**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**33017

##### Comparison between Beta Wavelets Neural Networks, RBF Neural Networks and Polynomial Approximation for 1D, 2DFunctions Approximation

**Authors:**
Wajdi Bellil,
Chokri Ben Amar,
Adel M. Alimi

**Abstract:**

This paper proposes a comparison between wavelet neural networks (WNN), RBF neural network and polynomial approximation in term of 1-D and 2-D functions approximation. We present a novel wavelet neural network, based on Beta wavelets, for 1-D and 2-D functions approximation. Our purpose is to approximate an unknown function f: Rn - R from scattered samples (xi; y = f(xi)) i=1....n, where first, we have little a priori knowledge on the unknown function f: it lives in some infinite dimensional smooth function space and second the function approximation process is performed iteratively: each new measure on the function (xi; f(xi)) is used to compute a new estimate Ôêºf as an approximation of the function f. Simulation results are demonstrated to validate the generalization ability and efficiency of the proposed Beta wavelet network.

**Keywords:**
Beta wavelets networks,
RBF neural network,
training algorithms,
MSE,
1-D,
2D function approximation.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1056514

**References:**

[1] S.Li and E. Leiss, "On Noise-immune RBF Networks, in Radial Basis Function Neural Networks: Recent Developments in Theory and Applications", Editors: Howlett, R. J. and Jain, L.C., Springer Verlag, pp. 95-124, 2001.

[2] P. Cristea, R. Tuduce, and A. Cristea, "Time Series Prediction with Wavelet Neural Networks", Proceedings of IEEE Neural Network Applications in Electrical Engineering, pp. 5-10, 2000.

[3] V. Kreinovich, O. Sirisaengtaksin. S. Cabrera, "Wavelets compress better than all other methods: a 1-dimensional theorem University of Texas at El Paso", Computer Science Department, Technical Report, June 1992.

[4] D.Lee, "An application of wavelet networks in condition monitoring", IEEE Power Engineering Review 19, 69-70, 1999.

[5] M.Thuillard, "Applications of wavelets and wavenets in soft computing illustrated with the example of fire detectors", SPIE Wavelet Applications VII, April 24-28 2000.

[6] J.Fernando Marar, E.C.B.Carvalho Filho, G.C.Vasconcelos, "Function Approximation by Polynomial Wavelets Generated from Powers of Sigmoids", SPIE 2762, 365-374. 1996.

[7] Y. Oussar, I. Rivals, L. Personnaz & G. Dreyfus, "Training Wavelet Networks for Nonlinear Dynamic Input-Output Modeling", Neurocomputing, 1998.

[8] J.Echauz, "Strategies for Fast Training of Wavelet Neural Networks, 2nd International Symposium on Soft Computing for Industry", 3rd World Automation Congress, Anchorage, Alaska, May 1998.

[9] V.Kruger, A. Happe, G. Sommer, "Affine real-time face tracking using a wavelet network", Proc. Workshop on Recognition, Analysis, and Tracking of Faces and Gestures in Real-Time Systems, 26-27 Sept. 1999 (Corfu), IEEE, 141-8, 1999.

[10] Q. Zhang, "Using Wavelet Network in Nonparametric Estimation", IEEE Trans. Neural Network, Vol. 8, pp.227-236, 1997.

[11] M.A Alimi, "The Beta System: Toward a Change in Our Use of Neuro- Fuzzy Systems", International Journal of Management, Invited Paper, no. June, pp. 15-19, 2000.

[12] M.A Alimi, "The Beta-Based Neuro-Fuzzy System: Genesis and Main Properties", TASK Quarterly Journal, Special Issue on "Neural Networks" edited by W. Duch and D. Rutkowska, vol. 7, no. 1, pp. 23- 41, 2003.

[13] C. Aouiti, M.A Alimi, and A. Maalej, "Genetic Designed Beta Basis Function Neural Networks for Multivariable Functions Approximation, Systems Analysis, Modeling, and Simulation", Special Issue on Advances in Control and Computer Engineering, vol. 42, no. 7, pp. 975- 1005, 2002.

[14] W.Bellil, C.Ben Amar et M.Adel Alimi, "Beta Wavelet Based Image Compression", International Conference on Signal, System and Design, SSD03, Tunisia, vol. 1, pp. 77-82, Mars, 2003.

[15] W.Bellil, C.Ben Amar, M.Zaied and M.Adel Alimi, "La fonction Beta et ses dérivées : vers une nouvelle famille d-ondelettes", First International Conference on Signal, System and Design, SCS-04, Tunisia, vol. 1, P. 201-207; Mars 2004.

[16] S.Qian and D.Chen, "Signal representation using adaptative normalized gaussian function", Signal prossing, vol.36, 1994.

[17] S.Chen, C. Cowan, and P. Grant, "Orthogonal least squares learning algorithm for radial basis function networks", IEEE Trans. On Neural Networks, vol. 2, pp.302-309, March 1989.

[18] S.Chen, S. Billings, and W. Luo, "Orthogonal least squares learning methods and their application to non-linear system identification", Ind. J. Control,vol. 50, pp.1873-1896, 1989.

[19] N. Draper and H. Smith, Applied regression analysis, Series in probability and Mathematical Statistics", Wiley, Second edi. 1981.

[20] K. Hornik, M. Stinchcombe, H. White and P. Auer, "Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives", Neural Computation, 6 (6), 1262-1275, 1994.

[21] C.S.Chang, Weihui Fu, Minjun Yi, "Short term load forecasting using wavelet networks", Engineering Intelligent Systems for Electrical Engineering and Communications 6, 217-23,1998.

[22] J.Frieman and W. Stuetzle, "Projection pursuit regression", J. Amer. Stal. Assoc., vol. 76, pp.817-823, 1981.

[23] P. Huber, Projection pursuit", Ann. Statist., vol. 13, pp 435, 1985.

[24] Q. Zhang and A. Benveniste, "Wavelet Networks", IEEE Trans. on Neural Networks 3 (6) 889-898, 1992.

[25] J. Zhang, G. G. Walter, Y. Miao and W. N. Wayne Lee, "Wavelet Neural Networks For Function Learning", IEEE Trans. on Signal Processing 43 (6), 1485-1497, 1995.

[26] Q. Zhang, "Using Wavelet Network in Nonparametric Estimation", IEEE Trans. on Neural Networks, Vol. 8, No. 2, pp. 227-236, 1997.