Doctoral Degrees (Mathematical Statistics and Actuarial Science)
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Browsing Doctoral Degrees (Mathematical Statistics and Actuarial Science) by Author "Groenewald, P. C. N."
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Item Open Access Aspects of Bayesian change-point analysis(University of the Free State, 2000-11) Schoeman, Anita Carina; Groenewald, P. C. N.English: In chapter one we looked at the nature of structural change and defined structural change as a change in one or more parameters of the model in question. Bayesian procedures can be applied to solve inferential problems of structural change. Among the various methodological approaches within Bayesian inference, emphasis is put on the analysis of the posterior distribution itself, since the posterior distribution can be used for conducting hypothesis testing as well as obtaining a point estimate. The history of structural change in statistics, beginning in the early 1950's, is also discussed. Furthermore the Bayesian approach to hypothesis testing was developed by Jeffreys (1935, 1961), where the centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. According to Kass and Raftery (1993) this posterior odds = Bayes factor x prior odds and the Bayes factor is the ratio of the posterior odds of Hl to its prior odds, regardless of the value of the prior odds. The intrinsic and fractional Bayes factors are defined and some advantages and disadvantages of the IBF's are discussed. In chapter two changes in the multivariate normal model are considered. Assuming that a change has taken place, one will want to be able to detect the change and to estimate its position as well as the other parameters of the model. To do a Bayesian analysis, prior densities should be chosen. Firstly the hyperparameters are assumed known, but as this is not. usually true, vague improper priors are used (while the number of change-point.s is fixed). Another way of dealing with the problem of unknown hyperparameters is to use a hierarchical model where the second stage priors are vague. We also considered Gibbs sampling and gave the full conditional distributions for all the cases. The three cases that are studied is (1) a change in the mean with known or unknown variance, (2) a change in the mean and variance by firstly using independent prior densities on the different variances and secondly assuming the variances to be proportional and (3) a change in the variance. The same models above are also considered when the number of change-points are unknown. In this case vague priors are not appropriate when comparing models of different dimensions. In this case we revert to partial Bayes factors, specifically the intrinsic and fractional Bayes factors, to obtain the posterior probabilities of the number of change-points. Furthermore we look at component analysis, i.e. determining which components of a multivariate variable are mostly responsible for the changes in the parameters. The univariate case is then also considered in more detail, including multiple model comparisons and models with auto correlated errors. A summary of approaches in the literature as well as four examples are included. In chapter three changes in the linear model, with (1) a change in the regression coefficient and a constant variance, (2) a change in only the variance and (3) a change in the regression coefficient and the variance, are considered. Bayes factors for the above mentioned cases, multiple change-points, component analysis, switchpoint (continuous change-point) and auto correlation are included, together with seven examples. In chapter four changes in some other standard models are considered. Bernoulli type experiments include the Binomial model, the Negative binomial model, the Multinomial model and the Markov chain model. Exponential type models include the Poisson model, the Gamma model and the Exponential model. Special cases of the Exponential model include the left truncated exponential model and the Exponential model with epidemic change. In all cases the partial Bayes factor is used to obtain posterior probabilities when the number of change-points is unknown. Marginal posterior densities of all parameters under the change-point model are derived. Eleven examples are included. In chapter five change-points in the hazard rate are studied. This includes an abrupt change in a constant hazard rate as well as a change from a decreasing hazard rate to a constant hazard rate or a change from a constant hazard rate to an increasing hazard rate. These hazard rates are obtained from combinations of Exponential and Weibull density functions. In the same way a bathtub hazard rate can also be constructed. Two illustrations are given. Some concluding remarks are made in chapter six, with discussions of other approaches in the literature and other possible applications not dealt with in this study.Item Open Access Hierarchical Bayesian modelling for the analysis of the lactation of dairy animals(University of the Free State, 2006-03) Lombaard (née Viljoen), Carolina Susanna; Groenewald, P. C. N.English: This thesis was written with the aim of modelling the lactation process in dairy cows and goats by applying a hierarchical Bayesian approach. Information on cofactors that could possibly affect lactation is included in the model through a novel approach using covariates. Posterior distributions of quantities of interest are obtained by means of the Markov chain Monte Carlo methods. Prediction of future lactation cycle(s) is also performed. In chapter one lactation is defined, its characteristics considered, the factors that could possibly influence lactation mentioned, and the reasons for modelling lactation explained. Chapter two provides a historical perspective to lactation models, considers typical lactation curve shapes and curves fitted to the lactation composition traits fat and protein of milk. Attention is also paid to persistency of lactation. Chapter three considers alternative methods of obtaining total yield and producing Standard Lactation Curves (SLAC’s). Attention is paid to methods used in fitting lactation curves and the assumptions about the errors. In chapter four the generalised Bayesian model approach used to simultaneous ly model more than one lactation trait, while also incorporating information on cofactors that could possibly influence lactation, is developed. Special attention is paid not only to the model for complete data, but also how modelling is adjusted to make provision for cases where not all lactation cycles have been observed for all animals, also referred to as incomplete data. The use of the Gibbs sampler and the Metropolis-Hastings algorithm in determining marginal posterior distributions of model parameters and quantities that are functions of such parameters are also discussed. Prediction of future lactation cycles using the model is also considered. In chapter five the Bayesian approach together with the Wood model, applied to 4564 lactation cycles of 1141 Jersey cows, is used to illustrate the approach to modelling and prediction of milk yield, percentage of fat and percentage of protein in milk composition in the case of complete data. The incorporation of cofactor information through the use of the covariate matrix is also considered in greater detail. The results from the Gibbs sampler are evaluated and convergence there-of investigated. Attention is also paid to the expected lactation curve characteristics as defined by Wood, as well as obtaining the expected lactation 254 curve of one of the levels of a cofactor when the influence of the other cofactors on the lactation curve has be eliminated. Chapter six considers the use of the Bayesian approach together with the general exponential and 4-parameter Morant model, as well as an adaptation of a model suggested by Wilmink, in modelling and predicting milk yield, fat content and protein content of milk for the Jersey data. In chapter seven a diagnostic comparison by means of Bayes factors of the results from the four models in the preceding two chapters, when used together with the Bayesian approach, is performed. As a result the adapted form of the Wilmink model fared best of the models considered! Chapter eight illustrates the use of the Bayesian approach, together with the four lactation models considered in this study, to predict the lactation traits for animals similar to, but not contained in the data used to develop the respective models. In chapter nine the Bayesian approach together with the Wood model, applied to 755 lactation cycles of 493 Saanen does collected during either or both of two consecutive year, is used to illustrate the approach to modelling and predicting milk yield, percentage of fat and percentage of protein in milk in the case of incomplete data. Chapter ten provides a summary of the results and a perspective of the contribution of this research to lactation modelling.