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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES
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Chaos 22, 013105 (2012); http://dx.doi.org/10.1063/1.3673238 (13 pages)

Transcripts: An algebraic approach to coupled time series

José M. Amigó1, Roberto Monetti2, Thomas Aschenbrenner2, and Wolfram Bunk2

1Centro de Investigación Operativa, Universidad Miguel Hernandez, Avda. de la Universidad s/n, 03202 Elche, Spain
2Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748 Garching, Germany

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(Received 31 August 2011; accepted 8 December 2011; published online 13 January 2012)

Ordinal symbolic dynamics is based on ordinal patterns. Its tools include permutation entropy (in metric and topological versions), forbidden patterns, and a number of mathematical results that make this sort of symbolic dynamics appealing both for theoreticians and practitioners. In particular, ordinal symbolic dynamics is robust against observational noise and can be implemented with low computational cost, which explains its increasing popularity in time series analysis. In this paper, we study the perhaps less exploited aspect so far of ordinal patterns: their algebraic structure. In a first part, we revisit the concept of transcript between two symbolic representations, generalize it to N representations, and derive some general properties. In a second part, we use transcripts to define two complexity indicators of coupled dynamics. Their performance is tested with numerical and real world data.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. CONCEPT AND PROPERTIES OF TRANSCRIPTS
  3. TRANSCRIPTS FOR N SYMBOLIC SERIES AND ENTROPY
  4. APPLICATIONS TO COMPLEXITY
  5. NUMERICAL EXPERIMENTS ON SYNTHETIC AND REAL WORLD DATA
  6. CONCLUSION

RELATED DATABASES

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KEYWORDS and PACS

Keywords

algebra, entropy, numerical analysis, time series, topology

PACS

  • 05.70.Ce

    Thermodynamic functions and equations of state

  • 05.45.Tp

    Time series analysis

  • 02.40.Re

    Algebraic topology

  • 02.60.-x

    Numerical approximation and analysis

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3673238

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (click on thumbnails to view enlargements)

FIG.1
Upper row: Complexity measures versus the coupling k for the Rössler system given by Eqs. ( 40 ). Middle row: Complexity measures versus the entropy of transcripts H(p4T). Lower row. Left: The entropy of transcripts H(p4T) versus the coupling constant k. Right: The mutual information I(α, β) versus the coupling constant k. Curves which are plotted using different line types indicate results for the component pairs x1,2, y1,2, and z1,2. All results were obtained using L = 4 and δ = 150.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
(Color) Left: Complexity diagram given by C1. Right: Complexity diagram given by C2. The color coding indicates increasing complexity values from deep blue to light yellow according to the respective color bar. Results were obtained using L = 4 and δ = 1. Complexity values were calculated for increments Δk1,2 = 10−3 and time series of length 217 data points.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Portraits of the delayed coupled logistic map (Eq. ( 41 )) x1 versus x2 for six different pairs (k1,k2) selected according to increasing values of the complexity measure C1. The coupling strengths and the values of both complexity measures are shown for every state. For the sake of comparison, the complexity values have been normalized by the respective maximum complexity.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Upper left: Time evolution of the spatial mean of the entropy of transcripts. Upper right: Time evolution of the spatial mean of the mutual information of ordinal patterns. Lower left: Time course of the spatial mean of the coupling complexity C1. Lower right: Time course of the spatial mean of the coupling complexity C2. At about 402 s an epileptic seizure starts. All results were obtained using L = 4 and a delay δ = 0.048 s.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint



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