Advertisement

Symbolic Shadowing and the Computation of Entropy for Observed Time Series

  • Diana A. Mendes
  • Vivaldo M. Mendes
  • Nuno Ferreira
  • Rui Menezes
Conference paper

Abstract

Order, disorder and recurrence are common features observed in complex time series that can be encountered in many fields, like finance, economics, biology and physiology. These phenomena can be modelled by chaotic dynamical systems and one way to undertake a rigorous analysis is via symbolic dynamics, a mathematical-statistical technique that allows the detection of the underlying topological and metrical structures in the time series. Symbolic dynamics is a powerful tool initially developed for the investigation of discrete dynamical systems. The main idea consists in constructing a partition, that is, a finite collection of disjoint subsets whose union is the state space. By identifying each subset with a distinct symbol, we obtain sequences of symbols that correspond to each trajectory of the original system. One of the major problems in defining a “good” symbolic description of the corresponding time series is to obtain a generating partition, that is, the assignment of symbolic sequences to trajectories that is unique, up to a set of measure zero. Unfortunately, this is not a trivial task, and, moreover, for observed time series the notion of a generating partition is no longer well defined in the presence of noise. In this paper we apply symbolic shadowing, a deterministic algorithm using tessellations, in order to estimate a generating partition for a financial time series (PSI20) and consequently to compute its entropy. This algorithm allows producing partitions such that the symbolic sequences uniquely encode all periodic points up to some order. We compare these results with those obtained by considering the Pesin’s identity, that is, the metric entropy is equal to the sum of positive Lyapunov exponents. To obtain the Lyapunov exponents, we reconstruct the state space of the PSI20 data by applying an embedding process and estimate them by using the Wolf et al. algorithm.

Keywords

Lyapunov Exponent Word Length Topological Entropy Original Time Series Symbolic Dynamic 
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.

Notes

Acknowledgements

Financial support from the Fundação Ciência e Tecnologia, Lisbon, is grateful acknowledged by the authors, under the contracts No PTDC/GES/73418/2006 and No PTDC/GES/70529/2006.

References

  1. 1.
    Abarbanel H (1996) Analysis of observed chaotic data. Springer, New YorkCrossRefMATHGoogle Scholar
  2. 2.
    Anosov DV (1967) Geodesic flows and closed Riemannian manifolds with negative curvature. Proc Steklov Inst Math 90:1–235Google Scholar
  3. 3.
    Aziz W, Arif M (2006) Complexity analysis of stride interval time series by threshold dependent symbolic entropy. Eur J Appl Physiol 98:30–40CrossRefGoogle Scholar
  4. 4.
    Bolt EM, Stanford T, Lai Y-C, Zyczkowski K (2001) What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series. Physica D 154(3–4):259–286MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Bowen R (1975) ω-Limit sets for axiom A diffeomorphisms. J Diff Eqs 18:333–339MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brida JG, Gómez DM, Risso WA (2009) Symbolic hierarchical analysis in currency markets: an application to contagion in currency crises. Expert Syst Appl 36:7721–7728CrossRefGoogle Scholar
  7. 7.
    Brock WA (1986) Distinguishing random and deterministic systems: abridged version. In: Grandmont J-M (ed) Nonlinear economic dynamics. Academic, New York, pp 168–195Google Scholar
  8. 8.
    Brock WA, Dechert W, Scheinkman J (1987) A test for independence based on the correlation dimension. Working paper, University of Winconsin at Madison, University of Houston, and University of ChicagoGoogle Scholar
  9. 9.
    Cochrane J (1994) Shocks. Carnegie-Rochester Conf Ser Public Policy 41:295–364CrossRefGoogle Scholar
  10. 10.
    Cvitanovic P, Gunaratne GH, Procaccia I (1988) Topological and metric properties of Hénon-type strange attractors. Phys Rev A 38(3):1503–1520MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Darbellay G (1998) Predictability, an information-theoretic perspective. In: Prochazka A, Uhlır J, Rayner PJW, Kingsbury NG (eds) Signal analysis and prediction. Birkhauser, Boston, pp 249–262Google Scholar
  12. 12.
    Dionísio A, Menezes R, Mendes DA (2006) Entropy-based independence test. Nonlinear Dyn 44(1–4):351–357CrossRefMATHGoogle Scholar
  13. 13.
    Eckmann J-P, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys 57:617–656MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Eguia MC, Rabinovich MI, Abarbanel HD (2000) Information transmission and recovery in neural communications channels. Phys Rev E 62(5B):7111–7122ADSCrossRefGoogle Scholar
  15. 15.
    Doyne Farmer J, Sidorowich JJ (1991) Optimal shadowing and noise reduction. Physica D 47:373–392MathSciNetADSCrossRefMATHGoogle Scholar
  16. 16.
    Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140MathSciNetADSCrossRefMATHGoogle Scholar
  17. 17.
    Grassberger P, Kantz H (1985) Generating partitions for the dissipative Henon map. Phys Lett A 113(5):235–238MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Grassberger P, Procaccia I (1983) Characterization of strange attractors. Phys Rev Lett 50:346–349MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Gu R (2008) On ergodicity of systems with the asymptotic average shadowing property. Comput Math Appl 55:1137–1141MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hammel SM, Yorke JA, Grebogi C (1987) Do numerical orbits of chaotic processes represent true orbits? J Complexity 3:136–145MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hirata Y, Judd K, Kilminster D (2004) Estimating a generating partition from observed time series: symbolic shadowing. Phys Rev E 70:016215MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Hirata Y, Judd K (2005) Constructing dynamical systems with specified symbolic dynamics. Chaos 15:033102MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Iseri M, Caglar H, Caglar N (2008) A model proposal for the chaotic structure of Istanbul stock exchange. Chaos Solitons Fractals 36:1392–1398ADSCrossRefGoogle Scholar
  24. 24.
    Kantz H, Schreiber TH (1997) Nonlinear time series analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  25. 25.
    Katok A, Hasselblat B (1999) An introduction to the modern theory of dynamical systems. Cambridge University Press, CambridgeGoogle Scholar
  26. 26.
    Kennel MB, Buhl M (2003) Estimating good discrete partitions from observed data: symbolic false nearest neighbors. Phys Rev Lett 91:084102ADSCrossRefGoogle Scholar
  27. 27.
    Koscielniak P, Mazur M (2007) Chaos and the shadowing property. Topol Appl 154:2553–2557MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Liebert W, Schuster HG (1988) Proper choice of the time delay for the analysis of chaotic time series. Phys Lett A, 142:107–111MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  30. 30.
    Mantegna RN, Stanley HE (1999) An introduction to econophysics: correlations and complexity in finance. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  31. 31.
    Maasoumi E, Racine J (2002) Entropy and predictability of stock market returns. J Econom 107:291–312MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mendes DA, Sousa Ramos J (2004) Kneading theory for triangular maps. Int J Pure Appl Math 10(4):421–450MathSciNetMATHGoogle Scholar
  33. 33.
    Milnor J, Thurston W (1988) On iterated maps of the interval. In: Alexander J (ed) Dynamical systems, Proceedings of Special Year at the University of Maryland. Lecture Notes in Mathematics, vol 1342. Springer, Berlin, pp 465–563Google Scholar
  34. 34.
    Nakamura T, Small M (2006) Nonlinear dynamical system identification with dynamic noise and observational noise. Physica D 223:54–68MathSciNetADSCrossRefMATHGoogle Scholar
  35. 35.
    Pearson DW (2001) Shadowing and prediction of dynamical systems. Math Comput Model 34:813–820CrossRefMATHGoogle Scholar
  36. 36.
    Pompe B (1993) Measuring statistical dependences in a time series. J Stat Phys 73:587–610MathSciNetADSCrossRefMATHGoogle Scholar
  37. 37.
    Serletis A, Gogas P (1997) Chaos in East European black market exchange rates. Res Econ 51:359–385CrossRefMATHGoogle Scholar
  38. 38.
    Small M, Tse CK (2003) Evidence for deterministic nonlinear dynamics in financial time series data. CIFEr 2003, Hong KongGoogle Scholar
  39. 39.
    Small M, Tse CK (2003) Determinism in financial time series. Stud Nonlinear Dyn Econom 7(3):5Google Scholar
  40. 40.
    Takens F (1981) Detecting strange attractors in turbulence. In: Rand D, Young L (eds) Dynamical sysytems and turbulence. Springer, Berlin, pp 366–381Google Scholar
  41. 41.
    Wessel N, Schwarz U, Saparin PI, Kurths J (2007) Symbolic dynamics for medical data analysis. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7153
  42. 42.
    Wolf A, Swift J, Swinney H, Vastano J (1985) Determining Lyapunov exponents from a time series. Physica D 16:285–292MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Diana A. Mendes
    • 1
  • Vivaldo M. Mendes
    • 2
  • Nuno Ferreira
    • 1
  • Rui Menezes
    • 1
  1. 1.Department of Quantitative MethodsISCTE-IUL and UNIDE, Avenida Forças ArmadasLisbonPortugal
  2. 2.Department of EconomicsISCTE-IUL and UNIDELisbonPortugal

Personalised recommendations