## 3 Results  
### Santiago Metropolitan Region Mobility Patterns  
The Grandata mobility data were used to characterize the mobility patterns of the communes (basic geographic units/sectors in our analysis) of the MR. We recall from Section 2 that this mobility data contain logs of geolocation events tracking mobility patterns of the smartphone by using Mobile Advertising ID hashes [42]. Mobility data were constructed for hexagon shapes using the H3 geospatial system [48]. The geolocation events were summed for each area covered by hexagons. The score for each hexagon area consists of the hexagon's percentile placement in the mobility distribution (\"out-of-home\" event distribution) for each day and taking into account all other hexagons in the MR. Figure 1C shows the scores or percentiles for hexagons covering the MR. The MR is made up of 52 communes. We assume that each hexagon is a kind of \"sensor\" to measure the mobility of some commune area. Each commune is covered by a specific number of sensor-hexagons (the MR is covered by a total of 11.853 hexagons) that measure the mobility of the area for 249 days (time interval between 2 March 2020 and 1 November 2020), thus generating a time series with mobility scores. We average the daily scores of all the sensor-hexagons that make up each commune as a proxy for the mobility of the communes during the day. In this way, we obtain 52 time series corresponding to the average scores of each commune during the study time interval. Figure 2 shows these average scores (for simplicity, we will continue to call this value score although it is understood that it is constructed as the average of the scores of the hexagons that cover the commune) time series for a sample of 16 MR communes (the choice of a sample of size 16 is only to obtain a better visualization, and the analysis is done with all the communes of the MR). We constructed the time series from the average of the scores of all the hexagons that cover each commune daily. This mean score by construction may contain space-time fluctuations. Moreover, its values over time may be out of phase concerning the scores of other communes, even although globally these communes could have similar mobility patterns. To discover similar patterns in the time series of the commune mobility scores, it is important to define a measure of similarity/distance between these series that considers  
Figure 2: Time series for the average mobility scores of a sample of 16 Metropolitan Region communes. this possible distortion in the time axis. A distance measure called Dynamic Time Warping (DTW) is a mapping of points between a pair of time series, designed to minimize the pairwise Euclidean distance. This method allows non-linear alignments between two series to accommodate similar sequences, and it is becoming recognized as one of the most accurate similarity measures for time series data [54; 55; 56; 57]. Indeed, DTW is a popular technique for comparing time series, providing both a distance measure that is insensitive to local compression and stretches and the warping, which optimally deforms one of the two input series onto the other. The rationale behind DTW is, given two time series, to stretch or compress them locally to make one resemble the other as much as possible (another advantage, concerning the methods based on the Euclidean distance, is that DTW allows the comparison between time series that may eventually have different numbers of components). To capture the mobility patterns of the MR, we used cluster analysis for time series with DTW distance. For this analysis, we used the R package **dtw**[58]. Figure 3 shows a dendrogram, the result of applying a hierarchical cluster analysis with DTW distance and average linkage method, where the five-clusters solution has been highlighted (we also used the PAM and Fuzzy c-means method included in the **dtwclust** package [59], but the Hierarchical method showed better regularity in the structure of the clusters).