Linear-wavelet networks

Galvão, Roberto ; Becerra, Victor ; Calado, João ; Silva, Pedro

International Journal of Applied Mathematics and Computer Science, Tome 14 (2004), p. 221-232 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper proposes a nonlinear regression structure comprising a wavelet network and a linear term. The introduction of the linear term is aimed at providing a more parsimonious interpolation in high-dimensional spaces when the modelling samples are sparse. A constructive procedure for building such structures, termed linear-wavelet networks, is described. For illustration, the proposed procedure is employed in the framework of dynamic system identification. In an example involving a simulated fermentation process, it is shown that a linear-wavelet network yields a smaller approximation error when compared with a wavelet network with the same number of regressors. The proposed technique is also applied to the identification of a pressure plant from experimental data. In this case, the results show that the introduction of wavelets considerably improves the prediction ability of a linear model. Standard errors on the estimated model coefficients are also calculated to assess the numerical conditioning of the identification process.

Détail

Publié le : 2004-01-01

Zbl 1076.93014

EUDML-ID : urn:eudml:doc:207693

Bibliographie

[000] Aborhey S. and Williamson D. (1978): State and parameter estimation of microbial growth processes. - Automatica, Vol. 14, No. 5, pp. 493-498. | Zbl 0425.93034

[001] Benveniste A., Juditsky A., Delyon B., Zhang Q. and Glorennec P.Y.(1994): Wavelets in identification. - Proc. 10th IFAC Symp. Syst. Identification, Copenhagen, pp. 27-48.

[002] Cannon M. and Slotine J.-J.E. (1995): Space-frequency localized basis function networks for nonlinear system estimation and control. - Neurocomput., Vol. 9, No. 3, pp. 293-342. | Zbl 0951.93515

[003] D'Ans G., Gottlieb D. and Kokotovic P.(1972): Optimal control of bacterial growth. -Automatica, Vol. 8, No. 6, pp. 729-736.

[004] Daubechies I. (1992): Ten Lectures on Wavelets. - Philadelphia: SIAM. | Zbl 0776.42018

[005] Draper N.R. and Smith H. (1981): Applied Regression Analysis, 2nd Ed. - New York: Wiley. | Zbl 0548.62046

[006] Ezekiel M. and Fox K.A. (1959): Methods of Correlation and Regression Analysis, 3rd Ed. - New York: Wiley. | Zbl 0086.35504

[007] Galv ao R.K.H. and Becerra V.M. (2002): Linear-wavelet models applied to the identification of a two-link manipulator. - Proc. 21st IASTED Int. Conf. Modelling, Identification and Control, Innsbruck, pp. 479-484.

[008] Galv ao R.K.H., Yoneyama T. and Rabello T.N. (1999): Signal representation by adaptive biased wavelet expansions. -Digital Signal Process., Vol. 9, No. 4, pp. 225-240.

[009] Haykin S.S. (1998): Neural Networks: A Comprehensive Foundation.- Upper Saddle River: Prentice-Hall.

[010] Jang J.-S. R. and Sun C.-T. (1995): Neuro-fuzzy modelling and control. - Proc. IEEE, Vol. 83, No. 3, pp. 378-406.

[011] Kan K.-C. and Wong K.-W. (1998): Self-construction algorithm for synthesis of wavelet networks. - Electronic Lett., Vol. 34, No. 20, pp. 1953-1955.

[012] Lawson C.L. and Hanson R.J. (1974): Solving Least Squares Problems. - Englewood Cliffs: Prentice-Hall. | Zbl 0860.65028

[013] Li K.C. (1986): Asymptotic optimality of c_l and generalized cross-validation in ridge regression and application to the spline smoothing. - Ann. Statist., Vol. 14, No. 3, pp. 1101-1112. | Zbl 0629.62043

[014] Liu G.P., Billings S.A. and Kadirkamanathan V. (2000): Nonlinear system identification using wavelet networks.- Int. J. Syst. Sci., Vol. 31, No. 12, pp. 1531-1541. | Zbl 1080.93577

[015] Ljung L. (1999): System Identification: Theory for the User. - Upper Saddle River: Prentice-Hall. | Zbl 0615.93004

[016] Naes T. and Mevik B.H. (2001): Understanding the collinearity problem in regression and discriminant analysis. - J. Chemometr., Vol. 15, No. 4, pp. 413-426.

[017] Naradaya E. (1964): On estimating regression. - Theory Prob. Applicns., Vol. 9, pp. 141-142.

[018] Narendra K.S. and Parthasarathy K. (1990): Identification and control of dynamical systems using neural networks. - IEEE Trans. Neural Netw., Vol. 1, No. 1, pp. 4-27.

[019] Poggio T. and Girosi F. (1990): Networks for approximation and learning. - Proc.IEEE, Vol. 78, No. 9, pp. 1481-1497. | Zbl 1226.92005

[020] Rissanen J.(1978): Modeling by shortest data description. -Automatica, Vol. 14, No. 5, pp. 465-471. | Zbl 0418.93079

[021] Rugh W.J. (1981): Nonlinear Systems Theory. The Volterra Wiener Approach. - Baltimore: Johns Hopkins University Press. | Zbl 0666.93065

[022] SchumakerL.L. (1981): Spline Functions: Basic Theory. - Chichester: Wiley.

[023] Souza Jr. C., Hemerly E.M. and Galv ao R.K.H. (2002): Adaptive control for mobile robot using wavelet network.- IEEE Trans. Syst. Man Cybern., Part B, Vol. 32, No. 4, pp. 493-504.

[024] Takagi T. and Sugeno M. (1985): Fuzzy identification of systems and its applications to modelling and control. - IEEE Trans. Syst. Man Cybern., Vol. 15, No. 1, pp. 116-132. | Zbl 0576.93021

[025] Watson G.S. (1964): Smooth regression analysis. - Sankhya, Ser. A, Vol. 26, No. 4, pp. 359-372. | Zbl 0137.13002

[026] Zhang J., Walter G.G., Miao Y. and Lee W.N.W. (1995): Wavelet neural networks for function learning. - IEEE Trans. Signal Process., Vol. 43, No. 6, pp. 1485-1496.

[027] Zhang Q. (1997): Using wavelet network in nonparametric estimation. -IEEE Trans. Neural Netw., Vol. 8, No. 2, pp. 227-236.

[028] Zhang Q. and Benveniste A. (1992): Wavelet networks. - IEEE Trans. Neural Netw., Vol. 3, No. 6, pp. 889-898.