## 3 Results  
The results of elemental concentrations determined by WD-XRF (Table 2) correspond with a particular case of the projective \(d+1\)-dimensional space where the projective points are projected into the simplex \(\mathbb{S}^{d}\). Points are represented by homogeneous coordinates that have a constant sum \(k\) (\(k\in\mathbb{R}+\)),  
\(\mathbb{C}\) (**w**) = **x** = [\(x_{1}\),..., \(x_{d}\), \(x_{d+1}\)] \(\mid\)\(x_{i}\) \(\geq\) 0 (\(i\) = 1,..., \(d\), \(d+1\)), \(x_{1}+\cdots+x_{d}+x_{d+1}=k\),  
(in this current case, \(k\) = 100). The projective points' vector space is the positive orthant. Hence, for the statistical data treatment, the raw concentrations have been alr (additive log-ratio) transformed according to  
\[\textbf{x}\ \in\ \mathbb{S}^{d}\ \rightarrow\textbf{y}=\ln\biggl{(}\frac{ \textbf{x}_{d}}{x_{d+1}}\biggr{)}\ \in\ \mathbb{R}^{d} \tag{1}\]  
being \(\mathbb{S}^{d}\) the d-dimensional simplex, and \(\textbf{x}_{d}\) = [\(x_{1}\),...,\(x_{d}\)]. They have also been clr (centred log-ratio) transformed following the equation  
\[\textbf{x}\ \in\ \mathbb{S}^{d}\ \rightarrow\textbf{z}=\ln\biggl{(}\frac{ \textbf{x}}{\textbf{g}(\textbf{x})}\biggr{)}\ \in\mathbb{H}\subset\mathbb{R}^{d+1}, \tag{2}\]  
being \(\mathbb{S}^{d}\) the d-dimensional simplex, g(**x**) the geometric mean of all \(d+1\) components of **x** and \(\mathbb{H}\subset\mathbb{R}^{d}\ {}^{+1}\) a hyperplane vector subspace of \(\mathbb{R}^{d}\ {}^{+1}\)[32; 36; 37; 38; 39]. The statistical treatment of the chemical data was performed using R [40]. The first analysis carried out was to calculate the variation matrix that completely determines the covariance structure of compositional data and provides the total variation (tv) of the analysed ceramic assemblage (Figure 4) [36; 41]. The overall data set exhibits a total variation of 2.28, a high value that indicates a polygenic group's existence. If the total variation quantifies how different the chemical data are, the information entropy, or Shannon index, quantifies how evenly the chemical differences are related to the retained components. Here, the chemical differences are related to many components (\(H_{2}\) = 3.87 Sh; \(H_{2}\) % = 84.41). As can be seen in the compositional evenness graph, the elements that introduce more variability are MnO and CaO (tv/\(\tau_{j}\) < 0.3) and, to a lesser extent, Na\({}_{2}\)O, Cu, K\({}_{2}\)O, Rb, and Zn (0.3 < tv/\(\tau_{j}\) < 0.5), but no components introduce low variability (tv/\(\tau_{j}\) > 0.9). Thus, total variation and compositional evenness indicate a complex structure in the data set. To summarise the data treatment, we present the results of the cluster analysis, using the squared Euclidean distance and the centroid agglomerative algorithm, performed on the clr transformed components: Na\({}_{2}\)O, MgO, Al\({}_{2}\)O\({}_{3}\), SiO\({}_{2}\), K\({}_{2}\)O, CaO, TiO\({}_{2}\), V, Cr, MnO, Fe\({}_{2}\)O\({}_{3}\), Ni, Cu, Zn, Ga, Rb, Sr, Y, Zr, Nb, Ba, Ce, Pb, and Th. The study of the dendrogram (Figure 5) indicates a complex structure with ten meaningful ceramic groups (or paste compositional reference units), defined as clusters of two or more individuals (plus group CGEIV, according to [22]), that account for just 33 individuals, plus 25 ungrouped  
Figure 4: Compositional evenness graph of the 58 studied individuals. \(H_{2}\): information entropy (in Shanons, Sh); \(H_{2}\) %: percentage of the maximum possible attainable; tv: total variation. \(\tau_{j}\): trace of the variance-covariance matrix following the alr transformation using element \(j\) as the divisor. Vertical dotted lines express different tv/\(\tau_{j}\) values. individuals because of their compositional singularities in the present data set. A higher total variation should be envisaged for such a set of compositions, but the picture suggests a complex situation of different groups (Table 3) sharing similar geological backgrounds in some cases.