## 3 Methods  
FDA aims to analyze information regarding curves. It has been used to examine temperature variation, gene expression signals, and human-movement comparisons [14]. The advantage of the FDA is that it derives significant sources of pattern and variation between the collected data. It is commonly used to detect outlying curves, based on a functional boxplot by [15]. This method is based on the notion of the modified band depth [16]. To understand the modified band depth, it is necessary to comprehend the concept of data depth.  
Data depth is a robust statistic with which to measure the centrality of the underlying data. Data depth orders data by their degree of centrality [17; 18]. A variety of data depth functions has been proposed: simplicial depth, projection depth, and Mahalanobis depth [19; 20; 21].  
The simplicial depth partitions the underlying data into a set of \(\left(\begin{array}{c}n\\ d+1\end{array}\right)\) unique \((d\ +\ 1)\)-simplices. Figure 1 illustrates the concept of the simplicial depth. A subset of the \(\left(\begin{array}{c}n\\ 3\end{array}\right)\) 3-simplicies (triangles) in \(\mathbb{R}^{2}\) is defined by a set of objects \((x_{1},x_{2},x_{3})\in X\), where \(x\in\mathbb{R}^{d}\) regarding the underlying data \(X=\{x_{1},\ldots,x_{n}\}\). The point triangle in Figure 1 is regarded as the deepest point within the underlying data because many simplices (triangles) contain it. However, the square symbol represents the smallest depth in the underlying data. Any simplices (triangles) that three points make do not contain it. Simplices depth presents the centrality of each point by describing how much each point lies in the cloud of points.  
The band depth extends the concept of data depth for FDA. Let a function \(y(t)\) be the subset of the plane \(G(y)=\{(t,y(t)):t\in T\}\), where \(t\) is time or a value of the function. The band is the subset of the plane that lies between the two curves \(y_{i1}\) and \(y_{i2}\) such that \(BD(y)\ =\ \{(t,x):t\in T,min(y_{i1}(t),y_{2}(t))\leq x\leq max(y_{i1}(t),y_{i2}(t))\}\)[22]. If the band depth of \(y\) is high, \(y\) is deeply nested by other curves.  
While the band depth does not take into account the proportion of time that a curve \(y(t)\) lies between \(y_{i1}(t)\) and \(y_{i2}(t)\), the modified band depth considers the proportion of time that \(y(t)\) contains a value in the band [16]. For example, there are six possible bands determined by two curves in Figure 2. \(BD(y_{4})=3/6=0.5\). \(y(4)\) is only contained in the bands delimited by itself and other curves (i.e., \(y(1)\) and \(y(4)\), \(y(2)\) and \(y(4)\), and \(y(3)\) and \(y(4)\)). However, let us consider the modified band  
Figure 1: Simplicial data depth.  
depth, \(MBD(y_{4})=(3+0.5+0.5)/6=0.67\). The modified band depth considers the proportion of times that the curve \(y_{4}\) in the band. For instance, the modified band depth of \(y_{4}\) in the band delimited by \(y_{1}\) and \(y_{3}\) is 0.5. Similarly the modified band depth of \(y_{4}\) in the band delimited by \(y_{2}\) and \(y_{3}\) is also 0.5.  
To apply FDA to AIS data, we anchor one of the directions. For example, Figure 3 represents vessels moving in the north and south directions. If we consider the north and south direction as the basis of FDA, such as \(y(t)\), the values of \(y(t)\) is the mean of the east-west locations. This study uses a robust statistic, such as trim mean (20%), to calculate the variation in the east-west locations of vessels. In terms of the interval of the north-south axis, this study uses 500 m owing to the granularity of the AIS data.  
Then, a functional boxplot [15] is applied to detect outliers from the vessel trajectories, and the identified vessel trajectories are removed. Finally, we describe the shipping routes based on the centerline from the function data analysis.  
Figure 3: Functional data analysis for shipping routes.  
Figure 2: Modified band depth.