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Abstract
This dissertation addresses two fundamental areas in statistics: the design of optimal physical experiments and the development of surrogate models for complex computer experiments, with a focus on feature importance and uncertainty quantification. In the first part, we investigate locally D-optimal crossover designs for generalized linear models. Model parameters and their variances are estimated using generalized estimating equations (GEEs). We identify optimal allocations of experimental units across treatment sequences and demonstrate through simulations that these allocations are reasonably robust to various choices of the correlation structure. Furthermore, we show that a two-stage design—employing our locally $D$-optimal design in the second stage—yields greater efficiency than a uniform design, particularly in the presence of intra-subject correlation. The second part of the dissertation extends the principles of Design of Experiments (DoE) to improve reinforcement learning techniques for computer experiments (CEs), which are essential tools for studying phenomena where physical experimentation is infeasible, such as the spread of COVID-19. Building accurate computer models often involves a high-dimensional input space, making the identification of active variables critical. We propose a novel variable selection approach integrated with Active Learning for Lasso Regression, using a weighted distance function to sequentially guide variable selection. Additionally, we introduce a multi-objective optimization framework to construct efficient Sequential MaxPro Designs and Sequential Orthogonal-MaxPro Designs. Finally, we explore an extension of this reinforcement learning framework to Deep Gaussian Process (DGP) models, enabling more flexible modeling and a deeper understanding of feature importance under uncertainty.