Complex-Valued Chemometrics in Spectroscopy: Classical Least Squares Regression.

Area of Science:

• Spectroscopy
• Chemometrics
• Physical Chemistry

Background:

• Classical least squares (CLS) regression is a standard method in spectroscopy.
• Spectra are often analyzed using only the real (refractive index, n) or imaginary (absorption index, k) parts of complex refractive index spectra.
• Deviations from Beer's law can complicate spectral analysis.

Purpose of the Study:

• To implement and evaluate complex-valued classical least squares (CLS) regression in spectroscopy.
• To assess the performance of complex-valued CLS compared to traditional methods using real or imaginary spectral components.
• To investigate the error detection and self-correction capabilities of the complex-valued CLS approach.

Main Methods:

• Implementation of complex-valued classical least squares (CLS) regression.
• Analysis of mid-infrared (MIR) spectra of binary mixtures (benzene-toluene, benzene-cyclohexane, benzene-carbon tetrachloride).
• Comparison of mean absolute error (MAE) using complex-valued CLS, k-spectra CLS, and n-spectra CLS.

Main Results:

• Complex-valued CLS did not consistently outperform methods using only the k or n spectra.
• A self-correction mechanism in complex-valued CLS reduced MAE by up to 26% (vs. k spectra) and 46% (vs. n spectra) for benzene-toluene mixtures.
• For benzene-cyclohexane mixtures, MAE reductions were up to 75% (vs. k) and 58% (vs. n).
• No improvement was observed for benzene-carbon tetrachloride mixtures, which show large Beer's law deviations.

Conclusions:

• Complex-valued CLS regression offers a valuable error detection and self-correction feature in spectroscopy.
• The effectiveness of complex-valued CLS depends on the spectral complexity and adherence to Beer's law.
• Further research is needed to optimize complex-valued CLS for systems with significant Beer's law deviations.