# Linear-wavelet networks

Roberto Galvão; Victor Becerra; João Calado; Pedro Silva

International Journal of Applied Mathematics and Computer Science (2004)

- Volume: 14, Issue: 2, page 221-232
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topGalvão, Roberto, et al. "Linear-wavelet networks." International Journal of Applied Mathematics and Computer Science 14.2 (2004): 221-232. <http://eudml.org/doc/207693>.

@article{Galvão2004,

abstract = {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.},

author = {Galvão, Roberto, Becerra, Victor, Calado, João, Silva, Pedro},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {regression analysis; wavelet networks; nonlinear models; system identification},

language = {eng},

number = {2},

pages = {221-232},

title = {Linear-wavelet networks},

url = {http://eudml.org/doc/207693},

volume = {14},

year = {2004},

}

TY - JOUR

AU - Galvão, Roberto

AU - Becerra, Victor

AU - Calado, João

AU - Silva, Pedro

TI - Linear-wavelet networks

JO - International Journal of Applied Mathematics and Computer Science

PY - 2004

VL - 14

IS - 2

SP - 221

EP - 232

AB - 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.

LA - eng

KW - regression analysis; wavelet networks; nonlinear models; system identification

UR - http://eudml.org/doc/207693

ER -

## References

top- Aborhey S. and Williamson D. (1978): State and parameter estimation of microbial growth processes. - Automatica, Vol. 14, No. 5, pp. 493-498. Zbl0425.93034
- 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.
- 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. Zbl0951.93515
- D'Ans G., Gottlieb D. and Kokotovic P.(1972): Optimal control of bacterial growth. -Automatica, Vol. 8, No. 6, pp. 729-736.
- Daubechies I. (1992): Ten Lectures on Wavelets. - Philadelphia: SIAM. Zbl0776.42018
- Draper N.R. and Smith H. (1981): Applied Regression Analysis, 2nd Ed. - New York: Wiley. Zbl0548.62046
- Ezekiel M. and Fox K.A. (1959): Methods of Correlation and Regression Analysis, 3rd Ed. - New York: Wiley. Zbl0086.35504
- 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.
- 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.
- Haykin S.S. (1998): Neural Networks: A Comprehensive Foundation.- Upper Saddle River: Prentice-Hall.
- Jang J.-S. R. and Sun C.-T. (1995): Neuro-fuzzy modelling and control. - Proc. IEEE, Vol. 83, No. 3, pp. 378-406.
- 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.
- Lawson C.L. and Hanson R.J. (1974): Solving Least Squares Problems. - Englewood Cliffs: Prentice-Hall. Zbl0860.65028
- 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. Zbl0629.62043
- 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. Zbl1080.93577
- Ljung L. (1999): System Identification: Theory for the User. - Upper Saddle River: Prentice-Hall. Zbl0615.93004
- 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.
- Naradaya E. (1964): On estimating regression. - Theory Prob. Applicns., Vol. 9, pp. 141-142.
- 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.
- Poggio T. and Girosi F. (1990): Networks for approximation and learning. - Proc.IEEE, Vol. 78, No. 9, pp. 1481-1497. Zbl1226.92005
- Rissanen J.(1978): Modeling by shortest data description. -Automatica, Vol. 14, No. 5, pp. 465-471. Zbl0418.93079
- Rugh W.J. (1981): Nonlinear Systems Theory. The Volterra Wiener Approach. - Baltimore: Johns Hopkins University Press. Zbl0666.93065
- SchumakerL.L. (1981): Spline Functions: Basic Theory. - Chichester: Wiley.
- 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.
- 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. Zbl0576.93021
- Watson G.S. (1964): Smooth regression analysis. - Sankhya, Ser. A, Vol. 26, No. 4, pp. 359-372. Zbl0137.13002
- 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.
- Zhang Q. (1997): Using wavelet network in nonparametric estimation. -IEEE Trans. Neural Netw., Vol. 8, No. 2, pp. 227-236.
- Zhang Q. and Benveniste A. (1992): Wavelet networks. - IEEE Trans. Neural Netw., Vol. 3, No. 6, pp. 889-898.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.