A Jacobian approach for calculating the Lyapunov exponents of short time series using support vector regression

In order to characterize a system and to analyze the predictability of the time series under investigation, the detection of chaos and fractal behavior in experimental data is essential. In this work, support vector regression with two different kernel types namely, the linear kernel and sigmoid kernel, has been utilized for the calculation of the Lyapunov exponents of the given time series. The developed technique for the estimation of Lyapunov exponents has been validated with the help of time series generated from well known chaotic maps and also by comparing the Lyapunov exponents obtained using Rosenstein method. The results of this work reveal that the proposed technique is capable of producing accurate positive exponents for all the considered chaotic maps.

[1]  Michael R. Lyu,et al.  Local Support Vector Regression for Financial Time Series Prediction , 2006, The 2006 IEEE International Joint Conference on Neural Network Proceedings.

[2]  A. Gallant,et al.  Finding Chaos in Noisy Systems , 1992 .

[3]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[4]  Soo-Yong Kim,et al.  Nonlinear Dynamic Analysis of the EEG in Patients with Alzheimer’s Disease and Vascular Dementia , 2001, Journal of clinical neurophysiology : official publication of the American Electroencephalographic Society.

[5]  M. Ataei,et al.  Estimating the Lyapunov exponents of chaotic time series: A model based method , 2003, 2003 European Control Conference (ECC).

[6]  J. Murillo-Escobar,et al.  Forecasting concentrations of air pollutants using support vector regression improved with particle swarm optimization: Case study in Aburrá Valley, Colombia , 2019, Urban Climate.

[7]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[8]  F. Girosi,et al.  Nonlinear prediction of chaotic time series using support vector machines , 1997, Neural Networks for Signal Processing VII. Proceedings of the 1997 IEEE Signal Processing Society Workshop.

[9]  Alexander J. Smola,et al.  Support Vector Method for Function Approximation, Regression Estimation and Signal Processing , 1996, NIPS.

[10]  Wei Chen,et al.  Support vector regression with modified firefly algorithm for stock price forecasting , 2018, Applied Intelligence.

[11]  J. Sprott.  Chaos and time-series analysis , 2001 .

[12]  Kamalanand Krishnamurthy.  A Method for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise Using Extended Kalman Filter , 2012 .

[13]  Alvin Shrier,et al.  Chaos in neurobiology , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[14]  W. Ricker.  Stock and Recruitment , 1954 .

[15]  D. Basak,et al.  Support Vector Regression , 2008 .

[16]  Bernhard Schölkopf,et al.  Learning with kernels , 2001 .

[17]  Mototsugu Shintani,et al.  Nonparametric Neutral Network Estimation of Lyapunov Exponents and a Direct Test for Chaos , 2002 .

[18]  Bakiya Ambikapathy,et al.  Analysis of electromyograms recorded using invasive and noninvasive electrodes: a study based on entropy and Lyapunov exponents estimated using artificial neural networks , 2018 .

[19]  Dimitris Kugiumtzis,et al.  Chaotic time series. Part II. System Identification and Prediction , 1994, chao-dyn/9401003.

[20]  K. Kamalanand,et al.  Unscented Transformation for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise , 2013 .

[21]  Sawada,et al.  Measurement of the Lyapunov spectrum from a chaotic time series. , 1985, Physical review letters.

[22]  Lijuan Cao,et al.  Support vector machines experts for time series forecasting , 2003, Neurocomputing.

[23]  S. Gunn.  Support Vector Machines for Classification and Regression , 1998 .

[24]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[25]  V. Nourani,et al.  A hybrid support vector regression–firefly model for monthly rainfall forecasting , 2018, International Journal of Environmental Science and Technology.

[26]  M. Perc,et al.  Detecting chaos from a time series , 2005 .

[27]  Christopher Essex,et al.  Chaotic time series analyses of epileptic seizures , 1990 .

[28]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[29]  C. Beck,et al.  Thermodynamics of chaotic systems : an introduction , 1993 .

[30]  A. Gallant,et al.  Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression , 1992 .

[31]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[32]  Ping Chen,et al.  Empirical and theoretical evidence of economic chaos , 1988 .

[33]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[34]  R. Eykholt,et al.  Estimating the Lyapunov-exponent spectrum from short time series of low precision. , 1991, Physical review letters.

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

[36]  Andrew Nicola Edmonds.  Time series prediction using supervised learning and tools from chaos theory , 1996 .

[37]  F. Tay,et al.  Application of support vector machines in financial time series forecasting , 2001 .

[38]  R. Eckmiller,et al.  Predicting the nonlinear dynamics of biological neurons using support vector machines with different kernels , 2001, IJCNN'01. International Joint Conference on Neural Networks. Proceedings (Cat. No.01CH37222).