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

Thomas G Mayerhöfer1,2, Oleksii Ilchenko3,4, Andrii Kutsyk4

  • 1Leibniz Institute of Photonic Technology (IPHT), Jena, Germany.

Applied Spectroscopy
|August 6, 2025
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Summary
This summary is machine-generated.

Complex-valued inverse least squares (ILS) regression improves mixture analysis by using complex refractive index data. This advanced method significantly reduces errors in concentration calculations for chemical mixtures.

Keywords:
Ideal binary liquid mixturescomplex refractive index spectrainverse least squares regression

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Area of Science:

  • Analytical Chemistry
  • Spectroscopy
  • Chemometrics

Background:

  • Classical least squares (CLS) regression requires prior knowledge of mixture components.
  • Inverse least squares (ILS) regression overcomes this limitation.
  • Complex-valued ILS further enhances ILS by utilizing complex refractive index data.

Purpose of the Study:

  • To demonstrate the effectiveness of complex-valued ILS in determining component concentrations without prior knowledge.
  • To showcase performance enhancements in thermodynamically ideal mixtures.
  • To explore error reduction strategies using complex-valued data.

Main Methods:

  • Application of complex-valued inverse least squares (ILS) regression.
  • Utilizing the complex refractive index function.
  • Employing leave-one-out cross-validation (LVOOCV) scheme.
  • Leveraging correlations between errors and imaginary concentration components.

Main Results:

  • Achieved over 50% reduction in mean absolute error for benzene-toluene and benzene-cyclohexane mixtures.
  • Demonstrated significant error reduction by incorporating complex refractive index data.
  • Identified additional error reduction through error-imaginary component correlations.

Conclusions:

  • Complex-valued ILS offers superior performance for mixture analysis compared to traditional methods.
  • The complex refractive index function, obtainable via infrared spectroscopy and Kramers-Kronig relations, is key to this improvement.
  • Complex-valued machine learning holds significant potential for advancing analytical applications.