## 3 Results
### Santiago Metropolitan Region Mobility Patterns
In essence, the analysis recognizes similar behaviors of communes according to the value of the mobility scores. The cluster validity indices (CVI) are shown in Table 1. The solution with two clusters has the best indices. However, basically, this solution indicates that there are communes with low scores (rural communes) and communes with medium/high scores (urban communes). Thus, to consider a more interesting situation, the next best solution is the one corresponding to five clusters. Figure 4 shows each of the clusters with the time series of their respective members. It is observed that each cluster includes communes with close mobility scores. Thus, _Cluster_1 is made up of the communes with the lowest mobility scores that belong to the approximate score interval (20-40). _Cluster_2 (40-50) and _Cluster_3 (80-100) are made up of the communes with the highest mobility scores. _Cluster_4 (60-80) and _Cluster_5 (55-65) have some overlap. Figure 3: Hierarchical clustering using DTW for mobility time series. The solution with five clusters is highlighted. To better appreciate the value for the analysis of the cluster structure found from the time series of the mobility scores of each commune, Figure 5 shows a sample of 10 communes in the MR and their corresponding membership to a cluster according to their mobility score values. The two-cluster solution clearly shows that the urban communes (Nunoa, Santiago, Estacion Central, Vitacura, Puente Alto, and Las Condes) have high mobility scores compared to rural communes (Paine, Pirque, Tlitil and Penaflor), which have low mobility scores. However, the structure of the cluster shows some more interesting details that this graphic helps to highlight. For example, among the communes belonging to the cluster with low scores, Penaflor clearly stands out with higher scores. This fact could be explained by the presence of several industries in the area that clearly did not stop operating during the months of the pandemic. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Num. of Clusters** & **Sil** & **SF** & **CH** & **DB** & **DBstar** & **D** & **COP** \\ \hline
2 & 0.8 & 0.0 & 158.6 & 0.2 & 0.2 & 0.1 & 0.1 \\ \hline
3 & 0.6 & 0.0 & 94.0 & 0.2 & 0.3 & 0.1 & 0.1 \\ \hline
4 & 0.7 & 0.0 & 134.1 & 0.3 & 0.4 & 0.1 & 0.1 \\ \hline
5 & 0.7 & 0.0 & 140.6 & 0.3 & 0.3 & 0.1 & 0.0 \\ \hline
6 & 0.6 & 0.0 & 128.5 & 0.3 & 0.4 & 0.1 & 0.0 \\ \hline \hline \end{tabular}  
**Sil: Silhouette index [60]**; (to be maximized), **SF**: score function [61] (to be maximized), **CH**: Calinski–Harabasz index [60] (to be maximized), **DB**: Davies–Bouldin index [60] (to be minimized), **DBstar**: Modified Davies–Bouldin index (DB’) [62] (to be minimized), **D**: Dunn index [60] (to be maximized), **COP**: COP index [60] (to be minimized).  
\end{table} 
Table 1: Cluster validity indices. Figure 4: The time series of the commune mobility scores that belong to the five cluster solution. **(1)**_Cluster\({}_{1}\)_; **(2)**_Cluster\({}_{2}\)_; **(3)**_Cluster\({}_{3}\)_; **(4)**_Cluster\({}_{4}\)_; **(5)**_Cluster\({}_{5}\)_. The central communes of the MR (Santiago, Nunoa, and Estacion Central) are the ones with the highest mobility scores. This can be explained fundamentally by the population density and/or the large number of services in the area. An observation of relevance for our subsequent analysis is given by the fact that communes with very dissimilar incomes, such as Las Condes, Vitacura (two of the communes with the highest income in the country), and Puente Alto (a commune with very low income) have very similar mobility scores. Puente Alto is the commune with more COVID-19 cases in the MR, as indicated by the analyzed COVID-19 dataset. However, the communes with the highest income, as mentioned above, have significantly fewer cases. This fact indicates that mobility is not the only factor to consider in the COVID-19 spread.