Permutation complexity via duality between values and orderings |
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Authors: | Taichi Haruna Kohei Nakajima |
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Institution: | a Department of Earth & Planetary Sciences, Graduate School of Science, Kobe University, 1-1 Rokkodaicho, Nada, Kobe 657-8501, Japanb PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japanc Artificial Intelligence Laboratory, Department of Informatics, University of Zurich, Andreasstrasse 15, 8050 Zurich, Switzerland |
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Abstract: | We study the permutation complexity of finite-state stationary stochastic processes based on a duality between values and orderings between values. First, we establish a duality between the set of all words of a fixed length and the set of all permutations of the same length. Second, on this basis, we give an elementary alternative proof of the equality between the permutation entropy rate and the entropy rate for a finite-state stationary stochastic processes first proved in J.M. Amigó, M.B. Kennel, L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D 210 (2005) 77-95]. Third, we show that further information on the relationship between the structure of values and the structure of orderings for finite-state stationary stochastic processes beyond the entropy rate can be obtained from the established duality. In particular, we prove that the permutation excess entropy is equal to the excess entropy, which is a measure of global correlation present in a stationary stochastic process, for finite-state stationary ergodic Markov processes. |
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Keywords: | Permutation entropy Excess entropy Duality Stationary stochastic processes Ergodic Markov processes |
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