Permutation Complexity and Coupling Measures in Hidden Markov Models

TL;DR

This paper extends the duality between values and orderings to hidden Markov models, demonstrating that permutation-based measures like transfer entropy and directed information accurately reflect their traditional counterparts, thus linking permutation entropy with information theory.

Contribution

It introduces the extension of permutation complexity and coupling measures to hidden Markov models, establishing their equivalence to classical information theoretic measures.

Findings

Permutation analogues of transfer entropy are equivalent to transfer entropy as rates.

Directed information can be captured by permutation entropy.

Results apply to hidden Markov models with ergodic internal processes.

Abstract

In [Haruna, T. and Nakajima, K., 2011. Physica D 240, 1370-1377], the authors introduced the duality between values (words) and orderings (permutations) as a basis to discuss the relationship between information theoretic measures for finite-alphabet stationary stochastic processes and their permutation analogues. 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 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.