We convert the longitude and latitude into the Universal Transverse Mercator (UTM) coordinates, and then group the AIS points by the proposed nearest-neighbor clustering method. The next-point connection (NPC) clustering algorithm uses the distance defined as

where K is the index set of the AIS points in interval [t₀, t] and d_st is the space-time distance which is the Euclidean distance using all spatial and temporal features, [t₀, t] is a preset search range of time (the interval length 1, 000 s used in our analysis), E and O stand for the estimated location and observed location at time t, respectively, and s is the set of variables used for finding the closest training points. The proposed clustering method contains the following steps:

Although the NPC method is similar to the minimum spanning tree (MST) and single linkage cluster analysis (SLCA) [18, 19] that combine two clusters with the closest pair of points, NPC uses the estimated position E to measure the distance instead of the observed positions and NPC only searches AIS points in a nearby time interval. When the labels of the AIS points in K are known, the NPC method becomes a classification method and some points from the same vessel can be removed and only the AIS point with time closest to t will be used. The NPC methods that use the nearest neighbors to predict a vessel VID at each time t, they have the weaknesses: (1) NPC classification requires known labels which may not be available; (2) NPC clustering may merge different vessels and some feature with large values may dominate the distance. Therefore, we focus on the clustering method and propose the following algorithm to solve these issues.

We propose an clustering algorithm which is based on trajectory reconstruction and thus called CBTR, which builds the trajectories of vessels by using local physical information. CBTR is based on an NPC clustering method which uses doubly checked distance to improve accuracy. For each point is the data set, we select another point as its best possible next point (BPNP) and put them in the same cluster. Like MST and SLCA, BPNP groups all points into a dendrogram with several tree-type clusters because two distinct points might have the same BPNP. The trajectory can be visualized by connecting all points with its BPNP with a line segment. See Figure 6 for an illustration.

Given an AIS data set, in which points are ordered by their recording time t_i, we denote each point by x_i. The two-dimensional positions of x_i on the earth are denoted as p_i = [LAT(x_i), LON(x_i)] where LAT and LON stand for latitude and longitude of the AIS point x, and their speeds and courses are denoted by v_i and c_i, respectively. For every x_i, we use its velocity, namely speed and course, to predict its future position and look for the best possible next point x_j of it. If there is no point inside a reasonable searching range, then we consider x_i as an endpoint of a trajectory. We trace each trail till an endpoint occurs and thus finish the clustering. So the algorithm of CBTR is designed as follows:

We evaluated the results by the correct-neighbor rate that is defined as ∑i=1nI(Yi=Yj)/n, where Y_j is the label of the closest neighbor of Y_i. In the CBTR algorithm, every AIS point is either assigned a BPNP or determined as an endpoint of a trajectory. We sum up all mistakes made in this process, say M, and compute the correct-neighbor rate as 1−M/n. The proposed method does not aim to find a correct sequential pattern of a trajectory. The definition of the accuracy used in this article only considers the correct clustered labels in the beginning of Section 3.1. It means that it is possible that the proposed method groups one vessel's AIS points in the order of (1, 3, 2) although the true order is (1, 2, 3). However, in this case, it is considered as a correct clustering result.

We compare the CBTR with other methods including the LSTM recurrent neural network (RNN) architecture [27–29] and the EM clustering algorithm [30, 31] which assumes mixed Gaussian distributed clusters. The Expectation-Maximization (EM) algorithm using a Gaussian mixture model estimates the probability of each observation iteratively through the E-step and M-step. Each EM cluster is determined by its mean and variance, so that it is suitable for vessels that are anchored or moving randomly in a fixed location. Since most vessels are moving with varying speeds and directions, the EM clustering does not perform well in the datasets. The comparisons of their correct-neighbor rates are listed in Tables 2, 3.

The time complexity of the proposed CBTR method is O(nr) with the sample size n and the neighborhood size r. See the computational time for each method in Table 4.

In Figures 2, 3, one sees that CBTR is able to regroup most of the trajectories correctly. We leave the detailed explanation of these plots in the Supplementary material. One may evaluate the performance of CBTR by two numbers: jumps and merges. The former counts the total breaks of trajectories done wrongly by CBTR and the later counts how many wrong groupings CBTR makes. Instead of counting how many points are connected to wrong next point, the sum of jumps and merges shows the performance of CBTR more faithfully. Since each jump creates a new clustering and each merge cancels a group, the difference between them is exactly the difference between the number of vessels of our data and the number of clusters via CBTR. Namely we have the following identity: merges − jumps = #{predicted clusters}−#{actual number of vessels}.

In order to evaluate the robustness of CBTR, we conducted experiments by removing points in the data sets so that the trajectories become harder to be tracked. Indeed, we consider validation sets by method 1: removing each fifth point of every five points (i.e., the fifth, tenth, etc.) and method 2 removing each second point of every two points (i.e., the second, fourth, etc.). In sum, we take out 20 and 50% points, respectively, in each validation set and apply CBTR to predict the trajectories. For the downsampled AIS datasets 1, 2, 3 using method 1, the correct rates of the estimated neighbors are 0.9977, 0.9977, and 0.9966, respectively. For the downsampled AIS datasets 1, 2, 3 using method 2, the correct rates of estimated neighbors are 0.9947, 0.9943, and 0.9913, respectively. As we anticipated, the more points are removed, the lower the correct rates of estimated neighbors are. However, CBTR still performs very well whereas large amounts of points are removed. Furthermore, we remark that there is a trade-off between the reduction of the number of jumps and the increment of the number of mergers. If the upper bound for time interval is lager than 1,000 in Step 1, it may lead to more candidates used for selecting BPNP and fewer jumping points while increases the number of merges.

The predicted trajectories of all vessels in data set 1 by using CBTR are shown in Figure 4. From the left-hand boundary of this picture, we know the data set contains some incomplete trajectories and it is impossible to cluster them correctly. One can see a zoomed-in picture of this boundary phenomenon in Figure 5.

Figure 6 shows another mistake made by CBTR. This kind of mistakes happens at the endpoint of some trajectories. To be precise, when a vessel goes back and parks at a pier, it will turn off the AIS signal transmitter. The last position reported shall be the endpoint of the trajectory. But sometimes CBTR finds a false next point for this endpoint and continues the trial. For example, in Figure 6, vessel no. 7 (colored purple) left toward west, came back, and parked to the east of vessel no. 5. At that moment, vessel no. 5 was reporting its last location before turning off its signal transmitter. CBTR found some point of vessel no. 7 to be a possible next point of the last point of vessel no. 5. So it makes a wrong connection from the circled point to the squared point. This is called a terminal-type mistake and counted as a merge.

These terminal-type mistakes only happen when two vessels are anchored close to each other. We can prevent this terminal-type mistakes by using more restrictive connecting criterion, but this will break some trajectories of moving vessels because AIS points in moving trajectories are much sparser than AIS points in steady vessels moored to the piers. In this case, the speed and angle of a vessel randomly change by wave drift forces, so the variances of the white noises in our model (3) may be larger than the signals (speed). These terminal-type mistakes are not that serious because the AIS data is mainly used to recognize moving vessels. Except the boundary phenomenon and the terminal-type mistakes, CBTR performs perfectly on generic situations and is a reliable method to predict trajectories. Table 5 shows the performance of CBTR for vessels with different speeds. One can see that slow vessels are the most challenging ones for CBTR.

Long Short-Term Memory [27] is a type of Recurrent Neural Networks (RNNs). LSTMs are an example of a recurrent neural network which has feedback loops allowing time-dependent problems to be solved. That is, the outputs (i.e., previous outputs) can be used as an input to help model the current output. More generally, problems that have a fundamental order can be solved. LSTMs are capable of modeling sequences of different lengths, and this is ideal as vessel paths often have a different number of points [32].

LSTMs have been used for predicting vessel trajectories with AIS data [33–35] as they can naturally be adapted to multi-target learning and are capable of learning both simple and complex patterns. Here we can think of the timestamp, latitude, longitude, speed, and direction, all at time t, as response variables whereas the predictor variables (i.e., inputs to the LSTM) are the timestamp, latitude, longitude, speed, and direction at time t−1, t−2, ⋯ , t−k. We train an LSTM using lagged versions of the timestamp, latitude, longitude, speed, and direction (i.e., time t−1, t−2.⋯ , t−k) in order to predict the timestamp, latitude, longitude, speed, and direction at one time point in the future (i.e., time t). The goal here is to attempt to predict all characteristics of a vessel automatically using previous information. The architecture and tuning were accomplished via trial an error using a random 20% validation sample.

The characteristics of the LSTM are the following: an input dimension of 5 (i.e., timestamp, latitude, longitude, speed, and direction are lagged by k = 1 time unit), 1 hidden layer, 250 hidden units using the Rectified Linear Unit (ReLU) activation function: max(0, x), and 5 output nodes (i.e., timestamp, latitude, longitude, speed, and direction at time t). Additional values for the number of lags were tried, but the performance was essentially unchanged and different activation functions were tried and tended to produce inferior results. The software used was the keras library in Python [36].

The results from the LSTM using all five variables as outputs seem to indicate that this approach is unable to distinguish the different vessel trajectories due to several reasons including the initial value and the training set of LSTM [33], the changes of courses and speeds [34] in the given prediction time range, and the normalization method x-minmax-min [35] which may over-compress trajectory data since the some trajectories have large ranges but others do not.

The performance of the LSTM next point prediction method is fundamentally dependent on the historical trajectories with labels used to train the LSTM model to predict the properties of the node at the next time point [32, 37]. That is, the training set with labeled trajectories are needed to accurately predict the timestamp, latitude, longitude, speed, and direction at some future point in time. However, to make a fair comparison, only the current AIS point is used in training a LSTM model for predicting the next point, and this makes the recurrent neurons not able to sufficiently learn the latent features in the AIS datasets and leads to inaccurate prediction [38, 39]. LSTM models are known to require a large amount of data in order to be effective, so the relatively small size of the individual AIS training datasets also is a contributing factor to the LSTM's performance.

An inspection of the LSTM predictions and the resulting nearest neighbor search indicate that most of the errors are related primarily to two factors: some vessels rapidly change their speed and direction while simultaneously other vessels that were previously similar to the rapidly changing vessel do not change their speed or direction suddenly and this results in misclassification, for example, vessels no. 5–8. The second source of error may be that the predicted AIS points by LSTM have large variations [37] and in combination with a larger number of candidates within each time window (i.e., the time window in the nearest neighbor search), mistakes are accumulated.