UBC Theses and Dissertations
How to improve prediction accuracy in the analysis of computer experiments, exploitation of low-order effects and dimensional analysis Rodríguez Arelis, Gilberto Alexi
A wide range of natural phenomena and engineering processes make physical experimentation hard to apply, or even impossible. To overcome these issues, we can rely on mathematical models that simulate these systems via computer experiments. Nonetheless, if the experimenter wants to explore many runs, complex computer codes can be excessively resource and time-consuming. Since the 1980s, Gaussian Stochastic Processes have been used in computer experiments as surrogate models. Their objective can be predicting outputs at untried input runs, given a model fitted with a training design coming from the computer code. We can exploit different modelling strategies to improve prediction accuracy, e.g., the regression component or the correlation function. This thesis makes a comprehensive exploitation of two additional strategies, which the existing literature has not fully addressed in computer experiments. One of these strategies is implementing non-standard correlation structures in model training and testing. Since the beginning of factorial designs for physical experiments in the first half of the 20th century, there have been basic guidelines for modelling from three effect principles: Sparsity, Heredity, and Hierarchy. We explore these principles in a Gaussian Stochastic Process by suggesting and evaluating novel correlation structures. Our second strategy focuses on output and input transformations via Dimensional Analysis. This methodology pays attention to fundamental physical dimensions when modelling scientific and engineering systems. It goes back at least a century but has recently caught statisticians' attention, particularly in the design of physical experiments. The core idea is to analyze dimensionless quantities derived from the original variables. While the non-standard correlation structures depict additive and low-order interaction effects, applying the three principles above relies on a proper selection of effects. Similarly, the implementation of Dimensional Analysis is far from straightforward; choosing the derived quantities is particularly challenging. Hence, we rely on Functional Analysis of Variance as a variable selection tool for both strategies. With the "right" variables, the Gaussian Stochastic Process' prediction accuracy improves for several case studies, which allows us to establish new modelling frameworks for computer experiments.
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