## 3 Results and Discussion
The extremely complex quaternary dataset (non-normalized) was taken as an example to build non-negative least squares model for predicting the concentrations of different oils in the mixture. The developed non-negative least squares model fit the dataset very well with small residuals (\(\pm\)5.000 arb.units). The concentrations of EVOO, rapeseed oil, corn oil and soybean oil were estimated with the root mean square errors of 3.8%, 6.5%, 5.2% and 6.9%, respectively (Figure 4). The estimated concentrations of four different oils for seven samples are shown in Figure 5 and their actual concentrations are given in Table 1 for reference. Figure 2: Pure Raman spectra of EVOO, soybean oil, corn oil and rapeseed oil. Figure 4: Actual concentration vs. estimated concentration of extra virgin olive oil (EVOO), corn oil, rapeseed oil and soybean oil calculated from quaternary mixtures of EVOO, corn, rapeseed and soybean oil (\(\
u/\
u\)%) using non-negative least squares. The solid line represents the zero-error curve. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample No** & **EVOO (\%)** & **Corn (\%)** & **Rapeseed (\%)** & **Soybean (\%)** \\ \hline
1 & 0 & 33.33 & 33.33 & 33.33 \\
2 & 10 & 30 & 30 & 30 \\
3 & 30 & 10 & 50 & 10 \\
4 & 40 & 20 & 20 & 20 \\
5 & 50 & 50 & 0 & 0 \\
6 & 70 & 10 & 10 & 10 \\
7 & 90 & 0 & 0 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The actual concentrations of EVOO, corn, rapeseed and soybean oil used in the bar plot (Figure 5). Figure 3: Raman spectra of EVOO adulterated with rapeseed oil, soybean oil and corn oil with different volume percentage. Inlet: The characteristic peak of EVOO at 1527 cm\({}^{-1}\) is enlarged for better visualization. In a real case scenario, there can be many adulterants in EVOO and the nature of the adulterants are unknown. Hence, it is easier to develop a single test that can estimate the purity of EVOO instead of determining what the adulterant is and its concentration. The PLS model along with 10-fold cross validation built on the binary, ternary, quaternary calibration datasets and the corresponding PLS regression coefficients are shown in Figure 6. The model identified the spectral changes around the major Raman peaks (844, 869, 972, 1023, 1263, 1303, 1441, 1527, 1635 and 1657 cm\({}^{-1}\)) due to the adulteration of EVOO (Figure 6). The PLS model provided root mean square errors of 3.6%, 2.8%, and 4.3%, respectively, for the binary, ternary and quaternary calibration datasets. The robustness of developed PLS model was tested on the binary, ternary and quaternary test datasets and root mean square errors of 5.8%, 1.9%, and 6.0% are obtained to estimate the purity of spiked EVOO (Figure 7). We further developed a PLS model on the combined dataset (binary + ternary + quaternary) and the respective regression coefficient is also displayed in Figure 6. The purity of spiked EVOO samples has been estimated with the root mean square error of prediction 5.2% for the calibration dataset and 5.1% for the test set (Figure 7). The PLS model developed from the combined dataset can estimate the purity of samples in spite of them being spiked with either one or more cheaper additives and being measured with different measurement parameters (power levels and acquisition times), and sample conditions (binary, ternary, quaternary oil mixtures). The amount of impurity or adulteration can also be calculated from the purity. As the purity or impurity of spiked EVOO with one or more adulterants can be estimated using a single PLS model, this study enables Raman spectroscopy to screen the quality of oil labelled as EVOO in the market without having prior knowledge of the adulterants. The ways food can be made fraudulent is in principle infinite, however, adulteration with cheap oils like corn, rapeseed or soybean oil is known to have been used [6; 8; 13; 17; 18]. The detection limit of adulteration was estimated as 13% for EVOO-rapeseed binary mixture, as the rapeseed oil is much closer to EVOO compared to other oils used in this study.