Quantifying Spectral Nonlinearity Using Complex-valued Classical Least Squares Regression and 2D-Correlation analysis

The Beer-Lambert law is regarded as a foundational concept in spectroscopy, establishing a linear relationship between absorbance, molar concentration, and the optical path length within a sample. However, the law's original formulation predates Maxwell’s equations - developed at a time when light was not yet understood as electromagnetic radiation capable of inducing polarization in matter. This oversight means that the law's prediction of linearity is only an approximation; the polarization of matter by light introduces subtle but systematic deviations.¹

In earlier work, we demonstrated this deviation qualitatively using two-dimensional (2D) correlation analysis.² By extending this approach with "smart error sums," we are now able to quantify the effect.³ These findings are compared with results from complex-valued Classical Least Squares (CV-CLS) regression,⁴ using model spectra derived from the Lorentz-Lorenz relation. The resulting systematic errors closely resemble those found in real-world mixtures.

A notable feature of CV-CLS is that such systematic errors appear as complex-valued volume fractions.⁴ The imaginary component-arising directly from these deviations-can be leveraged to reduce the total prediction error by 25% or more when CV-CLS is applied instead of conventional CLS to experimental spectra. Methods like complex-valued chemometric regression and 2D-correlation analysis have the potential to significantly expand the spectroscopic toolbox, introducing phase-sensitive, complex-valued approaches that offer deeper insight into spectral data.

References

1. Mayerhöfer, T. G., Wave Optics in Infrared Spectroscopy - Theory, Simulation and Modeling. Elsevier: Philadelphia, 2024.

2. Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Beyond Beer’s Law: Quasi-Ideal Binary Liquid Mixtures. Appl. Spectrosc. 2022, 76 (1), 92-104.

3. Mayerhöfer, T. G.; Noda, I.; Pahlow, S.; Heintzmann, R.; Popp, J. Correcting systematic errors by hybrid 2D correlation loss functions in nonlinear inverse modelling. PLOS ONE 2023, 18 (4), e0284723.

4. Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Complex-Valued Chemometrics in Spectroscopy: Classical Least Squares Regression. Appl. Spectrosc. 2025, accepted.

Acknowledgements

Financial support from the EU, the ”Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft”, the ”Thüringer Aufbaubank”, the Federal Ministry of Education and Research, Germany (BMBF), the German Science Foundation, the “Fonds der Chemischen Industrie” and the Carl-Zeiss Foundation is gratefully acknowledged.