Gutjahr Tim, Keller Karsten
Institute of Mathematics, University of Lübeck, D-23562 Lübeck, Germany.
Entropy (Basel). 2021 Aug 23;23(8):1097. doi: 10.3390/e23081097.
Ordinal patterns classifying real vectors according to the order relations between their components are an interesting basic concept for determining the complexity of a measure-preserving dynamical system. In particular, as shown by C. Bandt, G. Keller and B. Pompe, the permutation entropy based on the probability distributions of such patterns is equal to Kolmogorov-Sinai entropy in simple one-dimensional systems. The general reason for this is that, roughly speaking, the system of ordinal patterns obtained for a real-valued "measuring arrangement" has high potential for separating orbits. Starting from a slightly different approach of A. Antoniouk, K. Keller and S. Maksymenko, we discuss the generalizations of ordinal patterns providing enough separation to determine the Kolmogorov-Sinai entropy. For defining these generalized ordinal patterns, the idea is to substitute the basic binary relation ≤ on the real numbers by another binary relation. Generalizing the former results of I. Stolz and K. Keller, we establish conditions that the binary relation and the dynamical system have to fulfill so that the obtained generalized ordinal patterns can be used for estimating the Kolmogorov-Sinai entropy.
根据实向量各分量之间的序关系对实向量进行分类的序模式，是确定保测动力系统复杂性的一个有趣的基本概念。特别地，正如C. 班特、G. 凯勒和B. 庞贝所表明的，基于此类模式概率分布的置换熵在简单一维系统中等于柯尔莫哥洛夫 - 西奈熵。大致来说，这样做的一般原因是，从实值“测量排列”得到的序模式系统具有很高的分离轨道的潜力。从A. 安东尼奥克、K. 凯勒和S. 马克西缅科稍有不同的方法出发，我们讨论序模式的推广，这些推广提供了足够的分离度以确定柯尔莫哥洛夫 - 西奈熵。为了定义这些广义序模式，思路是用另一个二元关系取代实数上的基本二元关系≤。推广I. 斯托尔兹和K. 凯勒之前的结果，我们建立了二元关系和动力系统必须满足的条件，以便所得到的广义序模式可用于估计柯尔莫哥洛夫 - 西奈熵。
