Minimum energy designs: Extensions, algorithms, and applications
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Minimum Energy Design (MED) is a recently proposed technique for generating deterministic samples from any arbitrary probability distribution. Most space-filling designs look for uniformity in the region of interest. In MED, some weights are assigned in the optimal design criterion so that some areas are preferred over the other areas. With a proper choice of the weights, the MED can asymptotically represent the target distribution. In this dissertation, we improve and extend MED in three different aspects. In Chapter 1, we propose an efficient approach that uses MED to construct proposals for an independence sampler in Monte Carlo Markov Chain (MCMC). Between two adjacent temperatures, MED points are selected to keep and transfer the mixing information. In Chapter 2, when evaluations on the posterior distribution become expensive, traditional MC/MCMC methods are infeasible because of the requirement of large samples. MED is a good way to overcome this problem. It can be viewed as a ``deterministic’’ sampling method that avoids repeated sampling in the same places, which dramatically decreases the number of required samples. The MED criterion is generalized and a fast construction algorithm is developed. Finally, in Chapter 3, we propose a new type of MEDs and a new modeling method for robust parameter design in computer experiments. In the design part, a new design based on the generalized MED criterion is proposed, where different tuning parameters are used for control and noise factors. In the modeling part, we propose a simple but efficient nonstationary Gaussian process that takes into account of the experimental design structure.