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 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.

A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T

“…This research note illustrates the need of carefully picking the embedding dimension in LDE modeling and the potential aid of FOLDE therein. When applying the data-driven method by Hu et al (2014) to determine the optimal embedding dimension, SOLDE left us uncertain about which D to choose, whereas FOLDE yielded an optimal D. While extending the FOLDE model used by Boker et al to include multiple observed indicators and multiple subjects, we complemented previous, simulation-based research on FOLDE performance under ideal conditions (Boker et al 2020) with a perspective from a practical point of view under imperfect conditions using empirical data. Unlike most of the previous methodological studies that build upon time delay embedding, we not only studied effects of D on the frequency and damping parameters, but we also specifically inspected the wavelength, which is a function of those two parameters, as a substantively meaningful quantity.…”

“…Incorporating these higher-order derivatives gives us an extra source of information that has the potential to reduce bias in the parameters of the DLO. Performance of the FOLDE model under simulated, ideal conditions yielded an advantage over the SOLDE model for most cases except when the number of measurement occasions was small (i.e., ; Boker et al 2020 ). In this case, the frequency parameter as well as the likelihood ratio tests benefit from FOLDE, whereas the damping parameter exhibits larger bias in FOLDE than in SOLDE.…”

“…Constrained fourth-order LDE (FOLDE) models are an alternative to approximating second-order systems. This is possible because FOLDE builds upon three mathematically equivalent sets of second-order equations, in which the second derivative is regressed on the zeroth and first derivatives, the third is regressed on the first and second derivatives, and the fourth is regressed on the second and third derivatives, with each set having regression parameters constrained to and , respectively (for a more detailed derivation of these relationships see Boker et al 2020 , p. 205). Incorporating these higher-order derivatives gives us an extra source of information that has the potential to reduce bias in the parameters of the DLO.…”

“…Therefore, the recommendation of which model to choose is ‘not so clear’ in this situation. Instead, ‘a safe recommendation is to fit both [SO]LDE and FOLDE to ones data’ (Boker et al 2020 , p. 215).…”

“… Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al 2020 ) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., ), the recommendation of which model to use is not clear (Boker et al 2020 ). Based on a data set, which consists of observations of daily stress for individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T .…”