Complex-Valued Chemometrics for Analyzing Absorbance or Raman Spectra

Complex-valued chemometrics offers a promising extension of classical regression methods by exploiting both real and imaginary spectral components. Here, we show that conventional absorbance (χ⁽¹⁾) and Raman (χ⁽³⁾) spectra can be transformed into complex-valued forms by combining the measured intensities as imaginary parts with their Kramers-Kronig-derived real parts. We benchmark four regression methods─classical least squares (CLS), inverse least squares (ILS), principal component regression (PCR), and partial least-squares regression (PLSR)─across four representative systems: the quasi-ideal benzene-toluene and benzene-cyclohexane mixtures, the nonideal acetone-chloroform mixture, and blood plasma spiked with glucose and urea. Compared to conventional chemometrics, complex-valued approaches consistently reduce prediction errors (MAE, RMSE, and R²). Implementation is computationally inexpensive, since the Kramers-Kronig transform of absorbance or Raman spectra can be obtained within seconds using FFT-based routines, even for large data sets. Software implementation is straightforward, and programs can be adapted within minutes using standard environments such as Mathematica. Surprisingly, complex-valued ILS matches or surpasses complex-valued PLSR, echoing earlier results in infrared spectroscopy, using the complex refractive index function, and suggesting a re-evaluation of regression hierarchies when complex spectra are available. These findings demonstrate that complex-valued chemometrics is broadly applicable, physically grounded, and capable of enhancing both classical and modern regression strategies in analytical spectroscopy.