Baseline and interferent correction by the Tikhonov regularization framework for linear least squares modeling

Summary

Spectroscopic data are usually perturbed by noise from various sources that should be removed prior to model calibration. After conducting a preprocessing step to eliminate unwanted multiplicative effects (effects that scale the pure signal in a multiplicative manner), we discuss how to correct a model for unwanted additive effects in the spectra. Our approach is described within the Tikhonov regularization (TR) framework for linear regression model building, and our focus is on ignoring the influence of noninformative polynomial trends. This is obtained by including an additional criterion in the TR problem penalizing the resulting regression coefficients away from a selected set of possibly disturbing directions in the sample space. The presented method builds on the extended multiplicative signal correction, and we compare the two approaches on several real data sets showing that the suggested TR‐based method may improve the predictive power of the resulting model. We discuss the possibilities of imposing smoothness in the calculation of regression coefficients as well as imposing selection of wavelength regions within the TR framework. To implement TR efficiently in the model building, we use an algorithm that is heavily based on the singular value decomposition. Because of some favorable properties of the singular value decomposition, it is possible to explore the models (including their generalized cross‐validation error estimates) associated with a large number of regularization parameter values at low computational cost.