3.1.2. Agglomerative Clustering

Figure 6 shows the same results but based on the use of Agglomerative Clustering (AC) to group the time-series water depth data. As with the K-means results (Figure 5), the CHI value increase with the number of clusters for all scenarios from short-duration to long-duration rainfall. Again, it is difficult to identify an optimal number of clusters, and this suggests that the CHI is not suitable for ascertaining the best clustering solution with these data. In contrast, the SCI and DBI show clear peaks in their values. Figure 6a shows that 16 clusters result in the maximum SCI close to 0.76 and minimum DBI with 0.38. Figure 5c shows a peak in SCI values (~0.6) for eight clusters, with a corresponding

minimum in the DBI value (<0.4). However, Figure 6b shows that eight clusters could produce the largest SCI (~0.62) and the lowest DBI (~0.40) with the two year-12 h rainfall duration scenario (left subplot), but that 16 clusters are the optimal solution for the two year-12 h rainfall (SCI ~0.58 and DBI ~0.38; right subplot). In summary, the best cluster solutions AC algorithms are 16, eight, and eighteen under 3 h, 12 h, and 48-h duration rainfalls, respectively. Comparing the left subplots with the right subplots (Figure 6) provides evidence that the cluster number for the best AC performance remains the same, although the return period has been shifted from two-year to five-year. The rainfall return period (annual exceedance probability) was found to be less related to the number of clusters.

**Figure 6.** Performance evaluation for Agglomerative Clustering with different cluster numbers under synthetic rainfall scenarios including (**a**) 3-h (left 2-year and right 5-year), (**b**) 12-h (left 2-year and right 5-year), and (**c**) 48-h duration (left 2-year and right 5-year).

## 3.1.3. Spectral Clustering

Figure 7 shows the results obtained for different cluster numbers using Spectral Clustering to group the time-series water depth data. In contrast to the two previous methods, the SCI values decrease as the number of clusters increase. For the 12 and 48 h scenarios, this index identifies solutions at about 6–7 clusters, but no clear optimal solution is identified in the shorter scenarios (panel a). This suggests that this index is unsuitable for assessing this algorithm. The DBI values show greater variation as the number of clusters change, although minima can be observed at 6 to 7 clusters for most scenarios. The CHI values no longer show a linear increase, but show clear peaks, although usually for higher numbers of clusters than the DBI identifies. The highest CHI values (275 for 2 year-12 h and 190 for 5 year-12 h) are all generated by the SC with 13 clusters. For the for two year-48 h and five year-48 h scenarios, the largest CHI values are approximately 200 and 270, respectively, in both cases for 12 clusters.

**Figure 7.** Performance evaluation for Spectral Clustering with different cluster numbers under synthetic rainfall scenarios including (**a**) 3-h (left 2-year and right 5-year), (**b**) 12-h (left 2-year and right 5-year), and (**c**) 48-h duration (left 2-year and right 5-year).
