Complex-Valued Chemometrics in Spectroscopy: Classical Least Squares Regression

We present the first implementation of complex-valued classical least squares (CLS) regression in spectroscopy. Although the results indicate that complex-valued CLS does not outperform methods that utilize only the more suitable part of the complex refractive index spectra, it includes an error detection feature that enables a self-correction mechanism. This mechanism decreases the mean absolute error (MAE) to approximately 26% relative to using only the mid-infrared (MIR) absorption index (k) spectra for CLS, and to about 46% relative to using only the MIR refractive index (n) spectra of benzene–toluene mixtures. For benzene–cyclohexane mixtures, the MAE was reduced to approximately 75% relative to the k spectra and 58% relative to the n spectra. In contrast, for benzene–carbon tetrachloride (CCl₄) mixtures, i.e., a system that exhibits particularly large deviations from Beer’s law, no improvement over the n spectra was observed; the n-based MAE was 81% relative to the k spectra. These percentages may further vary based on the complexity of the system, the spectral regions selected for CLS and the corresponding deviations from Beer’s approximation.