Taking the Guess Work Out of the Initial Guess: A Solution Interval Method for Least-Squares Parameter Estimation in Nonlinear Models

Abstract

Fitting a specified model to data is critical in many science and engineering fields. A major task in fitting a specified model to data is to estimate the value of each parameter in the model. Iterative local methods, such as the Gauss–Newton method and the Levenberg–Marquardt method, are often employed for parameter estimation in nonlinear models. However, practitioners must guess the initial value for each parameter to initialize these iterative local methods. A poor initial guess can contribute to non-convergence of these methods or lead these methods to converge to a wrong or inferior solution. In this paper, a solution interval method is introduced to find the optimal estimator for each parameter in a nonlinear model that minimizes the squared error of the fit. To initialize this method, it is not necessary for practitioners to guess the initial value of each parameter in a nonlinear model. The method includes three algorithms that require different levels of computational power to find the optimal parameter estimators. The method constructs a solution interval for each parameter in the model. These solution intervals significantly reduce the search space for optimal parameter estimators. The method also provides an empirical probability distribution for each parameter, which is valuable for parameter uncertainty assessment. The solution interval method is validated through two case studies in which the Michaelis–Menten model and Fick’s second law are fit to experimental data sets, respectively. These case studies show that the solution interval method can find optimal parameter estimators efficiently. A four-step procedure for implementing the solution interval method in practice is also outlined.