Detection of Determinism

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


In Chap. 2 we have illustrated the applications of ordinal patterns with four examples. In this chapter we present a further application, this time to the detection of determinism in noisy time series. Following the common usage of the term in applied science, “determinism” is meant here as the opposite to statistical independence, hence it includes colored noise as well. This application hinges on two basic properties of ordinal patterns: existence of forbidden patterns in the orbits of maps (Sects. 1.2, 3.3, and 7.7) and robustness to observational noise (Sects. 3.4.3, and 9.1). We shall actually present two detection methods.

Method I is based on the number of missing ordinal patterns. It proceeds by (i) counting the number of missing ordinal patterns in sliding, overlapping windows of size L along the data sequence, (ii) randomizing the sequence, and (iii) repeating (i) with the randomized sequence. Is the result of step (iii) clearly greater than the result of step (i), so may we conclude that the original noisy sequence has a deterministic component.


Time Series Gaussian White Noise Colored Noise Multivariate Time Series Permutation Entropy 
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.


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. 38.
    W.A. Brock, W.D. Dechert, J.A. Scheinkman, and B. LeBaron, A test for independence based on the correlation dimension, Econometrics Reviews 15 (1996) 197–235.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 83.
    J. García-Ojalvo, J.M. Sancho, and L. Ramírez-Piscina, Generation of spatiotemporal colored noise, Physical Review A 46 (1992) 4670–4675.CrossRefADSGoogle Scholar
  4. 85.
    A. Golestani, M.R. Jahed Motlagh, K. Ahmadian, A.H. Omidvarnia, and N. Mozayani, A new criterion to distinguish stochastic and deterministic time series with the Poincaré section and fractal dimension, Chaos 19 (2009) 013137.CrossRefMathSciNetADSGoogle Scholar
  5. 112.
    H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge University Press, Cambridge, 1997.Google Scholar
  6. 113.
    N.J. Kasdin, Discrete simulation of colored noise and stochastic processes and 1/f α power law noise generation, Proceedings of the IEEE 83 (1995) 802–827.CrossRefGoogle Scholar
  7. 119.
    M.B. Kennel and S. Isabelle, Method to distinguish possible chaos from colored noise and to determine embedding parameters, Physical Review A 46 (1992) 3111–3118.CrossRefADSGoogle Scholar
  8. 135.
    A.M. Law and W.D. Kelton, Simulation, Modeling, and Analysis, 3rd edition. McGraw-Hill, Boston, 2000.Google Scholar
  9. 136.
    B. LeBaron, A fast algorithm for the BDS statistics, Studies in Nonlinear Dynamics & Econometrics 2 (1997) 53–59.CrossRefGoogle Scholar
  10. 140.
    T. Liu, C.W.J. Granger, and W.P. Heller, Using the correlation exponent to decide whether an economic series is chaotic. Journal of Applied Econometrics, Supplement: Special Issue on Nonlinear Dynamics and Econometrics (Dec., 1992) S25–S39.Google Scholar
  11. 152.
    M.E. Mera and M. Morán, Geometric noise reduction for multivariate time series, Chaos 16 (2006) 013116.CrossRefMathSciNetADSGoogle Scholar
  12. 164.
    G.J. Ortega and E. Louis, Smoothness implies determinism in time series: A measure based approach, Physical Review Letters 81 (1998) 4345–4348.CrossRefADSGoogle Scholar
  13. 176.
    O.A. Rosso, H.A. Larrondo, M.T. Martin, A. Platino, and M.A. Fuentes, Distinguishing noise from chaos, Physical Review Letters 99 (2007) 154102.CrossRefADSGoogle Scholar
  14. 193.
    J.C. Sprott, Chaos and Time-Series Analysis. Oxford University Press, Oxford, 2003.Google Scholar
  15. 194.
    J.C. Sprott, High-dimensional dynamics in the delayed Hénon map. Electronic Journal of Theoretical Physics 3 (2006) 19–35.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Centro de Investigacion OperativaUniversidad Miguel HernandezElcheSpain

Personalised recommendations