A self-guided search for good local minima of the sum-of-squared-error in nonlinear least squares regression

摘要

Hard modeling of nonlinear chemical or biological systems is highly relevant as a model function together with values for model parameters provides insights in the systems' functionalities. Deriving values for said model parameters via nonlinear regression, however, is challenging as usually one of the numerous local minima of the sum-of-squared errors (SSEs) is determined; furthermore, for different starting points, different minima may be found. Thus, nonlinear regression is prone to low accuracy and low reproducibility. Therefore, there is a need for a generally applicable, automated initialization of nonlinear least squares algorithms, which reaches a good, reproducible solution after spending a reasonable computation time probing the SSE-hypersurface. For this purpose, a three-step methodology is presented in this study. First, the SSE-hypersurface is randomly probed in order to estimate probability density functions of initial model parameter that generally lead to an accurate fit solution. Second, these probability density functions then guide a high-resolution sampling of the SSE-hypersurface. This second probing focuses on those model parameter ranges that are likely to produce a low SSE. As the probing continues, the most appropriate initial guess is retained and eventually utilized in a subsequent nonlinear regression. It is shown that this guided random search derives considerably better regression solutions than linearization of model functions, which has so far been considered the best-case scenario. Examples from infrared spectroscopy, cell culture monitoring, reaction kinetics, and image analyses demonstrate the broad and successful applicability of this novel method. Copyright (c) 2014 John Wiley & Sons, Ltd.