## 3 Results and Discussion
### Calibration and Validation
However, a comprehensive assessment of the predictive capabilities requires considering all quality parameters, as the evaluation of the correlation coefficient alone does not always provide the complete picture. For example, despite the relatively high correlation coefficient for Be (R\({}^{2}\)val = 0.63) in the validation phase, the PLS-VIP model cannot be considered fully validated because only two samples in the validation set exhibited concentrations significantly above 1 ppm. As a general result, the models obtained for trace elements must be regarded as tentative. Regarding the number of variables selected, it is worth noting that for Be, 92% of the total spectral variables (21,584 out of 23,401) were identified as significant in the PLS model. This is another indication that the spectra obtained from the standard samples poorly correlates with the Be concentration. The referenced scientific literature on PLS regression models (see, e.g., [79; 80]) states that the number of samples in a dataset should be sufficiently large in relation to the number of predictor variables to ensure the stability of the model. Although no specific values have been universally defined (apart from the obvious requirement that the number of samples must exceed the number of predictors), this is a crucial aspect that must be carefully considered. In the case presented in this study, the number of standard samples was seventeen, a statistically meaningful quantity, yet not sufficient to build stable models with an excessive number of predictor variables (i.e., latent variables). In Table 3, the number of latent variables retained in each VIP-PLS model is reported. Based on the analysis of these values, it can be stated that the dataset size was adequate to build stable models in all cases where fewer than six latent variables were retained. For those elements where more than five latent variables were retained (i.e., Mn, Zn, Sr, Mo), the dataset size is just sufficient to consider the model reliable. However, in order to increase model robustness, cross-validation was applied during the calibration step. The models that appeared to be less stable--those including six latent variables--are precisely those developed for trace elements, suggesting that the calibration of the h-LIBS device to quantitatively determine the concentration of trace elements with a sufficient precision level must be performedwith more than twenty standard samples. The poor predictive ability obtained for trace elements also depended on this aspect. The LOD values, estimated during the calibration phase, are consistently below 10,000 ppm (1%), with the exception of Si (15,246 ppm, \(\sim\)1.5%), which is the most abundant element in the standard samples, and Fe (17,000 ppm, \(\sim\)1.7%). Notably, the LOD for Fe is relatively high and not fully satisfying. As was previously mentioned above, two different baseline pre-treatments were tested in combination with a SNV normalization, namely, the Whittaker filter (later indicated as Whittaker+SNV) and airPLS (later airPLS+SNV). A numerical comparison of the fitting results, in terms of RMSE and R\({}^{2}\), both in the calibration and validation phases, is provided in Table 4 for the four elements taken as example (i.e., Na, Si, Al, K). Relative diagrams reporting predicted vs. measured concentration are reported as Supporting Information. The comparison indicates that Whittaker+SNV generally provides better performance across the calibration and validation phases. In particular, where the model for Na is concerned, all the parameters considered perform better in the case of Whittaker+SNV (i.e., lower RMSE and higher R\({}^{2}\)). In the case of Si, the model obtained with a Whittaker+SNV pre-treatment achieved lower RMSE and higher R\({}^{2}\) values for the calibration phase compared to airPLS, while the contrary is observed for the validation step. A reversed pattern is shown for K and Al, for which airPLS+SNV performed better in the calibration step, while the opposite was observed for the validation step.