Paper:

# Improved Estimation of Embedding Parameters of Nonlinear Time Series by Structural Learning of Neural Network with Fuzzy Regularizer

## Yusuke Manabe and Basabi Chakraborty

Graduate School of Software and Information Science, Iwate Prefectural University, 152-52 Sugo, Takizawa-mura, Iwate 020-0193, Japan

This work proposes an improved refinement scheme of estimation of optimal embedding parameters of a nonlinear time series by a feed-forward neural network trained by structural learning with a fuzzy regularizer (FR). The newly proposed fuzzy rules for tuning regularization parameter enables automatic selection of optimal model with lesser computational load than the basic refinement scheme with RNS proposed by authors earlier. From the simulation results, it has been found that the proposed scheme is very efficient in estimation of optimal embedding parameters in lesser computational time. The short term prediction results also show that the estimated embedding parameters produce better and stable one step prediction.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.6, pp. 600-609, 2007.

- [1] F. Takens, “Detecting strange attractors in turbulence,” In Lecture notes in mathematics, Vol.898, Dynamical systems and turbulence, pp. 366-381, Springer, 1981.
- [2] H. D. I. Abarbanel, Analysis of observed chaotic data, Springer-Verlag, New York, 1996.
- [3] R. Reed, “Pruning Algorithms – A Survey,” IEEE Trans. on Neural Networks, Vol.4, No.5, pp. 740-747, 1993.
- [4] J. Sietsma and R. J. F. Dow, “Creating Artificial Neural Networks That Generalize,” Neural Networks, Vol.4, pp. 67-79, 1991.
- [5] M. Ishikawa, “Structural Learning with Forgetting,” Neural Networks, Vol.9, No.3, pp. 509-521, 1996.
- [6] S. Kikuchi and M. Nakanishi, “Recurrent neural network with short-term memory and fast structural learning method,” Systems and Computers in Japan, Vol.34, No.6, pp. 69-79, 2003.
- [7] T. Matsui, T. Iizaka, and Y. Fukuyama, “Peak load forecasting using analyzable structured neural network,” IEEE Power Engineering Society Winter Meeting, pp. 405-410, 2001.
- [8] Y. Manabe, B. Chakraborty, and H. Fujita, “Structural Learning of Multilayer Feed Forward Neural Networks for Continuous Valued Functions,” in Proceedings of IEEE-MWSCAS 2004, pp. III77-III80, July, 2004.
- [9] B. Chakraborty and Y. Manabe, “Structural Learning of Neural Network for Continuous Valued Output: Effect of Penalty Term to Hidden Units,” in Lecture Notes in Computer Science 3316, Springer, pp. 599-605, 2004.
- [10] D. J. C. MacKay, “Bayesian non-linear modeling for the energy prediction competition,” ASHRAE Transactions, Vol.100, Part 2, pp. 1053-1062, 1994.
- [11] R. M. Neal, “Bayesian learning for neural networks,” Springer-Verlag, New York, 1996.
- [12] V. N. Vapnik, “The Nature of Statistical Learning Theory,” Springer, 1995.
- [13] D. J. C. MacKay, “Bayesian interpolation,” Neural Computation, Vol.4, pp. 415-447, 1992.
- [14] Y. Manabe and B. Chakraborty, “Estimating Embedding Parameters using Structural Learning of Neural Network,” IEEE International Workshop on NSIP 2005, Sapporo, Japan, May, 2005.
- [15] Y. Manabe and B. Chakraborty, “Estimation of Embedding Parameters by Neural Network with Pruning Algorithm,” Technical Report of IEICE, Vol.105, No.206, pp. 45-50, July, 2005 (in Japanese).
- [16] Y. Manabe and B. Chakraborty, “A Novel Approach for Estimation of Optimal Embedding Parameters of Nonlinear Time Series by Structural Learning of Neural Network,” Neurocomputing, Vol.70, issue 7-9, pp. 1360-1371, March 2007.
- [17] K. Alligood, T. Sauer, and J. A. Yorke, “Chaos: An Introduction to Dynamical Systems,” Springer-Verlag, New York, 1997.
- [18] H. Kantz and T. Schreiber, “Nonlinear Time Series Analysis,” Cambridge University Press, 1997.
- [19] A. M. Fraser and H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A, Vol.33, pp. 1134-1140, 1986.
- [20] M. B. Kennel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Phys. Rev. A, Vol.45, pp. 3403-3411, 1992.
- [21] A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, and P. E. Rapp, “Singular-value decomposition and the Grassberger-Procaccia algorithm,” Phys. Rev. A, Vol.38, pp. 3017-3026, 1988.
- [22] J. C. Principe, L. Wang, and M. A. Motter, “Local Dynamic Modelmodeling with Self-Organizing Maps and Applications to Nonlinear System Identification and Control,” Proceedings of the IEEE, Vol.86, No.11. pp. 2240-2258, November 1998.
- [23] C. J. Cellucci, A. M. Albano, and P. E. Rapp, “Comparative Study of Embedding Methods,” Phys. Rev. E, Vol.67, 066210-1-13, 2003.
- [24] S. Kim, R. Eykholt, and J. D. Salas, “Delay time window and plateau onset of the correlation for small data sets,” Phys. Rev. E, Vol.58, pp. 5676-5682, 1998.
- [25] K. Judd and A. Mees, “Embedding as a Modelling Problem,” Physica D, Vol.120, pp. 273-286, 1998.
- [26] M. Small and C. K. Tse, “Optimal embedding parameters: a modelling paradigm,” Physica D, Vol.194, pp. 283-296, 2004.
- [27] K. Judd, M. Small, and A. I. Mees, “Achieving Good Nonlinear Models: Keep it Simple, Vary the Embedding, and Get the Dynamics Right,” in Nonlinear Dynamics and Statistics, Ed., A. I. Mees, pp. 65-80, Birkhauser Boston, 2001.
- [28] T. Matsumoto, Y. Nakajima, M. Saito, J. Sugi, and H. Hamagishi, “Reconstructions and Predictions of Nonlinear Dynamical Systems: A Hierarchical Bayesian Approach,” IEEE Trans. on Signal Processing, Vol.49, No.9, pp. 2138-2155, 2001.
- [29] R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: the TISEAN package,” CHAOS 9, pp. 413-435, 1999.
- [30] Time Series Data Library,

http://www-personal.buseco.monash.edu.au/˜hyndman/TSDL/

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