## 3 Results and Discussion
### Quantification of the Level of the Adulterant Using Partial Least Squares Regression (PLSR)  
Having established with PCA that there were indeed consistent trends which could be largely related to the level of adulteration in EVOO using Raman spectroscopy and SORS, we aimed to quantify the level of adulteration. PLSR is a powerful multivariate regression algorithm that can be used to associate input data (Raman and SORS spectra) with the level of adulteration. This supervised learning algorithm requires a calibration (learning) phase before it can be tested for its predictive ability. We again exemplify using the pomace OO as the adulterant in EVOO where data were collected using a handheld 1064 nm Raman spectrometer. This data set comprised 0-100% pomace OO in 100-0% EVOO in 5% steps (i.e., 21 mixtures) and five replicate spectra were collected from each mixture (105 spectra in total). As detailed above in the  
Figure 3: PCA scores plots of Raman spectral data collected using a handheld instrument with 1064 nm excitation showing the adulteration of authentic EVOO with pomace olive oil. The adulteration has been performed in increments of 5% and a rainbow colour gradient from blue for pure EVOO to red for pure pomace OO is shown. The values in parentheses on the axes labels are the percentage of the total explained variance (TEV) for each principal component (PC). data analysis section for PLSR model calibration, we used a resampling technique based on bootstrapping. In this sampling process, samples to be used for model calibration are generated by choosing a sample randomly from the total data set (21 samples \(\times\) five replicates), which is placed in the training set along with all five replicates. The sample is then replaced into the total data set and selection proceeds until there are the same number of samples in the training data as there are in the whole data set (i.e., 105 Raman spectra (\(x\)-data) with their corresponding % pomace OO (\(y\)-data)). This consequently leads to the generation of a series of training sets, which on average includes 63.2% of all samples, and a series of test sets, which includes the remaining 36.8% of samples [36]. For this and all studies, PLSR was conducted using \(n=1000\) bootstrap selections, with an internal cross-validation step to optimise the number of latent variables (PLS factors) to use for PLSR. All 1000 models were then challenged using the 1000 test sets and we report the data from these training set and test set combinations in terms of linearity where the predictions are regressed against the known adulterant levels using coefficients of determination for the training (\(R^{2}\)) and test sets (\(Q^{2}\)). This is in addition to the root mean squared error of calibration (RMSEC) for the training set and RMSE of prediction (RMESP) for the test set, which are expressed as % RMSE. Finally, we also calculated the limit of detection (LOD) from these PLS models. The results of the PLSR prediction for pomace OO adulteration are shown in Figure 4, where we have plotted the results from the 1000 test sets only and show the average and standard deviations of these predictions. It is clear from this plot that the predictions are excellent as they are very close to the expected \(y=x\) line for a perfect model, and the best fit line for these models is \(y=-0.7054+1.0272^{\circ}x\). The linearity is excellent with an \(R^{2}\) of 0.97 and \(Q^{2}\) of 0.95 (a perfect correlation would be 1) and the error is also low with RMSEC and RMSEP calculated as just 5.32% and 6.50%, respectively, for the calibration and test sets. Finally, the LOD was 11.50%. These figures of merit are shown in Table 1 along with all the PLSR analyses where data were collected within vials with the three different excitation wavelengths. Additional PLSR results plots for these models are provided in the Supplementary Materials (Figures S8-S10).