## 2 Materials and Methods
### Statistical Analysis Methods  
Proper statistical analysis tools proved suitable to achieve the study objectives. Specifically, the Principal Component Analysis (PCA) was adopted for historic buildings to reduce the dimensionality of the five-parameter data set describing the main formal features of the historic buildings, with the minimum loss of information about the variability observed in the data set. Moreover, cluster analysis was developed to identify groups of historic buildings characterized by common formal features. The resulting clusters were compared with the typological classes on which the sampling was based, defined on the basis of the original functions (Section 2.2). Finally, a specific comparison analysis procedure was defined to compare traditional and contemporary buildings through quantitative analyses on the basis of the architectural parameters. The PCA procedure consisted in transforming the original parameters into a new set of variables, which represents the so-called principal components, having the peculiarity of being uncorrelated. The computation of principal components consists in the solution of an eigenvalue-eigenvector problem for a positive-semidefinite symmetric matrix [39]. The principal components were ordered as usual, according to the decreasing quantity of variability of the original variables each component retains. Thus, the first few principal components provide a numerical description of the most significant architectural characters expressed by the original parameters. Then, the sample of historic rural building was subdivided into clusters based on the values of the architectonic parameters of each building. The k-mean process was adopted for clustering purpose, as it is meant for partitioning an N-dimensional population into k sets on the basis of a sample. It proved to give partitions which are reasonably efficient in terms of a small within-class variance [40]. The k-means method was adopted to obtain groups of similar buildings, useful for a qualitative and quantitative understanding of the whole sample of historic buildings on the basis of the respective set of five descriptive architectural parameters. This clustering procedure begins by choosing k cluster centers to coincide with k randomly-chosen patterns; then, each pattern is assigned to the closest cluster center and the cluster centers are recomputed using the current cluster memberships. Convergence is assumed when no (or minimal) reassignment of patterns to new cluster centers occurs, or minimal decrease in squared error is experienced [41]. The results of the clustering procedure applied to historic buildings were analyzed under a typological-architectural point of view, by interpreting the mean parameters of each in terms of formal features. On such a basis, the sampled contemporary buildings were compared to the clusters of historic ones. A synthetic diagram of the methodological process is provided in Figure 3.  
An appropriate composite indicator was developed, based on literature review [42; 43], for measuring the level of similarity of the building with respect to all the five parameters simultaneously. This comparison was carried out based on the Euclidean distance computed in the 5-dimensional space defined by the variables (x, y, z, t, s) coinciding with the architectural parameters, with the following correspondence of coordinates: \(x\)LHW, \(y\)LHL, \(z\)LEV, \(r\)LEW, \(s\)LBFO.