An approach to comparing Kolmogorov-Sinai and permutation entropy

TL;DR

This paper explores the relationship between the conditional entropy of ordinal patterns and Kolmogorov-Sinai entropy, demonstrating that the former can effectively estimate the latter in various dynamical systems.

Contribution

It introduces the conditional entropy of ordinal patterns as a simple, effective estimator for Kolmogorov-Sinai entropy, especially in periodic and Markov shift systems.

Findings

Conditional entropy estimates Kolmogorov-Sinai entropy accurately in many cases.

Conditional entropy coincides with Kolmogorov-Sinai entropy for periodic dynamics.

The method is computationally simple and applicable to real-world data.

Abstract

In this paper we investigate a quantity called conditional entropy of ordinal patterns, akin to the permutation entropy. The conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given ordinal pattern. We observe that this quantity provides a good estimation of the Kolmogorov-Sinai entropy in many cases. In particular, the conditional entropy of ordinal patterns of a finite order coincides with the Kolmogorov-Sinai entropy for periodic dynamics and for Markov shifts over a binary alphabet. Finally, the conditional entropy of ordinal patterns is computationally simple and thus can be well applied to real-world data.