Kalman-Takens filtering in the presence of dynamical noise
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.
Unable to display preview. Download preview PDF.
- 2.E. Kalnay, Atmospheric modeling, data assimilation, and predictability (Cambridge Univ. Press, 2003) Google Scholar
- 3.G. Evensen, Data assimilation: the ensemble Kalman filter (Springer, Heidelberg, 2009) Google Scholar
- 6.S. Schiff, Neural control engineering (MIT Press, 2012) Google Scholar
- 15.F. Takens, in Lecture Notes in Math (Springer-Verlag, Berlin, 1981), Vol. 898 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) Google Scholar
- 20.H. Abarbanel, Analysis of observed chaotic data (Springer-Verlag, New York, 1996) Google Scholar
- 21.H. Kantz, T. Schreiber, Nonlinear time series analysis (Cambridge University Press, 2004) Google Scholar
- 27.T. Sauer, in Time series prediction: forecasting the future and understanding the past (Addison-Wesley, 1994), pp. 175–193 Google Scholar
- 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.F. Hamilton, T. Berry, T. Sauer, Phys. Rev. X 6, 011021 (2016) Google Scholar
- 39.D. Simon, Optimal state estimation: Kalman, H∞ and nonlinear approaches (John Wiley and Sons, 2006) Google Scholar
- 42.E.N. Lorenz, in Proceedings: Seminar on predictability (AMS, Reading, UK, 1996), pp. 1–18 Google Scholar