Reducing the Search Space for Global Minimum: A Focused Regions Identification Method for Least Squares Parameter Estimation in Nonlinear Models

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 two case studies in which a model based on Newton’s law of cooling and the Michaelis–Menten model are fitted to two experimental data sets, respectively. These case studies show that the focused regions identification method can find the optimal parameter estimators and the corresponding global minimum effectively and efficiently.