A Conjugate Model for Dimensional Analysis

Dimensional analysis (DA) is a methodology widely used in physics and engineering. The main idea is to extract key variables based on physical dimensions. Its overlooked importance in statistics has been recognized recently. However, most literature treats DA as merely a preprocessing tool, leading to multiple statistical issues. In particular, there are three critical aspects: (a) the nonunique choice of basis quantities and dimensionless variables; (b) the statistical representation and testing of DA constraints; (c) the spurious correlations between post-DA variables. There is an immediate need for an appropriate statistical methodology that integrates DA and the quantitative modeling. In this article, we propose a power-law type of “DA conjugate” model that is useful for incorporating dimensional information and analyzing post-DA variables. Adapting the similar idea of “conjugacy” in Bayesian analysis, we show that the proposed modeling technique not only produces flexible and effective results, but also provides good solutions to the above three issues. A modified projection pursuit regression analysis is implemented to fit the additive power-law model. A numerical study on ocean wave speed is discussed in detail to illustrate and evaluate the advantages of the proposed procedure. Supplementary materials for this article are available online.