| Abstract: | Several approaches exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA; Ramsay &; Silverman, 2005 Ramsay, J. O., &; Silverman, B. W. (2005). Functional data analysis (2nd ed.). New York, NY: Springer-Verlag. Google Scholar]), generalized local linear approximation (GLLA; Boker, Deboeck, Edler, &; Peel, 2010 Boker, S. M., Deboeck, P. R., Edler, C., &; Peel, P. K. (2010). Generalized local linear approximation of derivatives from time series. In S. Chow, E. Ferrer, &; F. Hsieh (Eds.), Statistical methods for modeling human dynamics: An interdisciplinary dialogue (pp. 161–178). New York, NY: Taylor &; Francis. Google Scholar]), and generalized orthogonal local derivative approximation (GOLD; Deboeck, 2010 Deboeck, P. R. (2010). Estimating dynamical systms: Derivative estimation hints from Sir Ronald A. Fisher. Multivariate Behavioral Research, 45, 725–745. doi:10.1080/00273171.2010.498294Taylor &; Francis Online], Web of Science ®] , Google Scholar]). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo (MC) study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children’s self-regulation. |
