Clusterability Analysis Module

Overview

The Clusterability module assesses the presence of meaningful non-random structure within a dataset. Specifically, it evaluates whether the multidimensional points in a sliding window exhibit clustering tendencies or are merely uniformly distributed noise.

Theoretical Interpretation

• Input Requirements: Expects a 3D tensor of motion data over a window.
• Value Interpretation: The module computes the Hopkins Statistic (\(H\)).
  • \(H \approx 1.0\): Indicates highly clustered data (the true data points are significantly closer to each other than uniformly generated random points are). This implies highly structured, repeating, or localized postures.
  • \(H \approx 0.5\): Indicates a completely random, uniform distribution in space (no discernible structure).
  • \(H \approx 0.0\): Indicates regularly spaced data (e.g., a perfect grid), which is rare in human movement.

Algorithm Details & Mathematics

The feature employs the Hopkins Statistic¹ based on Nearest Neighbor (NN) distances.

Given a dataset \(X\) of size \(N\), the algorithm:

1. Randomly selects a subset of \(m\) real data points (where \(m = N \times \text{subset\fraction}\)).
2. Generates \(m\) uniformly distributed simulated artificial points within the spatial bounding box of \(X\).
3. For each real point \(x_i\), it computes the distance \(w_i\) to its nearest real neighbor in \(X\).
4. For each artificial point \(y_i\), it computes the distance \(u_i\) to its nearest real neighbor in \(X\).

The Hopkins statistic (\(H\)) is calculated as the relative magnitude of the artificial nearest neighbor distances versus the total distances:

(Note: In some literature the numerator is \(w_i\). The Python implementation uses \(u_i\) in the numerator to ensure that highly clustered data (where real points are dense resulting in small \(w_i\), but random points fall in empty space resulting in large \(u_i\)) approaches \(1.0\).)

References