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In sufficient dimension reduction (Li, 1991; Cook, 1998b), original predictors are replaced by their low-dimensional linear combinations while preserving all of the conditional information of the response given the predictors. Sliced inverse regression [SIR; Li, 1991] and principal Hessian directions [PHD; Li, 1992] are two popular sufficient dimension reduction methods, and both SIR and PHD estimators involve all of the original predictor variables. To deal with the cases when the linear combinations involve only a subset of the original predictors, we propose a Bayesian model averaging (Raftery et al., 1997) approach to achieve sparse sufficient dimension reduction. We extend both SIR and PHD under the Bayesian framework. The superior performance of the proposed methods is demonstrated through extensive numerical studies as well as a real data analysis. / Statistics

The standard methodology when building statistical models has been to use one of several algorithms to systematically search the model space for a good model. If the number of variables is small then all possible models or best subset procedures may be used, but for data sets with a large number of variables, a stepwise procedure is usually implemented. The stepwise procedure of model selection was designed for its computational efficiency and is not guaranteed to find the best model with respect to any optimality criteria. While the model selected may not be the best possible of those in the model space, commonly it is almost as good as the best model. Many times there will be several models that exist that may be competitors of the best model in terms of the selection criterion, but classical model building dictates that a single model be chosen to the exclusion of all others. An alternative to this is Bayesian model averaging (BMA), which uses the information from all models based on how well each is supported by the data. Using BMA allows a variance component due to the uncertainty of the model selection process to be estimated. The variance of any statistic of interest is conditional on the model selected so if there is model uncertainty then variance estimates should reflect this. BMA methodology can also be used for variable assessment since the probability that a given variable is active is readily obtained from the individual model posterior probabilities. The multivariate methods considered in this research are principal components analysis (PCA), canonical variate analysis (CVA), and canonical correlation analysis (CCA). Each method is viewed as a particular multivariate extension of univariate multiple regression. The marginal likelihood of a univariate multiple regression model has been approximated using the Bayes information criteria (BIC), hence the marginal likelihood for these multivariate extensions also makes use of this approximation. One of the main criticisms of multivariate techniques in general is that they are difficult to interpret. To aid interpretation, BMA methodology is used to assess the contribution of each variable to the methods investigated. A second issue that is addressed is displaying of results of an analysis graphically. The goal here is to effectively convey the germane elements of an analysis when BMA is used in order to obtain a clearer picture of what conclusions should be drawn. Finally, the model uncertainty variance component can be estimated using BMA. The variance due to model uncertainty is ignored when the standard model building tenets are used giving overly optimistic variance estimates. Even though the model attained via standard techniques may be adequate, in general, it would be difficult to argue that the chosen model is in fact the correct model. It seems more appropriate to incorporate the information from all plausible models that are well supported by the data to make decisions and to use variance estimates that account for the uncertainty in the model estimation as well as model selection. / Ph. D.

The Multiset Sampler [Leman et al., 2009] has previously been deployed and developed for efficient sampling from complex stochastic processes. We extend the sampler and the surrounding theory to model selection problems. In such problems efficient exploration of the model space becomes a challenge since independent and ad-hoc proposals might not be able to jointly propose multiple parameter sets which correctly explain a new pro- posed model. In order to overcome this we propose a multiset on the model space to en- able efficient exploration of multiple model modes with almost no tuning. The Multiset Model Selection (MSMS) framework is based on independent priors for the parameters and model indicators on variables. We show that posterior model probabilities can be easily obtained from multiset averaged posterior model probabilities in MSMS. We also obtain typical Bayesian model averaged estimates for the parameters from MSMS. We apply our algorithm to linear regression where it allows easy moves between parame- ter modes of different models, and in probit regression where it allows jumps between widely varying model specific covariance structures in the latent space of a hierarchical model. The Storytelling algorithm [Kumar et al., 2006] constructs stories by discovering and con- necting latent connections between documents in a network. Such automated algorithms often do not agree with user's mental map of the data. Hence systems that incorporate feedback through visual interaction from the user are of immediate importance. We pro- pose a visual analytic framework in which such interactions are naturally incorporated in to the existing Storytelling algorithm through a redefinition of the latent topic space used in the similarity measure of the network. The document network can be explored us- ing the newly learned normalized topic weights for each document. Hence our algorithm augments the limitations of human sensemaking capabilities in large document networks by providing a collaborative framework between the underlying model and the user. Our formulation of the problem is a supervised topic modeling problem where the supervi- sion is based on relationships imposed by the user as a set of inequalities derived from tolerances on edge costs from inverse shortest path problem. We show a probabilistic modeling of the relationships based on auxiliary variables and propose a Gibbs sampling based strategy. We provide detailed results from a simulated data and the Atlantic Storm data set. / Ph. D.

La théorie statistique univariée des valeurs extrêmes se généralise au cas multivarié mais l'absence d'un cadre paramétrique naturel complique l'inférence de la loi jointe des extrêmes. Les marges d'erreur associée aux estimateurs non paramétriques de la structure de dépendance sont difficilement accessibles à partir de la dimension trois. Cependant, quantifier l'incertitude est d'autant plus important pour les applications que le problème de la rareté des données extrêmes est récurrent, en particulier en hydrologie. L'objet de cette thèse est de développer des modèles de dépendance entre extrêmes, dans un cadre bayésien permettant de représenter l'incertitude. Le chapitre 2 explore les propriétés des modèles obtenus en combinant des modèles paramétriques existants, par mélange bayésien (Bayesian Model Averaging BMA). Un modèle semi-paramétrique de mélange de Dirichlet est étudié au chapitre suivant : une nouvelle paramétrisation est introduite afin de s'affranchir d'une contrainte de moments caractéristique de la structure de dépendance et de faciliter l'échantillonnage de la loi à posteriori. Le chapitre 4 est motivé par une application hydrologique : il s'agit d'estimer la structure de dépendance spatiale des crues extrêmes dans la région cévenole des Gardons en utilisant des données historiques enregistrées en quatre points. Les données anciennes augmentent la taille de l'échantillon mais beaucoup de ces données sont censurées. Une méthode d'augmentation de données est introduite, dans le cadre du mélange de Dirichlet, palliant l'absence d'expression explicite de la vraisemblance censurée. Les conclusions et perspectives sont discutées au chapitre 5 / Uni-variate extreme value theory extends to the multivariate case but the absence of a natural parametric framework for the joint distribution of extremes complexifies inferential matters. Available non parametric estimators of the dependence structure do not come with tractable uncertainty intervals for problems of dimension greater than three. However, uncertainty estimation is all the more important for applied purposes that data scarcity is a recurrent issue, particularly in the field of hydrology. The purpose of this thesis is to develop modeling tools for the dependence structure between extremes, in a Bayesian framework that allows uncertainty assessment. Chapter 2 explores the properties of the model obtained by combining existing ones, in a Bayesian Model Averaging framework. A semi-parametric Dirichlet mixture model is studied next : a new parametrization is introduced, in order to relax a moments constraint which characterizes the dependence structure. The re-parametrization significantly improves convergence and mixing properties of the reversible-jump algorithm used to sample the posterior. The last chapter is motivated by an hydrological application, which consists in estimating the dependence structure of floods recorded at four neighboring stations, in the ‘Gardons’ region, southern France, using historical data. The latter increase the sample size but most of them are censored. The lack of explicit expression for the likelihood in the Dirichlet mixture model is handled by using a data augmentation framework

In this thesis Generalized Additive Neural Networks (GANNs) are studied in the context of predictive Data Mining. A GANN is a novel neural network implementation of a Generalized Additive Model. Originally GANNs were constructed interactively by considering partial residual plots. This methodology involves subjective human judgment, is time consuming, and can result in suboptimal results. The newly developed automated construction algorithm solves these difficulties by performing model selection based on an objective model selection criterion. Partial residual plots are only utilized after the best model is found to gain insight into the relationships between inputs and the target. Models are organized in a search tree with a greedy search procedure that identifies good models in a relatively short time. The automated construction algorithm, implemented in the powerful SAS® language, is nontrivial, effective, and comparable to other model selection methodologies found in the literature. This implementation, which is called AutoGANN, has a simple, intuitive, and user-friendly interface. The AutoGANN system is further extended with an approximation to Bayesian Model Averaging. This technique accounts for uncertainty about the variables that must be included in the model and uncertainty about the model structure. Model averaging utilizes in-sample model selection criteria and creates a combined model with better predictive ability than using any single model. In the field of Credit Scoring, the standard theory of scorecard building is not tampered with, but a pre-processing step is introduced to arrive at a more accurate scorecard that discriminates better between good and bad applicants. The pre-processing step exploits GANN models to achieve significant reductions in marginal and cumulative bad rates. The time it takes to develop a scorecard may be reduced by utilizing the automated construction algorithm. / Thesis (Ph.D. (Computer Science))--North-West University, Potchefstroom Campus, 2006.

In this thesis Generalized Additive Neural Networks (GANNs) are studied in the context of predictive Data Mining. A GANN is a novel neural network implementation of a Generalized Additive Model. Originally GANNs were constructed interactively by considering partial residual plots. This methodology involves subjective human judgment, is time consuming, and can result in suboptimal results. The newly developed automated construction algorithm solves these difficulties by performing model selection based on an objective model selection criterion. Partial residual plots are only utilized after the best model is found to gain insight into the relationships between inputs and the target. Models are organized in a search tree with a greedy search procedure that identifies good models in a relatively short time. The automated construction algorithm, implemented in the powerful SAS® language, is nontrivial, effective, and comparable to other model selection methodologies found in the literature. This implementation, which is called AutoGANN, has a simple, intuitive, and user-friendly interface. The AutoGANN system is further extended with an approximation to Bayesian Model Averaging. This technique accounts for uncertainty about the variables that must be included in the model and uncertainty about the model structure. Model averaging utilizes in-sample model selection criteria and creates a combined model with better predictive ability than using any single model. In the field of Credit Scoring, the standard theory of scorecard building is not tampered with, but a pre-processing step is introduced to arrive at a more accurate scorecard that discriminates better between good and bad applicants. The pre-processing step exploits GANN models to achieve significant reductions in marginal and cumulative bad rates. The time it takes to develop a scorecard may be reduced by utilizing the automated construction algorithm. / Thesis (Ph.D. (Computer Science))--North-West University, Potchefstroom Campus, 2006.

This paper uses Bayesian model comparison methods to simultaneously specify both the spatial weight structure and explanatory variables for a spatial growth regression involving 255 NUTS 2 regions across 25 European countries. In addition, a correct interpretation of the spatial regression parameter estimates that takes into account the simultaneous feed- back nature of the spatial autoregressive model is provided. Our findings indicate that incorporating model uncertainty in conjunction with appropriate parameter interpretation decreased the importance of explanatory variables traditionally thought to exert an important influence on regional income growth rates. (authors' abstract)

Pharmacometric model-based analysis using nonlinear mixed-effects models (NLMEM) has to date mainly been applied to learning activities in drug development. However, such analyses can also serve as the primary analysis in confirmatory studies, which is expected to bring higher power than traditional analysis methods, among other advantages. Because of the high expertise in designing and interpreting confirmatory studies with other types of analyses and because of a number of unresolved uncertainties regarding the magnitude of potential gains and risks, pharmacometric analyses are traditionally not used as primary analysis in confirmatory trials. The aim of this thesis was to address current hurdles hampering the use of pharmacometric model-based analysis in confirmatory settings by developing strategies to increase model compliance to distributional assumptions regarding the residual error, to improve the quantification of parameter uncertainty and to enable model prespecification. A dynamic transform-both-sides approach capable of handling skewed and/or heteroscedastic residuals and a t-distribution approach allowing for symmetric heavy tails were developed and proved relevant tools to increase model compliance to distributional assumptions regarding the residual error. A diagnostic capable of assessing the appropriateness of parameter uncertainty distributions was developed, showing that currently used uncertainty methods such as bootstrap have limitations for NLMEM. A method based on sampling importance resampling (SIR) was thus proposed, which could provide parameter uncertainty in many situations where other methods fail such as with small datasets, highly nonlinear models or meta-analysis. SIR was successfully applied to predict the uncertainty in human plasma concentrations for the antibiotic colistin and its prodrug colistin methanesulfonate based on an interspecies whole-body physiologically based pharmacokinetic model. Lastly, strategies based on model-averaging were proposed to enable full model prespecification and proved to be valid alternatives to standard methodologies for studies assessing the QT prolongation potential of a drug and for phase III trials in rheumatoid arthritis. In conclusion, improved methods for handling residual error, parameter uncertainty and model uncertainty in NLMEM were successfully developed. As confirmatory trials are among the most demanding in terms of patient-participation, cost and time in drug development, allowing (some of) these trials to be analyzed with pharmacometric model-based methods will help improve the safety and efficiency of drug development.