Optimal wavelet filter construction using X and Y data

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.