## 3 Results
### Variables Label Assignment  
First, we set intervals in an objective way, in order to suitably define labels for the variables. To this aim, we set interval extremals in correspondence of quartile percentages Q1, Q2, and Q3 as indicated in Table 1.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Intervals** & **Label** \\ \hline first element: Q1 & Low \\ \hline Q1: Q2 & Medium-to-low \\ \hline Q2: Q3 & Medium-to-high \\ \hline Q3: last element & High \\ \hline \hline \end{tabular}
\end{table}
Table 1: Labels and intervals for cluster definition.  
Figure 10: Grouping of the Emilia Romagna municipalities into four clusters.  
The chosen labels referred, respectively, to the presence of low, medium-to-low, medium-to-high and high amounts regarding that specific variable. Such subdivision was allowed because the variables were quantitative types and sorted by normal distribution. Furthermore, sorting out variables, the information within them was unaffected.  
We analyzed the variable with the greater value from the previous analysis, as a component that defined a risk, with the risk as the combined result of three factors, hazard, exposure, and vulnerability. We illustrate how to assign a label to each cluster for each variable considered, among the most relevant ones.  
We first considered the variable \(a_{g,max}\), i.e., the peak ground acceleration for the site, with a return period of 475 years. The first step was the extrapolation of observables in the initial dataset. Subsequently, we associated each observation with the respective cluster indexes and the respective values of \(a_{g,max}\). Then, we rearranged the observables in ascending order of \(a_{g,max}\), and defined the quartile as the extreme point of the interval. The cluster composition in terms of \(a_{g,max}\) is reported in Table 2, together with the resulting assigned labels.  
The labels were assigned based on the percentage prevalence of the cluster for each quartile. A prevalence allocated in the fourth quartile for one of the clusters indicated that the selected cluster gathered the most dangerous municipalities in terms of \(a_{g,max}\) On the other hand, a prevalence in the first quartile indicated that the cluster gathered the less dangerous municipalities in terms of seismic hazard.  
The same operation was carried out for the hydraulic risk component IDR_POPP2, the prevailing seismic vulnerability variable, i.e., the percentage of buildings under poor maintenance conditions E_30, and the main exposure variable, i.e., density population DENS_POP (see Tables 3-5).  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **\%Q1** & **\%Q2** & **\%Q3** & **\%Q4** & **Label** \\ \hline CL1 & 41 & 27 & 22 & 9 & Low \\ \hline CL2 & 25 & 30 & 29 & 15 & Medium-to-low \\ \hline CL3 & 18 & 29 & 37 & 16 & Medium-to-high \\ \hline CL4 & 0 & 4 & 11 & 86 & High \\ \hline \end{tabular}
\end{table}
Table 4: Quartile distribution of the E_30 variable in the four clusters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **\%Q1** & **\%Q2** & **\%Q3** & **\%Q4** & **Label** \\ \hline CL1 & 74 & 20 & 6 & 0 & Low \\ \hline CL2 & 0 & 25 & 25 & 50 & Medium-to-high \\ \hline CL3 & 3 & 18 & 68 & 11 & Medium-to-low \\ \hline CL4 & 7 & 16 & 13 & 64 & High \\ \hline \end{tabular}
\end{table}
Table 2: Quartile distribution of the \(a_{g,max}\) variable in the four clusters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **\%Q1** & **\%Q2** & **\%Q3** & **\%Q4** & **Label** \\ \hline CL1 & 77 & 20 & 54 & 9 & Medium-to-low \\ \hline CL2 & 46 & 41 & 12 & 1 & Low \\ \hline CL3 & 3 & 5 & 18 & 74 & Medium-to-high \\ \hline CL4 & 4 & 5 & 9 & 82 & High \\ \hline \end{tabular}
\end{table}
Table 3: Quartile distribution of the IDR_POPP2 variable in the four clusters.