## 2 MAP-DBSCAN Method
### Performance Evaluation
#### 2.4.3 Evaluation  
The ETAS earthquake catalog is divided into either single events or mother events with their descendants. We define by \(\mathbf{X}=\{\mathbf{X}_{k}\}_{k=1,\ldots,N_{k}}\) the true partition of the catalog, where \(N_{c}\) corresponds to the number of clusters, and with \(\mathbf{Y}_{i}=\{\mathbf{Y}_{n}\}_{n=1,\ldots,N_{k_{i}}},\ i=1,\ldots,K\), the partition after the implementation of the MAP-DBSCAN and the other \(K-1\) methods, including different tuning of the algorithm's parameters. \(N_{y_{i}}\) is the number of clusters after the implementation of method \(i\). Next, we define the Jaccard index [46], which is a measure to quantify the overlap between two partitions, in our case, the true one of the ETAS catalog, \(\mathbf{X}\), and the one of the \(i\)-th implemented algorithm, \(\mathbf{Y}_{i}\). The Jaccard index is expressed by \(J_{1}(\mathbf{X},\mathbf{Y})=a_{11}/(a_{11}+a_{10}+a_{01})\), where \(a_{11}\) indicates the number of pairs of elements which are correctly assigned into the same cluster (true links), \(a_{01}\)--the number of pairs of elements which are in the same cluster in the ETAS catalog and in different clusters in the estimated one (missed links) and \(a_{10}\)--the number of pairs of elements which are wrongly identified as clustered events (false links). If all the initial clusters are correctly identified by the implemented method, then \(a_{10}=0=a_{01}\) and \(J_{1}(\mathbf{X},\mathbf{Y})=1\). Conversely, if all pairs are wrongly identified as clustered or independent, then \(a_{11}=0\) and, as a consequence, \(J_{1}(\mathbf{X},\mathbf{Y})=0\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & \multicolumn{2}{c}{**Parameter**} \\ \hline \(K\) & 0.1 & \(d\) & \(2.41\times 10^{-5}\) \\ \(a\) & 2.19 & \(q\) & \(1.805\) \\ \(p\) & 1.13 & \(\gamma\) & \(0.59\) \\ \(c\) & 0.024 (days) & \(\mu\) (events/day) & \(4.50\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: ETAS parameters used for the synthetic earthquake catalog with \(m_{c}=2.5\). The target area is Corinth Gulf, Greece, with \(\Sigma x[t_{in},T]=[21.3^{\circ}E-23.2^{\circ}E]\times[37.9^{\circ}N-38.6^{ \circ}N]\times[2,20]\). The number of clustered events is \(N=4253\) and the number of mother events is \(N_{bg}=1595\). In addition, we introduce a generalization of the Jaccard index, \(J_{2}(\mathbf{X},\mathbf{Y})=b_{11}/(b_{11}+b_{10}+b_{01})\), to identify the partition \(\mathbf{Y}\) with the best discrimination between the background seismicity and clustered elements, following the definition in Lippiello and Bountzis [47]. We consider as background seismicity single events and the mother events of each cluster, i.e., the one that initiated a cascade of events. Here, \(b_{11}\) represents the number of common background events in the two partitions, \(b_{10}\) is the number of elements wrongly identified as mother events in the partition \(\mathbf{Y}\), whereas \(b_{01}\) corresponds to the number of true mother events identified as clustered elements in the partition \(\mathbf{Y}\). In Table 2, we show the Jaccard index values (\(J_{i},i=1,2\)) after the implementation of the different clustering algorithms. In particular, for the MAP-DBSCAN algorithm we show the one with the best results in terms of the Jaccard index (MAP-DBSCAN27). In Appendix B, the results for all input parameters are given. The window-based method removes all the events within \(d(M)\) kilometers and \(t(M)\) days after a main shock with magnitude \(M\). We used three different temporal and spatial intervals, given by Equations (A2) (GK1), (A3) (GK2) and (A4) (GK3). The Reasenberg algorithm combines a deterministic spatial window and a probabilistic temporal one, determined by the Omori law. We used three sets of parameters in the ZMAP tool. RB1 (Table A1) corresponds to the original parameters proposed in Reasenberg [12].