# A Benchmark for Entropy Estimators

## TL;DR

This paper evaluates how well different entropy estimation methods work on complex systems with known entropy values, revealing which methods are most accurate.

## Contribution

The study introduces a benchmark using certified entropy values from dynamical systems to evaluate entropy estimators.

## Key findings

- Approximate Entropy and symbolic methods provided accurate estimates within error bounds across all systems.
- Sample Entropy systematically underestimated entropy, while Permutation Entropy showed large biases for certain maps.

## Abstract

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.