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.

The main objective of our study is to employ copula methodology to develop Bayesian hierarchical models to study the dependencies exhibited by temporal, spatial and spatio-temporal processes. We develop hierarchical models for both discrete and continuous outcomes. In doing so we expect to address the dearth of copula based Bayesian hierarchical models to study hydro-meteorological events and other physical processes yielding discrete responses. First, we present Bayesian methods of analysis for longitudinal binary outcomes using Generalized Linear Mixed models (GLMM). We allow flexible marginal association among the repeated outcomes from different time-points. An unique property of this copula-based GLMM is that if the marginal link function is integrated over the distribution of the random effects, its form remains same as that of the conditional link function. This unique property enables us to retain the physical interpretation of the fixed effects under conditional and marginal model and yield proper posterior distribution. We illustrate the performance of the posited model using real life AIDS data and demonstrate its superiority over the traditional Gaussian random effects model. We develop a semiparametric extension of our GLMM and re-analyze the data from the AIDS study. Next, we propose a general class of models to handle non-Gaussian spatial data. The proposed model can deal with geostatistical data that can accommodate skewness, tail-heaviness, multimodality. We fix the distribution of the marginal processes and induce dependence via copulas. We illustrate the superior predictive performance of our approach in modeling precipitation data as compared to other kriging variants. Thereafter, we employ mixture kernels as the copula function to accommodate non-stationary data. We demonstrate the adequacy of this non-stationary model by analyzing permeability data. In both cases we perform extensive simulation studies to investigate the performances of the posited models under misspecification. Finally, we take up the important problem of modeling multivariate extreme values with copulas. We describe, in detail, how dependences can be induced in the block maxima approach and peak over threshold approach by an extreme value copula. We prove the ability of the posited model to handle both strong and weak extremal dependence and derive the conditions for posterior propriety. We analyze the extreme precipitation events in the continental United States for the past 98 years and come up with a suite of predictive maps.

This report focuses on introducing two statistical models for dealing with data involving complex social structures. Appropriate handling of data structures is a concern in the context of educational settings. From base single-level data to complex hierarchical with cross-classifications and multiple-memberships, we explain and demonstrate their distinction and establish appropriate regression models. Real data from the National Center for Education Statistics (NECS) is used to demonstrate different way of handling a cross-classified data structure as well as appropriate models. Results will be presented and compared to examine the practical operation for each model. / text

Environmental studies often employ data collected over large spatial regions. Although it is convenient, the conventional single model approach may fail to accurately describe the relationships between variables. Two alternative modeling approaches are available: one applies separate models for different regions; the other applies hierarchical models. The separate modeling approach has two major difficulties: first, we often do not know the underlying clustering structure of the entire data; second, it usually ignores possible dependence among clusters. To deal with the first problem, we propose a model-based clustering method to partition the entire data into subgroups according to the empirical relationships between the response and the predictors. To deal with the second, we propose Bayesian hierarchical models. We illustrate the use of the Bayesian hierarchical model under two situations. First, we apply the hierarchical model based on the empirical clustering structure. Second, we integrate the model-based clustering result to help determine the clustering structure used in the hierarchical model. The nature of the problem is classification since the response is categorical rather than continuous and logistic regression models are used to model the relationship between variables. / Ph. D.

A Markov Chain model can be used to model loan defaults because loans move through delinquency states as the borrower fails to make monthly payments. The transition matrix contains in each location a probability that a borrower in a given state one month moves to the possible delinquency states the next month. In order to use this model, it is necessary to know the transition probabilities, which are unknown quantities. A Bayesian hierarchical model is postulated because there may not be sufficient data for some rare transition probabilities. Using a hierarchical model, similarities between types or families of loans can be taken advantage of to improve estimation, especially for those probabilities with little associated data. The transition probabilities are estimated using MCMC and the Metropolis-Hastings algorithm.

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.

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.

A main objective of Departments of Transportation is to improve the safety of the roadways over which they have jurisdiction. Safety projects, such as cable barriers and raised medians, are utilized to reduce both crash frequency and crash severity. The efficacy of these projects must be evaluated in order to use resources in the best way possible. Five models are proposed for the evaluation of traffic projects: (1) a Bayesian Poisson regression model; (2) a hierarchical Poisson regression model building on model (1) by adding hyperpriors; (3) a similar model correcting for overdispersion; (4) a dynamic linear model; and (5) a traditional before-after study model. Evaluation of these models is discussed using various metrics including DIC. Using the models selected for analysis, it was determined that cable barriers are quite effective at reducing severe crashes and cross-median crashes on Utah highways. Raised medians are also largely effective at reducing severe crashes. The results of before and after analyses are highly valuable to Departments of Transportation in identifying effective projects and in determining which roadway segments will benefit most from their implementation.