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Validation Against PyRQA

This document presents the numerical comparison between gait-dynamics-features and PyRQA (v8.1.0), a well-established Python RQA library.

Test Setup

Parameters:

  • Embedding dimension: 3
  • Time delay: 5
  • Threshold: 0.5 (fixed)
  • Theiler window: 1
  • Min diagonal length: 2
  • Min vertical length: 2
  • Distance metric: Euclidean

Test signals:

  1. Sine - Simple sine wave (300 points)
  2. Harmonics - Sine + 2nd/3rd harmonics (300 points)
  3. Lorenz - Lorenz attractor x-component (300 points, chaotic)
  4. Noise - White Gaussian noise (300 points)

Results

Sine Wave

Measure gait-dynamics-features PyRQA Diff
RR 0.204923 0.204923 0.000000
DET 0.999056 0.999056 0.000000
L 24.748538 24.748538 0.000000
Lmax 289 289 0
ENTR 3.642857 3.642856 0.000001
LAM 0.999884 0.999884 0.000000
TT 25.119534 25.119534 0.000000
Vmax 49 49 0

Harmonics

Measure gait-dynamics-features PyRQA Diff
RR 0.167872 0.167872 0.000000
DET 0.999132 0.999132 0.000000
L 38.592179 38.592179 0.000000
Lmax 289 289 0
ENTR 3.509815 3.509815 0.000000
LAM 0.999858 0.999858 0.000000
TT 23.844595 23.844595 0.000000
Vmax 40 40 0

Lorenz (Chaotic)

Measure gait-dynamics-features PyRQA Diff Note
RR 0.004495 0.004495 0.000000
DET 0.886364 0.886364 0.000000
L 13.000000 13.000000 0.000000
Lmax 21 21 0
ENTR 1.098612 1.098612 0.000000
LAM 0.000000 0.000000 0.000000
TT 0.000000 nan - See note 1
Vmax 0 1 1 See note 2

Note 1 (TT): When no vertical lines meet min_vert_length >= 2, we return 0.0 while PyRQA returns NaN. Both are valid representations of an undefined mean over an empty set.

Note 2 (Vmax): PyRQA reports Vmax=1 (length of isolated vertical points), while we report 0 because no vertical lines meet the minimum length threshold of 2. This is a difference in how min_vert_length is applied to the Vmax output.

White Noise

Measure gait-dynamics-features PyRQA Diff
RR 0.014602 0.014602 0.000000
DET 0.012793 0.012793 0.000000
L 2.000000 2.000000 0.000000
Lmax 2 2 0
ENTR 0.000000 0.000000 0.000000
LAM 0.048860 0.048860 0.000000
TT 2.068966 2.068966 0.000000
Vmax 3 3 0

Summary

30 out of 32 measure comparisons show exact numerical agreement (within floating-point precision).

The 2 remaining differences occur only for the Lorenz system under extremely sparse recurrence (RR = 0.45%), where no vertical lines of length >= 2 exist. These are edge-case definitional differences, not computational errors.

Performance Comparison

Signal gait-dynamics-features PyRQA Speedup
Sine (N=300) 0.007s 0.091s 13.9x
Harmonics (N=300) 0.006s 0.020s 3.2x
Lorenz (N=300) 0.006s 0.019s 3.1x
Noise (N=300) 0.006s 0.021s 3.3x

Our pure NumPy/SciPy implementation is 3-14x faster than PyRQA (which uses OpenCL) for typical gait-sized data (N < 1000). PyRQA's OpenCL backend may be advantageous for very large N (> 10000).

Scaling Characteristics

N Time
100 0.001s
300 0.006s
500 0.017s
1000 0.066s

Scaling is approximately O(N^2), dominated by distance matrix computation.

Implementation Differences

The following table documents the key implementation details that were aligned during validation:

Aspect gait-dynamics-features PyRQA
RR denominator N*N (configurable via rr_exclude_theiler) N*N (always)
DET denominator Sum of all diagonal line points (>=1) Same
LAM denominator Sum of all vertical line points (>=1) Same
Theiler window in RM Not applied to stored matrix; applied internally for diagonal analysis Applied in matrix construction kernel
Vertical lines and Theiler Full RM used (not broken by Theiler) Same

Reproducing

pip install PyRQA
python examples/validate_against_pyrqa.py