Complex-Valued Chemometrics for Composition Analysis

In this tutorial, Thomas G. Mayerhöfer and Jürgen Popp introduce complex-valued chemometrics as a more physically grounded alternative to traditional intensity-based spectroscopy measurement methods. By incorporating both the real and imaginary parts of the complex refractive index of a sample, this approach preserves phase information and improves linearity with sample analyte concentration. The result is more robust and interpretable multivariate models, especially in systems affected by nonlinear effects or strong solvent and analyte interactions.

Conventional chemometric techniques typically rely on real-valued input data, most often absorbance or transmission spectra (1–4). These methods are inherently limited by their reliance on intensity-based measurements, which are subject to systematic errors stemming from reflection losses, interfacial effects, local field effects, chemical interactions and wavelength-dependent effective depths (especially in ATR). To mitigate such non-linear effects, data are often preprocessed using empirical methods like baseline correction, normalization, or multiplicative scatter correction—all of which introduce additional assumptions and can reduce physical interpretability.

Complex-valued chemometrics offers a physically grounded alternative by incorporating not only the imaginary part of the complex refractive index (related to absorption) but also the real part, which accounts for dispersion. This complex representation captures the full electromagnetic response of a material and retains the phase relationships that are discarded in intensity-only approaches. As a result, in particular also because of wave optics-based treatment, the data become inherently more linear with respect to volume or concentration—one of the key assumptions of linear chemometric models like classical least squares (CLS), inverse least squares (ILS), principal component analysis (PCA), or partial least square (PLS).

A major advantage is that systematic deviations from linearity—due to local field effects or chemical interactions that vary with concentration and cannot be corrected analytically—are significantly reduced. The inclusion of the real part often reveals inconsistencies in conventional models and improves robustness in multivariate regression, especially for complex or interacting systems.

How to Obtain Complex Refractive Index Spectra

Complex-valued spectra can be obtained in several ways:

1. Although complex refractive index spectra may seem challenging to acquire, modern techniques like spectroscopic ellipsometry enable direct measurement of the complex refractive index function.⁵ In addition, emerging approaches—such as infrared refraction and dispersion spectroscopy or field-resolved infrared spectroscopy—are making the direct determination of the refractive index function increasingly accessible (6–12).
2. Wave Optics-Based Corrections: For broader applicability, complex spectra can be generated from conventional intensity spectra (e.g., absorbance or reflectance) using wave optics and dispersion theory. Specifically, the Kramers–Kronig transformation allows the derivation of the real part from the imaginary part, assuming causality and certain boundary conditions. Alternatively, iterative wave-optics models based on Fresnel equations or transfer matrix methods can be used to invert measured transmission, transflection, and attenuated total reflection spectra and retrieve the complex refractive index, accounting for realistic sample geometries and measurement setups (13–22). An overview can be found in reference (23).

This approach replaces empirical preprocessing with a physically meaningful correction, laying the foundation for a new generation of physically consistent, complex-valued chemometrics (24,25). An example is presented in Figure 1, where the ATR absorbance spectra of the thermodynamically ideal system benzene-toluene are presented together with the corresponding n- and k-spectra.

Data Setup

Computing the Calibration Matrix

Predicting Unknown Compositions

Correcting Systematic Errors

Inverse Modeling for Unknown Components

Important Considerations

By following this framework, you can accurately perform complex-valued classical and inverse least square analysis, leveraging both the real and imaginary parts of your spectral data to enhance robustness and reduce systematic error. An example for CLS of the system benzene-toluene is provided in Figure 2.

Extensions to PCA and PLS are in preparation as well as an extension rendering it possible to employ everything also to Raman spectra.

References

(1) Wolberg, J. Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments; Springer: Berlin, 2006.

(2) Olivieri, A. Introduction to Multivariate Calibration: A Practical Approach; Springer International Publishing: Cham, 2024.

(3) Kramer, R. Chemometric Techniques for Quantitative Analysis; CRC Press: Boca Raton, FL, 1998.

(4) Mark, H.; Workman, J. Chemometrics in Spectroscopy, Rev. 2nd ed.; Academic Press: Cambridge, MA, 2021.

(5) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland: Amsterdam, 1987.

(6) Mayerhöfer, T. G.; Costa, W. D. P.; Popp, J. Combining Infrared Refraction and Attenuated Total Reflection Spectroscopy. Appl. Spectrosc. 2024, 78 (6), 00037028241283050. https://doi.org/10.1177/00037028241283050

(7) Mayerhöfer, T. G.; Ivanovski, V.; Popp, J. Infrared Refraction Spectroscopy – Kramers-Kronig Analysis Revisited. Spectrochim. Acta A 2022, 270, 120799. https://doi.org/10.1016/j.saa.2021.120799

(8) Mayerhöfer, T. G.; Ivanovski, V.; Popp, J. Infrared Refraction Spectroscopy. Appl. Spectrosc. 2021, 75 (12), 1526–1531. https://doi.org/10.1177/00037028211036761

(9) Dabrowska, A.; Schwaighofer, A.; Lendl, B. Mid-Infrared Dispersion Spectroscopy as a Tool for Monitoring Time-Resolved Chemical Reactions on the Examples of Enzyme Kinetics and Mutarotation of Sugars. Appl. Spectrosc. 2024, 78 (9), 982–992. https://doi.org/10.1177/00037028241258109

(10) Dabrowska, A.; Lindner, S.; Schwaighofer, A.; Lendl, B. Mid-IR Dispersion Spectroscopy – A New Avenue for Liquid Phase Analysis. Spectrochim. Acta A 2023, 286, 122014. https://doi.org/10.1016/j.saa.2022.122014

(11) Zorin, I.; Gattinger, P.; Ricchiuti, G.; Lendl, B.; Heise, B.; Brandstetter, M. All-Mirror Wavefront Division Interferometer for Fourier Transform Spectrometry across Multiple Spectral Ranges. Opt. Express 2025, 33 (1), 867–884. https://doi.org/10.1364/OE.545267

(12) Pupeza, I.; Huber, M.; Trubetskov, M.; Schweinberger, W.; Hussain, S. A.; Hofer, C.; Fritsch, K.; Poetzlberger, M.; Vamos, L.; Fill, E.; et al. Field-Resolved Infrared Spectroscopy of Biological Systems. Nature 2020, 577 (7788), 52–59. https://doi.org/10.1038/s41586-019-1850-7

(13) Bertie, J. E.; Eysel, H. H. Infrared Intensities of Liquids I: Determination of Infrared Optical and Dielectric Constants by FT-IR Using the CIRCLE ATR Cell. Appl. Spectrosc. 1985, 39 (3), 392–401.

(14) Ohta, K.; Ishida, H. Efficient Method for Optical Constant Determination by FTIR-ATR. In Proceedings of the 8th International Conference on Fourier Transform Spectroscopy; Heise, H. M., Korte, E. H., Siesler, H. W., Eds.; 1992; pp 604–605.

(15) Mayerhöfer, T. G.; Costa, W. D. P.; Popp, J. Sophisticated Attenuated Total Reflection Correction within Seconds for Unpolarized Incident Light at 45°. Appl. Spectrosc. 2024, 78 (3), 321–328. https://doi.org/10.1177/00037028231219528

(16) Mayerhöfer, T. G.; Popp, J. Understanding Advanced Attenuated Total Reflection Correction: The Low Absorbance Assumption. Appl. Spectrosc. 2025, 79 (2), 298–305. https://doi.org/10.1177/00037028241268024

(17) Mayerhöfer, T. G.; Popp, J. Developing Correction Methods by Revisiting the Concept of Effective Thickness in Attenuated Total Reflection Spectroscopy. Appl. Spectrosc. 2025, 79 (3), 465–472. https://doi.org/10.1177/00037028241290838

(18) Mayerhöfer, T. G.; Popp, J. Understanding and Employing (Non-)Linearities in Attenuated Total Reflection Spectroscopy. Appl. Spectrosc. 2025, 0 (0), 00037028251317540. https://doi.org/10.1177/00037028251317540

(19) Mayerhöfer, T. G.; Popp, J. Attenuated Total and Internal Reflection Correction Based on Fresnel’s Equations Beyond the Low Absorption Assumption. Appl. Spectrosc. 2025, accepted.

(20) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from the Infrared Transflectance Spectra of Thin Films on Metallic Substrates: A Fast and Wave Optics Conform Solution. Analyst 2018, 143 (13), 3164–3175. https://doi.org/10.1039/C8AN00526E

(21) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from Infrared Spectra – Interference Fringes Re-Revisited. Analyst 2020, 145 (9), 3385–3394. https://doi.org/10.1039/D0AN00062K

(22) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. CaF₂: An Ideal Substrate Material for Infrared Spectroscopy? Anal. Chem. 2020, 92 (13), 9024–9031. https://doi.org/10.1021/acs.analchem.0c01158

(23) Mayerhöfer, T. G. Wave Optics in Infrared Spectroscopy – Theory, Simulation and Modeling; Elsevier: Philadelphia, 2024.

(24) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Pop, J. Complex-Valued Chemometrics in Spectroscopy: Classical Least Squares Regression. Appl. Spectrosc. 2025, 0 (0), 00037028251343908. https://doi.org/10.1177/00037028251343908

(25) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Complex-Valued Chemometrics in Spectroscopy: Inverse Least Square Regression. Submitted 2025.

(26) Mayerhöfer, T. G.; Pahlow, S.; Popp, J. The Bouguer–Beer–Lambert Law: Shining Light on the Obscure. ChemPhysChem 2020, 21 (18), 2029–2046. https://doi.org/10.1002/cphc.202000464

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