D-MORPH regression for modeling with fewer unknown parameters than observation data

Abstract

D-MORPH regression is a procedure for the treatment of a model prescribed as a linear superposition of basis functions with less observation data than the number of expansion parameters. In this case, there is an infinite number of solutions exactly fitting the data. D-MORPH regression provides a practical systematic means to search over the solutions seeking one with desired ancillary properties while preserving fitting accuracy. This paper extends D-MORPH regression to consider the common case where there is more observation data than unknown parameters. This situation is treated by utilizing a proper subset of the normal equation of least-squares regression to judiciously reduce the number of linear algebraic equations to be less than the number of unknown parameters, thereby permitting application of D-MORPH regression. As a result, no restrictions are placed on model complexity, and the model with the best prediction accuracy can be automatically and efficiently identified. Ignition data for a H2/air combustion model as well as laboratory data for quantum-control-mechanism identification are used to illustrate the method.