A Focused Regions Identification Method for Nonlinear Least Squares Curve Fitting Problems

Abstract

Important for many science and engineering fields, meaningful nonlinear models result from fitting such models to data by estimating the value of each parameter in the model. Since parameters in nonlinear models often characterize a substance or a system (e.g., mass diffusivity), it is critical to find the optimal parameter estimators that minimize or maximize a chosen objective function. In practice, iterative local methods (e.g., Levenberg-Marquardt method) and heuristic methods (e.g., genetic algorithms) are commonly employed for least squares parameter estimation in nonlinear models. However, practitioners are not able to know whether the parameter estimators derived through these methods are the optimal parameter estimators that correspond to the global minimum of the squared error of the fit. In this paper, a focused regions identification method is introduced for least squares parameter estimation in nonlinear models. Using expected fitting accuracy and derivatives of the squared error of the fit, this method rules out the regions in parameter space where the optimal parameter estimators cannot exist. Practitioners are guaranteed to find the optimal parameter estimators through an exhaustive search in the remaining regions (i.e., focused regions). The focused regions identification method is validated through a case study in which the Michaelis-Menten model is fitted to an experimental data set. The case study shows that the focused regions identification method can find the optimal parameter estimators and the corresponding global minimum effectively and efficiently.