## 3 Results
### Statistical Analysis  
For evidencing the relationship between the elements and possible similarities among the fractions, a statistical approach was performed, in particular, a source discrimination analysis followed by a chemometric methodology allowed us to gain some information on the fractions sampled. The uncertainties due to variation in counting geometry is a typical error in the INAA methodology, depending on the equipment and sample-holder used. It can be minimized by means of the Aspinall protocol [60], in other words, the application of the source discrimination approach in relation to the Sc (considered as the internal standard due to its high accuracy in the determination), calculated according to the following formula:  
\[\text{Source Discrimination}= \frac{1}{[\text{Sc}]}\times\left([\text{Cs}]+[\text{Ta}]+\frac{[ \text{Rb}]}{100}+\frac{[\text{Th}]+[\text{La}]+[\text{Ce}]}{10}\right) \tag{1}\]  
Figure 2: Enrichment Factors of all the elements investigated in this study in the three PM fractions. La was used as the normalized agent. This procedure, also used when the sample numbers are few, allows for very close differences to be discriminated among the samples. Figure 3 shows the source discrimination equation applied to the samples investigated in this study: as can be seen, all the samples could be grouped in a single cluster. This means that the sampled aerosol, PM\({}_{10}\), PM\({}_{2.5}\), and PM\({}_{1}\), come from a single common source. Although this analysis allows us to draw such information, the elements involved (i.e., Ce, Cs, La, Rb, Sc, Ta and Th) were not enough to discriminate the samples satisfactorily. These data should be considered preliminary: in fact, the results are subjected to large uncertainty. To overcome these doubts, advanced statistical methods, quite similar to those used in cultural heritage studies (e.g., provenance) or in the quality control of goods, were used. Cluster analysis (CA) and multivariate analysis [61; 62] manage to describe a source apportionment character through a wide \"chemical composition profile\" [37]. Statistical analysis was carried out with Tanagra open-source software [63] using the centroid merge method and the Euclidean distance as a measure of proximity [64; 65]. First, application of the hierarchical cluster analysis (HAC) gives a dendrogram showing that the three-fraction aerosol samples can be grouped in the following three clusters:  
1 st Cluster (eleven members): W, Zn, Ba, Cr, Sc, Ag, Rb, Ce, Th, Sb, Ta. 2 nd Cluster: (ten members): Br, Co, Eu, Sm, Mo, Se, Fe, Ni, Hf, Tb. 3 rd Cluster (six members): As, Ca, Na, Cs, La, Au. Starting from this consideration, the chemometric investigation was addressed to validate the information about the samples with a high percentage of accuracy. Particularly, considering the three different size granulometric fractions as the main characteristics and the elements determined in the samples as variables, the discriminating factorial analysis was applied. The main purpose of the discriminant function analysis is to determine whether groups (i.e., the three aerosol fractions) differ with regard to the mean of a variable, and then to use that variable to predict group membership. The application of the principal component analysis (PCA) to the entire dataset managed to obtain three principal components describing 99% of the total variance of the data matrix (only with two components was almost 81% of the data described). Afterward, using the discriminating factorial analysis, the composition linear models of the discriminant functions were found. The statistical test showed the separation of the samples in two groups (Figure 4): the principal one was formed by the PM\({}_{10}\) and PM\({}_{2.5}\) fractions whereas the second cluster was made of PM\({}_{1}\) aerosol filters. These  
Figure 3: Multiple regression analysis: plot of the discrimination factor (source discrimination) versus Fe/Sc abundance ratio (blue: PM\({}_{10}\); orange: PM\({}_{2.5}\); grey: PM\({}_{1}\)).