# Discovering Moving Flock Patterns in Movement Data: A Reeb Graph-Based Approach
## 5 Case Study
### Results and Analysis  
As mentioned above, four parameters are involved in the proposed approach, which can result in different moving flock patterns depending on different parameter values. To examine how these parameter values affect the number of detected moving flock patterns, we analyzed the relationship between varying a single parameter and the resulting number of patterns, while keeping the other three parameters at default values. The initial default values of the four parameters are listed in Table 4. The initial default values were chosen for two main reasons: (1) each parameter's default value was balanced (neither too large nor too small) based on our preliminary experiments, and (2) a relatively large number of moving flock patterns could be discovered, based on which substantial and useful information was provided. The relationships between each parameter value and the number of moving flock patterns discovered are shown in Figure 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**The Parameters** & **The Default Values (Units)** \\ \hline \(r\) & 10 (meters) \\ \(m\) & 3 (players) \\ \(k\) & 3 (seconds) \\ \(d\) & 0.5 (meters) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The default values of the four parameters. Figure 4: Visualization of the movement data: (**a**) the discrete points, and (**b**) the corresponding trajectories. Figure 5: The relationships between the values of the parameters and the number of moving flock patterns detected: (**a**) \(r\); (**b**) \(m\); (**c**) \(k\) and (**d**) \(d\). Figure 5 shows that for parameter \(r\), the number of moving flock patterns generally rises as \(r\) increases, with an outlier at \(r=9\). According to our improved definition of moving flock, a flock must be divided into smaller moving flocks if the spatial extent between two specific consecutive timestamps is less than \(d\). This leads to an increase of detected moving flock patterns, making the sharp rise when \(r=9\) understandable. Based on our improved definition, many finer moving flocks could be detected instead of a single large moving flock, so the number of moving flock patterns does not necessarily increase continuously with \(r\). This is different from the previous results of [PERSON] et al. [32], who observed a direct correlation between increasing \(r\) and the number of patterns. This distinction is a significant finding in our research. Additionally, when \(r\) equals to 9, the number of moving flock patterns is suitable (neither too large nor too small), so 9 seems to be a favorable value for \(r\). For the parameters \(m\), \(k\) and \(d\), the number of moving flock patterns decreases as these values increase. However, when values exceed certain thresholds, no moving flock patterns are detected. Thus, the initial default values for \(m\), \(k\) and \(d\) listed in Table 4 appear to be well-chosen. The relationships between the parameter values and the number of detected moving flock patterns (types A-D) are shown in Figure 6. From Figure 6, it can be seen that the number of type A is obviously large when each parameter meets certain requirements (e.g., \(r\geq 9/m\leq 4/k\leq 5/d\leq 1\)). In contrast, the number of type D is the smallest among the four types for each parameter. For types B and C, the number is usually between types A and D. Therefore, we can see that type A generally always has a larger number than the other three types (when \(r\) exceeds a certain threshold or \(m/k/d\) is below certain values), and the number of type D is always quite small or even zero. This observation is intuitive since types A and D represent two extremes. Hence, Figure 6 shows that the results of the four types of detected moving flock patterns align well with expectations. In addition to individual parameter values, the combination of different parameter values can also influence the number of moving flock patterns being discovered. Therefore, it is essential to find a suitable value for each parameter.