## 3 Results and Discussion  
### Characterization of the Historic Study Case  
#### 3.1.1 Formal Characters of Rural Historic Buildings: Clusters  
PCA was performed on the sample of historic building as a way to obtain a synthetic and clear first representation and quantification of the mutual differences of the buildings according to the architectural parameters. First, a Pearson correlation matrix was computed (Table 2). This matrix shows that the considered population of buildings shows a strong correlation between EV an EW and significant correlation among the pairs of parameters HW-HL, HW-EV, EV-BFO, EW-BFO. The relationship between architectonic parameters and principal components (F1,..., F5) resulted from PCA is expressed by Equation (2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **HW** & **HL** & **EV** & **EW** & **BFO** \\ \hline HW & 1 & 0.472 & \(-\)0.275 & \(-\)0.128 & 0.057 \\ HL & 0.472 & 1 & \(-\)0.053 & \(-\)0.038 & \(-\)0.147 \\ EV & \(-\)0.275 & \(-\)0.053 & 1 & 0.890 & 0.350 \\ EW & \(-\)0.128 & \(-\)0.038 & 0.890 & 1 & 0.377 \\ BFO & 0.057 & \(-\)0.147 & 0.350 & 0.377 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pearson Correlation matrix. Coefficients in bold are different from zero with significance level 0.05. \[\left( \begin{array}{c}\text{HW}\\ \text{HL}\\ \text{EV}\\ \text{EW}\\ \text{BFO}\end{array} \right)=\left( \begin{array}{c c c c c c}-0.389&0.778&0.285&0.400&0.049\\ -0.287&0.796&-0.370&-0.382&-0.022\\ 0.923&0.170&-0.263&0.040&0.219\\ 0.901&0.273&-0.189&0.186&-0.207\\ 0.563&0.228&0.743&-0.282&-0.005\end{array} \right)\left( \begin{array}{c} \text{F1}\\ \text{F2}\\ \text{F3}\\ \text{F4}\\ \text{F5}\end{array} \right) \tag{3}\]  
Table 3 shows the eigenvalues of the principal components, the respective rate of explained variability and the cumulated variability. An effective graphic representation of Equation (2), i.e., the relationship between architectonic parameters and principal components is provided in the diagrams of Figure 4. In particular the diagrams highlight the high correlation between EV and EW, which are very close in the three Cartesian planes defined by the first three principal components (corresponding to 90% of variability). At the same time, the diagrams show the significant correlation between HW and HL, which are extremely close in F1-F2 plane, but clearly separated in the other two. A series of k-means cluster analyses with different numbers of clusters have firstly been performed in order to identify a number of clusters characterized with significant differences of the architectural parameters. The significance of these differences was defined on the basis of the capability of the clusters of identifying groups of buildings with common architectonic features within each cluster and diversified architectonic features among the clusters. Three resulted the optimal cluster number for the considered sample. A graphic representation of this result was developed adopting the first two principal components, through the Cartesian diagram provided in Figure 5. This diagram clearly shows the similarities of the buildings belonging to the same cluster and the differences among the three clusters, providing a synthetic view of their distribution in the parameters' space. The diagram of Figure 5 shows that cluster 1 occupies the area with high F1 and small or negative F2 values, cluster 2 the area with high F2, cluster 3 has prevailing small F2 and negative F1. Based on the relationship between the parameters and the first two principal components represented in Figure 5, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **F1** & **F2** & **F3** & **F4** & **F5** \\ \hline Eigenvalue & 2.215 & 1.395 & 0.875 & 0.422 & 0.094 \\ Variability (\%) & 44.296 & 27.894 & 17.495 & 8.445 & 1.870 \\ \% cumulated & 44.296 & 72.190 & 89.685 & 98.130 & 100.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Eigenvalues and rates of variability of the principal components.