Complex-Valued Chemometrics in Spectroscopy: Partial Least Squares Regression

Complex-valued chemometrics utilizes both the absorption index and refractive index spectra. Through application of a Kramers–Kronig transformation, it can also be extended to absorbance and Raman spectra. In this work, we expand complex-valued chemometrics to include partial least squares (PLS) regression. Several strategies for implementing complex-valued PLS are explored. One approach builds on the nonlinear iterative partial least squares (NIPALS) formalism to compute real and imaginary components of the PLS solutions in parallel. Additionally, as both the real and imaginary parts can assume positive or negative values, this results in 2^N possible solutions for N components. In this case, the optimal solutions are selected using a brute-force approach combined with a nested leave-one-out (LOO) scheme. Additionally, single-value decomposition (SVD) can be directly applied to the complex matrix product of the spectral and concentration matrices. We compare these approaches using complex refractive index spectra of mixtures from the thermodynamically ideal systems benzene–toluene, benzene–cyclohexane, and benzene–carbon tetrachloride (CCl₄). In particular, when the high-wavenumber refractive index differs between the neat components, complex-valued PLS achieves errors more than an order of magnitude lower than conventional PLS based solely on the imaginary part.