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The European Physical Journal Special Topics

, Volume 226, Issue 15, pp 3239–3250 | Cite as

Kalman-Takens filtering in the presence of dynamical noise

  • Franz Hamilton
  • Tyrus Berry
  • Timothy Sauer
Regular Article
Part of the following topical collections:
  1. Challenges in the Analysis of Complex Systems: Prediction, Causality and Communication

Abstract

The use of data assimilation for the merging of observed data with dynamical models is becoming standard in modern physics. If a parametric model is known, methods such as Kalman filtering have been developed for this purpose. If no model is known, a hybrid Kalman-Takens method has been recently introduced, in order to exploit the advantages of optimal filtering in a nonparametric setting. This procedure replaces the parametric model with dynamics reconstructed from delay coordinates, while using the Kalman update formulation to assimilate new observations. In this article, we study the efficacy of this method for identifying underlying dynamics in the presence of dynamical noise. Furthermore, by combining the Kalman-Takens method with an adaptive filtering procedure we are able to estimate the statistics of the observational and dynamical noise. This solves a long-standing problem of separating dynamical and observational noise in time series data, which is especially challenging when no dynamical model is specified.

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References

  1. 1.
    R. Kalman, J. Basic Eng. 82, 35 (1960) CrossRefGoogle Scholar
  2. 2.
    E. Kalnay, Atmospheric modeling, data assimilation, and predictability (Cambridge Univ. Press, 2003) Google Scholar
  3. 3.
    G. Evensen, Data assimilation: the ensemble Kalman filter (Springer, Heidelberg, 2009) Google Scholar
  4. 4.
    S. Julier, J. Uhlmann, H. Durrant-Whyte, IEEE Trans. Automat. Control 45, 477 (2000) MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Julier, J. Uhlmann, H. Durrant-Whyte, Proc. IEEE 92, 401 (2004) CrossRefGoogle Scholar
  6. 6.
    S. Schiff, Neural control engineering (MIT Press, 2012) Google Scholar
  7. 7.
    R.H. Reichle, R.D. Koster, Geophys. Res. Lett. 32, 102404 (2005) ADSCrossRefGoogle Scholar
  8. 8.
    H. Hersbach, A. Stoffelen, S. De Haan, J. Geophys. Res.: Oceans (1978–2012) 112, C03006 (2007) ADSCrossRefGoogle Scholar
  9. 9.
    H. Arnold, I. Moroz, T. Palmer, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 371, 20110479 (2013) ADSCrossRefGoogle Scholar
  10. 10.
    T. Berry, J. Harlim, Proc. R. Soc. A: Math. Phys. Eng. Sci. 470, 20140168 (2014) ADSCrossRefGoogle Scholar
  11. 11.
    E.N. Lorenz, K.A. Emanuel, J. Atmos. Sci. 55, 399 (1998) ADSCrossRefGoogle Scholar
  12. 12.
    T.N. Palmer, Q. J. R. Met. Soc. 127, 279 (2001) ADSGoogle Scholar
  13. 13.
    F. Hamilton, T. Berry, T. Sauer, Phys. Rev. E 92, 010902 (2015) ADSCrossRefGoogle Scholar
  14. 14.
    T. Berry, J. Harlim, J. Comput. Phys. 308, 305 (2016) ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    F. Takens, in Lecture Notes in Math (Springer-Verlag, Berlin, 1981), Vol. 898 Google Scholar
  16. 16.
    N. Packard, J. Crutchfield, D. Farmer, R. Shaw, Phys. Rev. Lett. 45, 712 (1980) ADSCrossRefGoogle Scholar
  17. 17.
    T. Sauer, J. Yorke, M. Casdagli, J. Stat. Phys. 65, 579 (1991) ADSCrossRefGoogle Scholar
  18. 18.
    T. Sauer, Phys. Rev. Lett. 93, 198701 (2004) ADSCrossRefGoogle Scholar
  19. 19.
    E. Ott, T. Sauer, J.A. Yorke, Coping with chaos: analysis of chaotic data and the exploitation of chaotic systems (Wiley, New York, 1994) Google Scholar
  20. 20.
    H. Abarbanel, Analysis of observed chaotic data (Springer-Verlag, New York, 1996) Google Scholar
  21. 21.
    H. Kantz, T. Schreiber, Nonlinear time series analysis (Cambridge University Press, 2004) Google Scholar
  22. 22.
    J. Farmer, J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987) ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Casdagli, Physica D: Nonlinear Phenom. 35, 335 (1989) ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Sugihara, R.M. May, Nature 344, 734 (1990) ADSCrossRefGoogle Scholar
  25. 25.
    L.A. Smith, Physica D: Nonlinear Phenom. 58, 50 (1992) ADSCrossRefGoogle Scholar
  26. 26.
    J. Jimenez, J. Moreno, G. Ruggeri, Phys. Rev. A 45, 3553 (1992) ADSCrossRefGoogle Scholar
  27. 27.
    T. Sauer, in Time series prediction: forecasting the future and understanding the past (Addison-Wesley, 1994), pp. 175–193 Google Scholar
  28. 28.
    G. Sugihara, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 348, 477 (1994) ADSCrossRefGoogle Scholar
  29. 29.
    C. G. Schroer, T. Sauer, E. Ott, J.A. Yorke, Phys. Rev. Lett. 80, 1410 (1998) ADSCrossRefGoogle Scholar
  30. 30.
    D. Kugiumtzis, O. Lingjaerde, N. Christophersen, Physica D: Nonlinear Phenom. 112, 344 (1998) ADSCrossRefGoogle Scholar
  31. 31.
    G. Yuan, M. Lozier, L. Pratt, C. Jones, K. Helfrich, J. Geophys. Res. 109, C08002 (2004) ADSGoogle Scholar
  32. 32.
    C.-H. Hsieh, S.M. Glaser, A.J. Lucas, G. Sugihara, Nature 435, 336 (2005) ADSCrossRefGoogle Scholar
  33. 33.
    C.C. Strelioff, A.W. Hubler, Phys. Rev. Lett. 96, 044101 (2006) ADSCrossRefGoogle Scholar
  34. 34.
    S. Regonda, B. Rajagopalan, U. Lall, M. Clark, Y.-I. Moon, Nonlinear Proc. Geophys. 12, 397 (2005) ADSCrossRefGoogle Scholar
  35. 35.
    B. Schelter, M. Winterhalder, J. Timmer, Handbook of time series analysis: recent theoretical developments and applications (John Wiley and Sons, 2006) Google Scholar
  36. 36.
    F. Hamilton, T. Berry, T. Sauer, Phys. Rev. X 6, 011021 (2016) Google Scholar
  37. 37.
    T. Berry, D. Giannakis, J. Harlim, Phys. Rev. E 91, 032915 (2015) ADSCrossRefGoogle Scholar
  38. 38.
    T. Berry, J. Harlim, Physica D 320, 57 (2016) ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    D. Simon, Optimal state estimation: Kalman, H∞ and nonlinear approaches (John Wiley and Sons, 2006) Google Scholar
  40. 40.
    T. Berry, T. Sauer, Tellus A 65, 20331 (2013) CrossRefGoogle Scholar
  41. 41.
    E. Lorenz, J. Atmos. Sci. 20, 130 (1963) ADSCrossRefGoogle Scholar
  42. 42.
    E.N. Lorenz, in Proceedings: Seminar on predictability (AMS, Reading, UK, 1996), pp. 1–18 Google Scholar
  43. 43.
    A. Sitz, U. Schwarz, J. Kurths, H. Voss, Phys. Rev. E 66, 16210 (2002) ADSCrossRefGoogle Scholar
  44. 44.
    J. Stark, D.S. Broomhead, M.E. Davies, J. Huke, J. Nonlinear Sci. 13, 519 (2003) ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.North Carolina State UniversityRaleighUSA
  2. 2.George Mason UniversityFairfaxUSA

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