Unscented Transformation for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise
Many systems in the natural world exhibit chaos or non-linear behavior, the complexity of which is so great that they appear to be random. Identification of chaos in experimental data is essential for characterizing the system and for analyzing the predictability of the data under analysis. The Lyapunov exponents provide a quantitative measure of the sensitivity to initial conditions and are the most useful dynamical diagnostic for chaotic systems. However, it is difficult to accurately estimate the Lyapunov exponents of chaotic signals which are corrupted by a random noise. In this work, a method for estimation of Lyapunov exponents from noisy time series using unscented transformation is proposed. The proposed methodology was validated using time series obtained from known chaotic maps. In this paper, the objective of the work, the proposed methodology and validation results are discussed in detail.
<p>[1] N. L. Gerr, and J. C. Allen, “Stochastic versions of chaotic time series:
generalized logistic and Henon time series models,” Physics D, vol. 68,
1993, pp. 232-249
[2] E. Takens, “Detecting nonlinearities in stationary time series,”
International Journal of Bifurcation and Chaos, vol. 3, 1993, pp. 241-
256
[3] D. F. McCaffrey, S. Ellner, A. R. Gallant and D. W. Nychka,
“Estimating the Lyapunov exponent of a chaotic system with
nonparametric regression,” Journal of the Amer. Sta.l Assoc., vol. 87,
1992, pp. 682-695
[4] X. Zeng, R. Eykholt and R. A. Pielke, “Estimating the lyapunovexponent
spectrum from short time series of low precision,” Phys. Rev.
Let., vol. 66, 1991
[5] S. Mototsugu and L. Oliver, “Nonparametric neural network estimation
of Lyapunov exponents and a direct test for chaos,” Journal of
Econometrics, vol. 120, 2004, pp. 1–33
[6] A. N. Edmonds, “Time series prediction using supervised learning and
tools from chaos theory,” Faculty of Science and Computing, University
of Luton, Ph.D Thesis, 1996
[7] M. Ataei, A. Khaki-Sedigh, B. Lohmann and C. Lucas, “Estimating the
lyapunov exponents of chaotic time series: a model based method,”
Proceedings of European Control Conference, 2003
[8] H. Kantz, “A robust method to estimate the maximum Lyapunov
exponent of a time series,” Phys. Lett. A, vol. 185, 1994, pp. 77-87
[9] M. Sano and Y. Sawada, “Measurement of the Lyapunov spectrum from
a chaotic time series,” Phys. Rev. Let., vol. 55, 1985, pp. 1082-1085
[10] M. T. Rosenstein, J. J. Collins and C. J. De Luca, “A practical method
for calculating largest Lyapunov exponents from small data sets,”
Physica D, vol. 65, 1993, pp. 117-134
[11] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, “Determining
Lyapunov exponents from a time series,” Physica D, vol. 16, 1985, pp.
285-317
[12] J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange
attractors,” Reviews of Modern Physics, vol. 57, 1985, pp. 617–656
[13] J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, “Liapunov
exponents from time series,” Physical Review A, vol. 34, 1986, pp.
4971–4979
[14] D. Nychka, S. Ellner, A. R. Gallant and D. McCaHrey, “Finding chaos
in noisy system,” Journal of the Royal Statistical Society, Series B, vol.
54, 1992, pp. 399–426
[15] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear
estimation,” Proceedings of the IEEE, vol. 92, 2004, pp. 401-422
[16] Wan, E. and R. Van der Merwe, “The unscented Kalman filter,” Wiley
Publishing, 2001
[17] Li Hongli, J. Wang, Y. Che, H. Wang and Y. Chen, “On neural network
training algorithm based on the unscented Kalman filter,” Proceedings of
29th Chinese Control Conference, 2010, pp. 1447-1450
[18] J. C. Sprott, “Chaos and time series analysis,” oxford university press,
2003
[19] R. L. Devaney, “An introduction to chaotic dynamical systems,” 2nd
edition, Addison Wesley, Redwood City, CA, 1989
[20] S. H. Strogatz, “Nonlinear dynamics and chaos,” Perseus publishing,
1994
[21] W. E. Ricker, “Stock and recruitment,” J. Fisheries Res. Board Can.,
vol. 11, 1954, pp. 559-623
[22] G. W. Frank, T. Lookman, M. A. H. Nerenberg, C. Essex, J. Lemieux
and W. Blume, “Chaotic time series analysis of epileptic seizures,”
Physica D, vol. 46, 1990, pp. 427
[23] P. Chen, “Empirical and theoretical evidence of economic chaos,” Sys.
Dyn. Rev., vol. 4, 1988, pp. 81</p>
<p>[1] N. L. Gerr, and J. C. Allen, “Stochastic versions of chaotic time series:
generalized logistic and Henon time series models,” Physics D, vol. 68,
1993, pp. 232-249
[2] E. Takens, “Detecting nonlinearities in stationary time series,”
International Journal of Bifurcation and Chaos, vol. 3, 1993, pp. 241-
256
[3] D. F. McCaffrey, S. Ellner, A. R. Gallant and D. W. Nychka,
“Estimating the Lyapunov exponent of a chaotic system with
nonparametric regression,” Journal of the Amer. Sta.l Assoc., vol. 87,
1992, pp. 682-695
[4] X. Zeng, R. Eykholt and R. A. Pielke, “Estimating the lyapunovexponent
spectrum from short time series of low precision,” Phys. Rev.
Let., vol. 66, 1991
[5] S. Mototsugu and L. Oliver, “Nonparametric neural network estimation
of Lyapunov exponents and a direct test for chaos,” Journal of
Econometrics, vol. 120, 2004, pp. 1–33
[6] A. N. Edmonds, “Time series prediction using supervised learning and
tools from chaos theory,” Faculty of Science and Computing, University
of Luton, Ph.D Thesis, 1996
[7] M. Ataei, A. Khaki-Sedigh, B. Lohmann and C. Lucas, “Estimating the
lyapunov exponents of chaotic time series: a model based method,”
Proceedings of European Control Conference, 2003
[8] H. Kantz, “A robust method to estimate the maximum Lyapunov
exponent of a time series,” Phys. Lett. A, vol. 185, 1994, pp. 77-87
[9] M. Sano and Y. Sawada, “Measurement of the Lyapunov spectrum from
a chaotic time series,” Phys. Rev. Let., vol. 55, 1985, pp. 1082-1085
[10] M. T. Rosenstein, J. J. Collins and C. J. De Luca, “A practical method
for calculating largest Lyapunov exponents from small data sets,”
Physica D, vol. 65, 1993, pp. 117-134
[11] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, “Determining
Lyapunov exponents from a time series,” Physica D, vol. 16, 1985, pp.
285-317
[12] J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange
attractors,” Reviews of Modern Physics, vol. 57, 1985, pp. 617–656
[13] J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, “Liapunov
exponents from time series,” Physical Review A, vol. 34, 1986, pp.
4971–4979
[14] D. Nychka, S. Ellner, A. R. Gallant and D. McCaHrey, “Finding chaos
in noisy system,” Journal of the Royal Statistical Society, Series B, vol.
54, 1992, pp. 399–426
[15] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear
estimation,” Proceedings of the IEEE, vol. 92, 2004, pp. 401-422
[16] Wan, E. and R. Van der Merwe, “The unscented Kalman filter,” Wiley
Publishing, 2001
[17] Li Hongli, J. Wang, Y. Che, H. Wang and Y. Chen, “On neural network
training algorithm based on the unscented Kalman filter,” Proceedings of
29th Chinese Control Conference, 2010, pp. 1447-1450
[18] J. C. Sprott, “Chaos and time series analysis,” oxford university press,
2003
[19] R. L. Devaney, “An introduction to chaotic dynamical systems,” 2nd
edition, Addison Wesley, Redwood City, CA, 1989
[20] S. H. Strogatz, “Nonlinear dynamics and chaos,” Perseus publishing,
1994
[21] W. E. Ricker, “Stock and recruitment,” J. Fisheries Res. Board Can.,
vol. 11, 1954, pp. 559-623
[22] G. W. Frank, T. Lookman, M. A. H. Nerenberg, C. Essex, J. Lemieux
and W. Blume, “Chaotic time series analysis of epileptic seizures,”
Physica D, vol. 46, 1990, pp. 427
[23] P. Chen, “Empirical and theoretical evidence of economic chaos,” Sys.
Dyn. Rev., vol. 4, 1988, pp. 81</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58362", author = "K. Kamalanand and P. Mannar Jawahar", title = "Unscented Transformation for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise", abstract = "Many systems in the natural world exhibit chaos or non-linear behavior, the complexity of which is so great that they appear to be random. Identification of chaos in experimental data is essential for characterizing the system and for analyzing the predictability of the data under analysis. The Lyapunov exponents provide a quantitative measure of the sensitivity to initial conditions and are the most useful dynamical diagnostic for chaotic systems. However, it is difficult to accurately estimate the Lyapunov exponents of chaotic signals which are corrupted by a random noise. In this work, a method for estimation of Lyapunov exponents from noisy time series using unscented transformation is proposed. The proposed methodology was validated using time series obtained from known chaotic maps. In this paper, the objective of the work, the proposed methodology and validation results are discussed in detail.
", keywords = "Lyapunov exponents, unscented transformation,
chaos theory, neural networks.", volume = "7", number = "4", pages = "610-6", }
