1. 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, Athens, Greece cTheoretical and Physical Chemistry Institute, National Hellenic Research Foundation,
48 Vassileos Constantinou Ave., Athens, Greece * Corresponding author. Tel.: ; fax address: (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
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)

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
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
yi = 0.807xi 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 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,