Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2014, Article ID 193758, 6 pages
http://dx.doi.org/10.1155/2014/193758
Research Article

Levenberg-Marquardt Algorithm for Mackey-Glass Chaotic Time Series Prediction

1School of Mathematics, Liaocheng University, Liaocheng 252059, China
2School of Science, Huzhou University, Huzhou 313000, China
3School of Automation, Southeast University, Nanjing 210096, China

Received 9 August 2014; Accepted 11 October 2014; Published 11 November 2014

Academic Editor: Rongni Yang

Copyright © 2014 Junsheng Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977. View at Publisher · View at Google Scholar · View at Scopus
  2. L. Glass and M. C. Mackey, “Mackey-Glass equation,” Scholarpedia, vol. 5, no. 3, p. 6908, 2010. View at Google Scholar
  3. J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. View at MathSciNet
  4. J. Mallet-Paret and R. D. Nussbaum, “A differential-delay equation arising in optics and physiology,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 249–292, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. O. Walther, “The 2-dimensional attractor of dx/dt=μxt+fxt1,” Memoirs of the American Mathematical Society, vol. 113, no. 544, p. 76, 1995. View at Google Scholar
  6. J. Mallet-Paret and G. R. Sell, “The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay,” Journal of Differential Equations, vol. 125, no. 2, pp. 441–489, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. B. Lani-Wayda and H.-O. Walther, “Chaotic motion generated by delayed negative feedback, part II: construction of nonlinearities,” Mathematische Nachrichten, vol. 180, pp. 181–211, 2000. View at Google Scholar
  8. G. Röst and J. H. Wu, “Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback,” Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, vol. 463, no. 2086, pp. 2655–2669, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. M. Awad, H. Pomares, I. Rojas, O. Salameh, and M. Hamdon, “Prediction of time series using RBF neural networks: a new approach of clustering,” International Arab Journal of Information Technology, vol. 6, no. 2, pp. 138–143, 2009. View at Google Scholar · View at Scopus
  10. M. Awad, “Chaotic time series prediction using wavelet neural network,” Journal of Artificial Intelligence: Theory and Application, vol. 1, no. 3, pp. 73–80, 2010. View at Google Scholar
  11. I. López-Yáñez, L. Sheremetov, and C. Yáñez-Márquez, “A novel associative model for time series data mining,” Pattern Recognition Letters, vol. 41, pp. 23–33, 2014. View at Publisher · View at Google Scholar
  12. A. C. Fowler, “Respiratory control and the onset of periodic breathing,” Mathematical Modelling of Natural Phenomena, vol. 9, no. 1, pp. 39–57, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. N. Wang, M. J. Er, and M. Han, “Generalized single-hidden layer feedforward networks for regression problems,” IEEE Transactions on Neural Networks and Learning Systems, 2014. View at Publisher · View at Google Scholar
  14. N. Wang, “A generalized ellipsoidal basis function based online self-constructing fuzzy neural network,” Neural Processing Letters, vol. 34, no. 1, pp. 13–37, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. H. K. Wei, Theory and Method of the Neural Networks Architecture Design, National Defence Industry Press, Beijing, China, 2005.
  16. H. Wei and S.-I. Amari, “Dynamics of learning near singularities in radial basis function networks,” Neural Networks, vol. 21, no. 7, pp. 989–1005, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. H. Wei, J. Zhang, F. Cousseau, T. Ozeki, and S.-I. Amari, “Dynamics of learning near singularities in layered networks,” Neural Computation, vol. 20, no. 3, pp. 813–843, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Quarterly of Applied Mathematics, vol. 2, pp. 164–168, 1944. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 989–993, 1994. View at Publisher · View at Google Scholar · View at Scopus
  20. W. Guo, H. Wei, J. Zhao, and K. Zhang, “Averaged learning equations of error-function-based multilayer perceptrons,” Neural Computing and Applications, vol. 25, no. 3-4, pp. 825–832, 2014. View at Publisher · View at Google Scholar · View at Scopus
  21. P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM Journal on Numerical Analysis, vol. 15, no. 5, pp. 977–992, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM Journal on Applied Mathematics, vol. 11, no. 2, pp. 431–441, 1963. View at Google Scholar · View at MathSciNet