Complex-Valued Chemometrics in Spectroscopy: Principal Component Regression

We have begun introducing complex-valued principal component regression (PCR) into spectroscopy. Unlike traditional methods that are constrained to either the real or imaginary axis, this approach allows principal components (PCs) to span the entire complex plane. While this added flexibility enhances modeling capabilities, it also introduces challenges, as existing tools often fail to identify optimal solutions. To address this, we explored two different strategies for computing eigenvectors. The most natural approach is to apply singular value decomposition (SVD) directly to the matrix of complex refractive index spectra. As an alternative, we combined the eigenvectors of the imaginary parts determined by SVD with their Kramers-Kronig transforms, which resulted in 2<i>^N</i> possible superpositions for <i>N</i> PCs. Although the optimal solution may still be unknown, the proposed second method for complex-valued PCR consistently outperformed conventional PCR in the systems investigated. This highlights its potential to enhance data analysis in infrared and Raman spectroscopy.