Optimal parameterization of a mathematical model for solving parameter estimation problems

Abstract

This article presents a method to solve parameter estimation problems by finding an optimal parameterization of the mathematical model. The pi-theorem of dimensional analysis is used to establish a formulation of the model based on dimensionless products and scaling parameters, together with the rules of a parameterization change. Logarithmic parameters are introduced for the purpose of working in a linear parameter space in which a given base corresponds to a specific parameterization. The optimal parameterization is assumed to yield uncorrelated estimators. A statistical independence criterion based on the Fisher information matrix is derived for maximum-likelihood estimators. The method allows one to solve inverse problems with highly correlated model parameters by identifying well-resolved parameters, leading to a solution expressed in terms of correlation laws between physical quantities.