## 2 Materials and Methods
### Initial Exploratory Analysis
#### 2.2.2 PCA
One of PCA's main purposes is to delete the noise due to non-useful data, which is evaluated in terms of how much information and how much variance they carry inside the dataset. Figure 4 represents variance for each principal component before variable reduction. Loading plots have been generated as histograms representing the weight of the variables transformed after the PCA and are reported in Figures 5 and 6. The variables reported along the abscissa have been selected among all the available data for being the most meaningful as per the multi-risk evaluation. For instance, AGMAX_50 denotes the maximum ground acceleration (fiftieth percentile) calculated on a grid with a 0.02\({}^{\circ}\) step, with the maximum and minimum of the values of the grid points falling within the municipal area. IDR_POPP3 indicates the resident population at risk in areas with high hydraulic hazard (P3). From Figures 5 and 6, the variables with the highest coefficients have been extrapolated, the higher the coefficient of the variable, the higher the weight of the variable on the principal component. Along the first principal component, the difference between observations will be led by the different values referred to the variables with highest coefficient in the histogram depicted in Figure 5.  
We chose to assess the weight of the coefficient of the variables referring to the first two principal components only, because they explicated more than 70% of the variance and are the most significant of the combined risk assessment. Figure 7 depicts the variance explicated by the first 10 principal components after the PCA. Figure 4: Variance for each principal component before variable reduction. Fundamental to the visualization of both observations and the relation between the variables is the biplot in Figure 8.  
Figure 5: Loading plot of the variable coefficients along the first principal component (see Appendix A for an explanation of the acronyms). Figure 6: Loading plot of the variables coefficients along the second principal component (see Appendix A for an explanation of the acronyms). This plot allows catching at an early stage any pattern within the dataset, such as the separation between observations and deep relation among variables. In general:  
Figure 8: Biplot along the first two principal components. Figure 7: Variance of the first 10 principal components after variable reduction. the projection of the values on each principal component shows how much weight those values have on that principal component;
* when two vectors are close, in terms of angle, the two represented variables have a positive correlation;
* if two vectors create a 90 angle, the respective variables are not correlated;
* when they diverge and create an angle of almost 180, they are negatively correlated. Outliers differ from the other observations in terms of vulnerability and the population at hydraulic risk. It is reasonable because, remembering the outliers are the provincial administrative centers, they present higher values in terms of population and built environment. Moreover, along the vertical axis the observations differ in terms of seismic hazard and exposition. Moreover, vulnerability and exposure to hydraulic risk variables are quite correlated and differentiate the observations along the horizontal axis, whereas seismic hazard and exposition variables are not correlated with the variables representing surfaces at hydraulic risk. These remarks will come in handy later, at a post-clustering stage, a level of multi-risk will be attributed to each cluster.