Constrained Fourth Order Latent Differential Equation Reduces Parameter Estimation Bias for Damped Linear Oscillator Models

Abstract

Second-order linear differential equations can be used as models for regulation since under a range of parameter values they can account for a return to equilibrium as well as potential oscillations in regulated variables. One method that can estimate parameters of these equations from intensive time series data is the method of Latent Differential Equations (LDE). However, the LDE method can exhibit bias in its parameters if the dimension of the time delay embedding and thus the width of the convolution kernel is not chosen wisely. This article presents a simulation study showing that a constrained fourth-order Latent Differential Equation (FOLDE) model for the second-order system almost completely eliminates bias as long as the width of the convolution kernel is less than two-thirds the period of oscillations in the data. The FOLDE model adds two degrees of freedom over the standard LDE model but significantly improves model fit.

Keywords:

• Dynamical systems
• latent differential equations
• damped linear oscillator
• continuous time

Additional information

Funding

This work was provided in part by NIH Grant R01DA018673 and fellowships from the University of Zurich Research Priority Program on Dynamics of Healthy Aging and the Max Planck Institute for Human Development. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Max Planck Society or of the National Institutes of Health.