Chemometrics Beyond Real Numbers: Reducing Systematic Errors with Complex-valued Regression Methods

In fifteen words or less, explain the significance of this contribution (Novel Aspect).: Introduces complex-valued chemometrics for vibrational spectroscopy, reducing systematic errors beyond traditional real-valued methods.

Abstract Text:

Chemometric analysis plays a central role in the interpretation of vibrational spectra, enabling the extraction of both quantitative and qualitative information from infrared and Raman measurements. Traditionally, these methods rely on real-valued data. However, complex-valued chemometrics—which, in the context of infrared spectroscopy, incorporates both the real and imaginary parts of the refractive index—offers a more comprehensive analytical framework. This approach becomes particularly valuable when spectra are evaluated using wave optics-based models. ¹

In this talk, I will explore complex-valued extensions of classical least squares (CLS),² inverse least squares (ILS),³ principal component regression (PCR), and partial least squares regression (PLSR) as applied to vibrational spectroscopy. Each method will be discussed in terms of its mathematical foundation, physical interpretability, and practical implementation. Special emphasis will be placed on how incorporating complex-valued information can significantly reduce systematic errors caused by unavoidable deviations from Beer’s approximation—often to below 25%.

Several case studies based on experimental data will be presented to demonstrate the benefits and potential challenges of working in the complex domain. This talk aims to provide a comprehensive introduction for spectroscopists and data scientists interested in expanding their chemometric toolbox with phase-sensitive, complex-valued methodologies.

 

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., Complex-valued Chemometrics in Spectroscopy: Classical Least Squares Regression. Appl. Spectrosc. 2025, accepted.

3.    Mayerhöfer, T. G.;  Ilchenko, O.;  Kutsyk, A.; Popp, J., Complex-valued Chemometrics in Spectroscopy: Inverse Least Square Regression. submitted 2025.

 

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.