Complex-Valued Chemometrics in Spectroscopy: Inverse Least Squares Regression

Inverse least squares (ILS) regression is an advancement of classical least squares (CLS) regression, enabling the calculation of concentrations without requiring prior knowledge of the number of components in a mixture. Complex-valued ILS further enhances the performance of ILS by incorporating the complex refractive index function, as demonstrated in the thermodynamically ideal mixtures of benzene–toluene and benzene–cyclohexane. In both systems, the mean absolute error can be reduced by over 50% using the leave-one-out cross-validation (LVOOCV) scheme with complex-valued ILS. Additional error reduction is achievable by leveraging correlations between the errors and the imaginary components of the concentrations or volume fractions. Since the complex refractive index function can be conveniently determined using conventional infrared spectroscopy through the Kramers–Kronig relations, we believe that complex-valued machine learning has the potential to significantly advance analytical applications.