### Abstract

It has been recently shown that the predictive ability of wavelet models in multivariate calibration problems can be improved by optimizing the filters employed in the Discrete Wavelet Transform (DWT) with respect to the statistics of the matrix of instrumental responses. However, no attempt has been made at exploiting the statistics of the matrix of predicted parameters in the optimization process. This work addresses this issue and proposes a novel strategy for wavelet filter optimization that aims at directly minimizing the prediction error of a wavelet regression model with respect to a given validation set. Moreover, some theoretical and algorithmic aspects of the angular parameterization of wavelet filters needed for the optimization procedure are clarified. The parameterization is explained in a simple graphical manner. A requirement on the sum of the angular parameters and its implications in the optimization strategy are discussed. Finally, a procedure for obtaining the angular parameters associated to any traditional mother wavelet is provided, both as an algorithm and as a Matlab code. The proposed strategy is illustrated in a simulated multivariate calibration example involving two analytes and also in a problem of total sulphur determination in diesel samples by near-infrared (NIR) absorption spectrometry. Both the simulated and the real examples show that the proposed filter optimization procedure improves the prediction ability of wavelet regression models. Moreover, in the sulphur determination problem, the wavelet models result in a smaller prediction error than a traditional Partial Least Squares Regression (PLS) model. (C) 2003 Elsevier B.V All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 110-119 |

Number of pages | 10 |

Journal | Chemometrics and intelligent laboratory systems |

Volume | 70 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 28 Jan 2004 |

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### Keywords

- wavelet transform optimization
- quadrature-mirror filter banks
- NIR spectrometry
- diesel analysis
- total sulphur determination
- multivariate calibration
- variable selection
- regression

### Cite this

*Chemometrics and intelligent laboratory systems*,

*70*(1-2), 110-119. https://doi.org/10.1016/j.chemolab.2003.09.001

}

*Chemometrics and intelligent laboratory systems*, vol. 70, no. 1-2, pp. 110-119. https://doi.org/10.1016/j.chemolab.2003.09.001

**Optimal wavelet filter construction using X and Y data.** / Galvao, R K H ; Jose, G E ; Dantas, H A ; Araujo, M C U ; da Silva, E C ; Paiva, H M ; Saldanha, T C B ; de Souza, E S O N .

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optimal wavelet filter construction using X and Y data

AU - Galvao, R K H

AU - Jose, G E

AU - Dantas, H A

AU - Araujo, M C U

AU - da Silva, E C

AU - Paiva, H M

AU - Saldanha, T C B

AU - de Souza, E S O N

PY - 2004/1/28

Y1 - 2004/1/28

N2 - It has been recently shown that the predictive ability of wavelet models in multivariate calibration problems can be improved by optimizing the filters employed in the Discrete Wavelet Transform (DWT) with respect to the statistics of the matrix of instrumental responses. However, no attempt has been made at exploiting the statistics of the matrix of predicted parameters in the optimization process. This work addresses this issue and proposes a novel strategy for wavelet filter optimization that aims at directly minimizing the prediction error of a wavelet regression model with respect to a given validation set. Moreover, some theoretical and algorithmic aspects of the angular parameterization of wavelet filters needed for the optimization procedure are clarified. The parameterization is explained in a simple graphical manner. A requirement on the sum of the angular parameters and its implications in the optimization strategy are discussed. Finally, a procedure for obtaining the angular parameters associated to any traditional mother wavelet is provided, both as an algorithm and as a Matlab code. The proposed strategy is illustrated in a simulated multivariate calibration example involving two analytes and also in a problem of total sulphur determination in diesel samples by near-infrared (NIR) absorption spectrometry. Both the simulated and the real examples show that the proposed filter optimization procedure improves the prediction ability of wavelet regression models. Moreover, in the sulphur determination problem, the wavelet models result in a smaller prediction error than a traditional Partial Least Squares Regression (PLS) model. (C) 2003 Elsevier B.V All rights reserved.

AB - It has been recently shown that the predictive ability of wavelet models in multivariate calibration problems can be improved by optimizing the filters employed in the Discrete Wavelet Transform (DWT) with respect to the statistics of the matrix of instrumental responses. However, no attempt has been made at exploiting the statistics of the matrix of predicted parameters in the optimization process. This work addresses this issue and proposes a novel strategy for wavelet filter optimization that aims at directly minimizing the prediction error of a wavelet regression model with respect to a given validation set. Moreover, some theoretical and algorithmic aspects of the angular parameterization of wavelet filters needed for the optimization procedure are clarified. The parameterization is explained in a simple graphical manner. A requirement on the sum of the angular parameters and its implications in the optimization strategy are discussed. Finally, a procedure for obtaining the angular parameters associated to any traditional mother wavelet is provided, both as an algorithm and as a Matlab code. The proposed strategy is illustrated in a simulated multivariate calibration example involving two analytes and also in a problem of total sulphur determination in diesel samples by near-infrared (NIR) absorption spectrometry. Both the simulated and the real examples show that the proposed filter optimization procedure improves the prediction ability of wavelet regression models. Moreover, in the sulphur determination problem, the wavelet models result in a smaller prediction error than a traditional Partial Least Squares Regression (PLS) model. (C) 2003 Elsevier B.V All rights reserved.

KW - wavelet transform optimization

KW - quadrature-mirror filter banks

KW - NIR spectrometry

KW - diesel analysis

KW - total sulphur determination

KW - multivariate calibration

KW - variable selection

KW - regression

U2 - 10.1016/j.chemolab.2003.09.001

DO - 10.1016/j.chemolab.2003.09.001

M3 - Article

VL - 70

SP - 110

EP - 119

JO - Chemometrics and intelligent laboratory systems

JF - Chemometrics and intelligent laboratory systems

SN - 0169-7439

IS - 1-2

ER -