Identification of the chaotic behavior of Lorenz equations using vector support machines
Introduction: The article is derived from the research Characterization of complex signals with Computational Intelligence techniques that has been ongoing since 2016 within the investigation Group Modelación en Ingeniería de Sistemas MIS of the Universidad Distrital Francisco José de Caldas.
Objective: Determine the chaotic behavior in equations of Lorenz by using data directly taken from the time domain without prior processing, in order to classify a chaotic system. Methodology: Firstly, through simulation training data is acquired; later, it is used validation data to observe the system response. Finally, it is presented a discussion together with a set of conclusions regarding the data obtained.
Results: In most of the implemented vector support machines a positive classification is prevailing.
Conclusion: The data set used for the classification of the chaotic behavior in Lorenz equations was achieved implementing vector support machines, so they may be an alternative to obtaining behavior classification where data are directly taken from the time domain with none prior processing.
Originality: This paper is expected to serve to further developments as in diagnosis of patients using biological signals. This work is aimed to the observation of the characteristics manifested in the vector support machines to chaotic system classification.
Limitations: None preliminary data processing is performed wherefore such classification is subjected by the values obtained directly from the simulation.
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A. Stefanski, T. Kapitaniak y J. Brindley, “Dynamics of coupled Lorenz systems and its geophysical implications”, Physica D: Nonlinear Phenomena, vol. 98, no. 2-4, pp. 594-598, 1996, doi: https://doi.org/10.1016/0167-2789(96)00106-6
Y. Zhang, J. Yang, K. Wang, Y. Wang y Z. Wang, “Improved wind prediction based on the Lorenz system”, Renewable Energy, vol. 81, pp. 219-226, 2015, doi: https://doi.org/10.1016/j.renene.2015.03.039
S. Strogatz, Nonlinear dynamics and chaos with applications to physics, biology, chemistry and engineering, Cambridge, Perseus Books, 1994.
P. Vadasz, “Analytical prediction of the transition to chaos in Lorenz equations”, Applied Mathematics Letters, vol. 23, no. 5, pp. 503-507, 2010, doi: https://doi.org/10.1016/j.aml.2009.12.012
C. Li, J.C. Sprott, W. Thio, “Linearization of the Lorenz system”, Physics Letters A, vol. 379, pp. 888-893, 2015, doi: https://doi.org/10.1016/j.physleta.2015.01.003
B. Munmuangsaen, B. Srisuchinwong, “A hidden chaotic attractor in the classical Lorenz system”, Chaos, Solitons and Fractals, vol. 107, pp. 61-66, 2018, doi: https://doi.org/10.1016/j.chaos.2017.12.017
M.A. Hearst, S.T. Dumais, E. Osuna, J. Platt y B. Scholkopf, “Support vector machines”, IEEE Intelligent Systems and their Applications, vol. 13, no. 4, 1998, doi: https://doi.org/10.1109/5254.708428
D. Tomar y S. Agarwal, “Twin Support Vector Machine: A review from 2007 to 2014”, Egyptian Informatics Journal, vol. 16, pp. 55-69, 2015, doi: https://doi.org/10.1016/j.eij.2014.12.003
R. Khemchandania, K. Goyal y S. Chandrab, “TWSVR: Regression via Twin Support Vector Machine”, Neural Networks, vol. 74 pp. 14-21, 2016, doi: https://doi.org/10.1016/j.neunet.2015.10.007
J. Tang y Y. Tian, “A multi-kernel framework with nonparallel support vector machine”, Neurocomputing, vol. 266, pp. 226-238, 2017, doi: https://doi.org/10.1016/j.neucom.2017.05.036
S.V. Dudul, “Prediction of a Lorenz chaotic attractor using two-layer perceptron neural network”, Applied Soft Computing, vol 5, no. 4, pp. 333-355, 2005, doi: https://doi.org/10.1016/j.asoc.2004.07.005
U. Köse y A. Arslan, “An Adaptive Neuro-Fuzzy Inference System-Based Approach to Forecast Time Series of Chaotic Systems”, Chaos, Complexity and Leadership, pp. 17-22, 2012, doi: https://doi.org/10.1007/978-94-007-7362-2_3
L. Morales, H. Espitia y J. Soriano, “Propuesta de un sistema NEURO-DBR y su aplicación en la predicción de la serie de tiempo de lorenz”, Ciencia e Ingeniería Neogranadina, vol. 20, no. 2, pp. 31-51, 2010, doi: https://doi.org/10.18359/rcin.275
L.A. Dmitrieva, Y.A. Kuperin, N.M. Smetanin y G.A. Chernykh, “Method of calculating Lyapunov exponents for time series using artificial neural networks committees”, Proceedings of the International Conference DAYS on DIFFRACTION, pp. 127-132, 2016, doi: https://doi.org/10.1109/DD.2016.7756827
L. Zhang, “Artificial Neural Network Model Design and Topology Analysis for FPGA Implementation of Lorenz Chaotic Generator”, IEEE 30th Canadian Conference on Electrical and Computer Engineering (CCECE), pp. 1-4, 2017, doi: https://doi.org/10.1109/CCECE.2017.7946635
J.S.H. Tsai, J.S. Fang, J.J. Yan, M.C. Dai, S.M. Guo y L.S. Shieh, “Hybrid robust discrete sliding mode control for generalized continuous chaotic systems subject to external disturbances”, Nonlinear Analysis: Hybrid Systems, vol. 29, pp. 74-84, 2018, doi: https://doi.org/10.1016/j.nahs.2018.01.001
H. Wendt, Support Vector Machines for Regression Estimation and their Application to Chaotic Time Series Prediction, Thesis, Vienna University of Technology Faculty of Electrical Engineering and Information Technology, 2005, doi: https://pdfs.semanticscholar.org/7aa7/9531f16af48bd5132bc5b174a8502d6b15d9.pdf
F. Tay y L. Cao, “Modified support vector machines in financial time series forecasting”, Neurocomputing, vol. 48, no. 1-4, pp. 847-861, 2002, doi: https://doi.org/10.1016/S0925-2312(01)00676-2
L. Ralaivola y F. d’Alche Buc, “Dynamical modeling with kernels for nonlinear time series prediction”, Proceedings of the 16th International Conference on Neural Information Processing Systems, NIPS’03, pp. 129-136, 2003, doi: http://dl.acm.org/citation.cfm?id=2981345.2981362
J. Gómez y J. Martínez, “Support Vector Machines”, Springer Computational Intelligence, pp. 147-191, 2007, doi: https://doi.org/10.1007/0-387-37452-3_7
I. Steinwart y A. Christmann, Support Vector Machines, Information Science and Statistics, Springer Science & Business Media, 2008, doi: https://doi.org/10.1007/978-0-387-77242-4
B. Scholkopf y A.J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press Cambridge, MA, USA, 2001, doi: https://mitpress.mit.edu/books/learning-kernels
G. Betancourt, “Las máquinas de soporte vectorial (SVMs)”, Scientia et Technica, vol. 1, no. 27, 2005, doi: http://revistas.utp.edu.co/index.php/revistaciencia/article/view/6895
C.C. Chang y C.J. Lin, “LIBSVM: a library for support vector machines”, ACM Transactions on Intelligent Systems and Technology, vol. 2, no. 27, pp. 1-27, abr. 2011, doi: https://dl.acm.org/citation.cfm?doid=1961189.1961199
D. Meyer, “Support Vector Machines”, R-News, vol. 1, no. 3, pp. 23-26, September 2001, doi: https://cran.r-project.org/doc/Rnews/Rnews_2001-3.pdf
