Skip to main content
Log in

Kalman-Takens filtering in the presence of dynamical noise

The European Physical Journal Special Topics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Kalman, J. Basic Eng. 82, 35 (1960)

    Article  Google Scholar 

  2. E. Kalnay, Atmospheric modeling, data assimilation, and predictability (Cambridge Univ. Press, 2003)

  3. G. Evensen, Data assimilation: the ensemble Kalman filter (Springer, Heidelberg, 2009)

  4. S. Julier, J. Uhlmann, H. Durrant-Whyte, IEEE Trans. Automat. Control 45, 477 (2000)

    Article  MathSciNet  Google Scholar 

  5. S. Julier, J. Uhlmann, H. Durrant-Whyte, Proc. IEEE 92, 401 (2004)

    Article  Google Scholar 

  6. S. Schiff, Neural control engineering (MIT Press, 2012)

  7. R.H. Reichle, R.D. Koster, Geophys. Res. Lett. 32, 102404 (2005)

    Article  ADS  Google Scholar 

  8. H. Hersbach, A. Stoffelen, S. De Haan, J. Geophys. Res.: Oceans (1978–2012) 112, C03006 (2007)

    Article  ADS  Google Scholar 

  9. H. Arnold, I. Moroz, T. Palmer, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 371, 20110479 (2013)

    Article  ADS  Google Scholar 

  10. T. Berry, J. Harlim, Proc. R. Soc. A: Math. Phys. Eng. Sci. 470, 20140168 (2014)

    Article  ADS  Google Scholar 

  11. E.N. Lorenz, K.A. Emanuel, J. Atmos. Sci. 55, 399 (1998)

    Article  ADS  Google Scholar 

  12. T.N. Palmer, Q. J. R. Met. Soc. 127, 279 (2001)

    ADS  Google Scholar 

  13. F. Hamilton, T. Berry, T. Sauer, Phys. Rev. E 92, 010902 (2015)

    Article  ADS  Google Scholar 

  14. T. Berry, J. Harlim, J. Comput. Phys. 308, 305 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  15. F. Takens, in Lecture Notes in Math (Springer-Verlag, Berlin, 1981), Vol. 898

  16. N. Packard, J. Crutchfield, D. Farmer, R. Shaw, Phys. Rev. Lett. 45, 712 (1980)

    Article  ADS  Google Scholar 

  17. T. Sauer, J. Yorke, M. Casdagli, J. Stat. Phys. 65, 579 (1991)

    Article  ADS  Google Scholar 

  18. T. Sauer, Phys. Rev. Lett. 93, 198701 (2004)

    Article  ADS  Google Scholar 

  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)

  20. H. Abarbanel, Analysis of observed chaotic data (Springer-Verlag, New York, 1996)

  21. H. Kantz, T. Schreiber, Nonlinear time series analysis (Cambridge University Press, 2004)

  22. J. Farmer, J. Sidorowich, Phys. Rev. Lett. 59, 845 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  23. M. Casdagli, Physica D: Nonlinear Phenom. 35, 335 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  24. G. Sugihara, R.M. May, Nature 344, 734 (1990)

    Article  ADS  Google Scholar 

  25. L.A. Smith, Physica D: Nonlinear Phenom. 58, 50 (1992)

    Article  ADS  Google Scholar 

  26. J. Jimenez, J. Moreno, G. Ruggeri, Phys. Rev. A 45, 3553 (1992)

    Article  ADS  Google Scholar 

  27. T. Sauer, in Time series prediction: forecasting the future and understanding the past (Addison-Wesley, 1994), pp. 175–193

  28. G. Sugihara, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 348, 477 (1994)

    Article  ADS  Google Scholar 

  29. C. G. Schroer, T. Sauer, E. Ott, J.A. Yorke, Phys. Rev. Lett. 80, 1410 (1998)

    Article  ADS  Google Scholar 

  30. D. Kugiumtzis, O. Lingjaerde, N. Christophersen, Physica D: Nonlinear Phenom. 112, 344 (1998)

    Article  ADS  Google Scholar 

  31. G. Yuan, M. Lozier, L. Pratt, C. Jones, K. Helfrich, J. Geophys. Res. 109, C08002 (2004)

    ADS  Google Scholar 

  32. C.-H. Hsieh, S.M. Glaser, A.J. Lucas, G. Sugihara, Nature 435, 336 (2005)

    Article  ADS  Google Scholar 

  33. C.C. Strelioff, A.W. Hubler, Phys. Rev. Lett. 96, 044101 (2006)

    Article  ADS  Google Scholar 

  34. S. Regonda, B. Rajagopalan, U. Lall, M. Clark, Y.-I. Moon, Nonlinear Proc. Geophys. 12, 397 (2005)

    Article  ADS  Google Scholar 

  35. B. Schelter, M. Winterhalder, J. Timmer, Handbook of time series analysis: recent theoretical developments and applications (John Wiley and Sons, 2006)

  36. F. Hamilton, T. Berry, T. Sauer, Phys. Rev. X 6, 011021 (2016)

    Google Scholar 

  37. T. Berry, D. Giannakis, J. Harlim, Phys. Rev. E 91, 032915 (2015)

    Article  ADS  Google Scholar 

  38. T. Berry, J. Harlim, Physica D 320, 57 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  39. D. Simon, Optimal state estimation: Kalman, H∞ and nonlinear approaches (John Wiley and Sons, 2006)

  40. T. Berry, T. Sauer, Tellus A 65, 20331 (2013)

    Article  Google Scholar 

  41. E. Lorenz, J. Atmos. Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  42. E.N. Lorenz, in Proceedings: Seminar on predictability (AMS, Reading, UK, 1996), pp. 1–18

  43. A. Sitz, U. Schwarz, J. Kurths, H. Voss, Phys. Rev. E 66, 16210 (2002)

    Article  ADS  Google Scholar 

  44. J. Stark, D.S. Broomhead, M.E. Davies, J. Huke, J. Nonlinear Sci. 13, 519 (2003)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Timothy Sauer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamilton, F., Berry, T. & Sauer, T. Kalman-Takens filtering in the presence of dynamical noise. Eur. Phys. J. Spec. Top. 226, 3239–3250 (2017). https://doi.org/10.1140/epjst/e2016-60363-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2016-60363-2

Navigation