An Initial Guess Free Method for Least Squares Parameter Estimation in Nonlinear Models

Abstract

Fitting models to data is critical in many science and engineering fields. A major task in fitting models to data is to estimate the value of each parameter in a given model. Iterative 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 in order to initialize these iterative methods. A poor initial guess can contribute to non-convergence of these methods or lead these methods to converge to a wrong solution. In this paper, an initial guess free method is introduced to find the optimal parameter estimators in a nonlinear model that minimizes the squared error of the fit. The method includes three algorithms that require different level 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 initial guess free method is validated through a case study in which Fick’s second law is fit to an experimental data set. This case study shows that the initial guess free method can find the optimal parameter estimators efficiently. A four-step procedure for implementing the initial guess free method in practice is also outlined.