## 4 Discussion  
In the previous sections, we have discussed the motives for using the new IUPAC formula for the calculation of the \(LOD\) of an analyte with a given experimental procedure, using a univariate or multivariate approach. The use of the old IUPAC formula should, thus, be deprecated, in \(LIBS\) as in other spectro-analytical applications. When the formulas for the \(LOD\) are applied to the case of \(LIBS\), we noticed several critical points that would partially invalidate the application of the new IUPAC definition for the \(LOD\) (as well as the old one). The failure of the Gaussian approximation for the \(LIBS\) intensity distribution and the non-homoscedasticity of the \(LIBS\) signals contrast with the hypotheses at the basis of the derivation of Equation (4). The method proposed by [PERSON] et al. [23], on the other hand, did not rely on the hypothesis of Gaussianity and homoscedasticity; however, it should be noted that the new IUPAC definition of \(LOD\) is based on the idea of quantifying the risk of false positives or false negatives for measurements performed near the \(LOD\), and the [PERSON] method (which can also be easily applied to a univariate calibration curve) does not provide information about these essential parameters. The definition of'stabilization' of the \(MRE\) with increasing the concentration of the standards is essential arbitrary; moreover, \begin{table}
\begin{tabular}{c c} \hline \hline
**Definition** & **LOD (w\%)** \\ \hline Univariate & \\ \hline Old IUPAC formula (Equation (3)) & 0.2 \\ \hline New IUPAC formula (Equation (4)) & 0.35 \\ \hline Multivariate & \\ \hline Pseudo-univariate IUPAC formula (Equation (4)) & 0.15 \\ \hline [PERSON] et al., method & 0.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Value of the \(LOD\) as calculated according to the different definitions. Figure 10: Variation of the Mean Relative Error of Cu as a function of the Cu concentration. at high concentrations of the analyte, other effects can produce errors and corresponding fluctuations of the _MRE_. On the other hand, a definition based on the analysis of the _MRE_ is interesting from an operative perspective, although it seems more related to the problem of the quantification of the analyte concentration, rather than just on its detection. The issue of determining the _LOD_ of an analyte by _LIBS_ is, thus, very complex, both in the univariate and in the multivariate case. The discussion and the examples shown in this paper should suggest some care in providing figures for the _LOD_ of the analytes under study without checking the validity of the Gaussian distribution and homoscedasticity of the _LIBS_ signals near the _LOD_. In particular, non-Gaussian profiles of the intensity distribution may arise from spectral selection treatments when, as in the example reported in this paper, the outlier spectra are removed. Since this kind of treatment has become customary in the analysis of large quantities of data, their effect on the calculation of the _LOD_ should be checked. As a general suggestion, in a multivariate as well as univariate quantitative approach, it is always appropriate to check the histogram of the relevant _LIBS_ signals, comparing them with the histogram of the blank signal. To guarantee that false positives and false negatives would have the same probability at the _LOD_ is on the other hand necessary to use for the calculation of only the data with a standard deviation very close to the one of the blank. Samples characterized by high concentrations of the analyte should be excluded from the calculation because of the possible non-linear effect, but also because the standard deviation of _LIBS_ signals increases with the square root of the signal. Moreover, points at high concentration in the calibration curve contribute to the increase in the leverage, which must be considered in the calculation of the _LOD_. Conceptualization, V.P. ; methodology, V.P. ; software, V.P. ; validation, V.P., F.P. and B.C. ([PERSON]); formal analysis, V.P. and B.C.