Dimensional analysis in statistics: Theories, methodologies and applications

Dimensional Analysis (DA) is a well-developed widely-employed methodology in the physical and engineering sciences. It features advantages such as (1) the reduction of the number of factors, (2) the interpretability of the variables, and (3) the scalability of results. However, few researches in statistical literature offer a comprehensive view on incorporating DA in statistical problems. This dissertation is among the first few which aim at a thorough study on both the practical benefits and theoretical development of statistical design and analysis in random and deterministic experiments. We first introduce the physical background and the formal procedure of DA. A literature review of DA applications in various disciplines is provided, including recent literature in statistics. The need for new statistical treatment incorporating DA is established. Next, the main body of the dissertation is divided into two parts: applications and theories. In the applications part, we begin from physical studies with random errors. We examine the benefits of implementing DA in statistical analysis and design of experiment, and conclude with the advantages and potential issues. Then, we move to deterministic experiments and conduct a study on applying DA in computer experiments. Both design and analysis phases are discussed with detailed case studies to show when DA works and when it does not. Through the applications, we clearly observe the merits of combining DA (as physical knowledge) in empirical studies. Meanwhile, the statistical mechanism behind DA transformation has still not been well understood. Therefore, in the theory part, we first present a new statistical perspective on DA through properties in terms of invariance, sufficiency and completeness. Then, in order to solve the potential issues in modeling DA variables, a new type of modeling methodology is proposed along with its corresponding inference procedures. The established solution unifies the physical knowledge from DA and the empirical knowledge from data in a clear and controllable way, providing both practical guidance and theoretical insights in general. Although far from an all-round study on applying DA in statistical problems, this dissertation contributes among the first few to formally treat DA from statistical points of view. Both design and analysis, random and deterministic experiments, applications and theories are covered and discussed in detail. It serves as a comprehensive investigation of DA in statistics discipline, which is helpful to both practitioners and researchers.