Data-Adaptive Dimensional Analysis for Accurate Interpolation and
  Extrapolation in Computer Experiments
G. Alexi Rodriguez-Arelis William J. Welch
Dimensional analysis (DA) pays attention to fundamental physical dimensions
such as length and mass when modelling scientific and engineering systems. It
goes back at least a century to Buckingham's Pi theorem, which characterizes a
scientifically meaningful model in terms of a limited number of dimensionless
variables. The methodology has only been exploited relatively recently by
statisticians for design and analysis of experiments, however, and computer
experiments in particular. The basic idea is to build models in terms of new
dimensionless quantities derived from the original input and output variables.
A scientifically valid formulation has the potential for improved prediction
accuracy in principle, but the implementation of DA is far from
straightforward. There can be a combinatorial number of possible models
satisfying the conditions of the theory. Empirical approaches for finding
effective derived variables will be described, and improvements in prediction
accuracy will be demonstrated. As DA's dimensionless quantities for a
statistical model typically compare the original variables rather than use
their absolute magnitudes, DA is less dependent on the choice of experimental
ranges in the training data. Hence, we are also able to illustrate sustained
accuracy gains even when extrapolating substantially outside the training data.