Abstract
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.
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H.D.I. Abarbanel, Analysis of Observed Chaotic Data. Springer, New York, 1996.
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.
J. García-Ojalvo, J.M. Sancho, and L. Ramírez-Piscina, Generation of spatiotemporal colored noise, Physical Review A 46 (1992) 4670–4675.
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.
H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge University Press, Cambridge, 1997.
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.
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.
A.M. Law and W.D. Kelton, Simulation, Modeling, and Analysis, 3rd edition. McGraw-Hill, Boston, 2000.
B. LeBaron, A fast algorithm for the BDS statistics, Studies in Nonlinear Dynamics & Econometrics 2 (1997) 53–59.
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.
M.E. Mera and M. Morán, Geometric noise reduction for multivariate time series, Chaos 16 (2006) 013116.
G.J. Ortega and E. Louis, Smoothness implies determinism in time series: A measure based approach, Physical Review Letters 81 (1998) 4345–4348.
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.
J.C. Sprott, Chaos and Time-Series Analysis. Oxford University Press, Oxford, 2003.
J.C. Sprott, High-dimensional dynamics in the delayed Hénon map. Electronic Journal of Theoretical Physics 3 (2006) 19–35.
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Amigó, J.M. (2010). Detection of Determinism. In: Permutation Complexity in Dynamical Systems. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04084-9_9
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DOI: https://doi.org/10.1007/978-3-642-04084-9_9
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