Nondimensionalizing physical and statistical models: a unified approach

This paper presents a version of the statistical invariance principle that incorporates constraints imposed by dimensional analysis. It does so by adapting Buckingham's Pi-theorem for deterministic models of physical processes based on vectors of attribute variables, to one for data models for stochastic processes based on random matrices of those attributes. Buckingham's method is meant for a priori modeling to reduce the number of attributes, thus simplifying experimental design, and reducing experimental costs.  In contrast, this paper's proposal allows for more flexibility in modeling, including a stochasatic component that brings with it a measure of that model's uncertainty. The theory is extended to incorporate both a priori and a posteriori Bayesian modeling. A simple example of that extension to regression modelling is given.