# Discovering Moving Flock Patterns in Movement Data: A Reeb Graph-Based Approach
## 5 Case Study
### Results and Analysis
To achieve this, we performed a hierarchical clustering on the four parameters, as shown in Figure 5. The reason is that hierarchical clustering helps reveal similarities and dissimilarities progressively, enabling us to suggest optimal parameter values that can be adjusted to meet specific requirements. The resulting hierarchical clustering is shown in Figure 7, in which a higher link shows that the corresponding elements are more different from each other. Thus, the optimal parameter values can be determined based on the desired degree of similarity. Since football is typically considered as a complex dynamic interaction among players, the optimal values require further refinement. For parameter \(r\), values 14 and 15 are the most dissimilar but  
Figure 6: The relationships between the parameter values and the number of the four different types of moving flock patterns detected: (**a**) \(r\); (**b**) \(m\); (**c**) \(k\), and (**d**) \(d\). are not considered optimal, as they produce an excessive number of moving flock patterns, which does not yield satisfactory results, as shown in Figure 4a. Therefore, values with lower dissimilarity level (i.e., 9 or 12) appear more suitable, and based on the outlier in Figure 4a, 9 was chosen as the optimal value for \(r\). Regarding parameter \(m\), a setting of 3 resulted in many moving flocks involving fewer players. However, since complex tactics generally involve larger groups of players, such flocks are more useful for performance evaluation. For this reason, we consider the optimal value for \(m\) to be 4 (i.e., the value with the second highest dissimilarity). For parameter \(k\), since we aimed to detect moving flock patterns involving more players for performance assessment, the duration of the moving flocks was less critical for us. Therefore, 3 was an appropriate value for \(k\). When the parameter \(d\) is 0, it means that the flocks are stationary. As we are interested in moving flocks, 0.5 seemed to be an optimal value for \(d\). Therefore, according to Figure 6, the optimal values for the four parameters are: \(r=9\), \(m=4\), \(k=3\) and \(d=0.5\). Note that the optimal values may be different for different scenarios, so one has to choose the appropriate parameter values depending on the specific objectives. We therefore tested the proposed approach with the optimal parameter combination: \(r=9\), \(m=4\), \(k=3\) and \(d=0.5\). Finally, 72 moving flock patterns were discovered. However, since not all parameters were of particular interest, we extracted the four distinct types of moving flock patterns, the total number of which was 21. The exact number of each type is listed in Table 5.  
From Table 5, it can be seen that the number of type A was comparatively large. This shows that many moving flock patterns involve the fewest players over the shortest time duration. In contrast, patterns with the minimum number of players and the longest time duration (type B) or the maximum number of players and the shortest time duration (type C) were less frequent. Furthermore, there was no pattern with both the maximum \begin{table}
\begin{tabular}{c c} \hline \hline
**Types of Moving Flock Patterns** & **Number of Moving Flock Patterns Being Discovered** \\ \hline A & 17 \\ B & 3 \\ C & 1 \\ D & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Number of the four distinct types of moving flock patterns detected under the parameter combination \(r=9\), \(m=4\), \(k=3\) and \(d=0.5\). Figure 7: Hierarchical clustering of the four parameters: (**a**) \(r\), (**b**) \(m\), (**c**) \(k\), and (**d**) \(d\). number of players and the longest time duration (type D). To examine the detected moving flocks more closely, we provides the detailed information (i.e., the players involved and the corresponding time intervals) for each type of moving flock pattern in Table 6. For brevity, we arbitrarily selected three moving flocks of type A for illustration.