Typical Spectral Multivariate Calibration

1. Food adulteration analysis without laboratory prepared or determined reference food adulterant values John H. Kalivasa*, Constantinos A. Georgioub, Marianna Moirab, IliasTsafarasb, Eleftherios A. Petrakisb, George A. MousdiscaDepartmentof Chemistry, Idaho State University, Pocatello, Idaho 83209, USAbChemistryLaboratory, Agricultural University of Athens, 75 IeraOdos, 118 55 Athens, GreececTheoreticaland Physical Chemistry Institute, National Hellenic Research Foundation, 48 VassileosConstantinou Ave., 116 35 Athens, Greece* Corresponding author. Tel.: +01 208-282-2726; fax +01 208-282-4373.E-mail address: kalijohn@isu.edu (J. Kalivas).

2. Typical Spectral Multivariate Calibration • y = Xb y = m x 1 vector of analyte reference values for m calibration samples X = m x n matrix of spectra for n wavelengths b = n x 1 regression vector Biased regression solutions such as Tikhonov regularization (TR), RR, PLS, and PCR or use MLR • Biased methods require tuning parameter selection • MLR requires m≥ p (variable selection)

3. Prediction Equation Analysis • Prediction • Pure component interferent and other matrix effect spectra are not always known X = Measured spectrum ya = analyte quantity ka = pure component analyte spectrum yN = interferent quantities KN = pure component interferent and other non-analyte spectra r = random noise • Assume a linear Beer-Lambert law type relationship N = Non-analyte spectra (KN scaled by yN)

4. Conditions for Accurate Prediction • Select respective tuning parameters to obtain • Typically not possible to satisfy all three conditions simultaneously by varying respective model tuning parameters

5. Compromise PCTR Model - • Minimizing the sum requires a tradeoff between the three conditions • The closer the three conditions are met, the more likely • Updating the non-matrix effected PC ka to predict in current conditions (spanned by N) • No reference values needed

6. Extra Virgin Olive Oil Adulteration • EVOO samples: Crete, Peloponnese, and Zakynthos • RR calibration y: 56 samples spiked 5, 10, and 15% (wt/wt) sunflower oil • ka: PC sunflower oil, 1 sample • N: PC EVOO, 25 samples • Validation: 22 spiked samples • Synchronous fluorescence spectra 270 to 340 nm at Δλ=20 nm Zakynthos

7. Model Updating From PC Sunflower • Updated PC models better than a full calibration yi = 0.442xi + 0.048 yi = 0.807xi - 0.0074 yi

8. Using PLS • PLS (and other methods) can also be used • With PLS, the PLS latent vectors (PLS factors) replace the η values PCTR PLS

9. Other TR Variants Kalivas, J.H. (2012). Overview of two-norm (L2) and one-norm (L1) Tikhonov regularization variants for full wavelength or sparse spectral multivariate calibration models or maintenance. Journal of Chemometrics,26, 218-230.