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Entropy 2013, 15(9), 3910-3930; doi:10.3390/e15093910

Permutation Complexity and Coupling Measures in Hidden Markov Models

Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University,Rokkodaicho, Nada, Kobe 657-8501, Japan
Department of Mechanical and Process Engineering, ETH Zurich, Leonhardstrasse 27, Zurich 8092,Switzerland
Author to whom correspondence should be addressed.
Received: 24 July 2013 / Revised: 29 August 2013 / Accepted: 11 September 2013 / Published: 16 September 2013
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Recently, the duality between values (words) and orderings (permutations) has been proposed by the authors as a basis to discuss the relationship between information theoretic measures for finite-alphabet stationary stochastic processes and their permutatio nanalogues. It has been used to give a simple proof of the equality between the entropy rate and the permutation entropy rate for any finite-alphabet stationary stochastic process and to show some results on the excess entropy and the transfer entropy for finite-alphabet stationary ergodic Markov processes. In this paper, we extend our previous results to hidden Markov models and show the equalities between various information theoretic complexity and coupling measures and their permutation analogues. In particular, we show the following two results within the realm of hidden Markov models with ergodic internal processes: the two permutation analogues of the transfer entropy, the symbolic transfer entropy and the transfer entropy on rank vectors, are both equivalent to the transfer entropy if they are considered as the rates, and the directed information theory can be captured by the permutation entropy approach.
Keywords: duality; permutation entropy; excess entropy; transfer entropy; directed information duality; permutation entropy; excess entropy; transfer entropy; directed information
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Haruna, T.; Nakajima, K. Permutation Complexity and Coupling Measures in Hidden Markov Models. Entropy 2013, 15, 3910-3930.

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