Rarity Analysis Module

Overview

The Rarity module quantifies the statistical novelty or anomaly of the most recent movement frame relative to the preceding history. It evaluates how unexpected a given value is by comparing it against a dynamically constructed probability distribution of the previous temporal window.

Theoretical Interpretation

  • Input Requirements: Expects a single 1D array representing a temporal sequence of aggregated scalar values (for instance, the output values of a specific sub-feature like Vertical Kinetic Energy weight or distance over time).
  • Value Interpretation:
    • High Rarity (approaching 1.0): The latest movement is highly anomalous or unusual compared to the immediate past.
    • Low Rarity (approaching 0.0): The latest movement falls squarely within the most common, frequent, and expected behavior observed inside the target window.

Algorithm Details & Mathematics

The Rarity \(R\) is calculated using a convex combination of two distinct metrics derived from the probability density of the recent sequence window:

  1. The contextual Unlikelihood (\(U\)) of the final point.
  2. The Normalized Shannon Entropy (\(H_{norm}\)) of the entire context window.

Given a sequence window \(S\) of length \(N\):

  1. Histogram Construction: The algorithm constructs a dynamic probability density function by segmenting the range [min(S), max(S)] into \(B\) bins, where the optimal bin count dynamically scales with the available data:
\[ B = \max(\lfloor\sqrt{N}\rfloor, 1) \]

The probability \(p_b\) of any bin \(b\) is the frequency of points falling into that bin divided by \(N\).

  1. Unlikelihood (\(U\)): The unlikelihood algorithm isolates the raw probability of the specific bin containing the most recent observation (\(p_{current}\)). A very infrequent behavior has a low probability.
\[ U = 1.0 - p_{current} \]
  1. Normalized Entropy (\(H_{norm}\)): Standard Shannon entropy evaluates the overall diversity or chaos of the window's distribution:
\[ H_{norm} = -\frac{1}{\log(B)} \sum_{b=1}^{B} p_b \log(p_b) \]
  1. Final Rarity Score: The rarity is returned as a weighted blend of the single-point unlikelihood and the holistic window unpredictability, governed by a configurable parameter \(\alpha \in [0, 1]\):
\[ R = \alpha \cdot U + (1 - \alpha) \cdot H_{norm} \]

References

  1. Niewiadomski, R., Mancini, M., Cera, A., Piana, S., Canepa, C., & Camurri, A. (2019). Does embodied training improve the recognition of mid-level expressive movement qualities sonification?. Journal on Multimodal User Interfaces, 13, 191-203. ↩