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

José M. Amigó, Roberto Dale, Piergiulio Tempesta

TL;DR

A new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy is revisited, which provides a unified framework to discuss chaotic and random behaviors.

Abstract

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 time serie 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 behaviours.

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

TL;DR

A new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy is revisited, which provides a unified framework to discuss chaotic and random behaviors.

Abstract

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 time serie 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 behaviours.
Paper Structure (21 sections, 2 theorems, 71 equations, 4 figures)

This paper contains 21 sections, 2 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Group entropy
  3. Beyond the Shannon-Khinchin formulation: the Composability Axiom
  4. Formal group laws
  5. Group logarithms and Z-entropies
  6. A general theorem
  7. Examples
  8. Weak group entropies
  9. Outlook of group entropies
  10. Group entropies and information geometry
  11. Group entropies and negativity
  12. Multivariate family
  13. Permutation complexity
  14. Ordinal patterns
  15. Asymptotic growth of allowed patterns
  16. ...and 6 more sections

Key Result

Theorem 5

Let $\mathcal{W}$ be the state space growth rate function, corresponding to a given universality class of a statistical system. Then there exists a group entropy which satisfies the extensivity requirement. This entropy is given by where $\alpha >0$, $\lambda =\frac{1}{(\mathcal{W}^{-1})^{\prime }(1)}$, $p=(p_{1},...,p_{W})$ and it is assumed $(\mathcal{W}^{-1})^{\prime }(1)\neq 0$.

Figures (4)

  • Figure 1: Ordinal partition $\mathcal{P}_{3}$ of $[0,1]$ generated by the ordinal 3-patterns of the full range logistic map $f(x)=4x(1-x)$. The solid line (bisector) corresponds to $y=f^{0}(x)=x$, the dashed line to $y=f(x)$ and the dotted line to $y=f^{2}(x)$. With the rule “ earlier is smaller” for the intersections of those lines, we find: $P_{(0,1,2)}=\left[ 0,\frac{1}{4}\right] \bigcup \left\{ \frac{3}{4}\right\}$, $P_{(0,2,1)}=\left( \frac{1}{4},\frac{5-\sqrt{5}}{8}\right]$, $P_{(2,0,1)}=\left( \frac{5-\sqrt{5}}{8},\frac{3}{4}\right)$, $P_{(1,0,2)}=\left( \frac{3}{4},\frac{5+\sqrt{5}}{8}\right]$, $P_{(1,2,0)}=\left( \frac{5+\sqrt{5}}{8},1\right]$ and $P_{(2,1,0)}=\emptyset .$
  • Figure 2: Example of a random process with forbidden patterns: the logistic parabola $f_{a}(x)=ax(1-x)$ with $3.83\leq a\leq 3.84$ (period-3 window) and additive white noise of amplitude $0.002$, see Equation (\ref{['dynamical noise']}).
  • Figure 3: Averages of the finite PC function $g(L,T)$ over 10 realizations of the random processes listed in the inset are plotted vs $T$ (the time series length) for $L=6$ and $6\leq T\leq 15,000$ (left panel) and $L=7$ and $7\leq T\leq 25,000$ (right panel). See the text for details.
  • Figure 4: Averages of $Z_{\text{fac},0.5}^{\ast }(X_{0}^{L})/L$ (a), $Z_{\text{fac},1}^{\ast }(X_{0}^{L})/L$ (b) and $Z_{\text{fac},1.5}^{\ast }(X_{0}^{L})/L$ (c) over 10 realizations of the random processes listed in the inset (see Section \ref{['sec:33']} for detail) are plotted vs $L$ for $3\leq L\leq 7$.

Theorems & Definitions (16)

  • Definition 1: Composability Axiom
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Example
  • Remark 6
  • Definition 7
  • Remark 8
  • Remark 9
  • ...and 6 more