[Paper Review] On the analysis of signals in a permutation Lempel-Ziv complexity - permutation Shannon entropy plane

Diego M. Mateos, Steeve Zozor+1 more|arXiv (Cornell University)|Jul 17, 2017

Chaos control and synchronization56 references3 citations

TL;DR

This paper introduces a permutation Lempel–Ziv complexity vs. permutation Shannon entropy plane to analyze time series by combining deterministic (complexity) and statistical (entropy) perspectives. It successfully distinguishes chaotic signals from stochastic processes, differentiates between Gaussian and non-Gaussian noises like FBM, FGN, and K-noise—even when they share the same spectrum—demonstrating robustness in characterizing complex dynamics from continuous-state signals via Bandt and Pompe's permutation approach.

ABSTRACT

The aim of this paper is to introduce the Lempel-Ziv permutation complexity vs permutation entropy plane as a tool to analyze time series of different nature. This two quantities make use of the Bandt and Pompe representation to quantify continuous-state time series. The strength of this plane is to combine two different perspectives to analyze a signal, one being statistic (the permutation entropy) and the other being deterministic (the Lempel-Ziv complexity). This representation is applied (i) to characterize non-linear chaotic maps, (ii) to distinguish deterministic from stochastic processes and (iii) to analyze and differentiate fractional Brownian motion from fractional Gaussian noise and K-noise given a same (averaged) spectrum. The results allow to conclude that this plane is "robust" to distinguish chaotic signals from random signals, as well as to discriminate between different Gaussian and nonGaussian noises.

Motivation & Objective

• To develop a unified framework for analyzing complex time series by combining algorithmic complexity and statistical entropy.
• To address the challenge of distinguishing chaotic dynamics from stochastic processes in continuous-state time series.
• To improve discrimination between different types of noise—especially those with similar spectra but different correlation structures—such as FBM, FGN, and K-noise.
• To evaluate the robustness of the permutation-based complexity–entropy plane in capturing subtle dynamical features beyond spectral analysis.
• To provide a complementary tool to existing information-theoretic and deterministic methods for time series characterization.

Proposed method

• Quantize continuous-state time series into permutation vectors using Bandt and Pompe's symbolic dynamics approach with embedding delay τ and embedding dimension d.
• Compute permutation Shannon entropy as the empirical Shannon entropy of the permutation patterns derived from the time series.
• Apply Lempel–Ziv complexity to the sequence of permutation symbols to quantify algorithmic complexity of the symbolic sequence.
• Construct a two-dimensional complexity–entropy plane using permutation Lempel–Ziv complexity on the x-axis and permutation Shannon entropy on the y-axis.
• Use the plane to analyze and compare chaotic maps, K-noise, fractional Brownian motion (FBM), and fractional Gaussian noise (FGN) across varying parameters.
• Fix embedding parameters at d=5 and τ=1 for consistency, and validate results across alternative d and τ values (e.g., d=4,6).

Experimental results

Research questions

• RQ1 Can the permutation Lempel–Ziv complexity vs. permutation Shannon entropy plane effectively distinguish chaotic time series from stochastic processes?
• RQ2 How well can this plane differentiate between Gaussian and non-Gaussian noise processes with identical power spectra, such as FBM, FGN, and K-noise?
• RQ3 Does the permutation complexity capture short-range correlation properties that standard entropy measures miss?
• RQ4 To what extent does the plane reveal dynamical characteristics such as persistency, anti-persistency, or deterministic structure in time series?
• RQ5 Can this method provide a more informative characterization than spectral analysis or single measures like entropy or complexity alone?

Key findings

• The complexity–entropy plane clearly separates chaotic maps from stochastic processes, with chaotic signals clustering in distinct regions based on their dynamical nature.
• K-noise sequences form a straight-line pattern in the plane, indicating a consistent relationship between complexity and entropy across correlation levels.
• FBM and K-noise occupy an intermediate-to-high complexity and entropy region, while FGN concentrates in a high-entropy, lower-complexity zone, enabling clear discrimination despite shared spectral properties.
• Permutation Lempel–Ziv complexity captures the distinction between persistent (H > 0.5) and anti-persistent (H < 0.5) correlations in FGN, whereas permutation entropy alone cannot differentiate them.
• The method successfully distinguishes FGN with increasing min(H, 1−H) from FBM and K-noise, even when their power-law spectra are identical, proving sensitivity to correlation structure beyond spectral content.
• The results are robust across different embedding parameters (d=4,5,6 and τ=1), confirming the stability of the method’s classification performance.