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A Benchmark for Entropy Estimators
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DOI：10.3390/e28030311.png)
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Abstract
En 中文
This study assessed the performance of several entropy estimators for numerical time series and symbolic data on non-trivial one-dimensional dynamical systems whose Kolmogorov-Sinai entropy is known with certified accuracy: recent computer-assisted proof techniques provide rigorous values together with explicit error bounds. We considered four classes of interval maps, including piecewise expanding maps with and without a Markov partition and an intermittent Pomeau-Manneville map, and generated long orbits for each system. We then compared the certified entropy with the output of widely used estimators: Approximate Entropy, Sample Entropy, Permutation Entropy, a symbolic Plug-In estimator of the entropy rate, and the Non-Sequential Recursive Pair Substitution (NSRPS) method (the latter two with Grassberger-type bias correction). Our experiments reveal substantial, dynamics-dependent differences in accuracy and robustness. In particular, Approximate Entropy and the symbolic methods (Plug-In and NSRPS) consistently yielded estimates within the rigorous error bars across all systems, whereas Sample Entropy showed a marked systematic underestimation, and Permutation Entropy exhibited large biases, especially for expanding maps without a Markov partition. The resulting benchmark provides a quantitative testbed for evaluating entropy estimation techniques in deterministic dynamical systems.
Keywords:
Kolmogorov-Sinai entropy
Lyapunov exponent
complexity
time series
Journal
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2
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919
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