In longitudinal studies, observational units (commonly referred to as individuals) drawn from some population of interest are followed prospectively over time, and measurements from each individual are taken repeatedly at different points in time with the ultimate goal of characterizing the important features of the population. Longitudinal data naturally arise in many areas of study, where the characterization of the population may be achieved by investigating the effects of covariates on a response. Two or more correlated responses from each individual are also common in longitudinal studies, giving rise to multivariate longitudinal data. For example, the National Longitudinal Survey of Children and Youth (NLSCY) is a long-term study to observe the development of Canadian children. In this survey, measurements about factors influencing a child's social, emotional and behavioral development are collected over time; anxiety and aggression reported for each child in this study may be considered as two response variables to characterize the emotional and behavioral development of children. Since in longitudinal studies, information is collected repeatedly from each individual over time, the occurrence of an event at a particular time point may increase/decrease the likelihood of the occurrence of another event in future. Failure to take into account this phenomenon in analyzing longitudinal data may lead to erroneous conclusion. Moreover, repeated responses (e.g., anxiety and aggression) from an individual may exhibit correlation over time. Separate analyses of such multivariate longitudinal responses ignore this correlation, and as a result, cannot reveal the potential association among the responses which could be of paramount importance in many applications. Therefore, analysis of multivariate longitudinal data requires substantial extension of the standard longitudinal methods. In this thesis, we describe a methodology based on the transition models for multivariate longitudinal binary data to address the transitional behavior between two states characterized by binary responses for two different responses (i.e., two processes). Transitional analysis of multivariate longitudinal binary data can address the longitudinal association within processes and enable marginal interpretation of covariate effects. In addition, estimation and inference of the association between the processes can also be achieved via such models. We illustrate this approach with an application to the NLSCY data, where anxiety and aggression (two correlated responses) are modeled as a function of covariates (gender, depression of person most knowledgeable, number of siblings and family status) to identify their effects on behavioral development of Canadian children. In addition, the extent and direction of the association between two responses are estimated. Gender of the child is found statistically significant for both directions of transition, i.e., from low to high and high to low, of aggression. On contrary, gender of the child is found statistically not significant for both transitions of anxiety. Meanwhile, depression of person most knowledgeable is found marginally significant in the high to low direction for aggression. For association parameters, all four directions of associations between anxiety and aggression are found statistically significant.

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This dissertation presents a sampling-based method that can be used for uncertainty quantification and deterministic or probabilistic optimization. The objective is to simultaneously address several difficulties faced by classical techniques based on response values and their gradients. In particular, this research addresses issues with discontinuous and binary (pass or fail) responses, and multiple failure modes. All methods in this research are developed with the aim of addressing problems that have limited data due to high cost of computation or experiment, e.g. vehicle crashworthiness, fluid-structure interaction etc.The core idea of this research is to construct an explicit boundary separating allowable and unallowable behaviors, based on classification information of responses instead of their actual values. As a result, the proposed method is naturally suited to handle discontinuities and binary states. A machine learning technique referred to as support vector machines (SVMs) is used to construct the explicit boundaries. SVM boundaries can be highly nonlinear, which allows one to use a single SVM for representing multiple failure modes.One of the major concerns in the design and uncertainty quantification communities is to reduce computational costs. To address this issue, several adaptive sampling methods have been developed as part of this dissertation. Specific sampling methods have been developed for reliability assessment, deterministic optimization, and reliability-based design optimization. Adaptive sampling allows the construction of accurate SVMs with limited samples. However, like any approximation method, construction of SVM is subject to errors. A new method to quantify the prediction error of SVMs, based on probabilistic support vector machines (PSVMs) is also developed. It is used to provide a relatively conservative probability of failure to mitigate some of the adverse effects of an inaccurate SVM. In the context of reliability assessment, the proposed method is presented for uncertainties represented by random variables as well as spatially varying random fields.In order to validate the developed methods, analytical problems with known solutions are used. In addition, the approach is applied to some application problems, such as structural impact and tolerance optimization, to demonstrate its strengths in the context of discontinuous responses and multiple failure modes.