abstract: The primary objective in time series analysis is forecasting. Raw data often exhibits nonstationary behavior: trends, seasonal cycles, and heteroskedasticity. After data is transformed to a weakly stationary process, autoregressive moving average (ARMA) models may capture the remaining temporal dynamics to improve forecasting. Estimation of ARMA can be performed through regressing current values on previous realizations and proxy innovations. The classic paradigm fails when dynamics are nonlinear; in this case, parametric, regime-switching specifications model changes in level, ARMA dynamics, and volatility, using a finite number of latent states. If the states can be identified using past endogenous or exogenous information, a threshold autoregressive (TAR) or logistic smooth transition autoregressive (LSTAR) model may simplify complex nonlinear associations to conditional weakly stationary processes. For ARMA, TAR, and STAR, order parameters quantify the extent past information is associated with the future. Unfortunately, even if model orders are known a priori, the possibility of over-fitting can lead to sub-optimal forecasting performance. By intentionally overestimating these orders, a linear representation of the full model is exploited and Bayesian regularization can be used to achieve sparsity. Global-local shrinkage priors for AR, MA, and exogenous coefficients are adopted to pull posterior means toward 0 without over-shrinking relevant effects. This dissertation introduces, evaluates, and compares Bayesian techniques that automatically perform model selection and coefficient estimation of ARMA, TAR, and STAR models. Multiple Monte Carlo experiments illustrate the accuracy of these methods in finding the "true" data generating process. Practical applications demonstrate their efficacy in forecasting. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018

Recent advances in technology for brain imaging and high-throughput genotyping have motivated studies examining the influence of genetic variation on brain structure. In this setting, high-dimensional regression for multi-SNP association analysis is challenging as the brain imaging phenotypes are multivariate and there is a desire to incorporate a biological group structure among SNPs based on their belonging genes. Wang et al. (Bioinformatics, 2012) have recently developed an approach for simultaneous estimation and SNP selection based on penalized regression with regularization based on a novel group l_{2,1}-norm penalty, which encourages sparsity at the gene level. A problem with the proposed approach is that it only provides a point estimate. We solve this problem by developing a corresponding Bayesian formulation based on a three-level hierarchical model that allows for full posterior inference using Gibbs sampling. For the selection of tuning parameters, we consider techniques based on: (i) a fully Bayes approach with hyperpriors, (ii) empirical Bayes with implementation based on a Monte Carlo EM algorithm, and (iii) cross-validation (CV). When the number of SNPs is greater than the number of observations we find that both the fully Bayes and empirical Bayes approaches overestimate the tuning parameters, leading to overshrinkage of regression coefficients. To understand this problem we derive an approximation to the marginal likelihood and investigate its shape under different settings. Our investigation sheds some light on the problem and suggests the use of cross-validation or its approximation with WAIC (Watanabe, 2010) when the number of SNPs is relatively large. Properties of our Gibbs-WAIC approach are investigated using a simulation study and we apply the methodology to a large dataset collected as part of the Alzheimer's Disease Neuroimaging Initiative. / Graduate

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A whole genome scan was conducted to identify and fine map QTL regions for ultrasound and carcass merit traits in beef cattle. A total of 465 steers and bulls, genotyped for 4592 SNPs, were analysed for 16 ultrasound and carcass merit traits using interval mapping, single marker regression and Bayesian shrinkage approaches. Thirty QTL and 22 SNPs associated with traits were identified by interval mapping and single marker regression respectively. In Bayesian shrinkage estimation, 218 QTL were identified, wherein 11 of the 30 QTL identified by interval mapping were confirmed. The proportions of QTL variance on the trait variations estimated by Bayesian shrinkage analysis were relatively small. They ranged from 0.1 to 4.8% compared to 6.1 to 11.7% in interval mapping because the QTL in Bayesian approach were adjusted to remove effects of other QTL in the genome. These results are useful for detection of underlying causative QTN variants. / Animal Science

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