Separating Long-Term Equilibrium Adaptation from Short-Term Self-Regulation Dynamics Using Latent Differential Equations

Abstract

Self-regulating systems change along different timescales. Within a given week, a depressed person’s affect might oscillate around a low equilibrium point. However, when the timeframe is expanded to capture the year during which they onboarded antidepressant medication, their equilibrium and oscillatory patterns might reorganize around a higher affective point. To simultaneously account for the meaningful change processes that happen at different time scales in complex self-regulatory systems, we propose a single model that combines a second-order linear differential equation for short timescale regulation and a first-order linear differential equation for long timescale adaptation of equilibrium. This model allows for individual-level moderation of short-timescale model parameters. The model is tested in a simulation study which shows that, surprisingly, the short and long timescales can fully overlap and the model still converges to the reasonable estimates. Finally, an application of this model to self-regulation of emotional well-being in recent widows is presented and discussed.

Keywords:

• Equilibrium dynamics
• multi-timescale analysis
• latent differential equations
• emotional well-being

Article information

Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported in part by a Jefferson Scholars Foundation Fellowship awarded to Katharine E. Daniel, and a pre-doctoral fellowship of the International Max Planck Research School on the Life Course awarded to Jannik H. Orzek. Steven Boker was supported in part by NIH grant 1R01 MH113752 and a fellowship from the MPI for Human Development in Berlin.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions or the funding agency is not intended and should not be inferred.

Open Scholarship

This article has earned the Center for Open Science badges for Open Data and Open Materials through Open Practices Disclosure. The data and materials are openly accessible at https://doi.org/10.17605/OSF.IO/QCYEH and https://doi.org/10.17605/OSF.IO/QCYEH.

Notes

1 We also considered using continuous time structural equation modeling (Ou et al., 2019; Oud & Jansen, 2000; Voelkle et al., 2012) and formulating the model as a stochastic differential equation as an alternative estimation approach. Chen et al. (2018) proposed models similar to the models fit here, but based on stochastic differential equations and a Kalman filtering approach. We attempted to fit the data simulated here using this approach, but the model appeared to be underidentified in that the damping in the second order model and the first order equilibrium change parameters would trade off against one another during optimization. For this article, we focus on LDE estimation of the second order process and an integrated first order process for the equilibrium since this approach led to stable estimates. However, it would be interesting to compare these different estimation frameworks in future work.

2 This modification does not change the model for the short term dynamics, it merely provides more precise estimates for the short term dynamics by taking advantage of the fact that in an ODE the same linear relationship between the second, first, and zeroth derivatives holds for the third, second, and first derivatives as well as for the fourth, third, and second derivatives.

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Funding

The author(s) reported there is no funding associated with the work featured in this article.