D-MORPH regression: Application to modeling with unknown parameters more than observation data

Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) is a model exploration method, originally developed for differential equations. We extend D-MORPH to regression treatment of a model described as a linear superposition of basis functions with unknown parameters being the expansion coefficients. The goal of D-MORPH regression is to improve prediction accuracy without sacrificing fitting accuracy. When there are more unknown parameters than observation data, the corresponding linear algebraic equation system is generally consistent, and has an infinite number of solutions exactly fitting the data. In this case, the solutions given by standard regression techniques can significantly deviate from the true system structure, and consequently provide large prediction errors for the model. D-MORPH regression is a practical systematic means to search over system structure within the infinite number of possible solutions while preserving fitting accuracy. An explicit expression is provided by D-MORPH regression relating the data to the expansion coefficients in the linear model. The expansion coefficients obtained by D-MORPH regression are particular linear combinations of those obtained by least-squares regression. The resultant prediction accuracy provided by D-MORPH regression is shown to be significantly improved in several model illustrations.