Analysis of high dimensional data often seeks to identify a subset of important features and assess their effects on the outcome. Furthermore, the ultimate goal is often to build a prediction model with these features that accurately assesses risk for future subjects. Such statistical challenges arise in the study of genetic associations with health outcomes. However, accurate inference and prediction with genetic information remains challenging, in part due to the complexity in the genetic architecture of human health and disease. A valuable approach for improving prediction models with a large number of potential predictors is to build a parsimonious model that includes only important variables. Regularized regression methods are useful, though often pose challenges for inference due to nonstandard limiting distributions or finite sample distributions that are difficult to approximate. In Chapter 1 we propose and theoretically justify a perturbation-resampling method to derive confidence regions and covariance estimates for marker effects estimated from regularized procedures with a general class of objective functions and concave penalties. Our methods outperform their asymptotic-based counterparts, even when effects are estimated as zero. In Chapters 2 and 3 we focus on genetic risk prediction. The difficulty in accurate risk assessment with genetic studies can in part be attributed to several potential obstacles: sparsity in marker effects, a large number of weak signals, and non-linear effects. Single marker analyses often lack power to select informative markers and typically do not account for non-linearity. One approach to gain predictive power and efficiency is to group markers based on biological knowledge such genetic pathways or gene structure. In Chapter 2 we propose and theoretically justify a multi-stage method for risk assessment that imposes a naive bayes kernel machine (KM) model to estimate gene-set specific risk models, and then aggregates information across all gene-sets by adaptively estimating gene-set weights via a regularization procedure. In Chapter 3 we extend these methods to meta-analyses by introducing sampling-based weights in the KM model. This permits building risk prediction models with multiple studies that have heterogeneous sampling schemes

This work focuses on automatic classification of images into semantic classes based on their contentc, especially in using SVM classifiers. The main objective of this work is to improve classification accuracy on large datasets. Both linear and nonlinear SVM classifiers are considered. In addition, the possibility of transforming features by Restricted Boltzmann Machines and using linear SVM is explored as well. All these approaches are compared in terms of accuracy, computational demands, resource utilization, and possibilities for future research.

We are living in an era in which many mysteries related to science, technologies and design can be answered by "learning" the huge amount of data accumulated over the past few decades. In the processes of those endeavors, highly-correlated high-dimensional data are frequently observed in many areas including predicting shelf life, controlling manufacturing processes, and identifying important pathways related with diseases. We define a "set" as a group of highly-correlated high-dimensional (HCHD) variables that possess a certain practical meaning or control a certain process, and define an "element" as one of the HCHD variables within a certain set. Such an elements-within-a-set structure is very complicated because: (i) the dimensions of elements in different sets can vary dramatically, ranging from two to hundreds or even thousands; (ii) the true relationships, include element-wise associations, set-wise interactions, and element-set interactions, are unknown; (iii) and the sample size (n) is usually much smaller than the dimension of the elements (p). The goal of this dissertation is to provide a systematic way to identify both the set effects and the element effects associated with survival outcomes from heterogeneous populations using Bayesian survival kernel models. By connecting kernel machines with semiparametric Bayesian hierarchical models, the proposed unified model frameworks can identify significant elements as well as sets regardless of mis-specifications of distributions or kernels. The proposed methods can potentially be applied to a vast range of fields to solve real-world problems. / PHD / We are living in an era in which many mysteries related to science, technologies and design can be answered by “learning” the huge amount of data accumulated over the past few decades. In the processes of those endeavors, highly-correlated high-dimensional data are frequently observed in many areas including predicting shelf life, controlling manufacturing processes, and identifying important pathways related with diseases. For example, for a group of 30 patients in a medical study, values for an immense number of variables like gender, age, height, weight, and blood pressure of each patient are recorded. High-dimensional means the number of variables (i.e. p) could be very large (e.g. p > 500), while the number of subjects or the sample size (i.e. n) is small (n = 30). We define a “set” as a group of highly-correlated high-dimensional (HCHD) variables that possess a certain practical meaning or control a certain process, and define an “element” as one of the HCHD variables within a certain set. Such an elements-within-a-set structure is very complicated because: (i) the dimensions of elements in different sets can vary dramatically, ranging from two to hundreds or even thousands; (ii) the true relationships, include element-wise associations, set-wise interactions, and element-set interactions, are unknown; (iii) and the sample size (n) is usually much smaller than the dimension of the elements (p). The goal of this dissertation is to provide a systematic way to identify both the set effects and the element effects associated with survival outcomes from heterogeneous populations using different proposed statistical models. The proposed models can incorporate prior knowledge to boost the model performance. The proposed methods can potentially be applied to a vast range of fields to solve real-world problems.

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A variety of complex data has broadened in many research fields such as epidemiology, genomics, and analytical chemistry with the development of science, technologies, and design scheme over the past few decades. For example, in epidemiology, the matched case-crossover study design is used to investigate the association between the clustered binary outcomes of disease and a measurement error in covariate within a certain period by stratifying subjects' conditions. In genomics, high-correlated and high-dimensional(HCHD) data are required to identify important genes and their interaction effect over diseases. In analytical chemistry, multiple time series data are generated to recognize the complex patterns among multiple classes. Due to the great diversity, we encounter three problems in analyzing those complex data in this dissertation. We have then provided several contributions to semiparametric and nonparametric methods for dealing with the following problems: the first is to propose a method for testing the significance of a functional association under the matched study; the second is to develop a method to simultaneously identify important variables and build a network in HDHC data; the third is to propose a multi-class dynamic model for recognizing a pattern in the time-trend analysis. For the first topic, we propose a semiparametric omnibus test for testing the significance of a functional association between the clustered binary outcomes and covariates with measurement error by taking into account the effect modification of matching covariates. We develop a flexible omnibus test for testing purposes without a specific alternative form of a hypothesis. The advantages of our omnibus test are demonstrated through simulation studies and 1-4 bidirectional matched data analyses from an epidemiology study. For the second topic, we propose a joint semiparametric kernel machine network approach to provide a connection between variable selection and network estimation. Our approach is a unified and integrated method that can simultaneously identify important variables and build a network among them. We develop our approach under a semiparametric kernel machine regression framework, which can allow for the possibility that each variable might be nonlinear and is likely to interact with each other in a complicated way. We demonstrate our approach using simulation studies and real application on genetic pathway analysis. Lastly, for the third project, we propose a Bayesian focal-area detection method for a multi-class dynamic model under a Bayesian hierarchical framework. Two-step Bayesian sequential procedures are developed to estimate patterns and detect focal intervals, which can be used for gas chromatography. We demonstrate the performance of our proposed method using a simulation study and real application on gas chromatography on Fast Odor Chromatographic Sniffer (FOX) system. / Doctor of Philosophy / A variety of complex data has broadened in many research fields such as epidemiology, genomics, and analytical chemistry with the development of science, technologies, and design scheme over the past few decades. For example, in epidemiology, the matched case-crossover study design is used to investigate the association between the clustered binary outcomes of disease and a measurement error in covariate within a certain period by stratifying subjects' conditions. In genomics, high-correlated and high-dimensional(HCHD) data are required to identify important genes and their interaction effect over diseases. In analytical chemistry, multiple time series data are generated to recognize the complex patterns among multiple classes. Due to the great diversity, we encounter three problems in analyzing the following three types of data: (1) matched case-crossover data, (2) HCHD data, and (3) Time-series data. We contribute to the development of statistical methods to deal with such complex data. First, under the matched study, we discuss an idea about hypothesis testing to effectively determine the association between observed factors and risk of interested disease. Because, in practice, we do not know the specific form of the association, it might be challenging to set a specific alternative hypothesis. By reflecting the reality, we consider the possibility that some observations are measured with errors. By considering these measurement errors, we develop a testing procedure under the matched case-crossover framework. This testing procedure has the flexibility to make inferences on various hypothesis settings. Second, we consider the data where the number of variables is very large compared to the sample size, and the variables are correlated to each other. In this case, our goal is to identify important variables for outcome among a large amount of the variables and build their network. For example, identifying few genes among whole genomics associated with diabetes can be used to develop biomarkers. By our proposed approach in the second project, we can identify differentially expressed and important genes and their network structure with consideration for the outcome. Lastly, we consider the scenario of changing patterns of interest over time with application to gas chromatography. We propose an efficient detection method to effectively distinguish the patterns of multi-level subjects in time-trend analysis. We suggest that our proposed method can give precious information on efficient search for the distinguishable patterns so as to reduce the burden of examining all observations in the data.

Model and variable selection have attracted considerable attention in areas of application where datasets usually contain thousands of variables. Variable selection is a critical step to reduce the dimension of high dimensional data by eliminating irrelevant variables. The general objective of variable selection is not only to obtain a set of cost-effective predictors selected but also to improve prediction and prediction variance. We have made several contributions to this issue through a range of advanced topics: providing a graphical view of Bayesian Variable Selection (BVS), recovering sparsity in multivariate nonparametric models and proposing a testing procedure for evaluating nonlinear interaction effect in a semiparametric model. To address the first topic, we propose a new Bayesian variable selection approach via the graphical model and the Ising model, which we refer to the ``Bayesian Ising Graphical Model'' (BIGM). There are several advantages of our BIGM: it is easy to (1) employ the single-site updating and cluster updating algorithm, both of which are suitable for problems with small sample sizes and a larger number of variables, (2) extend this approach to nonparametric regression models, and (3) incorporate graphical prior information. In the second topic, we propose a Nonnegative Garrote on a Kernel machine (NGK) to recover sparsity of input variables in smoothing functions. We model the smoothing function by a least squares kernel machine and construct a nonnegative garrote on the kernel model as the function of the similarity matrix. An efficient coordinate descent/backfitting algorithm is developed. The third topic involves a specific genetic pathway dataset in which the pathways interact with the environmental variables. We propose a semiparametric method to model the pathway-environment interaction. We then employ a restricted likelihood ratio test and a score test to evaluate the main pathway effect and the pathway-environment interaction. / Ph. D.