Food adulteration analysis without laboratory prepared or determined reference food adulterant values John H. Kalivasa*, Constantinos A. Georgioub, Marianna Moirab, Ilias Tsafarasb, Eleftherios A. Petrakisb, George A. Mousdisc aDepartment of Chemistry, Idaho State University, Pocatello, Idaho 83209, USA bChemistry Laboratory, Agricultural University of Athens, 75 Iera Odos, 118 55 Athens, Greece cTheoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vassileos Constantinou 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).

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)

Prediction Equation Analysis Assume a linear Beer-Lambert law type relationship Pure component interferent and other matrix effect spectra are not always known Prediction 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 N = Non-analyte spectra (KN scaled by yN)

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

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 -

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

Model Updating From PC Sunflower Method (No. Samples) RMSEV R2 η λ PCTR (26) 2.6e-7 0.031 0.882 9.1e3 0.0036 RR (56) 4.0e-7 0.028 0.649 1.9e5 - RR with PCTR samples (26) 0.141 0.320 1.5e5 Updated PC models better than a full calibration yi = 0.442xi + 0.048 yi = 0.807xi - 0.0074 yi

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

Other TR Variants Expression Comments RR when L = I; includes variable selection (sparse model) when L = diag of full model vector; approximates 1-norm Model updating to standardization set M; can include sparse model Sparse model updating with L = I; approximates 0-norm with diag L Model updating with robustness to M Adaptive LASSO and LASSO when L = I Claerbout et al., Geophysics, 38 (1973) 826 Elastic net 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.