This Article 
 Bibliographic References 
 Add to: 
Chaotic Time Series Prediction Using a Neuro-Fuzzy System with Time-Delay Coordinates
July 2008 (vol. 20 no. 7)
pp. 956-964
This paper presents an investigation into the use of the time delay coordinate embedding technique in the multi-input-multi-output-adaptive-network-based fuzzy inference system (MANFIS) for chaotic time series prediction. The inputs of the MANFIS are embedded-phase-space (EPS) vectors preprocessed from the time series under test while the output time series is extracted from the EPS vectors. With such EPS preprocessing, the prediction accuracy of the MANFIS is found to be significantly improved. The proposed system will be tested with a periodic and the Mackey-Glass chaotic time series by comparing the prediction accuracy with and without EPS preprocessing. A moving root-mean-square error is used to monitor the error along the prediction horizon and to tune the membership functions in the MANFIS.

[1] C. Chatfield, The Analysis of Time Series: An Introduction. Chapman & Hall/CRC, 2004.
[2] H. Abarbanel, Analysis of Observed Chaotic Data. Springer-Verlag, 1996.
[3] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge Univ. Press, 2004.
[4] C. Gao and J. Qian, “Evidence of Chaotic Behavior in Noise from Industrial Process,” IEEE Trans. Signal Processing, vol. 55, no. 6,Part 2, pp. 2877-2884, June 2007.
[5] A. Weigend and N. Gershenfeld, Time Series Prediction: Forecasting the Future and Understanding the Past. Addison-Wesley, 1994.
[6] Z. Ye and L. Gu, “A Fuzzy System for Trading the Shanghai Stock Market,” Trading on the Edge, Neural, Genetic, and Fuzzy Systems for Chaotic Financial Markets, G.J. Deboeck, ed., pp. 207-214, Wiley, 1994.
[7] E.S. Garcia-Trevino and V. Alarcon-Aquino, “Single-Step Prediction of Chaotic Time Series Using Wavelet-Networks,” Proc. Electronics, Robotics and Automotive Mechanics Conf. (CERMA '06), vol. 1, pp. 243-248, Sept. 2006.
[8] H. Leung, T. Lo, and S. Wang, “Prediction of Noisy Chaotic Time Series Using an Optimal Radial Basis Function Neural Network,” IEEE Trans. Neural Networks, vol. 12, no. 5, pp. 1163-1172, Sept. 2001.
[9] M. Han, J. Xi, S. Xu, and F. Yin, “Prediction of Chaotic Time Series Based on the Recurrent Predictor Neural Network,” IEEE Trans. Signal Processing, vol. 52, no. 12, pp. 3409-3416, Dec. 2004.
[10] W. Jiang and P. Wang, “Research on Interval Prediction of Nonlinear Chaotic Time Series Based on New Neural Networks,” Proc Sixth World Congress Intelligent Control and Automation (WCICA '06), vol. 1, pp. 2835-2839, June 2006.
[11] T. Matsumoto, Y. Nakajima, M. Saito, J. Sugi, and H. Hamagishi, “Reconstructions and Predictions of Nonlinear Dynamical Systems: A Hierarchical Bayesian Approach,” IEEE Trans. Signal Processing, vol. 49, no. 9, pp. 2138-2155, Sept. 2001.
[12] D. Kim and C. Kim, “Forecasting Time Series with Genetic Fuzzy Predictor Ensemble,” IEEE Trans. Fuzzy Systems, vol. 5, no. 4, pp.523-535, Nov. 1997.
[13] L. Chen and G. Chen, “Fuzzy Modeling, Prediction, and Control of Uncertain Chaotic Systems Based on Time Series,” IEEE Trans. Circuits and Systems I, vol. 47, no. 10, Oct. 2000.
[14] H. Kunhuang, “Heuristic Models of Fuzzy Time Series for Forecasting,” Fuzzy Sets and Systems, vol. 123, no. 3, pp. 369-386, Nov. 2001.
[15] H. Yu, “A Refined Fuzzy Time-Series Model for Forecasting,” Physica A: Statistical and Theoretical Physics, vol. 346, nos. 3-4, pp.657-681, Feb. 2005.
[16] W. Ibrahim and M. Morcos, “An Adaptive Fuzzy Self-Learning Technique for Prediction of Abnormal Operation of Electrical Systems,” IEEE Trans. Power Delivery, vol. 21, no. 4, pp. 1770-1777, Oct. 2006.
[17] C. Lee, A. Liu, and W. Chen, “Pattern Discovery of Fuzzy Time Series for Financial Prediction,” IEEE Trans. Knowledge and Data Eng., vol. 18, no. 5, pp. 613-625, May 2006.
[18] J.R. Jang, “ANFIS: Adaptive-Network-Based Fuzzy Inference System,” IEEE Trans. Systems, Man, and Cybernetics, vol. 23, pp.665-685, May 1993.
[19] W. Abdel-Hamid, A. Noureldin, and N. El-Sheimy, “Adaptive Fuzzy Prediction of Low-Cost Inertial-Based Positioning Error,” IEEE Trans. Fuzzy Systems, (in press).
[20] M. Marseguerra, E. Zio, and P. Avogadri, “Model Identification by Neuro-Fuzzy Techniques: Predicting the Water Level in a Steam Generator of a PWR,” Progress in Nuclear Energy, vol. 44, no. 3, pp.237-252, 2004.
[21] P.C. Nayak, K.P. Sudheer, D.M. Rangan, and K.S. Ramasastri, “A Neuro-Fuzzy Computing Technique for Modeling Hydrological Time Series,” J. Hydrology, vol. 291, nos. 1-2, pp. 52-66, May 2004.
[22] K. Ang and C. Quek, “Stock Trading Using RSPOP: A Novel Rough Set-Based Neuro-Fuzzy Approach,” IEEE Trans. Neural Networks, vol. 17, no. 5, pp. 1301-1315, Sept. 2006.
[23] Y. Yildirim and M. Bayramoglu, “Adaptive Neuro-Fuzzy Based Modelling for Prediction of Air Pollution Daily Levels in City of Zonguldak,” Chemosphere, vol. 63, no. 9, pp. 1575-1582, June 2006.
[24] Zaheeruddin and Garima, “A Neuro-Fuzzy Approach for Prediction of Human Work Efficiency in Noisy Environment,” Applied Soft Computing, vol. 6, no. 3, pp. 283-294, Mar. 2006.
[25] M.J.L. Aznarte, J. Manuel Benítez Sánchez, D. Nieto Lugilde, C. de Linares Fernández, C. Díaz de la Guardia, and F. Alba Sánchez, “Forecasting Airborne Pollen Concentration Time Series with Neural and Neuro-Fuzzy Models,” Expert Systems with Applications, vol. 32, no. 4, pp. 1218-1225, May 2007.
[26] O. Castillo and P. Melin, “Hybrid Intelligent Systems for Time Series Prediction Using Neural Networks, Fuzzy Logic, and Fractal,” IEEE Trans. Neural Networks, vol. 13, no. 6, pp. 1395-1408, Nov. 2002.
[27] S. Su and F.Y.P. Yang, “On the Dynamical Modeling with Neural Fuzzy Networks,” IEEE Trans. Neural Networks, vol. 13, no. 6, pp.1548-1553, Nov. 2002.
[28] W.L. Tung and C. Quek, “GenSoFNN: A Generic Self-Organizing Fuzzy Neural Network,” IEEE Trans. Neural Networks, vol. 13, no. 5, pp. 1075-1086, Sept. 2002.
[29] Y.G. Leu, W.Y. Wang, and T.T. Lee, “Observer-Based Direct Adaptive Fuzzy-Neural Control for Nonaffine Nonlinear Systems,” IEEE Trans. Neural Networks, vol. 16, no. 4, pp. 853-861, July 2005.
[30] C.F. Hsu, “Self-Organizing Adaptive Fuzzy Neural Control for a Class of Nonlinear Systems,” IEEE Trans. Neural Networks, vol. 18, no. 4, pp. 1232-1241, July 2007.
[31] N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, “Geometry from a Time Series,” Physical Rev. Letters, vol. 45, pp.712-716, 1980.
[32] F. Takens, Dynamical Systems and Turbulence, Warwick, D.A. Rand and L.S. Young, eds., p. 366. Springer, 1980.
[33] J.P. Eckmann and D. Ruelle, “Ergodic Theory of Chaos and Strange Attractor,” Rev. of Modern Physics, vol. 57, no. 3, pp. 617-656, 1985.
[34] A.K. Alparslan, M. Sayar, and A.R. Atilgan, “State-Space Prediction Model for Chaotic Time Series,” Physical Rev. E, vol. 58, no. 2, pp. 2640-2643, Aug. 1998.
[35] T. Oguchi and H. Nijmeijer, “Prediction of Chaotic Behavior,” IEEE Trans. Circuits and Systems I, vol. 52, no. 11, pp. 2464-2472, Nov. 2005.
[36] N. Xie and H. Leung, “Reconstruction of Piecewise Chaotic Dynamic Using a Genetic Algorithm Multiple Model Approach,” IEEE Trans. Circuits and Systems I, vol. 51, no. 6, pp. 1210-1222, June 2004.
[37] S. Guo, L. Shieh, G. Chen, and C. Lin, “Effective Chaotic Orbit Tracker: A Prediction-Based Digital Redesign Approach,” IEEE Trans. Circuits and Systems I, vol. 47, no. 11, pp. 1557-1570, Nov. 2000.
[38] A.M. Fraser and H.L. Swinney, “Independent Coordinates for Strange Attractors from Mutual Information,” Physical Rev. A, vol. 33, pp. 1134-1140, 1986.
[39] X. Yao, Y. Liu, and G. Lin, “Evolutionary Programming Made Faster,” IEEE Trans. Evolutionary Computation, vol. 3, no. 2, pp. 82-102, July 1999.
[40] E. Ott, T. Sauer, and J.A. York, Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems, pp. 1-13. John Wiley & Sons, 1994.
[41] L. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Prentice Hall, 1994.

Index Terms:
Chaotic time series prediction, neuro-fuzzy systems, time delay coordinate embedding
Jun Zhang, Henry Shu-Hung Chung, W. -L. Lo, "Chaotic Time Series Prediction Using a Neuro-Fuzzy System with Time-Delay Coordinates," IEEE Transactions on Knowledge and Data Engineering, vol. 20, no. 7, pp. 956-964, July 2008, doi:10.1109/TKDE.2008.35
Usage of this product signifies your acceptance of the Terms of Use.