Probability Commons^™

Statistical Modeling, Learning And Computing For Stochastic Dynamics Of Complex Systems, Mohammadmahdi Hajiha

Graduate Theses and Dissertations

With the recent advances in sensor technology, it is much easier to collect and store streams of system operational and environmental (SOE) data. These data can be used as input to model the underlying behavior of complex engineered systems and phenomenons if appropriate algorithms with well-defined assumptions are developed. This dissertation is comprised of the research work to show the applicability of SOE data when fed into proposed tailored algorithms. The first purposes of these algorithms are to estimate and analyze the reliability of a system as elaborated in Chapter 2. This chapter provides the derivation of closed-form expressions that …

Controlled Branching Processes With Continuous Time, Miguel Gonzalez, Manuel Molina, Ines Del Puerto, Nikolay Yanev, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

A class of controlled branching processes with continuous time is introduced and some limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.

A Computational Study Of Genotype-Phenotype Mutation Patterns, Kamaludin Dingle, Omar Tawfik, Ahmed Aldabagh

Undergraduate Research Symposium

Understanding properties of genotype-phenotype maps is important for understanding biology and evolution. In this project we make a computational study of the statistical effects of genetic mutations, in particular computing the probabilities of each phenotype transitioning to any other phenotype. We also investigate the importance of the local phenotypic environment of a single genotype, and its role in determining mutation transition probabilities. We use HP protein folding, RNA structure, and a simplified GRN matrix model to study these questions.

Bayesian Variable Selection Strategies In Longitudinal Mixture Models And Categorical Regression Problems., Md Nazir Uddin

Electronic Theses and Dissertations

In this work, we seek to develop a variable screening and selection method for Bayesian mixture models with longitudinal data. To develop this method, we consider data from the Health and Retirement Survey (HRS) conducted by University of Michigan. Considering yearly out-of-pocket expenditures as the longitudinal response variable, we consider a Bayesian mixture model with $K$ components. The data consist of a large collection of demographic, financial, and health-related baseline characteristics, and we wish to find a subset of these that impact cluster membership. An initial mixture model without any cluster-level predictors is fit to the data through an MCMC …

Glivenko-Cantelli Theorems For Integrated Functionals Of Stochastic Processes, Jia Li, Congshan Zhang, Yunxiao Liu

Research Collection School Of Economics

We prove a Glivenko-Cantelli theorem for integrated functionals of latent continuous-time stochastic processes. Based on a bracketing condition via random brackets, the theorem establishes the uniform convergence of a sequence of empirical occupation measures towards the occupation measure induced by underlying processes over large classes of test functions, including indicator functions, bounded monotone functions, Lipschitz-in-parameter functions, and Hölder classes as special cases. The general Glivenko-Cantelli theorem is then applied in more concrete high-frequency statistical settings to establish uniform convergence results for general integrated functionals of the volatility of efficient price and local moments of microstructure noise.

Identification And Characterization Of De Novo Germline Tp53 Mutation Carriers In Families With Li-Fraumeni Syndrome, Carlos C. Vera Recio

Dissertations and Theses (Open Access)

Li-Fraumeni syndrome (LFS) is an inherited cancer syndrome caused by a deleterious mutation in TP53. An estimated 48% of LFS patients present due to a de novo mutation (DNM) in TP53. The knowledge of DNM status, DNM or familial mutation (FM), of an LFS patient requires genetic testing of both parents which is often inaccessible, making de novo LFS patients difficult to study. Famdenovo.TP53 is a Mendelian Risk prediction model used to predict DNM status of TP53 mutation carriers based on the cancer-family history and several input genetic parameters, including disease-gene penetrance. The good predictive performance of Famdenovo.TP53 was demonstrated …

Knowledge Discovery From Complex Event Time Data With Covariates, Samira Karimi

Graduate Theses and Dissertations

In particular engineering applications, such as reliability engineering, complex types of data are encountered which require novel methods of statistical analysis. Handling covariates properly while managing the missing values is a challenging task. These type of issues happen frequently in reliability data analysis. Specifically, accelerated life testing (ALT) data are usually conducted by exposing test units of a product to severer-than-normal conditions to expedite the failure process. The resulting lifetime and/or censoring data are often modeled by a probability distribution along with a life-stress relationship. However, if the probability distribution and life-stress relationship selected cannot adequately describe the underlying failure …

Evaluating The Efficiency Of Markov Chain Monte Carlo Algorithms, Thuy Scanlon

Graduate Theses and Dissertations

Markov chain Monte Carlo (MCMC) is a simulation technique that produces a Markov chain designed to converge to a stationary distribution. In Bayesian statistics, MCMC is used to obtain samples from a posterior distribution for inference. To ensure the accuracy of estimates using MCMC samples, the convergence to the stationary distribution of an MCMC algorithm has to be checked. As computation time is a resource, optimizing the efficiency of an MCMC algorithm in terms of effective sample size (ESS) per time unit is an important goal for statisticians. In this paper, we use simulation studies to demonstrate how the Gibbs …

Applications Of Nonstandard Analysis In Probability And Measure Theory, Irfan Alam

LSU Doctoral Dissertations

This dissertation broadly deals with two areas of probability theory and investigates how methods from nonstandard analysis may provide new perspectives in these topics. In particular, we use nonstandard analysis to prove new results in the topics of limiting spherical integrals and of exchangeability.

In the former area, our methods allow us to represent finite dimensional Gaussian measures in terms of marginals of measures on hyperfinite-dimensional spheres in a certain strong sense, thus generalizing some previously known results on Gaussian Radon transforms as limits of spherical integrals. This first area has roots in the kinetic theory of gases, which is …

Markov Model Composition Of Balinese Reyong Norot Improvisations, Taylor Flanagan, Robert Rovetti

Honors Thesis

Markov models are mathematical structures that model the transition between possible states based on the probability of moving from one state to any other. Thus, given a distribution of starting points, the model produces a chain of states that are visited in sequence. Such models have been used extensively to generate music based on probabilities, as sequences of states can represent sequences of notes and rhythms. While music generation is a common application of Markov models, most existing work attempts to reconstruct the musical style of classical Western composers. In this thesis, we produce a series of Markov chains that …

Application Of Randomness In Finance, Jose Sanchez, Daanial Ahmad, Satyanand Singh

Publications and Research

Brownian Motion which is also considered to be a Wiener process and can be thought of as a random walk. In our project we had briefly discussed the fluctuations of financial indices and related it to Brownian Motion and the modeling of Stock prices.

Improving Bayesian Graph Convolutional Networks Using Markov Chain Monte Carlo Graph Sampling, Aneesh Komanduri

Computer Science and Computer Engineering Undergraduate Honors Theses

In the modern age of social media and networks, graph representations of real-world phenomena have become incredibly crucial. Often, we are interested in understanding how entities in a graph are interconnected. Graph Neural Networks (GNNs) have proven to be a very useful tool in a variety of graph learning tasks including node classification, link prediction, and edge classification. However, in most of these tasks, the graph data we are working with may be noisy and may contain spurious edges. That is, there is a lot of uncertainty associated with the underlying graph structure. Recent approaches to modeling uncertainty have been …

Statistical Analysis Of 2017-18 Premier League Match Statistics Using A Regression Analysis In R, Bergen Campbell

Undergraduate Theses and Capstone Projects

This thesis analyzes the correlation between a team’s statistics and the success of their performances, and develops a predictive model that can be used to forecast final season results for that team. Data from the 2017-2018 Premier League season is to be gathered and broken down within R to highlight what factors and variables are largely contributing to the success or downfall of a team. A multiple linear regression model and stepwise selection process is then used to include any factors that are significant in predicting in match results.

The predictions about the 17-18 season results based on the model …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

Markov Chains And Their Applications, Fariha Mahfuz

Math Theses

Markov chain is a stochastic model that is used to predict future events. Markov chain is relatively simple since it only requires the information of the present state to predict the future states. In this paper we will go over the basic concepts of Markov Chain and several of its applications including Google PageRank algorithm, weather prediction and gamblers ruin.

We examine on how the Google PageRank algorithm works efficiently to provide PageRank for a Google search result. We also show how can we use Markov chain to predict weather by creating a model from real life data.

Predicting Tumor Response To Radiotherapy Based On Estimation Of Non-Treatment Parameters, Yutian Huang, Allison L. Lewis

Spora: A Journal of Biomathematics

Though clinicians can now collect detailed information about a variety of tumor characteristics as a tumor evolves, it remains difficult to predict the efficacy of a given treatment prior to administration. Additionally, the process of data collection may be invasive and expensive. Thus, the creation of a framework for predicting patient response to treatment using only information collected prior to the start of treatment could be invaluable. In this study, we employ ordinary differential equation models for tumor growth and utilize synthetic data from a cellular automaton model for calibration. We investigate which parameters have the most influence upon treatment …

Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu

LSU Doctoral Dissertations

This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …

Continuous-Time Controlled Branching Processes, Ines Garcia, George Yanev, Manuel Molina, Nikolay Yanev, Miguel Velasco

School of Mathematical & Statistical Sciences Faculty Publications

Controlled branching processes with continuous time are introduced and limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.

On The Evolution Equation For Modelling The Covid-19 Pandemic, Jonathan Blackledge

Books/Book chapters

The paper introduces and discusses the evolution equation, and, based exclusively on this equation, considers random walk models for the time series available on the daily confirmed Covid-19 cases for different countries. It is shown that a conventional random walk model is not consistent with the current global pandemic time series data, which exhibits non-ergodic properties. A self-affine random walk field model is investigated, derived from the evolutionary equation for a specified memory function which provides the non-ergodic fields evident in the available Covid-19 data. This is based on using a spectral scaling relationship of the type 1/ωα where ω …

Bayesian Techniques For Relating Genetic Polymorphisms To Diffusion Tensor Images Of Cocaine Users, Tmader Alballa

Theses and Dissertations

Past investigations utilizing Diffusion Tensor Imaging (DTI) have demonstrated that cocaine use disorder (CUD) yields white matter changes. We proposed three Bayesian techniques in order to explore the relationship between Fractional Anisotropy (FA), genetic data, and years of cocaine use (YCU). CUD participants exhibit abnormality in different areas of the brain versus non-drug using controls, which is measured by DTI. This dissertation is motivated by a neuroimaging genetic study in cocaine dependence, which found that there were relationships between several genes such as GAD and 5-HT₂R and CUD subjects.

In the first chapter, there is background on the …