally symmetric spaces and the cohomology of the Weil representation(2019) Shi, Yousheng; Millson, John; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\R) $ and $\mathrm{O}^*(2n) $. These cycles are covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson (\cite{Anderson}), we show that Poincar\'e duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G $ to vector-valued automorphic functions associated to the groups $G'=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which are members of a dual pair with $G$ in the sense of Howe. The above three groups are all the groups that show up in real reductive dual pairs of type I whose symmetric spaces are of Hermitian type with the exception of $\mathrm{O}(p,2)$.Item Mathematical Models of Underlying Dynamics in Acute and Chronic Immunology(2019) Wyatt, Asia Alexandria; Levy, Doron; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)During an immune response, it is understood that there are key differences between the cells and cytokines that are present in a primary response versus those present in subsequent responses. Specifically, after a primary response, memory cells are present and drive the clearance of antigen in these later immune responses. When comparing acute infections to chronic infections, there are also differences in the dynamics of the immune system. In this dissertation, we develop three mathematical models to explore these differences in the immune response to acute and chronic infections through the creation, activation, regulation, and long term maintenance of T cells. We mimic this biological behavior through the use of delayed differential equation (DDE) models. The first model explores the dynamics of adaptive immunity in primary and secondary responses to acute infections. It is shown that while we observe similar amounts of antigen stimulation from both immune responses, with the incorporation of memory T cells, we see an increase in both the amount of effector T cells present and the speed of activation of the immune system in the secondary response. We conclude that our model is robust and can be applied to study different types of antigen from viral to bacterial. Extending our work to chronic infections, we develop our second and third models to explore breast cancer dormancy and T cell exhaustion. For our breast cancer dormancy model, we find that our model behaves similar to acute infections, but with constant antigen stimulation. Moreover, we observe the importance of immune protection on the long term survival of breast cancer cells. In our third model we find that while memory T cells play a major role in the effectiveness of the immune system in acute infection, in chronic infections, over long periods of time, T cell exhaustion prevents proper immune function and clearance of antigen. We also observe how the lack of long term maintenance of memory T cells plays an important role in the final outcome of the system. Finally, we propose two potential extensions to the three models developed: creating a simplified acute infection model and creating a combined breast cancer dormancy model with T cell exhaustion.Item ANALYTIC APPROACHES IN LAGRANGIAN GEOMETRY(2019) Dellatorre, Matthew; Rubinstein, Yanir A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The focus of this thesis is two equations that arise in special Lagrangian geometry: the degenerate special Lagrangian equation (DSL) and the Lagrangian mean curvature flow (LMCF). A significant part of this focus centers on Dirichlet duality, subequations, and viscosity solutions, the analytic framework which we use to formulate and study both equations. Given a Calabi--Yau manifold $(X, \omega, J, \Omega)$ and a model manifold $M$, one can construct a kind of moduli space of Lagrangians in $X$ called the space of positive Lagrangians. A Lagrangian $L\subset X$ belongs to this infinite-dimensional space if $L$ is diffeomorphic to $M$ and Re$(\Omega|_L) >0$. % i.e., the set of Lagrangians that are Hamiltonian deformations of a fixed Lagrangian, A Hamiltonian deformation class of the space of positive Lagrangians admits an $L^2$-type Riemannian metric which allows one to study this space from a geometric point of view. Geodesics in this space play a crucial role in a program initiated by Solomon \cite{S1, S2} to understand the existence and uniqueness of special Lagrangian submanifolds in Calabi–Yau manifolds. They also play a key role in a new approach to the Arnold conjecture put forth by Rubinstein–Solomon and in the development of a pluripotential theory for Lagrangian graphs \cite{RS, DR}. The DSL arises as the geodesic equation in the space of positive graph Lagrangians when $X= \mathbb{C}^n$ and $\omega$ and $\Omega$ are associated to the Euclidean structure \cite{RS}. Building on the results of Rubinstein--Solomon \cite{RS}, we show that the DSL induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth semi-flat Calabi--Yau torus fibration are governed by the Riemannian DSL on the product of the base manifold and an interval. The geodesic endpoint problem in this setting thus corresponds to solving the Dirichlet problem for the DSL. However, the DSL is a degenerate-elliptic, fully non-linear, second-order equation, and so the standard elliptic theory does not furnish solutions. Moreover, for Lagrangians with boundary the natural domains on which one would like to solve the Dirichlet problem are cylindrical and thus not smooth. These issues are resolved by Rubinstein--Solomon in the Euclidean setting by adapting the Dirichlet duality framework of Harvey--Lawson to domains with corners \cite{RS}. We further develop these analytic techniques, specifically modifications of the Dirichlet duality theory in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence, in certain settings, continuous geodesics in the space of positive Lagrangians. The uniqueness of solutions to the Dirichlet problem in the Euclidean formulation of Dirichlet duality theory relies on an important convex-analytic theorem of Slodkowski \cite{Slod}. Motivated by the significance of this result and the technical, geometric nature of its proof, we provide a detailed exposition of the proof. We then study some of the quantities involved using the Legendre transform, offering a dual perspective on this theorem. Given a Lagrangian submanifold in a Calabi--Yau, a fundamental and still open question is whether or not there is a special Lagrangian representative in its homology or Hamiltonian isotopy class. A natural approach to this problem is the Lagrangian mean curvature flow, which preserves not only the Lagrangian condition but also the homology and isotopy class. Assuming the flow exists for all time and converges, it will converge to a minimal (i.e., zero mean curvature) Lagrangian. In the Calabi--Yau setting these are precisely the special Lagrangian submanifolds. A major conjecture in this area is the Thomas--Yau conjecture \cite{TY}, which posits certain stability conditions on the initial Lagrangian under which the LMCF will exist for all time and converge to the unique special Lagrangian in that isotopy class. Thomas--Yau stated a variant of their conjecture for a related, more tractable flow, called the almost Lagrangian mean curvature flow (ALMCF). In the setting of highly symmetric Lagrangian spheres in Milnor fibers, and under some additional technical assumptions, they make significant progress towards a proof of this variant of the conjecture \cite{TY}. We study the flow of $2$-spheres from a slightly different perspective and provide a relatively short proof of the longtime existence of viscosity solutions under certain stability conditions, and their convergence to a special Lagrangian sphere.Item Branching diffusion processes in periodic media(2019) Hebbar, Pratima; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In the first part of this manuscript, we investigate the asymptotic behavior of solutions to parabolic partial differential equations (PDEs) in $\real^d$ with space-periodic diffusion matrix, drift, and potential. The asymptotics is obtained up to linear in time distances from the support of the initial function. Using this asymptotics, we describe the behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k-$th moment dominates the $k-$th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. In the second part of the manuscript, we obtain asymptotic expansions for the distribution functions of continuous time stochastic processes with weakly dependent increments in the domain of large deviations. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying H\"ormander condition on a $d$--dimensional compact manifold admit asymptotic expansions of all orders in the domain of large deviations.