A solution to the wavelet transform optimization problem in multicomponent analysis

The wavelet transform has been shown to be an efficient tool for data treatment in multivariate calibration. However, previous works had the limitation of using fixed wavelets, which must be chosen a priori, because adjusting the wavelets to the data set involves a complex constrained optimization problem. This difficulty is overcome here and the mathematical background involved is described in detail. The proposed approach maximizes the compression performance of the quadrature-mirror filter bank used to process the spectra. After the optimization phase, the recently proposed successive projections algorithm is used to select subsets of wavelet coefficients in order to minimize collinearity problems in the regression. To demonstrate the efficiency of the entire strategy, a low-resolution ICP-AES was deliberately chosen to tackle a hard multivariate calibration problem involving the simultaneous multicomponent determination of Mn, Mo, Cr, Ni and Fe in steel samples. This analysis is intrinsically complex, due to strong collinearity and severe spectral overlapping, problems that are aggravated by the use of low-resolution optics. Moreover, there are also several regions in the spectra where the signal-to-noise ratio is poor. The results demonstrate that the optimization leads to models with better parsimony and prediction ability when compared to the fixed-wavelet approach adopted in previous papers.