Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 983051, 9 pages.

Incomplete Phase Space Reconstruction Method Based on Subspace Adaptive Evolution Approximation

Tai-fu Li, Wei Jia, Wei Zhou, Ji-ke Ge, Yu-cheng Liu, and Li-zhong Yao

Full-text: Open access

Abstract

The chaotic time series can be expanded to the multidimensional space by phase space reconstruction, in order to reconstruct the dynamic characteristics of the original system. It is difficult to obtain complete phase space for chaotic time series, as a result of the inconsistency of phase space reconstruction. This paper presents an idea of subspace approximation. The chaotic time series prediction based on the phase space reconstruction can be considered as the subspace approximation problem in different neighborhood at different time. The common static neural network approximation is suitable for a trained neighborhood, but it cannot ensure its generalization performance in other untrained neighborhood. The subspace approximation of neural network based on the nonlinear extended Kalman filtering (EKF) is a dynamic evolution approximation from one neighborhood to another. Therefore, in view of incomplete phase space, due to the chaos phase space reconstruction, we put forward subspace adaptive evolution approximation method based on nonlinear Kalman filtering. This method is verified by multiple sets of wind speed prediction experiments in Wulong city, and the results demonstrate that it possesses higher chaotic prediction accuracy.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 983051, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394806121

Digital Object Identifier
doi:10.1155/2013/983051

Mathematical Reviews number (MathSciNet)
MR3115293

Zentralblatt MATH identifier
06950974

Citation

Li, Tai-fu; Jia, Wei; Zhou, Wei; Ge, Ji-ke; Liu, Yu-cheng; Yao, Li-zhong. Incomplete Phase Space Reconstruction Method Based on Subspace Adaptive Evolution Approximation. J. Appl. Math. 2013, Special Issue (2013), Article ID 983051, 9 pages. doi:10.1155/2013/983051. https://projecteuclid.org/euclid.jam/1394806121


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References

  • G. Kaddoum, F. Gagnon, P. Chargé, and D. Roviras, “A generalized BER prediction method for differential chaos shift keying system through different communication channels,” Wireless Personal Communications, vol. 64, no. 2, pp. 425–437, 2012.
  • B. Sivakumar, “Chaos theory in geophysics: past, present and future,” Chaos, Solitons & Fractals, vol. 19, no. 2, pp. 441–462, 2004.
  • P. A. Mastorocostas, J. B. Theocharis, and A. G. Bakirtzis, “Fuzzy modeling for short term load forecasting using the orthogonal least squares method,” IEEE Transactions on Power Systems, vol. 14, no. 1, pp. 29–36, 1999.
  • H. Y. Yang, H. Ye, G. Wang, J. Khan, and T. Hu, “Fuzzy neural very-short-term load forecasting based on chaotic dynamics reconstruction,” Chaos, Solitons & Fractals, vol. 29, no. 2, pp. 462–469, 2006.
  • B. Sivakumar, “A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers,” Journal of Hydrology, vol. 258, no. 1–4, pp. 149–162, 2002.
  • A. Porporato and L. Ridolfi, “Nonlinear analysis of river flow time sequences,” Water Resources Research, vol. 33, no. 6, pp. 1353–1367, 1997.
  • P. Zhao, L. Xing, and J. Yu, “Chaotic time series prediction: from one to another,” Physics Letters A, vol. 373, no. 25, pp. 2174–2177, 2009.
  • H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Springer, New York, NY, USA, 1996.
  • M. B. Kennel, R. Brown, and H. D. I. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Physical Review A, vol. 45, no. 6, pp. 3403–3411, 1992.
  • H. Ma and C. Han, “Selection of embedding dimension and delay time in phase space reconstruction,” Frontiers of Electrical and Electronic Engineering in China, vol. 1, no. 1, pp. 111–114, 2006.
  • B. Chen, G. Liu, J. Tang et al., “Research on chaotic sequence autocorrelation by phase space method,” Journal of the University of Electronic Science and Technology of China, vol. 39, no. 6, pp. 859–863, 2010.
  • A. Jiang, X. Huang, Z. Zhang, J. Li, Z.-Y. Zhang, and H.-X. Hua, “Mutual information algorithms,” Mechanical Systems and Signal Processing, vol. 24, no. 8, pp. 2947–2960, 2010.
  • N. Abu-Shikhah and F. Elkarmi, “Medium-term electric load forecasting using singular value decomposition,” Energy, vol. 36, no. 7, pp. 4259–4271, 2011.
  • C. Gao and X. Liu, “Chaotic identification of BF ironmaking process I: the calculation of saturated correlative dimension,” Acta Metallurgica Sinica, vol. 40, no. 4, pp. 347–350, 2004.
  • I. M. Carrión and E. A. Antúnez, “Thread-based implementations of the false nearest neighbors method,” Parallel Computing, vol. 35, no. 10-11, pp. 523–534, 2009.
  • J. Hite Jr., Learning in Chaos, Gulf Professional Publishing, Amsterdam, The Netherlands, 1999.
  • Y. Hu and T. Chen, “Phase-space reconstruction technology of chaotic attractor based on C-C method,” Journal of Electronic Measurement and Instrument, vol. 35, no. 10-11, pp. 425–430, 2012.
  • D. A. Fadare, “The application of artificial neural networks to mapping of wind speed profile for energy application in Nigeria,” Applied Energy, vol. 87, no. 3, pp. 934–942, 2010.
  • S. Salcedo-Sanz, A. M. Ángel M. Pérez-Bellido, E. G. Ortiz-García, A. Portilla-Figueras, L. Prieto, and D. Paredes, “Hybridizing the fifth generation mesoscale model with artificial neural networks for short-term wind speed prediction,” Renewable Energy, vol. 34, no. 6, pp. 1451–1457, 2009.
  • D. C. Dracopoulos, Evolutionary Learning Algorithms for Neural Adaptative Control, Springer, London, UK, 1997.
  • J. Xue and Z. Shi, “Short-time traffic flow prediction based on chaos time series theory,” Journal of Transportation Systems Engineering and Information Technology, vol. 8, no. 5, pp. 68–72, 2008.
  • H. Yang, J. Li, and F. Ding, “A neural network learning algorithm of chemical process modeling based on the extended Kalman filter,” Neurocomputing, vol. 70, no. 4–6, pp. 625–632, 2007.
  • R. S. Bucy and K. D. Senne, “Digital synthesis of non-linear filters,” Automatica, vol. 7, no. 3, pp. 287–298, 1971.
  • J. Wang and Y. Xie, “Solar radiation prediction based on phase space reconstruction of wavelet neural network,” Procedia Engineering, vol. 15, no. 1, pp. 4603–4607, 2011.
  • H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, UK, 1997.
  • T. E. Karakasidis and A. Charakopoulos, “Detection of low-dimensional chaos in wind time series,” Chaos, Solitons & Fractals, vol. 41, no. 4, pp. 1723–1732, 2009.
  • P. Louka, G. Galanis, N. Siebert et al., “Improvements in wind speed forecasts for wind power prediction purposes using Kalman filtering,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 96, no. 12, pp. 2348–2362, 2008.
  • M. Poncela, P. Poncela, and J. R. Perán, “Automatic tuning of Kalman filters by maximum likelihood methods for wind energy forecasting,” Applied Energy, vol. 108, no. 12, pp. 349–362, 2013.