## 4 Results and Discussion
### Multivariate Analyses on Semi-Quantitative Data  
As already stated, we choose to apply a MV approach so to consider as many variables as useful for a reliable discrimination. Indeed, appropriate statistical methods can be used to handle all the variables at a time on possibly large sample sets when dealing with a relatively high number of variables--for instance, chemical elements. These methods have already proved to work on concentration data [53], but also on semiquantitative data or on XRF spectra [48; 49]. Actually, they also permit the consideration of analytical results from different techniques in one unique elaboration to classify samples [42]. The fourteen selected elements detected in all the samples (namely: K, Ca, Ti, Mn, Fe, Cu, Zn, Rb, Sr, Y, Zr, Nb, Pb, and Th) were considered for MV elaborations; statistical elaboration was applied to all the measured samples with the addition of the five average virtual samples obtained, one for each class--namely, PNTav, PAav, MNav, PMav, and LPAv--as defined above through the relative concentration average values. In Figure 4, the result of HC elaboration for our obsidian samples is reported; original data have been auto-scaled, and average linking method coupled with Euclidean distance has been used. The scaling of data is mandatory as different chemical element peak areas can have different orders of magnitude. The five groups for the five provenances are easily identified and are highlighted in the dendrogram with the same color code as Figure 1. Comparable results have been obtained by applying PCA to the same set of samples and detected elements, as evident by the scatter plot reported in Figure 5, in which larger dots represent the group centroids, while small dots represent the samples; explained variance for the first three PCs was about 88%. The groups are indeed quite well separated already by the score plot of the two first PCs, explaining a variance of about 69% (see Figure 5). Figure 3: Relative concentration of Mn, Zr, and Fe; for each class, the average value of the reported elements is also shown (PNTav, PAav, MNav, PMav, LPAv). Samples’ codes are the following: Pantelleria (PNT), Palmarola (PA), Sardinia-Monte Arci (MN), Lipari (LP), Milos Island (PM). From the relative loading plot (not reported), correlation between the original variables and their loading were highlighted, allowing us to select the most significant variables and to increase variance up to 92% for the first three PCs. This result was obtained using a set of data (Data Set I) composed of some of the variables (namely, Fe, Cu, Rb, Sr, Zr, Y, Th) for all the samples. Score and loading plots for the first two PCs are reported in Figures 6 and 7, respectively; larger dots in the scatter plot represent the group centroids, while small dots represent the samples. For PC1 and PC2 alone, the explained variance is 76%, but the different groups are already evident. It is interesting to note that the dendrogram (not reported) obtained with the same parameters as that reported in Figure 4 (average linkage, Euclidean distance) for Data Set I is not as reliable as the one obtained on the complete dataset, as a few samples (namely PM4 and MN3) are misclassified in the wrong groups. Figure 4: Dendrogram obtained for obsidian samples—Pantelleria (PNT), Palmarola (PA), Sardinia-Monte Arci (MN), Lipari (LP), Milos Island (PM)—considering all the elements; data have been auto-scaled, and average linking method coupled with Euclidean distance has been used. Colors are the same as in Figure 1.  
Figure 5: Score plot obtained considering the complete dataset used for dendrogram reported in Figure 3. The variance accounted by the first two PC is about 69%. Colors are the same as in Figure 1; samples’ names are not reported for sake of clarity. We also considered a second set of variables (Data Set II) to undergo MV, selecting those elements used for bivariate graphs in the quoted papers, and namely K, Ca, Fe, Cu, Rb, Sr, and Zr.