## 4 Experiments
### Case 1: Jeju Island Unjin Port to Mara Island Saledok Port  
There are thirteen trajectories in the Case 1 dataset. We first detected outliers from the trajectories. The functional boxplot in Figure 7 presents the box, whiskers, median curve, and two outliers. The magenta box indicates the 50% central region, which is the region through which most trajectories pass. The whiskers with blue curves denote envelopes, which provide the 25% and 75% central regions. The red dashed lines indicate outliers, which are identified as 1.5 times the 50% central region [15].  
Figure 5: Case 2 dataset.  
Figure 6: System architecture.  
Figure 7 shows the two detected outliers with comparatively non-typical patterns. We divided the north-south axis into 500 m increments, as indicated by the \(x\)-axis in Figure 7. The \(y\)-axis shows the average longitude of the vessel locations along the east-west axis per each interval. The two identified outliers are represented on a map to the right of the graph.  
While the functional boxplot identified two trajectories as outliers, the Mahalanobis distance method detected one outlier in Figure 8. The Mahalanobis distance method is a multivariate outlier-detection method [14]. This study used two parameters (i.e., distance and time) for the Mahalanobis distance method. Interestingly, Figure 8 shows that one trajectory had a considerably longer time compared with the other trajectories, although this trajectory is one of the main patterns.  
A functional boxplot tends to be more concerned with trajectory patterns without explicit time information, whereas the Mahalanobis distance method suffers from identifying different trajectory patterns. It is worth noting that although it is easy and fast to calculate the Mahalanobis distance, this method is not robust because the high correlation of variables makes harder the calculation of the inverse of the correlation matrix in the Mahalanobis distance [24].  
Figure 8: Outlier detection using Mahalanobis distance method.  
Figure 7: Outlier detection using functional boxplot.  
In constructing the shipping routes, the previously identified outliers were removed, and the median curve was generated. The final shipping routes generated by the proposed method are shown in Figure 9a. Further, this study used an alternative method [10] to generate shipping routes. The algorithm in [10] uses a density-based technique to construct the map by varying the cell sizes. This study changed the cell sizes ranging from 119 m \(\times\) 380 m to 238 m \(\times\) 760 m in Figures 9b-e.  
In addition, all results of shipping routes construction were superimposed on the same map in Figure 10. In particular, Figure 10 clearly shows that the density-based approach is highly affected by cell size. Discretization in a regular grid generated irregular shipping routes. The following section describes another dataset that was evaluated using the same method.  
Figure 10: Comparison of all shipping-routes constructions in Case 1.  
Figure 9: Shipping routes construction: (**a**) functional data analysis; (**b**) density-based technique (cell size: 238 m \(\times\) 760 m); (**c**) density-based technique (cell size: 179 m \(\times\) 571 m); (**d**) density-based technique (cell size: 143 m \(\times\) 456 m); (**e**) density-based technique (cell size: 119 m \(\times\) 380 m).