Multi-Scale Entropy Dominance Analysis Module

Overview

The Multi-Scale Entropy (MSE) Dominance module implements a sophisticated non-linear analysis algorithm to detect leadership and dominance within ensemble performances based on movement complexity.

Theoretical Interpretation

• Input Requirements: Expects a lengthy sliding window of movement velocity data. Minimum sample threshold strongly applies (e.g., 500 samples) because entropy estimation is mathematically unstable on short sequences.
• Value Interpretation:
  • Complexity Index (CI): Represents the overall irregularity and unpredictability of a signal across multiple temporal resolutions. A signal is "complex" if it retains structural unpredictability across scales (unlike white noise which loses structure when scaled, or simple sine waves which are entirely predictable).
  • Dominance Score: Computes the relative dominance within a multi-agent group. The actor with the lowest movement complexity (lowest CI) is typically classified as the leader, acting as a periodic anchor around which other participants exhibit higher, reactive complexity.
  • Leader Identification: A direct argmin pointer to the index of the predicted group leader.

Algorithm Details & Mathematics

The algorithm leverages Multi-Scale Entropy, which consists of three sequential steps:

1. Coarse-Graining

The original signal \(x = \{x_1, x_2, \dots, x_N\}\) is segmented into non-overlapping windows of length \(\tau\) (the scale factor). The coarse-grained signal \(y^{(\tau)}\) at scale \(\tau\) is formed by averaging the data points within each window: $$ y_j^{(\tau)} = \frac{1}{\tau} \sum_{i=(j-1)\tau + 1}^{j\tau} x_i $$ This is repeated for all scales from \(\tau = 1\) to \(\tau = max\_scale\).

2. Sample Entropy (SampEn)

For each coarse-grained signal, the Sample Entropy is computed. SampEn quantifies the conditional probability that two sequences that are similar for \(m\) points remain similar at \(m+1\) points, given a tolerance \(r\). $$ \text{SampEn}(m, r) = -\ln\left(\frac{A}{B}\right) $$ where \(B\) is the number of template matches of length \(m\), and \(A\) is the number of strict template matches of length \(m+1\).

3. Complexity Index (CI)

The final Complexity Index for an individual signal is the numerical integration (area under the curve) of the Sample Entropy values across all considered scales: $$ CI = \int_{1}^{max_scale} \text{SampEn}(\tau) \, d\tau $$

4. Dominance Mapping

In a multi-participant context, the dominant actor is identified. The dominance score scales the \(CI\) relative to the group's maximum entropy: $$ \text{Dominance} = 1.0 - \frac{CI_i}{\max(CI)} $$

References