{"title":"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","volume":13,"journal":"International Journal of Aerospace and Mechanical Engineering","pagesStart":189,"pagesEnd":195,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/2132","abstract":"<p>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 &Ocirc;&ecirc;&ordm;f as an approximation of the function f. Simulation results are demonstrated to validate the generalization ability and efficiency of the proposed Beta wavelet network.<\/p>\r\n","references":"[1] S.Li and E. Leiss, \"On Noise-immune RBF Networks, in Radial Basis\r\nFunction Neural Networks: Recent Developments in Theory and\r\nApplications\", Editors: Howlett, R. J. and Jain, L.C., Springer Verlag,\r\npp. 95-124, 2001.\r\n[2] P. Cristea, R. Tuduce, and A. Cristea, \"Time Series Prediction with\r\nWavelet Neural Networks\", Proceedings of IEEE Neural Network\r\nApplications in Electrical Engineering, pp. 5-10, 2000.\r\n[3] V. Kreinovich, O. Sirisaengtaksin. S. Cabrera, \"Wavelets compress\r\nbetter than all other methods: a 1-dimensional theorem University of\r\nTexas at El Paso\", Computer Science Department, Technical Report,\r\nJune 1992.\r\n[4] D.Lee, \"An application of wavelet networks in condition monitoring\",\r\nIEEE Power Engineering Review 19, 69-70, 1999.\r\n[5] M.Thuillard, \"Applications of wavelets and wavenets in soft computing\r\nillustrated with the example of fire detectors\", SPIE Wavelet\r\nApplications VII, April 24-28 2000.\r\n[6] J.Fernando Marar, E.C.B.Carvalho Filho, G.C.Vasconcelos, \"Function\r\nApproximation by Polynomial Wavelets Generated from Powers of\r\nSigmoids\", SPIE 2762, 365-374. 1996.\r\n[7] Y. Oussar, I. Rivals, L. Personnaz & G. Dreyfus, \"Training Wavelet\r\nNetworks for Nonlinear Dynamic Input-Output Modeling\",\r\nNeurocomputing, 1998.\r\n[8] J.Echauz, \"Strategies for Fast Training of Wavelet Neural Networks, 2nd\r\nInternational Symposium on Soft Computing for Industry\", 3rd World\r\nAutomation Congress, Anchorage, Alaska, May 1998.\r\n[9] V.Kruger, A. Happe, G. Sommer, \"Affine real-time face tracking using a\r\nwavelet network\", Proc. Workshop on Recognition, Analysis, and\r\nTracking of Faces and Gestures in Real-Time Systems, 26-27 Sept. 1999\r\n(Corfu), IEEE, 141-8, 1999.\r\n[10] Q. Zhang, \"Using Wavelet Network in Nonparametric Estimation\",\r\nIEEE Trans. Neural Network, Vol. 8, pp.227-236, 1997.\r\n[11] M.A Alimi, \"The Beta System: Toward a Change in Our Use of Neuro-\r\nFuzzy Systems\", International Journal of Management, Invited Paper,\r\nno. June, pp. 15-19, 2000.\r\n[12] M.A Alimi, \"The Beta-Based Neuro-Fuzzy System: Genesis and Main\r\nProperties\", TASK Quarterly Journal, Special Issue on \"Neural\r\nNetworks\" edited by W. Duch and D. Rutkowska, vol. 7, no. 1, pp. 23-\r\n41, 2003.\r\n[13] C. Aouiti, M.A Alimi, and A. Maalej, \"Genetic Designed Beta Basis\r\nFunction Neural Networks for Multivariable Functions Approximation,\r\nSystems Analysis, Modeling, and Simulation\", Special Issue on\r\nAdvances in Control and Computer Engineering, vol. 42, no. 7, pp. 975-\r\n1005, 2002.\r\n[14] W.Bellil, C.Ben Amar et M.Adel Alimi, \"Beta Wavelet Based Image\r\nCompression\", International Conference on Signal, System and Design,\r\nSSD03, Tunisia, vol. 1, pp. 77-82, Mars, 2003.\r\n[15] W.Bellil, C.Ben Amar, M.Zaied and M.Adel Alimi, \"La fonction Beta et\r\nses d\u00e9riv\u00e9es : vers une nouvelle famille d-ondelettes\", First International\r\nConference on Signal, System and Design, SCS-04, Tunisia, vol. 1, P.\r\n201-207; Mars 2004.\r\n[16] S.Qian and D.Chen, \"Signal representation using adaptative normalized\r\ngaussian function\", Signal prossing, vol.36, 1994.\r\n[17] S.Chen, C. Cowan, and P. Grant, \"Orthogonal least squares learning\r\nalgorithm for radial basis function networks\", IEEE Trans. On Neural\r\nNetworks, vol. 2, pp.302-309, March 1989.\r\n[18] S.Chen, S. Billings, and W. Luo, \"Orthogonal least squares learning\r\nmethods and their application to non-linear system identification\", Ind.\r\nJ. Control,vol. 50, pp.1873-1896, 1989.\r\n[19] N. Draper and H. Smith, Applied regression analysis, Series in\r\nprobability and Mathematical Statistics\", Wiley, Second edi. 1981.\r\n[20] K. Hornik, M. Stinchcombe, H. White and P. Auer, \"Degree of\r\nApproximation Results for Feedforward Networks Approximating\r\nUnknown Mappings and Their Derivatives\", Neural Computation, 6 (6),\r\n1262-1275, 1994.\r\n[21] C.S.Chang, Weihui Fu, Minjun Yi, \"Short term load forecasting using\r\nwavelet networks\", Engineering Intelligent Systems for Electrical\r\nEngineering and Communications 6, 217-23,1998.\r\n[22] J.Frieman and W. Stuetzle, \"Projection pursuit regression\", J. Amer.\r\nStal. Assoc., vol. 76, pp.817-823, 1981.\r\n[23] P. Huber, Projection pursuit\", Ann. Statist., vol. 13, pp 435, 1985.\r\n[24] Q. Zhang and A. Benveniste, \"Wavelet Networks\", IEEE Trans. on\r\nNeural Networks 3 (6) 889-898, 1992.\r\n[25] J. Zhang, G. G. Walter, Y. Miao and W. N. Wayne Lee, \"Wavelet\r\nNeural Networks For Function Learning\", IEEE Trans. on Signal\r\nProcessing 43 (6), 1485-1497, 1995.\r\n[26] Q. Zhang, \"Using Wavelet Network in Nonparametric Estimation\",\r\nIEEE Trans. on Neural Networks, Vol. 8, No. 2, pp. 227-236, 1997.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 13, 2008"}