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Recursive prediction of chaotic time series

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Summary

Considerable progress has been made in recent years in the analysis of time series arising from chaotic systems. In particular, a variety of schemes for the short-term prediction of such time series has been developed. However, hitherto all such algorithms have used batch processing and have not been able to continuously update their estimate of the dynamics using new observations as they are made. This severely limits their usefulness in real time signal processing applications. In this paper we present a continuous update prediction scheme for chaotic time series that overcomes this difficulty. It is based on radial basis function approximation combined with a recursive least squares estimation algorithm. We test this scheme using simulated data and comment on its relationship to adaptive transversal filters, which are widely used in conventional signal processing.

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References

  • H. D. I. Abarbanel, R. Brown, and J. B. Kadtke, 1989, Prediction and System Identification in Chaotic Time Series with Broadband Fourier Spectra,Phys. Lett. A,138, 401–408.

    Article  Google Scholar 

  • H. D. I. Abarbanel, R. Brown, and J. B. Kadtke, 1990, Prediction in Chaotic Nonlinear Systems: Methods for Time Series with Broadband Fourier Spectra,Phys. Rev. A,41, 1782–1807.

    Article  MathSciNet  Google Scholar 

  • H. D. I. Abarbanel, R. Brown, and M. B. Kennel, 1991, Variation of Lyapunov Exponents on a Strange Attractor,J. Nonlin. Sci.,1, 175–199.

    Article  MATH  MathSciNet  Google Scholar 

  • S. T. Alexander, 1987,Adaptive Signal Processing, Theory and Applications, New York: Springer-Verlag.

    Google Scholar 

  • D. S. Broomhead and D. Lowe, 1988, Multivariable Functional Interpolation and Adaptive Networks,Complex Systems,2, 321–355.

    MATH  MathSciNet  Google Scholar 

  • J. R. Bunch and C. P. Nielsen, 1978, Updated the Singular Value Decomposition,Numerische Mathematik,31, 111–129.

    Article  MATH  MathSciNet  Google Scholar 

  • T. Buzug and G. Pfister, 1992, Optimal Delay Time and Embedding Dimension for Delay-Time Coordinates by Analysis of the Global Static and Local Dynamical Behaviour of Strange Attractors,Phys. Rev. A,45, 7073–7084.

    Article  Google Scholar 

  • M. Casdagli, 1989, Nonlinear Prediction of Chaotic Time Series,Physica D,35, 335–356.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Casdagli, 1992, Chaos and Deterministic versus Stochastic Non-linear Modelling,J. Roy. Stat. Soc. B,54, 303–328.

    MathSciNet  Google Scholar 

  • A. Čenys and K. Pyragas, 1988, Estimation of the Number of Degrees of Freedom from Chaotic Time Series,Phys. Lett. A,129, 227–230.

    Article  Google Scholar 

  • S. Chen, C. F. N. Cowan, and P. M. Grant, 1991, Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks,IEEE Trans. Neural Networks,2, 302–309.

    Article  Google Scholar 

  • J. P. Crutchfield and B. S. McNamara, 1987, Equations of Motion from a Data Series,Complex Systems,1, 417–452.

    MATH  MathSciNet  Google Scholar 

  • J.-P. Eckmann and D. Ruelle, 1985, Ergodic Theory of Chaos and Strange Attractors,Rev. Mod. Phys.,57, 617–656.

    Article  MATH  MathSciNet  Google Scholar 

  • J. D. Farmer and J. J. Sidorowich, 1987, Predicting Chaotic Time Series,Phys. Rev. Lett.,59, 845–848.

    Article  MathSciNet  Google Scholar 

  • J. D. Farmer and J. J. Sidorowich, 1991, Optimal Shadowing and Noise Reduction,Physica D,47, 373–392.

    Article  MATH  MathSciNet  Google Scholar 

  • A. M. Fraser and H. L. Swinney, 1986, Independent Coordinates for Strange Attractors from Mutual Information,Phys. Rev. A,33, 1134–1140.

    Article  MathSciNet  Google Scholar 

  • M. Giona, F. Lentini, and V. Cimagalli, 1991, Functional Reconstruction and Local Prediction of Chaotic Time Series,Phys. Rev. A,44, 3496–3502.

    Article  MathSciNet  Google Scholar 

  • P. Grassberger, T. Schreiber, and C. Schaffrath, 1992, Non-linear Time Sequence Analysis,Int. J. Bifurcation Chaos,1, 521–547.

    MathSciNet  Google Scholar 

  • S. M. Hammel, 1990, A Noise Reduction Method for Chaotic Systems,Phys. Lett. A,148, 421–428.

    Article  MathSciNet  Google Scholar 

  • J. Jiménez, J. A. Moreno, and G. J. Ruggeri, 1992, Forecasting on Chaotic Time Series: A Local Optimal Linear-Reconstruction Method,Phys. Rev. A,45, 3553–3558.

    Article  Google Scholar 

  • M. B. Kennel, R. Brown, and H. D. I. Abarbanel, 1992, Determining Embedding Dimension for Phase-Space Reconstruction Using a Geometrical Construction,Phys. Rev. A,45, 3403–3411.

    Article  Google Scholar 

  • E. J. Kostelich and J. A. Yorke, 1988, Noise Reduction in Dynamical Systems,Phys. Rev. A,38, 1649–1652.

    Article  MathSciNet  Google Scholar 

  • A. S. Lapedes and R. Farber, 1987, Nonlinear Signal Processing Using Neural Networks: Prediction and System Modelling, Technical Report, LA-UR-87-2662, Los Alamos National Laboratory.

  • C. L. Lawson and R. J. Hanson, 1974,Solving Least Squares Problems, Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • R. Lewin, 1992, Making Maths Make Money,New Scientist, 134, 11th May 1992 (No. 1816), 31–34.

  • W. Liebert and H. G. Schuster, 1989, Proper Choice of the Time Delay for the Analysis of Chaotic Time Series,Phys. Lett. A,142, 107–111.

    Article  MathSciNet  Google Scholar 

  • W. Liebert, K. Pawelzik, and H. G. Schuster, 1991, Optimal Embeddings of Chaotic Attractors from Topological Considerations,Europhysics Lett.,14, 521–526.

    MathSciNet  Google Scholar 

  • F. Ling, D. Manolakis, and J. G. Proakis, 1986, A Recursive Modified Gram-Schmidt Algorithm for Least-Squares Estimation,IEEE Trans. ASSP,34, 829–835.

    Article  Google Scholar 

  • P. S. Linsay, 1991, An Efficient Method of Forecasting Chaotic Time Series Using Linear Interpolation,Phys. Lett. A,153, 353–356.

    Article  Google Scholar 

  • P. F. Marteau and H. D. I. Abarbanel, 1991, Noise Reduction in Chaotic Time Series Using Scaled Probabilistic Methods,J. Nonlin. Sci.,1, 313–343.

    Article  MATH  MathSciNet  Google Scholar 

  • R. M. May, 1992, Discussion on the Meeting on Chaos,J. R. Statis. Soc. B,54, 451–452.

    Google Scholar 

  • L. Noakes, 1991, The Takens Embedding Theorem,Int. J. Bifurcation Chaos,1, 867–872.

    Article  MATH  MathSciNet  Google Scholar 

  • K. Pawelzik and H. G. Schuster, 1991, Unstable Periodic Orbits and Prediction,Phys. Rev. A,43, 1808–1812.

    Article  Google Scholar 

  • M. J. D. Powell, 1987, Radial Basis Functions for Multivariable Interpolation; A Review, inAlgorithms for Approximation, ed. J. C. Moxon, and M. G. Cox, Oxford, UK: Clarendon Press, pp. 143–147.

    Google Scholar 

  • W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, 1988,Numerical Recipes in C, The Art of Scientific Computing, Cambridge, UK: Cambridge University Press.

    MATH  Google Scholar 

  • T. Sauer, J. A. Yorke, and M. Casdagli, 1991, Embedology,J. Stat. Phys.,65, 579–616.

    Article  MATH  MathSciNet  Google Scholar 

  • T. Schreiber and P. Grassberger, 1991, A Simple Noise-Reduction Method for Real Data,Phys. Lett. A,160, 411–418.

    Article  MathSciNet  Google Scholar 

  • L. A. Smith, 1992, Identification and Prediction of Low Dimensional Dynamics, to appear inPhysica D.

  • J. Stark and B. Arumugam, 1992, Extracting Slowly Varying Signals from a Chaotic Background, to appear inInt. J. Bifurcation Chaos.

  • J. Stoer and R. Bulirsch, 1980,An Introduction to Numerical Analysis, New York: Springer-Verlag.

    Google Scholar 

  • G. Sugihara and R. M. May, 1990, Nonlinear Forecasting as a Way of Distinguishing Chaos from Measurement Error in Time Series,Nature,344, 734–741.

    Article  Google Scholar 

  • F. Takens, 1980, Detecting Strange Attractors in Turbulence, inDynamical Systems and Turbulence, Warwick, 1980, ed. D. A. Rand, and L.-S. Young, Lecture Notes in Mathematics, 898, New York: Springer-Verlag, pp. 366–381.

    Google Scholar 

  • W. W. Taylor, 1991, Quantifying Predictability for Applications in Signal Separation, to appear inSPIE Proceedings,1565.

  • A. A. Tsonis and J. B. Elsner, 1992, Nonlinear Prediction as a Way of Distinguishing Chaos from Random Fractal Sequences,Nature,358, 217–220.

    Article  Google Scholar 

  • R. C. L. Wolff, 1992, Local Lyapunov Exponents: Looking Closely at Chaos,J. R. Stat. Soc. B,54, 353–371.

    MathSciNet  Google Scholar 

  • P. Young, 1984,Recursive Estimation and Time-Series Analysis, An Introduction, New York: Springer-Verlag.

    MATH  Google Scholar 

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Author notes

  1. J. Stark

    Present address: Centre for Nonlinear Dynamics and Its Applications, University College London, Gower Street, WC1E 6BT, London, United Kingdom

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  1. Long Range Research Laboratory, GEC Hirst Research Centre, East Lane, HA9 7PP, Wembley, United Kingdom

    J. Stark

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  1. J. Stark
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Communicated by Stephen Wiggins

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Stark, J. Recursive prediction of chaotic time series. J Nonlinear Sci 3, 197–223 (1993). https://doi.org/10.1007/BF02429864

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  • DOI: https://doi.org/10.1007/BF02429864

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