## 3 Results
### LOD of Cu in Cast Iron from Multivariate Calibration  
In ref. [31], studying the same samples here described, the authors have demonstrated the substantial improvement in the analytical performances of the _LIBS_ technique that can be obtained using a multivariate approach, instead of limiting the analysis to a single predictor of the concentration, as is done using a univariate approach.  
Adopting the same strategy, we trained a simple Artificial Neural Network [15], using as input/predictors all the spectral points in the range previously considered (32 points between 324.5 nm and 328.0 nm). We used a single neuron in the hidden layer, with a sigmoid transfer function.  
In the framework of a multivariate analysis, the equivalent of a calibration curve is expressed by the regression graph, which represents the results of the ANN vs. the nominal concentrations. The curve obtained for the Cu concentration is shown in Figure 8.  
As expected, the regression curve obtained using a multivariate approach is more precise than the corresponding univariate calibration curve (Figure 5). Pushing forward this analogy, the _LOD_ corresponding to the regression curve can be calculated in the same way as for the calibration curve (Equation (4)). In this case (pseudo-univariate estimation), the _LOD_ for Cu improves substantially, due to the reduction of \(\sigma\) obtained using a multivariate approach. Applying Equation (4), we obtain a _LOD_ = 0.15 w% for the measurement of Cu in cast iron, which should be compared with the univariate _LOD_ = 0.35 w%.  
Additionally, in this case we can do a comparison between the distribution of the predicted values for a sample near the _LOD_, as shown in Figure 9.  
Figure 8: Regression curve of copper in cast iron.  
Similar to the univariate case, the assumption of a Gaussian distribution is not fulfilled for the predicted Cu concentrations. Moreover, the standard deviation of the zero signal is not equal to the one of the signal at the calculated _LOD_.  
Therefore, the actual limit of detection of Cu can be considered substantially lower than the one calculated using Equation (4).  
An alternative empirical method for determining the _LOD_ in multivariate analysis has been recently proposed by [PERSON] et al. [23]. The authors considered the Mean Relative Error (_MRE_) of the estimated concentration of a given sample \(i\), defined as:  
\[MRE_{i}=\frac{mean(\widetilde{c_{i}}-c_{i})}{c_{i}} \tag{7}\]  
where \(\widetilde{c_{i}}\) is the concentration estimated by the multivariate algorithm for each repetition of the measurement on the sample with certified concentration \(c_{i}\). We assume to have ordered the samples according to the growing concentration of the analyte, i.e., \(c_{i+1}\geq c_{i}\). Intuitively, the _MRE_ of the samples with analyte concentration lower than the _LOD_ would be higher than the ones for samples with concentration above the _LOD_. A practical definition of the _LOD_ can, thus, be, according to the authors, the value of c\({}_{i}\) after which the _MRE_ fluctuations.  
\[\Delta MRE=MRE_{i+1}-MRE_{i} \tag{8}\]  
which stabilizes below a given threshold.  
Applying this method to the ANN results for Cu, we obtain the results shown in Figure 10.  
The estimated _LOD_ (about 0.12 w%) is slightly lower than the value estimated through the pseudo-univariate method (Equation (4)); this is coherent with the failure of the approximation of Gaussian distribution and homoscedasticity of the results, which leads to an overestimation of the _LOD_ in Equation (4).  
Figure 9: Distribution of the predicted values for zero concentration (black) and for a sample (yellow) at the _LOD_ (Sample S9, Cu concentration = 0.15 w%).  
A summary of the results obtained with the different approaches for the \(LOD\) of Cu is reported in Table 2.