Combination of Models Obtained by Regression in the Wavelet Domain

L.A. Pinto, R.K.H. Galvão


The wavelet transform is a useful tool to preprocess and compress datasets for linear regression modelling. However, the prediction performance of the resulting model depends on the choice of wavelet filter and number of decomposition levels, which may not be a straightforward task. This paper proposes an alternative approach, which consists of combining models obtained from different wavelet decompositions of the dataset. For this purpose, a method is developed to convert wavelet regression models back to the original domain. The proposed approachis illustrated in a case study involving the determination of density in gasoline samples by using infrared spectroscopy. The results are favourably compared to those obtained by using individual wavelet decompositions.


K.R. Beebe, R.J. Pell, B. Seasholtz, “Chemometrics - A Practical Guide”, Wiley, New York, 1998.

G. Bohacs, Z. Ovadi, A. Salco, Prediction of gasoline properties with near infrared spectroscopy, J. Near Infrared Spectroscopy, 6 (1998), 341–348.

P.J. Brown, T. Fearn, M. Vannucci, Bayesian wavelet regression on curves with application to a spectroscopic calibration problem, J. Am. Stat. Assoc., 96 (2001), 398–408.

P. Chalus, S. Walter, M. Ulmschneider, Combined wavelet transform-artificial neural network use in tablet active content determination by near-infrared spectroscopy, Anal. Chim. Acta, 591 (2007), 219–224.

C.J. Coelho, R.K.H. Galvão, M.C.U. Araújo, M.F. Pimentel, E.C. Silva, A solution to the wavelet transform optimization problem in multicomponent analysis, Chemom. Intell. Lab. Syst., 66 (2003), 205–217.

I.E. Díez, J.M. Saiz, C. Pizarro, OWAVEC: a combination of wavelet analysis and an orthogonalization algorithm as a pre-processing step in multivariate calibration, Anal. Chim. Acta, 515 (2004), 31–41.

D. Donald, D. Coomans, Y. Everingham, D. Cozzolino, M. Gishen, T. Hancock, Adaptive wavelet modeling of a nested 3 factor experimental design in NIR chemometrics, Chemom. Intell. Lab. Syst., 82 (2006), 122–129.

N.R. Draper, H. Smith, “Applied Regression Analysis”, Wiley, New York, 1998.

R.K.H. Galvão, H.A. Dantas Filho, M.N. Martins, M.C.U. Araújo, C. Pasquini, Sub-optimal wavelet denoising of coaveraged spectra employing statistics from individual scans, Anal. Chim. Acta, 581 (2007), 159–167.

J.H. Kalivas, Pareto calibration with built-in wavelength selection, Anal. Chim. Acta, 505 (2004), 9–14.

K.R. Kanduc, J. Zupan, N. Majcen, Separation of data on the training and test set for modeling: a case study for modeling of five colours properties of a white pigment, Chemom. Intell. Lab. Syst., 65 (2003), 221–229.

R.W. Kennard, L.A. Stone, Computer aided design of experiment, Technometrics, 11 (1969), 137–148.

S.G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674–693.

B.M. Nicolai, K.I. Theron, J. Lammertyn, Kernel PLS regression on wavelet transformed NIR spectra for prediction of sugar content of apple, Chemom. Intell. Lab. Syst., 85 (2007), 243–252.

M.F. Pimentel, B.B. Neto, M.C.U. Araújo, C. Pasquini, Simultaneous multielemental determination using a low-resolution inductively coupled plasma spectrometer/diode array detection system, Spectrochimica Acta B., 52 (1997), 2151–2161.

L.A. Pinto, R.K.H. Galvão, Combinação de modelos obtidos por regressão no domínio wavelet, Anais do XXXII CNMAC, 2 (2009), 761–767.

M.J.C. Pontes, J. Cortez, R.K.H. Galvão, C. Pasquini, M.C.U. Araújo, R.M. Coelho, M.K. Chiba, M.F. Abreu, B.E. Madari, Classification of Brazilian soils by using LIBS and variable selection in the wavelet domain, Anal. Chim. Acta,

(2009), 12–18.

R.N.F. Santos, “Calibração Multivariada para Análise Espectrofotométricas Empregando Pacotes Wavelet e Mínimos-Quadrados Parciais”, Tese de Mestrado, ITA, São José dos Campos, SP, 2006.

D.A. Skoog, F.J. Holler, T.A. Nieman, “Princípios de Análise Instrumental”, Bookman, Porto Alegre, 2002.

F. Stout, M.R. Baines, J.H. Kalivas, Impartial graphical comparison of multivariate calibration methods and the harmony/parsimony tradeoff, J. Chemometrics, 20 (2006), 464–475.

G. Strang, T. Nguyen, “Wavelet and Filter Banks”, Cambridge Press, Wellesley, 1996.

J. Trygg, S. Wold, PLS regression on wavelet compressed NIR spectra, Chemom. Intell. Lab. Syst., 42 (1998), 209–220.

B. Walczak, D.L. Massart, Wavelet packet transform applied to a set of signals: A new approach to the best-basis selection, Chemom. Intell. Lab. Syst., 38 (1997), 39–50.


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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

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