## 3 Results and Discussion
### Multivariate Analysis  
Before applying the PLS model, the LIBS spectra were normalized using the SNV method and an SG filter to smooth and reduce noise. In multivariate analysis, all the elemental emission lines selected were used simultaneously together with a portion of the lateral background, and the number of components used in the model was optimized to achieve the smallest error.  
Figure 6 shows raw LIBS spectra for all the samples compared with detailed SNV and SNV + SG spectra in a specific spectral region (310-365 nm) and illustrates how the filter directly affects the transitions and the background attenuating fluctuations. As a verification criterion to estimate the noise reduction achieved by the applied filters, three spectral configurations were considered, i.e., raw without any treatment, SNV normalized, and SNV-SG corrected of the transition of Fe II (266.663 nm) (Figure 7). This transition was chosen because it was apparently not inferring interferences, and the concentration of Fe in the sample was very high and practically constant. To ensure an identical analytical condition for the three configurations, a baseline correction and normalization was performed to guarantee that the set of spectra remained within a minimum of zero and a maximum of one in intensity. This process  
Figure 5: Univariate calibration (**a**,**c**) and validation (**b**,**d**) curves for Si and Ti versus certified X-ray fluorescence (XRF) concentrations.  
did not affect the fluctuation among spectra (Figure 7), as it was applied to the set of spectra as a whole and not individually to each spectrum. A sum of pixel intensities was performed, which resulted in a transition intensity/area value (Riemann sum) for each spectrum. Then, the mean of these intensities and the standard deviation were calculated. The percentage value of the standard deviation related to the mean value was 31.5% for the raw spectra set, 3.7% for the SNV set, and 1.9% for the SNV-SG set. These results show that the process as a whole was able to reduce global spectral fluctuations and probably was a collaborative factor in improving the multivariate model.  
Figure 6: Raw, standard normal variate (SNV), and SNV + Savitzky–Golay (SG) LIBS spectra in a specific spectral region (310–365 nm).  
The calibration and the validation curves of the elements C, Mn, P, Si, and Ti constructed by applying PLS to the areas of the corresponding elemental lines versus the certified concentrations are shown in Figure 8. The R and the RMSE values achieved for the calibration curves and MAPE and LOD for the validation curves using univariate and multivariate models (Table 1) indicated a significant improvement achieved by PLS and PLS + SG data treatment. In particular, the R (above 0.93) and the  
Figure 7: Raw, SNV, and SNV + SG LIBS spectra of the transition of Fe II at 266.663 nm.  
RMSE (below 0.07, with the exception of C) values of the calibration curves constructed using the PLS approach were very satisfactory for all elements, and especially for Si and Ti. Additionally, the MAPE percentage values achieved in the validation by PLS (below 15% for all elements except Mn) were also satisfactory.  
Figure 8: _Cont._  
Although the \(h\)LIBS approach might present intrinsic laser fluctuations leading to an unpredictable interaction between radiation and matter, overfitting processes could be totally avoided by performing an efficient cross-validation procedure. Thus, the error due to equipment and laser matter interaction could be minimized, but the error associated with the reference technique could not.