Abstract
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.
Highlights
In 2002, Bandt and Pompe introduced so-called permutation entropy [1]
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
It is a crucial point that permutation entropy is theoretically justified by asymptotic results relating it to Kolmogorov–Sinai entropy (KS entropy, called metric entropy) which is the central complexity measure for dynamical systems
Summary
In 2002, Bandt and Pompe introduced so-called permutation entropy [1]. This entropy has been established in non-linear dynamical system theory and time series analysis, including applications in many fields from biomedicine to econophysics Under what conditions on a discriminating relation R the partitions given by the generalized ordinal patterns determine the KS entropy of a dynamical system? In contrast to classical symbolization approaches with symbolizing only in the range of single “measurements” x, the symbolization of pairs ( x, y) via the partition { R, R2 \ R} regards some kind of link between x and y if R lies “diagonal” in a certain sense
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