Contrasting stochasticity with chaos in a permutation Lempel–Ziv complexity — Shannon entropy plane

Abstract

This paper aims at introducing the Lempel–Ziv permutation complexity vs. permutation entropy plane (EC-plane) as a tool to analyze time series with diverse dynamical nature. These two quantities use the Bandt and Pompe representation to quantify a continuous-state time series. The main strength of this approach lies in the fact that this plane combines two different perspectives to study a signal, one being purely statistic (the permutation entropy) and the other being algorithmic (the Lempel–Ziv complexity). The results allow us to conclude that the EC-plane constitutes an appropriate framework for: (i) characterizing non-linear chaotic maps, (ii) distinguishing deterministic from stochastic processes and (iii) to discriminate between fractional Brownian motion, fractional Gaussian noise and K-noise.