Statistical versus optimal partitioning for block entropies

Statistical versus optimal partitioning for block entropies Purpose – Given a time‐series, what is the best partitioning of the state space in order to obtain reasonable values for the block entropies? The purpose of this paper is to provide a simple answer (an algorithm), although approximative, in connection with symbolic dynamics and statistical properties of 1‐d maps on the interval. Design/methodology/approach – The logistic map is examined as an archetype of a Complex System with different behaviors, namely: periodicity, order‐to‐chaos period‐doubling transition, weak chaos, parametric intermittent chaos, developed chaos and fully developed chaos. For the logistic map the generating partition is known, and allows comparison with other prescriptions in the literature. The partitioning of the phase space with the easy generated bipartition induced by the mean value of a curve in the plane, gives results in good agreement (roughly up to a 20 per cent difference) with the results of the generating partition, if the trajectory of the system is in parametric intermittent chaos and in developed chaos (DC). In the case of fully developed chaos (FDC), the agreement is perfect. Findings – The authors confirm that a statistical partitioning is almost equivalent with the exact partitioning for the logistic map. Originality/value – The paper updates previous results and proposes a better understanding on the partitioning for symbolic dynamics. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Kybernetes Emerald Publishing

Statistical versus optimal partitioning for block entropies

Kybernetes, Volume 42 (1): 20 – Jan 4, 2013

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Publisher
Emerald Publishing
Copyright
Copyright © 2013 Emerald Group Publishing Limited. All rights reserved.
ISSN
0368-492X
D.O.I.
10.1108/03684921311295466
Publisher site
See Article on Publisher Site

Abstract

Purpose – Given a time‐series, what is the best partitioning of the state space in order to obtain reasonable values for the block entropies? The purpose of this paper is to provide a simple answer (an algorithm), although approximative, in connection with symbolic dynamics and statistical properties of 1‐d maps on the interval. Design/methodology/approach – The logistic map is examined as an archetype of a Complex System with different behaviors, namely: periodicity, order‐to‐chaos period‐doubling transition, weak chaos, parametric intermittent chaos, developed chaos and fully developed chaos. For the logistic map the generating partition is known, and allows comparison with other prescriptions in the literature. The partitioning of the phase space with the easy generated bipartition induced by the mean value of a curve in the plane, gives results in good agreement (roughly up to a 20 per cent difference) with the results of the generating partition, if the trajectory of the system is in parametric intermittent chaos and in developed chaos (DC). In the case of fully developed chaos (FDC), the agreement is perfect. Findings – The authors confirm that a statistical partitioning is almost equivalent with the exact partitioning for the logistic map. Originality/value – The paper updates previous results and proposes a better understanding on the partitioning for symbolic dynamics.

Journal

KybernetesEmerald Publishing

Published: Jan 4, 2013

Keywords: Dynamics; Time series analysis; Dynamical systems; Symbolic dynamics; Block entropies; Lumping

References

  • Experimental mathematics and computational statistics
    Bailey, D.; Borwein, J.M.
  • Introduction to Nonlinear Science
    Nicolis, G.
  • Deterministic Chaos
    Schuster, H.G.
  • Entropy and optimal partition for data analysis
    Steuer, R.; Molgedey, L.; Ebeling, W.; Jimenez‐Montano, M.A.

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