## 3 Results and Discussion
### Calibration and Validation
The calibration phase started with a PLS multivariate regression between the \(X_{\text{cal}}\) vs. \(Y_{\text{cal}}\) matrices, performed sequentially for each individual element. This approach generated 19 distinct models, one for each element. Subsequently, a VIP analysis was conducted on each PLS model to identify the most correlated variables (i.e., emissions at a specific wavelength) with the column response (i.e., element by element concentration in the Y matrix) using VIP scores [75]. This analysis facilitated variable selection, enabling a refined PLS regression using only the significant variables. The performances of the VIP analysis  
Figure 4: Effect of the pre-treatments on the LIBS spectra of standard samples along the entire spectral range. Each color represents the spectral profile of a standard sample. and the number of Latent Variables (LV) to be retained in the PLS-VIP models were assessed by RMSECV. After this calibration step, the validation step consisted of projecting \(\mathrm{X_{val}}\) and \(\mathrm{Y_{val}}\) matrices of the standard samples, never used across calibration and cross-validation steps, onto the PLS-VIP models to test its prediction capabilities in terms of element concentration. A graphical representation of the agreement between predicted and measured concentration of Na, Si, K, and Al is provided in Figure 5. These elements were selected because they are among the major constituents of the standard samples and the corresponding predictive models reflect different performance conditions. Sodium (Na) was included because its model demonstrated the best overall performance, in terms of RMSE and R\({}^{2}\). Potassium (K) and Silicon (Si) were selected due to their strong performance in the calibration phase, although a decline was observed during validation step. Aluminum (Al), on the other hand, showed satisfactory calibration results, but exhibited a low correlation coefficient in the validation phase. In each plot, the standard samples used during the calibration phase are represented by grey circles, whereas those employed for validation are depicted as red diamonds. Moreover, the red line represents the regression slope of the predicted vs. measured concentrations during the validation phase, while the green line denotes the ideal 1:1 relationship, indicating perfect accuracy. Figure 5: Plot of predicted (obtained from the PLS-VIP model) vs. measured concentration for Na, Si, K, and Al. Grey circles: standard samples used in the calibration step; red diamonds: standard samples used for validation; red lines: fitting of predicted vs. measured concentrations values during validation; green lines: ideal fitting. Regarding Na, the fitted line (red) and the ideal line (green) are nearly superimposed, revealing excellent predictive capability. Furthermore, reliable predictions are achieved across the entire Na concentration range, as evidenced by the distribution of test samples (i.e., red diamonds) throughout the graph. A similar observation applies to Si, where the fitting line closely follows the ideal one, reflecting satisfying concentration predictions. This indicates the absence of bias (i.e., no systematic deviation from the expected value) during the validation phase. However, in this case, while the samples of validation set cover the full range of Si concentrations, their distribution is skewed towards lower values, with only one validation sample (GB07103) exceeding a Si concentration of 3\(\cdot\)10\({}^{5}\) ppm. As far as K and Al are concerned, red line and green line are not fully superimposed, indicating the presence of bias. However, if, for K, the validation samples are spread in a manner that traces Si, with more data points at low concentration and one at high concentration (GB07103), in the case of Al validation, the samples are clustered within a narrow concentration range (4 to 7\(\cdot\)10\({}^{4}\) ppm).