Entropy-Based Metrics for Quantifying Complexity in Chaotic Attractors

Povzetek

Quantifying the complexity of a chaotic attractor from a finite, noisy time series remains an unsettled problem, with
different entropy-based estimators often disagreeing about which of two systems is 'more complex.' We compare
four widely used entropy metrics, permutation entropy, sample entropy, approximate entropy, and a numerically
estimated Kolmogorov-Sinai (KS) entropy rate, across five canonical chaotic systems (Lorenz, Rossler, Chua's
circuit, the logistic map, and the Henon map) simulated at six noise levels (signal-to-noise ratio from infinite, i.e.
noise-free, down to 10 dB) and three time-series lengths (1,000, 5,000, and 20,000 points), for a total of 360 test
conditions replicated 20 times each. Permutation entropy showed the strongest agreement with the numerically
estimated KS entropy rate, with a Pearson correlation of 0.91 (95% CI: 0.88-0.93) across all noise-free conditions,
compared with 0.79 for sample entropy and 0.68 for approximate entropy. Under additive noise, permutation
entropy also degraded most gracefully, retaining a correlation of 0.84 with the noise-free KS-entropy ranking at 10
dB SNR, while approximate entropy's correlation fell to 0.41 at the same noise level, a 51% relative decline.
Time-series length had a smaller but still material effect: increasing length from 1,000 to 20,000 points reduced the
coefficient of variation of permutation entropy estimates across replicates from 14.2% to 3.8%, whereas sample
entropy's coefficient of variation declined more slowly, from 22.6% to 9.1% over the same range. Across the five
systems, the rank ordering of complexity produced by permutation entropy matched the KS-entropy-rate ordering in
89% of pairwise system comparisons at the longest series length, versus 71% for sample entropy and 58% for
approximate entropy. A sensitivity analysis over the permutation-entropy embedding dimension (3 to 7) showed that
embedding dimension 5 or 6 maximized agreement with KS entropy for series of 5,000 points or longer, while
dimension 4 was preferable for the shortest series tested. These results indicate that permutation entropy,
appropriately parameterized, is the most reliable of the four estimators tested for ranking attractor complexity from
realistic, noise-corrupted, finite-length time series, and we provide practical parameter guidance for its application to
experimental chaotic data.