Permutation Complexity via Duality between Values and Orderings

TL;DR

This paper explores the permutation complexity of finite-state stationary stochastic processes by establishing a duality between values and orderings, providing new proofs and insights into entropy measures.

Contribution

It introduces a duality between words and permutations, offers an elementary proof of entropy rate equality, and relates permutation excess entropy to excess entropy.

Findings

Permutation entropy rate equals entropy rate.

Permutation excess entropy equals excess entropy for ergodic Markov processes.

Duality provides new insights into the structure of stochastic processes.

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 [Amigo, J.M., Kennel, M. B., Kocarev, L., 2005. Physica D 210, 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.