Permutation Entropy of Weakly Noise-Affected Signals

Leonardo Ricci,Antonio Politi

doi:10.3390/e24010054

Abstract

We analyze the permutation entropy of deterministic chaotic signals affected by a weak observational noise. We investigate the scaling dependence of the entropy increase on both the noise amplitude and the window length used to encode the time series. In order to shed light on the scenario, we perform a multifractal analysis, which allows highlighting the emergence of many poorly populated symbolic sequences generated by the stochastic fluctuations. We finally make use of this information to reconstruct the noiseless permutation entropy. While this approach works quite well for Hénon and tent maps, it is much less effective in the case of hyperchaos. We argue about the underlying motivations.

Highlights

The analysis of ordinal patterns is one of the most used tools to characterize irregular signals [1]
The most relevant known quantifier that can be built by classifying ordinal patterns is the so-called permutation entropy (PE) [2], which quantifies the diversity of motifs, or symbolic sequences, by encoding m-dimensional sections of a time series into permutations
They are due to the entropy increase originating from the implicit refinement of the phase-space partition when the window length is increased

Summary

Introduction

The analysis of ordinal patterns is one of the most used tools to characterize irregular (whether stochastic or deterministic) signals [1]. The possibility of straightforwardly computing PE out of the observed occurrence frequency of symbolic sequences is balanced by it being heavily affected by finite-size corrections They are due to the entropy increase originating from the implicit refinement of the phase-space partition when the window length is increased (a phenomenon entirely different from the entropy increase associated with a finite KS entropy). We focus on deterministic signals affected by a weak observational, i.e., additive, noise, with the goal of unveiling general scenarios and possibly using them as a springboard to tackle more complex problems For this purpose, we study here four dynamical systems: (i) the Hénon map, as one of the prototypical examples of chaotic dynamics; (ii) the tent map, a simple model that offers chances to derive analytical results; (iii) the Hénon map with delay, referred to as the generalized 3D Hénon map, which makes up a hyperchaotic system to investigate the generality of the scenario; and (iv) the Mackey–Glass model, which provides a more realistic continuous time system. The attempted reconstruction of the deterministic value of entropy upon reassignment of noise-induced symbolic sequences is the topic of Section 4, while the remaining open problems are presented in the final Section 5

Permutation Entropy and Its Noise-Induced Scaling Behavior

Multifractal Analysis

Assessing the Entropy Increase via Distribution Reconstruction

Discussion and Open