# Sequential learning algorithm for PG-RBF network using regression weights for time series prediction

## Abstract

We propose a modified radial basis function network (RBF) in which the main characteristics are that: a) the gaussian function is modified using pseudo-gaussian (PG) in which two scaling parameters σ are introduced; b) the activation of the hidden neurons is normalized c) instead of using a single parameter for the output weights, these are functions of the input variables; d) a sequential learning algorithm is presented to adapt the structure of the network, in which it is possible to create a new hidden unit and also to detect and remove inactive units. It is shown that the modified PG-RBF can reduce the number of didden units significantly compared with the classical RBF network. The feasibility of the resulting algorithm for the neural network to evolve and learn is demonstrated by predicting time series.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]N. B. Karayiannis, G. Weiqun Mi, “Growing Radial Basis Neural Networks: Merging Supervised and Unsupervised Learning with Network Growth Techniques”, IEEE Transaction on Neural Networks, vol.8, no. 6, pp. 1492–1506, November 1997.CrossRefGoogle Scholar
- [2]J. E. Moody, C. J. Darke, “Fast learning in networks of locally-tuned processing units”, Neural Computa., vol. 1, pp. 281–294, 1989.CrossRefGoogle Scholar
- [3]J. Platt, “A resource allocating network for function interpolation”, Neural Computa., vol. 3, pp. 213–225, 1991.MathSciNetCrossRefGoogle Scholar
- [4]L. Yingwei, N. Sundarajan, P. Saratchandran, “Performance Evaluation of a Sequential Minimal Radial Basis Function (RBF) Neural Network Learning Algorithm”, IEEE Transactions on Neural Networks, vol. 9, no. 2, pp. 308–318, March 1998.CrossRefGoogle Scholar
- [5]S. Chen, C. F. N. Cowan, P. M. Grant, “Orthogonal least squares learning algorithm for radial basis function networks”, IEEE Trans. Neural Networks 2 (2), pp. 302–309, 1991.CrossRefGoogle Scholar
- [6]S. Haykin, Neural Networks-A Comprehensive Foundation, IEEE Press, New York, 1994.MATHGoogle Scholar
- [7]H. Surmann, A. Kanstein, K. Goser “Self-organizing and genetic algorithms for an automatic design of fuzzy control and decision systems”,
*Proc. of EUFIT 93*.Google Scholar - [8]T. Kohonen, Self-Organization and Associative Memory, Springer, New York, 1988.CrossRefMATHGoogle Scholar
- [9]A. Sherstinsky, R. W. Picard, “On the efficiency of the orthogonal least squares training method for radial basis function networks”, IEEE Trans. Neural Networks vol. 7, no. 1, pp. 195–200, 1996.CrossRefGoogle Scholar
- [10]D. S. Broomhead, D. Lowe, “Multivariable functional interpolation and adaptive networks”, Complex Syst., vol. 2, pp. 321–355, 1988.MathSciNetMATHGoogle Scholar
- [11]I. Rojas, M. Anguita, E. Ros, H. Pomares, O. Valenzuela, A. Prieto,: “What are the main factors involved in the design of a Radial Basis Function Network?”, 6
^{th}European Symposium on Artificial Neural Network, ESANN’98, pp. 1–6, April 22–24, 1998.Google Scholar - [12]F. Anouar, F. Badran, S. Thiria, “Probabilistic self-organizing maps and radial basis function networks”, Neurocomputing, vol. 20, pp. 83–96, 1998.CrossRefMATHGoogle Scholar
- [13]M. Benaim, “On functional approximation with normalized Gaussian units”, Neural Comput. Vol. 6, 1994Google Scholar
- [14]S. Nowlan, “Maximum likelihood competitive learning”, Proc. Neural Inform. Process. Systems, pp. 574–582, 1990.Google Scholar
- [15]J. S. R. Jang, C. T. Sun, E. Mizutani, “Neuro-Fuzzy and soft computing”, Prentice Hall, ISBN 0-13-261066-3, 1997.Google Scholar
- [16]G. Bugmann, “Normalized Gaussian Radial Basis Function Networks”, Neurocomputing, vol. 20, pp. 97–110, 1998.CrossRefGoogle Scholar
- [17]I. Rojas, O. Valenzuela, A. Prieto, “Statistical analysis of the main parameters in the definition of Radial Basis Function Networks”, Lecture Notes in Computer Science, Vol. 1240, pp. 882–891, Springer-Verlag, June 1997.CrossRefGoogle Scholar
- [18]S. Lee, R. M. Kil, “A Gaussian potential function network with hierarchically self-organizing learning”, Neural Network, vol. 2, pp. 207–224, 1991.CrossRefGoogle Scholar
- [19]M. T. Musavi, W. Ahmed, K. H. Chan, K. B. Faris, D. M. Hummels, “On the training of radial basis function classifier”, Neural Networks, vol. 5, pp. 595–603, 1992.CrossRefGoogle Scholar
- [20]V. Kadirkamanathan, M. Niranjan, “A function estimation approach to sequential learning with neural networks”, Neural Computa., vol. 5, pp. 954–975, 1993.CrossRefGoogle Scholar
- [21]B. A. Whitehead, Tinothy. D. Choate, “Cooperative-Competitive Genetic Evolution of Radial Basis Function Centers and Widths for Time Series Prediction”, IEEE Transaction on Neural Networks, vol. 7, no. 4, pp. 869–880, July, 1996.CrossRefGoogle Scholar
- [22]S.-H. Lee, I. Kim, “Time series analysis using fuzzy learning”,
*in Proc. Int. Conf. Neural Inform. Processing, Seoul, Korea.*, pp. 1577–1582, vol. 6, Oct. 1994.Google Scholar - [23]C. J. Lin, C. T. Lin, “An ART-Based fuzzy adaptive learning control network”,
*IEEE Transactions on Fuzzy Systems*vol. 5, no. 4, pp. 477–496, November 1997.CrossRefGoogle Scholar - [24]D. Kim, C. Kim, “Forecasting time series with genetic fuzzy predictor ensemble”,
*IEEE Transactions on Fuzzy Systems*vol. 5, no. 4, pp. 523–535, November 1997.CrossRefGoogle Scholar - [25]R. S. Crowder III, “Predicting the Mackey-Glass time series with cascadecorrelation learning”,
*In D. Touretzky, G. Hinton and T. Sejnowski, editors, Proceedings of the 1990 Connectionist Model Summer School, pages 117–123, Carnegic Mellon University, 1990*.Google Scholar - [26]K. B. Cho, B. H. Wang, “Radial basis function based adaptive fuzzy systems and their applications to system identification and prediction”,
*Fuzzy Sets and Systems*, vol. 83, pp. 325–339, 1995.MathSciNetCrossRefGoogle Scholar - [27]L. X. Wang, J. M. Mendel, “Generating fuzzy rules by learning from examples”,
*IEEE Trans. On Syst. Man and Cyber*vol. 22, no. 6, November/December, pp. 1414–1427, 1992.MathSciNetCrossRefGoogle Scholar