Modelling of Chaotic Systems with Novel Weighted Recurrent Least Squares Support Vector Machines

This paper discusses the use of Support Vector Machines(SVM) for dynamic modelling of the chaotic time series. Based on Recurrent Least Squares Support Vector Machines (RLS-SVM), a weighted term is introduced to the cost function to compensate the prediction errors resulting from the positive global Lyapunov exponent in context of the chaotic time series. For demonstrating the effectiveness of our algorithm, the dynamic invariants involves the Lyapunov exponent and the correlation dimension are used for criterions. Finally we apply our method to Santa Fe competition time series. The simulation results shows that the proposed method can capture the dynamics of the chaotic time series effectively.

[1]  A. Lapedes,et al.  Nonlinear Signal Processing Using Neural Networks , 1987 .

[2]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[3]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[4]  Ramazan Gençay,et al.  A statistical framework for testing chaotic dynamics via Lyapunov exponents , 1996 .

[5]  J. Suykens,et al.  Recurrent least squares support vector machines , 2000 .

[6]  L. Cao.  Practical method for determining the minimum embedding dimension of a scalar time series , 1997 .

[7]  A. N. Sharkovskiĭ.  Dynamic systems and turbulence , 1989 .

[8]  Li Ke-Ping,et al.  Phase Space Prediction Model Based on the Chaotic Attractor , 2002 .

[9]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[10]  Jon Juel Thomsen,et al.  Vibrations and Stability , 2003 .

[11]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[12]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[13]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[14]  Johan A. K. Suykens,et al.  Least Squares Support Vector Machines , 2002 .

[15]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[16]  Andreas S. Weigend,et al.  Time Series Prediction: Forecasting the Future and Understanding the Past , 1994 .