Advertisement

First Applications

  • José María AmigóEmail author
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

In this chapter we present four applications of permutation entropy and ordinal patterns: entropy estimation, complexity analysis, recovery of parameters from itineraries, and synchronization analysis of time series. The scope is to give the reader a multifaceted picture of ordinal analysis in action. Two more applications (to determinism detection and to space–time chaos) will be discussed at length in Chaps. 9 and 10, respectively.

Keywords

Invariant Measure Shannon Entropy Phase Synchronization Topological Entropy Symbolic Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

  1. 1.
    H.D.I. Abarbanel, Analysis of Observed Chaotic Data. Springer, New York, 1996.Google Scholar
  2. 5.
    G. Alvarez, M. Romera, G. Pastor, and F. Montoya, Gray codes and 1D quadratic maps, Electronic Letters 34 (1998) 1304–1306.CrossRefGoogle Scholar
  3. 6.
    J.M. Amigó, J. Szczepanski, E. Wajnryb, and M.V. Sanchez-Vives, Estimating the entropy of spike trains via Lempel-Ziv complexity, Neural Computation 16 (2004) 717–736.zbMATHCrossRefGoogle Scholar
  4. 21.
    D. Arroyo, G. Alvarez, and J.M. Amigó, Estimation of the control parameter from symbolic sequences: Unimodal maps with variable critical point, Chaos 19 (2009) 023125.CrossRefMathSciNetADSGoogle Scholar
  5. 24.
    N. Ay and J.P. Crutchfield, Reductions of hidden information sources, Journal of Statistical Physics 120 (2005) 659–684.zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 28.
    C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Physical Review Letters 88 (2002) 174102.CrossRefADSGoogle Scholar
  7. 30.
    C. Bandt and F. Shiha, Order patterns in time series, Journal of Time Series Analysis 28 (2007) 646–665.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 34.
    S. Boccaletti and D.L. Valladares, Characterization of intermittent lag synchronization, Physical Review E 62 (2000) 7497–7500.CrossRefADSGoogle Scholar
  9. 45.
    Y. Cao, W. Tung, J.B. Gao, V.A. Protopopescu, and L.M. Hively, Detecting dynamical changes in time series using the permutation entropy, Physical Review E 70 (2004) 046217.CrossRefMathSciNetADSGoogle Scholar
  10. 56.
    R.W. Clarke, M.P. Freeman, and N.W. Watkins, Application of computational mechanics to the analysis of natural data: An example in geomagnetism, Physical Review E 67 (2003) 016203.CrossRefADSGoogle Scholar
  11. 57.
    P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, 5th printing. Birkhäuser, Boston, 1997.Google Scholar
  12. 59.
    T.M. Cover and J.A. Thomas, Elements of Information Theory, 2nd edition. New York, John Wiley & Sons, 2006.zbMATHGoogle Scholar
  13. 60.
    J.P. Crutchfield and K. Young, Inferring statistical complexity, Physical Review Letters 63 (1989) 105–108.CrossRefMathSciNetADSGoogle Scholar
  14. 73.
    J.P. Eckmann, S.O. Kamphorst, and D. Ruelle, Recurrence plots of dynamical systems, Europhysics Letters 4 (1987) 973–977.CrossRefADSGoogle Scholar
  15. 78.
    A. Fernández, J. Quintero, R. Hornero, P. Zuluaga, M. Navas, C. Gómez, J. Escudero, N. García-Campos, J. Biederman, and T. Ortiz, Complexity analysis of spontaneous brain activity in attention-deficit/hyperactivity disorder: Diagnosis implications, Biological Psychiatry 65 (2009) 571–577.CrossRefGoogle Scholar
  16. 80.
    A.M. Fraser and H.L. Swinney, Independent coordinates for strange attractors from mutual information, Physical Review A 33 (1986) 1134–1140.CrossRefMathSciNetADSzbMATHGoogle Scholar
  17. 81.
    J.B. Gao and H.Q. Cai, On the structures and quantification of recurrence plots, Physics Letters A 270 (2000) 75–87.CrossRefADSGoogle Scholar
  18. 82.
    Y. Gao, I. Kontoyiannis, and E. Bienenstock, Estimating the entropy of binary time series: Methodology, some theory and a simulation study, Entropy 10 (2008) 71–99.zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 88.
    P. Grassberger, Finite sample corrections to entropy and dimension estimates, Physics Letters A 128 (1988) 369–373.CrossRefMathSciNetADSGoogle Scholar
  20. 90.
    F. Gu, X. Meng, E. Shen, and Z. Cai, Can we measure consciousness with EEG complexities?, International Journal of Bifurcations and Chaos 13 (2003) 733–742.zbMATHCrossRefGoogle Scholar
  21. 93.
    H. Herzel, Complexity of symbol sequences, Systems, Analysis, Modelling, Simulations 5 (1988) 435–444.zbMATHMathSciNetGoogle Scholar
  22. 94.
    H. Herzel, A.O. Schmitt, and W. Ebeling, Finite sample effects in sequence analysis, Chaos, Solitons & Fractals 4 (1994) 97–113.zbMATHCrossRefADSGoogle Scholar
  23. 98.
    M.W. Hirsch, S. Smale, and R.L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, San Diego, 2003.Google Scholar
  24. 105.
    O. Jenkinson and M. Pollicott, Entropy, exponents and invariant densities for hyperbolic systems: Dependence and computation. In: M. Brin, B. Hasselblatt, and Y. Pesin (Eds.), Modern Dynamical Systems and Applications. pp. 365–384 Cambridge University Press, Cambridge, 2004.Google Scholar
  25. 111.
    H. Kantz, Quantifying the closeness of fractal measures, Physical Review E 49 (1994) 5091–5097.CrossRefADSGoogle Scholar
  26. 112.
    H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge University Press, Cambridge, 1997.Google Scholar
  27. 116.
    K. Keller and K. Wittfeld, Distances of time series components by means of symbolic dynamics, International Journal of Bifurcation and Chaos 14 (2004) 693–703.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 118.
    K. Keller, H. Lauffer, and M. Sinn, Ordinal analysis of EEG time series, Chaos and Complexity Letters 2 (2007) 247–258.Google Scholar
  29. 120.
    M.B. Kennel, Statistical test for dynamical nonstationarity in observed time-series data, Physical Review E 56 (1997) 316–321.CrossRefADSGoogle Scholar
  30. 127.
    I. Kontoyiannis, P.H. Algoet, Y.M. Suhov, and A.J. Wyner, Nonparametric entropy estimation for stationary processes and random fields, with applications to English text. IEEE Transactions on Information Theory 44 (1998) 1319–1327.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 131.
    A.P. Kurian and S. Puttusserypady, Self-synchronizing chaotic stream ciphers, Signal Processing 88 (2008) 2442–2452.zbMATHCrossRefGoogle Scholar
  32. 134.
    A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society 186 (1973), 481–488.CrossRefMathSciNetGoogle Scholar
  33. 143.
    M.T. Martin, A. Plastino, and O.A. Rosso, Generalized statistical complexity measures: Geometrical and analytical properties, Physica A 369 (2006) 439–462.CrossRefADSGoogle Scholar
  34. 144.
    N. Marwan, M.C. Romano, M. Thiel, and J. Kurths, Recurrence plots for the analysis of complex systems, Physics Reports 438 (2007) 237–329.CrossRefMathSciNetADSGoogle Scholar
  35. 146.
    M. Matilla-García, A non-parametric test for independence based on symbolic dynamics, Journal of Economic Dynamic & Control 31 (2007) 3889–3903.zbMATHCrossRefGoogle Scholar
  36. 147.
    M. Matilla-García and M. Ruiz Marín, A non-parametric independence test using permutation entropy, Journal of Econometrics 144 (2008) 139–155.CrossRefMathSciNetGoogle Scholar
  37. 150.
    W. de Melo and S. van Strien, One-Dimensional Dynamics. Springer Verlag, Berlin, 1993.Google Scholar
  38. 153.
    N. Metropolis, M. Stein, and P. Stein, On finite limit sets for transformations on the unit interval, Journal of Combinatorial Theory, Series A 15, 25–44 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  39. 159.
    R. Monetti, W. Bunk, T. Aschenbrenner, and F. Jamitzky, Characterizing synchronization in time series using information measures extracted from symbolic representations, Physical Review E 79 (2009) 046207.CrossRefADSGoogle Scholar
  40. 163.
    E. Olbrich, N. Bertschinger, N. Ay, and J. Jost, How should complexity scale with system size?, The European Physical Journal B 63 (2008) 407–415.CrossRefMathSciNetADSzbMATHGoogle Scholar
  41. 166.
    N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, Geometry from a time series, Physical Review Letters 45 (1980) 712–716.CrossRefADSGoogle Scholar
  42. 167.
    L. Paninski, Estimation of entropy and mutual information, Neural Computation 15 (2003) 1191–1253.zbMATHCrossRefGoogle Scholar
  43. 173.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2007.Google Scholar
  44. 175.
    M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, Phase synchronization of chaotic oscillators, Physical Review Letters 76 (1997) 1804–1807.CrossRefADSGoogle Scholar
  45. 179.
    A.N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukrainian Mathematical Journal 16 (1964) 61–71.MathSciNetGoogle Scholar
  46. 180.
    P.R. Scalassara, M.E. Dajer, C. Dias Maciel, C. Capobianco Guido, and J.C. Pereira, Relative entropy measures applied to healthy and pathological voice characterization, Applied Mathematics and Computation 207 (2009) 95–108.zbMATHCrossRefGoogle Scholar
  47. 181.
    A.O. Schmitt, H. Herzel, and W. Ebeling, A new method to calculate higher-order entropies from finite samples, Europhysics Letters 23 (1993) 303–309.CrossRefADSGoogle Scholar
  48. 183.
    T. Schreiber, Detecting and analyzing nonstationarity in a time series using nonlinear cross predictions, Physical Review Letters 78 (1997) 843–846.CrossRefADSGoogle Scholar
  49. 185.
    C.R. Shalizi and J.P. Crutchfield, Computational mechanics: Pattern and prediction, structure and simplicity, Journal of Statistical Physics 104 (2001) 817–879.zbMATHCrossRefMathSciNetGoogle Scholar
  50. 190.
    M. Sinn and K. Keller, Estimation of ordinal pattern probabilities in fractional Brownian motion, arXiv:0801.1598.Google Scholar
  51. 195.
    S.P. Strong, R. Koberle, R.R. de Ruyter van Steveninck, and W. Bialek, Entropy and information in neural spike trains. Physical Review Letters 80 (1998) 197–200.CrossRefADSGoogle Scholar
  52. 196.
    J. Szczepanski, J.M. Amigó, E. Wajnryb, and M.V. Sanchez-Vives. Application of Lempel-Ziv complexity to the analysis of neural discharges, Network: Computation in Neural Systems 14 (2003) 335–350.CrossRefGoogle Scholar
  53. 197.
    F. Takens, Detecting strange attractors in turbulence, In: D. Rand and L.S. Young (Eds.), Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer, Berlin, 1981, pp. 366–381.Google Scholar
  54. 200.
    D.B. Vasconcelos, S.R. Lopes, R.L. Viana, and J. Kurths, Spatial recurrence plots, Physical Review E 73 (2006) 056207.CrossRefMathSciNetADSGoogle Scholar
  55. 203.
    L. Wang and N.D. Kazarinoff, On the universal sequence generated by a class of unimodal functions, Journal of Combinatorial Theory, Series A 46 (1987) 39–49.zbMATHCrossRefMathSciNetGoogle Scholar
  56. 208.
    X-S. Zhang, R.J. Roy, and E.W. Jensen, EEG complexity as a measure of depth anesthesia for patients, IEEE Transactions on Biomedical Engineering 48 (2001) 1424–1433.CrossRefGoogle Scholar
  57. 209.
    J. Zhang and M. Small, Complex networks from pseudoperiodic time series: Topology versus dynamics. Physical Review Letters 96 (2006) 238701.CrossRefADSGoogle Scholar
  58. 212.
    L. Zunino, D.G. Pérez, M.T. Martín, M. Garavaglia, A. Plastino, and O.A. Rosso, Permutation entropy of fractional Brownian motion and fractional Gaussian noise, Physics Letters A 372 (2008) 4768–4774.CrossRefADSGoogle Scholar
  59. 213.
    L. Zunino, D.G. Pérez, M.T. Martín, M. Garavaglia, A. Plastino, and O.A. Rosso, Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy, Physica A 387 (2008) 6057–6068.CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Centro de Investigacion OperativaUniversidad Miguel HernandezElcheSpain

Personalised recommendations