Basketball is a sport that has become increasingly popular world-wide. At the professional level it is a game in which each of the five positions has a specific responsibility that requires unique skills. It seems likely that it would be valuable for coaches to know which skills for each position are most conducive to winning. Knowing which skills to develop for each position could help coaches optimize each player's ability by customizing practice to contain drills that develop the most important skills for each position that would in turn improve the team's overall ability. Through the use of Bayesian hierarchical modeling and NBA box-score performance categories, this project will determine how each position needs to perform in order for their team to be successful.

Assessment of the “health” of an ecosystem is often of great interest to those interested in monitoring and conservation of ecosystems. Traditionally, scientists have quantified the health of an ecosystem using multimetric indices that are semi-qualitative. Recently, a statistical-based index called the Latent Health Factor Index (LHFI) was devised to address many inadequacies of the conventional indices. Relying on standard modelling procedures, unlike the conventional indices, accords the LHFI many advantages: the LHFI is less arbitrary, and it allows for straightforward model inference and for formal statistical prediction of health for a new site (using only supplementary environmental covariates). In contrast, with conventional indices, formal statistical prediction does not exist, meaning that proper estimation of health for a new site requires benthic data which are expensive and time-consuming to gather. As the LHFI modelling methodology is a relatively new concept, it has so far only been demonstrated (and validated) on freshwater ecosystems. The goal of this thesis is to apply the LHFI modelling methodology to estuarine ecosystems, particularly to the previously unassessed system in Richibucto, New Brunswick. Specifically, the aims of this thesis are threefold: firstly, to investigate whether the LHFI is even applicable to estuarine systems since estuarine and freshwater metrics, or indicators of health, are quite different; secondly, to determine the appropriate form that the LHFI model if the technique is applicable; and thirdly, to assess the health of the Richibucto system. Note that the second objective includes determining which covariates may have a significant impact on estuarine health. As scientists have previously used the AZTI Marine Biotic Index (AMBI) and the Infaunal Trophic Index (ITI) as measurements of estuarine ecosystem health, this thesis investigates LHFI models using metrics from these two indices simultaneously. Two sets of models were considered in a Bayesian framework and implemented using Markov chain Monte Carlo techniques, the first using only metrics from AMBI, and the second using metrics from both AMBI and ITI. Both sets of LHFI models were successful in that they were able to make distinctions between health levels at different sites.

Assessment of the “health” of an ecosystem is often of great interest to those interested in monitoring and conservation of ecosystems. Traditionally, scientists have quantified the health of an ecosystem using multimetric indices that are semi-qualitative. Recently, a statistical-based index called the Latent Health Factor Index (LHFI) was devised to address many inadequacies of the conventional indices. Relying on standard modelling procedures, unlike the conventional indices, accords the LHFI many advantages: the LHFI is less arbitrary, and it allows for straightforward model inference and for formal statistical prediction of health for a new site (using only supplementary environmental covariates). In contrast, with conventional indices, formal statistical prediction does not exist, meaning that proper estimation of health for a new site requires benthic data which are expensive and time-consuming to gather. As the LHFI modelling methodology is a relatively new concept, it has so far only been demonstrated (and validated) on freshwater ecosystems. The goal of this thesis is to apply the LHFI modelling methodology to estuarine ecosystems, particularly to the previously unassessed system in Richibucto, New Brunswick. Specifically, the aims of this thesis are threefold: firstly, to investigate whether the LHFI is even applicable to estuarine systems since estuarine and freshwater metrics, or indicators of health, are quite different; secondly, to determine the appropriate form that the LHFI model if the technique is applicable; and thirdly, to assess the health of the Richibucto system. Note that the second objective includes determining which covariates may have a significant impact on estuarine health. As scientists have previously used the AZTI Marine Biotic Index (AMBI) and the Infaunal Trophic Index (ITI) as measurements of estuarine ecosystem health, this thesis investigates LHFI models using metrics from these two indices simultaneously. Two sets of models were considered in a Bayesian framework and implemented using Markov chain Monte Carlo techniques, the first using only metrics from AMBI, and the second using metrics from both AMBI and ITI. Both sets of LHFI models were successful in that they were able to make distinctions between health levels at different sites.

In the oil and gas industry, statistical methods have been used for corrosion analysis for various asset systems such as pipelines, storage tanks, and so on. However, few industrial standards and guidelines provide comprehensive stepwise procedures for the usage of statistical approaches for corrosion analysis. For example, the UK HSE (2002) report "Guidelines for the use of statistics for analysis of sample inspection of corrosion" demonstrates how statistical methods can be used to evaluate corrosion samples, but the methods explained in the document are very basic and do not consider risk factors such as pressure, temperature, design, external factors and other factors for the analyses. Furthermore, often the industrial practice that uses linear approximation on localised corrosion such as pitting is considered inappropriate as pitting growth is not uniform. The aim of this research is to develop an approach that models the stochastic behaviour of localised corrosion and demonstrate how the influencing factors can be linked to the corrosion analyses, for predicting the remaining useful life of components in oil and gas plants. This research addresses a challenge in industry practice. Non-destructive testing (NDT) and inspection techniques have improved in recent years making more and more data available to asset operators. However, this means that these data need to be processed to extract meaningful information. Increasing computer power has enabled the use of statistics for such data processing. Statistical software such as R and OpenBUGS is available to users to explore new and pragmatic statistical methods (e.g. regression models and stochastic models) and fully use the available data in the field. In this thesis, we carry out extreme value analysis to determine maximum defect depth of an offshore conductor pipe and simulate the defect depth using geometric Brownian motion in Chapter 2. In Chapter 3, we introduce a Weibull density regression that is based on a gamma transformation proportional hazards model to analyse the corrosion data of piping deadlegs. The density regression model takes multiple influencing factors into account; this model can be used to extrapolate the corrosion density of inaccessible deadlegs with data available from other piping systems. In Chapter 4, we demonstrate how the corrosion prediction models in Chapters 2 and 3 could be used to predict the remaining useful life of these components. Chapter 1 sets the background to the techniques used, and Chapter 5 presents concluding remarks based on the application of the techniques.

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Dual response surface methodology (Vining and Myers (1990)) has been successfully used as a cost-effective approach to improve the quality of products and processes since Taguchi (Tauchi (1985)) introduced the idea of robust parameter design on the quality improvement in the United States in mid-1980s. The original procedure is to use the mean and the standard deviation of the characteristic to form a dual response system in linear model structure, and to estimate the model coefficients using least squares methods. In this dissertation, a Bayesian hierarchical approach is proposed to model the dual response system so that the inherent hierarchical variance structure of the response can be modeled naturally. The Bayesian model is developed for both univariate and multivariate dual response surfaces, and for both fully replicated and partially replicated dual response surface designs. To evaluate its performance, the Bayesian method has been compared with the original method under a wide range of scenarios, and it shows higher efficiency and more robustness. In applications, the Bayesian approach retains all the advantages provided by the original dual response surface modelling method. Moreover, the Bayesian analysis allows inference on the uncertainty of the model parameters, and thus can give practitioners complete information on the distribution of the characteristic of interest. / Ph. D.

Genetic data analysis has been capturing a lot of attentions for understanding the mechanism of the development and progressing of diseases like cancers, and is crucial in discovering genetic markers and treatment targets in medical research. This dissertation focuses on several important issues in genetic data analysis, graphical network modeling, feature selection, and covariance estimation. First, we develop a gene network modeling method for discrete gene expression data, produced by technologies such as serial analysis of gene expression and RNA sequencing experiment, which generate counts of mRNA transcripts in cell samples. We propose a generalized linear model to fit the discrete gene expression data and assume that the log ratios of the mean expression levels follow a Gaussian distribution. We derive the gene network structures by selecting covariance matrices of the Gaussian distribution with a hyper-inverse Wishart prior. We incorporate prior network models based on Gene Ontology information, which avails existing biological information on the genes of interest. Next, we consider a variable selection problem, where the variables have natural grouping structures, with application to analysis of chromosomal copy number data. The chromosomal copy number data are produced by molecular inversion probes experiments which measure probe-specific copy number changes. We propose a novel Bayesian variable selection method, the hierarchical structured variable se- lection (HSVS) method, which accounts for the natural gene and probe-within-gene architecture to identify important genes and probes associated with clinically relevant outcomes. We propose the HSVS model for grouped variable selection, where simultaneous selection of both groups and within-group variables is of interest. The HSVS model utilizes a discrete mixture prior distribution for group selection and group-specific Bayesian lasso hierarchies for variable selection within groups. We further provide methods for accounting for serial correlations within groups that incorporate Bayesian fused lasso methods for within-group selection. Finally, we propose a Bayesian method of estimating high-dimensional covariance matrices that can be decomposed into a low rank and sparse component. This covariance structure has a wide range of applications including factor analytical model and random effects model. We model the covariance matrices with the decomposition structure by representing the covariance model in the form of a factor analytic model where the number of latent factors is unknown. We introduce binary indicators for estimating the rank of the low rank component combined with a Bayesian graphical lasso method for estimating the sparse component. We further extend our method to a graphical factor analytic model where the graphical model of the residuals is of interest. We achieve sparse estimation of the inverse covariance of the residuals in the graphical factor model by employing a hyper-inverse Wishart prior method for a decomposable graph and a Bayesian graphical lasso method for an unrestricted graph.

This dissertation presents a Bayesian hierarchical model to combine two-resolution metrology data for inspecting the geometric quality of manufactured parts. The high- resolution data points are scarce, and thus scatter over the surface being measured, while the low-resolution data are pervasive, but less accurate or less precise. Combining the two datasets could supposedly make a better prediction of the geometric surface of a manufactured part than using a single dataset. One challenge in combining the metrology datasets is the misalignment which exists between the low- and high-resolution data points. This dissertation attempts to provide a Bayesian hierarchical model that can handle such misaligned datasets, and includes the following components: (a) a Gaussian process for modeling metrology data at the low-resolution level; (b) a heuristic matching and alignment method that produces a pool of candidate matches and transformations between the two datasets; (c) a linkage model, conditioned on a given match and its associated transformation, that connects a high-resolution data point to a set of low-resolution data points in its neighborhood and makes a combined prediction; and finally (d) Bayesian model averaging of the predictive models in (c) over the pool of candidate matches found in (b). This Bayesian model averaging procedure assigns weights to different matches according to how much they support the observed data, and then produces the final combined prediction of the surface based on the data of both resolutions. The proposed method improves upon the methods of using a single dataset as well as a combined prediction without addressing the misalignment problem. This dissertation demonstrates the improvements over alternative methods using both simulated data and the datasets from a milled sine-wave part, measured by two coordinate measuring machines of different resolutions, respectively.

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Intrinsic connectivity networks (ICNs) refer to brain functional networks that are consistently found under various conditions, during tasks or at rest. Some studies demonstrated that while some stimuli do not impact intrinsic connectivity, other stimuli actually activate intrinsic connectivity through suppression, excitation, moderation or modi cation. Most analyses of functional magnetic resonance imaging (fMRI) data use ad-hoc methods to estimate the latent structure of ICNs. Modeling the effects on ICNs has also not been fully investigated. Bayesian Intrinsic Connectivity Network (BICNet) captures the ICN structure with We propose a BICNet model, an extended Bayesian dynamic sparse latent factor model, to identify the ICNs and quantify task-related effects on the ICNs. BICNet has the following advantages: (1) It simultaneously identifies the individual and group-level ICNs; (2) It robustly identifies ICNs by jointly modeling resting-state fMRI (rfMRI) and task-related fMRI (tfMRI); (3) Compared to independent component analysis (ICA)-based methods, it can quantify the difference of ICNs amplitudes across different states; (4) The sparsity of ICNs automatically performs feature selection, instead of ad-hoc thresholding. We apply BICNet to the rfMRI and language tfMRI data from the Human Connectome Project (HCP) and identify several ICNs related to distinct language processing functions.