Ordinal Pattern Based Entropies and the Kolmogorov–Sinai Entropy: An Update

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.

[1]  K. Keller,et al.  Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras , 2013, 1304.4450.

[2]  Antonio Politi,et al.  Permutation entropy revisited , 2018, Chaos, Solitons & Fractals.

[3]  G. Keller,et al.  Entropy of interval maps via permutations , 2002 .

[4]  Karsten Keller,et al.  An approach to comparing Kolmogorov-Sinai and permutation entropy , 2013 .

[5]  Kohei Nakajima,et al.  Permutation Complexity via Duality between Values and Orderings , 2011, ArXiv.

[6]  K. Keller,et al.  Equality of kolmogorov-sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts , 2018, Discrete & Continuous Dynamical Systems - A.

[7]  Karsten Keller,et al.  On the relation of KS entropy and permutation entropy , 2012, 1407.6473.

[8]  B. Pompe,et al.  Permutation entropy: a natural complexity measure for time series. , 2002, Physical review letters.

[9]  E. Hewitt,et al.  On the fundamental ideas of measure theory , 1962 .

[10]  José M. Amigó,et al.  The equality of Kolmogorov–Sinai entropy and metric permutation entropy generalized , 2012 .

[11]  M. Einsiedler,et al.  Ergodic Theory: with a view towards Number Theory , 2010 .

[12]  Wolfram Koepf,et al.  The ordinal Kolmogorov-Sinai entropy: A generalized approximation , 2017, Commun. Nonlinear Sci. Numer. Simul..

[13]  M. Mirzakhani,et al.  Introduction to Ergodic theory , 2010 .

[14]  D. Newton.  AN INTRODUCTION TO ERGODIC THEORY (Graduate Texts in Mathematics, 79) , 1982 .

[15]  L. Kocarev,et al.  The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems , 2005, nlin/0503044.