Damped Linear Oscillators in Emotion Dynamics

The Challenge with ML Extracted Emotion Data

Original MAFW Video (Liu et al. 2022)

Video Emotion Timeseries

ML extracted emotion data is complex but has unknown latent structure

Solution: Controlled Simulation

Primary Objective

Simulate emotion-like time series with latent structure using Damped Linear Oscillators (DLOs)

Why DLO?

Oscillatory patterns mimic emotional rhythms
Controllable dynamics: frequency, damping, coupling
Realistic complexity from simple rules

Research Questions

Can DLO parameters systematically control emotion dynamics?
Can Dynamic EGA recover the known latent structure from DLO-generated time series?
What is the relationship between time series complexity and network estimation accuracy?

Damped Linear Oscillator (DLO) (Ollero et al. 2023; Tomašević, Golino, and Christensen 2024)

$\frac{d^{2} X}{d t^{2}} = η X + ζ \frac{d X}{d t} + q (t)$

Parameters:

$η$ (eta): Oscillation frequency
$ζ$ (zeta): Damping ratio
$q (t)$ : Stochastic forcing/noise term

DLO Parameter Effects

Realistic Oscilatory Time Series Generation

From Theory to Simulation: Original Framework (Tomašević, Golino, and Christensen 2024)

Two-Component Model

1. Latent Dynamics (DLO)

Latent scores
Parameters: $η$ (equilibrium), $ζ$ (resilience), $q (t)$ (noise)

2. Measurement Model

Factor loadings matrix $Λ$
Observable scores
Measurement error variance $Q$

Alternatve: Direct Dynamics

Use DLO to model observables, not latent scores

Direct dynamics

Latent structure emerges naturally from:

Similar parameters ( $η$ , $ζ$ )
Correlated noise processes
Oscillator coupling

Coupled DLOs

Simulation Design

Parameter Space

6 parameter sets: Different $ $, $ζ \leq 0$ combinations
Noise levels ( $σ_{q}$ ): [0.1, 0.4, 0.6, 0.8, 1.0]
Coupling strengths: [0.0, 0.1, 0.2, 0.3, 0.4, 0.5]
Output: 9 emotions × 300 time points
3 emotion groups/factors
Total: 37,500 simulated datasets

Dynamic EGA Analysis

Method

Time-delay embedding
Network estimation
Community detection
TEFI for model selection

Two Approaches

Zero-order: Position-based networks
First-order: Velocity-based networks

Period, Damping, Sample Entropy

Sample Entropy

Measures complexity/irregularity of time series
Higher values = more unpredictable behavior

Period Estimation

Autocorrelation method: Time lag of first positive peak
Links to DLO frequency parameter

Damping Ratio

Estimated from spectral analysis
Strong correlation with ζ parameter (r = -0.971)
Controls oscillation decay rate

Results: Parameter Control

DLO Parameters → Dynamical Properties

Near-perfect correlations:

$η$ ↔︎ natural frequency: $r = - 0.996$
$ζ$ ↔︎ damping ratio: $r = - 0.971$

Strong behavioral effects:

$η / ζ$ ↔︎ sample_entropy: $r = - 0.61$

Results: Complexity-Structure Link

Sample Entropy → Community Detection

Critical relationships:

entropy ↔︎ number of communities (0): r = 0.330
entropy ↔︎ number of communities (1): r = 0.358
EGA loves entropy

Results: TEFI

TEFI handles complexity of time series

Key insight: Small effect of sample entropy on TEFI

TEFI of 0-order network ↔︎ sample entropy: r = 0.071
TEFI of 1-order network ↔︎ sample entropy: r = 0.272

Temporal coherence matters

TEFI ↔︎ period measure: r= 0.33 in both cases
Very slow or very fast oscillations disrupt the ability to detect clean structure
TEFI is sensitive to whether the detected communities make temporal sense

Key Takeaways

DLOs generate realistic time series with full control over frequency ( $η$ ), damping ( $ζ$ ) and coupling
Latent structure emerges naturally from DLO parameters and coupling
Strong, linear links from microscopic parameters → macroscopic network structure
Complexity (entropy) and rhythmicity (period) shape community detection

Future Directions

Extend simulation to include full range of parameters
Empirical validation on real video-based emotion datasets
Parameter recovery: infer $η$ , $ζ$ , coupling from observed data

Thank You

Questions or feedback?
✉️ atomasevic@ipb.ac.rs

References

Liu, Yuanyuan, Wei Dai, Chuanxu Feng, Wenbin Wang, Guanghao Yin, Jiabei Zeng, and Shiguang Shan. 2022. “MAFW: A large-scale, multi-modal, compound affective database for dynamic facial expression recognition in the wild.” In Proceedings of the 30th ACM International Conference on Multimedia. ACM. https://doi.org/10.1145/3503161.3548190.

Ollero, Mar JF, Eduardo Estrada, Michael D Hunter, et al. 2023. “Characterizing Affect Dynamics with a Damped Linear Oscillator Model: Theoretical Considerations and Recommendations for Individual-Level Applications.” Psychological Methods.

Tomašević, Aleksandar, Hudson Golino, and Alexander P Christensen. 2024. “Decoding emotion dynamics in videos using dynamic Exploratory Graph Analysis and zero-shot image classification: A simulation and tutorial using the transforEmotion R package.” PsyArXiv. https://doi.org/10.31234/osf.io/hf3g7.