Abstract
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.
Highlights
The permutation-information theoretic approach to time series analysis proposed by Bandt and Pompe [1] has become popular in various fields [2]
If we turn our eyes to its theoretical side, few results are known for the permutation analogues of information theoretic measures, except the entropy rate
Being confronted with real-world time series data, we cannot take the limit of a large length of words
Summary
The permutation-information theoretic approach to time series analysis proposed by Bandt and Pompe [1] has become popular in various fields [2]. For quantities other than the entropy rate, three results for finite-alphabet stationary ergodic Markov processes have been shown by our previous work: the equality between the excess entropy and the permutation excess entropy [17], the equality between the mutual information expression of the excess entropy and its permutation analogue [18] and the equality between the transfer entropy rate and the symbolic transfer entropy rate [19]. For the modified permutation entropies defined by the partition of the set of words based on permutations and equalities between occurrences of symbols, which is finer than the partition obtained by permutations only, we have the corresponding equalities for general finite-alphabet stationary ergodic stochastic processes [20].
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