Statistical Moments Analysis Module

Overview

The Statistical Moment module provides real-time computation of foundational descriptive statistics over sliding windows of motion signals. It quantifies the shape, spread, and central tendency of continuous movement patterns.

Theoretical Interpretation

  • Input Requirements: Evaluates a temporal sequence (window) of \(N\)-dimensional data points. Computations are performed independently over every dimension or feature provided.
  • Value Interpretation:
    • Mean (\(\mu\)): The central baseline or average persistent posture/velocity of the limb across the window.
    • Standard Deviation (\(\sigma\)): The magnitude of the movement span. High variance denotes expansive, reaching, or highly varied effort. Low variance denotes rigidity or stillness around the mean.
    • Skewness (\(\gamma_1\)): The asymmetry of the movement. A positive skew denotes sudden forward thrusts with slow returns; negative skew denotes slow builds with sudden snaps back to resting.
    • Kurtosis (\(\gamma_2\)): The presence of extreme outliers or heavy tails. High kurtosis marks sudden, jerky, erratic anomalies within an otherwise regular pattern.

Algorithm Details & Mathematics

The module isolates each one-dimensional signal array \(X = \{x_1, \dots, x_N\}\) of length \(N\) inside the current window block, and continuously computes the requested moments.

1. Mean

The arithmetic average defining the center point:

\[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \]

2. Standard Deviation

The sample standard deviation (using \(N-1\) degrees of freedom to correct for sample size bias), representing typical dispersal distance from the mean:

\[ s = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2} \]

3. Skewness

The unadjusted Fisher-Pearson coefficient of skewness, tracking symmetrical distribution:

\[ \gamma_1 = \frac{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^3}{\left(\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\right)^{3/2}} \]

4. Kurtosis

Fisher's definition of excess kurtosis (subtracting 3), tracking the fatness of the tails relative to a Normal distribution:

\[ \gamma_2 = \frac{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^4}{\left(\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\right)^2} - 3.0 \]