Gutjahr Tim, Keller Karsten
Institute of Mathematics, University of Lübeck, D-23562 Lübeck, Germany.
Entropy (Basel). 2020 Jan 2;22(1):63. doi: 10.3390/e22010063.
Different authors have shown strong relationships between ordinal pattern based entropies and the Kolmogorov-Sinai entropy, including equality of the latter one and the permutation entropy, the whole picture is however far from being complete. This paper is updating the picture by summarizing some results and discussing some mainly combinatorial aspects behind the dependence of Kolmogorov-Sinai entropy from ordinal pattern distributions on a theoretical level. The paper is more than a review paper. A new statement concerning the conditional permutation entropy will be given as well as a new proof for the fact that the permutation entropy is an upper bound for the Kolmogorov-Sinai entropy. As a main result, general conditions for the permutation entropy being a lower bound for the Kolmogorov-Sinai entropy will be stated. Additionally, a previously introduced method to analyze the relationship between permutation and Kolmogorov-Sinai entropies as well as its limitations will be investigated.
不同的作者已经证明了基于序数模式的熵与柯尔莫哥洛夫-西奈熵之间存在紧密的关系，包括后者与排列熵的相等性，然而，整体情况远未完整。本文通过总结一些结果并在理论层面讨论柯尔莫哥洛夫-西奈熵依赖于序数模式分布背后的一些主要组合方面，来更新这一情况。本文不仅仅是一篇综述文章。将给出一个关于条件排列熵的新陈述，以及排列熵是柯尔莫哥洛夫-西奈熵的上界这一事实的新证明。作为主要结果，将阐述排列熵是柯尔莫哥洛夫-西奈熵的下界的一般条件。此外，将研究一种先前引入的分析排列熵与柯尔莫哥洛夫-西奈熵之间关系的方法及其局限性。
