It is the end of the spring semester and almost all our graduating Ph.D. in Mathematics and Masters in Statistics candidates have now submitted their theses. All Theses defended in 2011 are below.
Examining the Effectiveness of Multiple Imputation: A Case Study on HIV Risk Behaviors in Women with Substance Use Disorders
Raphiel Murden, Department of Mathematics, Washington University in St. Louis
May 2, 2011
1:00 pm - 3:00 pm
Master in Statistics Thesis Defense
Abstract: Women in the United States are becoming infected with HIV more quickly now than ever before; many of whom are at higher risk because of their substance use habits or that of their partners. (CDC, 2010) This study analyzes cross sectional data regarding the risk behaviors and addiction severity of a sample of women receiving treatment for substance use disorders (SUDs). The data was gathered between 2006 and 2010 at a women's substance use treatment center in St. Louis, Missouri (MO), the name of which cannot be disclosed. We develop a scale, the HIV Risk Scale (HRS), to quantify a woman's risk of contracting HIV at the time of presenting for rehabilitation based on self-reported sexual and drug behaviors. We then, using the seven interviewer-ratings of the Addiction Severity Index (ASI) as predictors of the HRS, examine the results of regression using two methods to adjust for missing data: (1) case-wise deletion and (2) multiple imputation. Results suggest that using several of the ASI, a tool already implemented in rehabilitation efforts, interventions can be tailored to address more closely all of the issues regarding the health and safety of substance abusing women seeking relief from addiction. Results show that specifically looking at the interviewer's assessment of how severely addiction impacts legal, drug-related, alcohol-related, employment-related and medical aspects of a woman's life may enable treatment centers to help her alleviate the HIV to which she maybe exposed.
Location: Cupples I, Room 199
Host: Prof. Ed Spitznagel
Markov Chains Derived from Lagrangian Mechanical Systems
Scott Cook, Department of Mathematics, Washington University in St. Louis
April 28, 2011
2:30 pm - 4:30 pm
Abstract: The theory of Markov chains with countable state spaces is a greatly developed and successful area of probability theory and statistics. There is much interest in continuing to develop the theory of Markov chains beyond countable state spaces. One needs good and well motivated model systems in this effort. In this thesis, we propose to produce such systems by introducing randomness into well-known deterministic systems. Hence, we can draw upon existing (deterministic) results to aid the analysis of our Markov chains. We will focus most heavily on models drawn from Lagrangian mechanical systems with collisions (billiards).
Location: Cupples I, Room 199
Host: Prof. Renato Feres
The Boundary Behavior of Holomorphic Functions
Baili Min, Department of Mathematics, Washington University in St. Louis
April 26, 2011
11:30 am - 1:30 pm
Abstract: In the theory of several complex variables, the Fatou type problems, the Lindel\"{o}f principle, and inner functions have been well studied for strongly pseudoconvex domains. In this thesis, we are going to study more generalized domains, those of finite type. In Chapter 2 we show that there is no Fatou's theorem for approach regions complex tangentially broader than admissible ones, in domains of finite type. In Chapter 3 discussing the Lindel\"{o}f principle, we provide some conditions which yield admissible convergence. In Chapter 4 we construct inner functions for a type of domains more general than strongly pseudoconvex ones. Discussion is carried out in $\CC^2$.
Location: Cupples I, Room 199
Host: Prof. Steven Krantz
Using Dirichlet Process Priors for Bayesian Mixture Clustering
Xiao Huang, Department of Mathematics, Washington University in St. Louis
April 25, 2011
1:00 pm - 3:00 pm
Abstract: We describe a non-parametric Bayesian model using genotype data to classify individuals among populations where the total number of populations is unknown. The model assumes that a population is characterized by a set of allele frequencies that follow multinomial distributions. The Dirichlet Process is applied as the prior distribution. The method estimates the number of populations together with the allele frequencies and the ancestry coefficients of each individual. Distance matrices and bootstrap support numbers based on MCMC runs are generated to create a phylogeny of the ancestral populations.
Location: Cupples I, Room 199
Host: Prof. Stanley Sawer
Predictive Alternatives in Bayesian Model Selection
Andrew Womack, Department of Mathematics, Washington University in St. Louis
April 19, 2011
9:30 am - 11:00 am
Abstract: We explore some commonly used techniques in Bayesian model selection and propose new tools for model selection and comparison. In particular, we are interested in criteria that use predictive densities and information integrals in two ways. The first tries to assess the fit and complexity of the model in order to create a criterion like the DIC. The second builds on intrinsic and fractional Bayes' Factors in order to create a family of criteria that behave asymptotically like the traditional Bayes' Factor. As well as presenting theoretical considerations, we will look at some computational examples and compare them to traditional Bayesian analysis.
Location: Cupples I, Room 199
Host: Professor Jeff Gill
On Bayesian Regression Regularization Methods
Qing Li, Department of Mathematics, Washington University in St. Louis
December 9, 2010
10:00 am - 11:30 am
Abstract: Regression regularization methods, where penalty terms on model parameters are usually added to the standard loss functions, are recently drawing increasing attention from researchers. Among others, elastic net is a flexible regularization and variable selection method that uses a mixture of L_1 and L_2 penalties. It is particularly useful when there are much more predictors than the sample size. The first part of this thesis proposes a Bayesian method to solve the elastic net model using a Gibbs sampler. While the marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian elastic net, the Bayesian elastic net has two major advantages. Firstly, as a Bayesian method, the distributional results on the estimates are straightforward, making the statistical inference easier. Secondly, it chooses the two penalty parameters simultaneously, avoiding the "double shrinkage problem" in the elastic net method. Real data examples and simulation studies show that the Bayesian elastic net behaves comparably in prediction accuracy but performs better in variable selection. The regularization methods also have been shown to be effective in quantile regressions in improving the prediction accuracy. The second part of this thesis studies regularization in quantile regressions from a Bayesian perspective. By proposing a hierarchical model framework, we give a generic treatment to a set of regularization approaches, including lasso, elastic net and group lasso penalties. Gibbs samplers are derived for all cases. This is the first work to discuss regularized quantile regression with the elastic net penalty and the group lasso penalty. Both simulated and real data examples show that Bayesian regularized quantile regression methods often outperform quantile regression without regularization and their non-Bayesian counterparts with regularization.
Cupples I, Room 6
Host: Prof. Nan Lin
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—Math news, stories, videos, and interviews by Marie C. Taris, http://www.math.wustl.edu/marietaris/math.html⇨
See also 2012 Theses in Mathematics and Statistics⇨ and 2013 Theses in Mathematics and Statistics⇨