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

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.