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An Entropy-initiated Coupled-Trait ODE Framework for Modeling Longitudinal Cohort Dynamics

Anderson M. Rodriguez

TL;DR

This paper introduces Entropy-Initiated Coupled-Trait ODEs (ECTO), a minimal, information-theoretic framework that maps high-dimensional longitudinal Likert data into a low-dimensional autonomous dynamical system. By compressing wave-specific item responses into a pooled Shannon entropy index $H^*(t)$ to initialize three coupled state variables $(N,P,E_{stress})$, ECTO captures broad cohort-level trajectories without requiring latent-variable models or time-varying inputs. The method is validated on SATSA and an external dental student dataset, showing stable out-of-sample performance and reasonable cross-cohort generalization, with coupling between states improving fit over uncoupled baselines. The work highlights entropy as a fixed initialization tool rather than a driver and positions the coupled ODE framework as an interpretable alternative to high-dimensional or black-box approaches, with promising avenues for extension to multivariate and stochastic formulations.

Abstract

This work introduces a minimal, information-theoretic dynamical framework for modeling longitudinal cohort data using an entropy-initiated system of coupled-trait ordinary differential equations (ECTO). For each survey wave, item-level Likert responses are compressed into a normalized Shannon entropy index that summarizes cross-sectional dispersion; this index is used to initialize the low-dimensional state variables of the autonomous ODE system. ECTO then tracks the interactions among a primary trait-like state, a secondary coupled state, and a latent environmental-stress component through phenomenological terms representing generic self-limitation, trade-offs, and feedback. Using data from the Swedish Adoption/Twin Study on Aging (SATSA), the framework reproduces broad cohort-level trajectories and is evaluated with leave-one-wave-out forecasting and comparisons against simple statistical baselines. A second longitudinal dataset of U.S. dental student data provides an external validation test, demonstrating that low-dimensional dynamics initialized from entropy measures can generalize across cohorts with different measurement instruments, demographic compositions, and timescales. Across both datasets, ECTO achieves stable out-of-sample performance, indicating that major cohort-level trends can be captured without assuming complex latent-variable models or time-varying causal inputs. Entropy here functions as a compact summary of population heterogeneity rather than a dynamical driver, and the coupled ODEs supply an interpretable alternative to high-dimensional or black box machine-learning approaches. This framework establishes a concise, transparent method for linking information-theoretic preprocessing with cohort-level dynamical modeling and provides a foundation for future multivariate or multi-cohort extensions.

An Entropy-initiated Coupled-Trait ODE Framework for Modeling Longitudinal Cohort Dynamics

TL;DR

This paper introduces Entropy-Initiated Coupled-Trait ODEs (ECTO), a minimal, information-theoretic framework that maps high-dimensional longitudinal Likert data into a low-dimensional autonomous dynamical system. By compressing wave-specific item responses into a pooled Shannon entropy index to initialize three coupled state variables , ECTO captures broad cohort-level trajectories without requiring latent-variable models or time-varying inputs. The method is validated on SATSA and an external dental student dataset, showing stable out-of-sample performance and reasonable cross-cohort generalization, with coupling between states improving fit over uncoupled baselines. The work highlights entropy as a fixed initialization tool rather than a driver and positions the coupled ODE framework as an interpretable alternative to high-dimensional or black-box approaches, with promising avenues for extension to multivariate and stochastic formulations.

Abstract

This work introduces a minimal, information-theoretic dynamical framework for modeling longitudinal cohort data using an entropy-initiated system of coupled-trait ordinary differential equations (ECTO). For each survey wave, item-level Likert responses are compressed into a normalized Shannon entropy index that summarizes cross-sectional dispersion; this index is used to initialize the low-dimensional state variables of the autonomous ODE system. ECTO then tracks the interactions among a primary trait-like state, a secondary coupled state, and a latent environmental-stress component through phenomenological terms representing generic self-limitation, trade-offs, and feedback. Using data from the Swedish Adoption/Twin Study on Aging (SATSA), the framework reproduces broad cohort-level trajectories and is evaluated with leave-one-wave-out forecasting and comparisons against simple statistical baselines. A second longitudinal dataset of U.S. dental student data provides an external validation test, demonstrating that low-dimensional dynamics initialized from entropy measures can generalize across cohorts with different measurement instruments, demographic compositions, and timescales. Across both datasets, ECTO achieves stable out-of-sample performance, indicating that major cohort-level trends can be captured without assuming complex latent-variable models or time-varying causal inputs. Entropy here functions as a compact summary of population heterogeneity rather than a dynamical driver, and the coupled ODEs supply an interpretable alternative to high-dimensional or black box machine-learning approaches. This framework establishes a concise, transparent method for linking information-theoretic preprocessing with cohort-level dynamical modeling and provides a foundation for future multivariate or multi-cohort extensions.

Paper Structure

This paper contains 27 sections, 27 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods for Data Cleaning and Comparative Approach Analysis
  3. Utilizing Shannon Entropy
  4. Calculating and Normalizing Shannon Entropy
  5. Normalization and Construction of the Pooled Entropy Index
  6. Interpretation and Use in the Dynamical Framework
  7. Comparable Approaches
  8. Methods for the Proposed Ordinary Differential Equation System (ECTO)
  9. State Variable $N(t)$: Selection Pressure Dynamics
  10. State Variable $P(t)$: Pleiotropic Dynamics
  11. State Variable $E_{\text{stress}}(t)$: Environmental Stress Dynamics
  12. Parameter Definitions and Model Structure
  13. Initialization and Simulation Procedure
  14. Dimensionality and Overfitting Considerations
  15. Results
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Shannon entropy of four traits isolated from the set under review: Hot-Tempered, Satisfaction, Indignant and Depressed, across six SATSA survey years (1984–2007).
  • Figure 2: ODE simulation with $\beta = 0.17117$. Fit to Hot-Tempered entropy improves further with higher $R^2$, while Worry entropy shows stable alignment with empirical data. This parameterization keeps the environmental stress term at baseline.
  • Figure 3: ODE model with added oscillatory $c_3$ environmental stress term ($c_3 = 12.6$). The fit shows notable influence from the Environmental stress term. Notably, 'Worry' entropy improves in demonstrative fit.