Permutation group entropy: A new route to complexity for real-valued processes.

This is a review of group entropy and its application to permutation complexity. Specifically, we revisit a new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviors.

A generalized permutation entropy for noisy dynamics and random processes.

Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.