3.1 Overall solution

First, to employ the topic model to explore AIS vessel data, vector quantisation is introduced, that is, trajectories are encoded into a set of codes so that a document collection in ‘code’ is obtained. Furthermore, the topic model can be applied to the code document collection and achieve a code distribution for each topic. Finally, the representative codes for each topic are reconstructed to longitude, latitude, heading and speed, from which emerges the vessels’ navigation pattern.

The step-by-step procedures that are incrementally and algorithmically implemented are as follows.

1. (i) Prepare moving vessel dataset:

   1. (a) remove data samples of anchored or moored vessels from the original dataset;

   2. (b) segment AIS data series into sails at longer temporal intervals, where the word ‘sail’ denotes a continuous vessel sailing without stops.

2. (ii) Prepare code document collection for the topic model:

   1. (a) an AIS data record is a vector associated with longitude, latitude, heading and speed data. By introducing vector quantisation to all these vectors of AIS data, a codebook can be generated;

   2. (b) by referencing the codebook, each AIS data record of a sail can be converted to a code, thereby a sail can be converted to a code document. Moreover, by converting all sails in the dataset, a code document collection can be obtained from the original AIS dataset.

3. (iii) Apply the topic model to a code document collection. By applying the LDA method of the topic model to the code document collection, a document-topic matrix and a topic-code matrix are obtained. Then:

   1. (a) with the document-topic matrix, split the matrix to a topic distribution vector, then similar sails can be identified in relation to the proportion of topics by calculating cosine similarity;

   2. (b) with the topic-code matrix, choose the top-ranked codes as a topic representative, and draw back the top codes on the map of the region of interest, then vessel navigation patterns are likely to emerge.

3.2 Vector quantisation: PQk-means algorithm

Scalar quantisation is a one-dimensional signal processing technique to represent a signal sample by absorbing each value to linearly divided discrete gradations and then valued by a code (mostly from a finite set of integer numbers). When applied to vector data, if one uses scalar quantisation to each dimension of the vector, it may be difficult to achieve efficient precision even after setting a very large value of gradation. Vector quantisation is a lossy algorithm for data compression used in speech and image processing. The basic idea of vector quantisation comes from Shannon's rate-distortion theory, which is only to quantise the observed data vectors rather than the whole space expanded by each dimension of a possible vector. It has been shown that better performance can be achieved by coding vectors rather than scalars (Linde et al., Reference Linde, Buzo and Gray1980; Gray, Reference Gray1984). Essentially, vector quantisation is implemented in two steps. The first step is to generate a codebook, which contains a finite set of code and its reference information. The second step is to quantise each vector by comparing its value with the reference information in the codebook and then find its corresponding code.

Many implementations of vector quantisation have been proposed so far (Linde et al., Reference Linde, Buzo and Gray1980; Lloyd, Reference Lloyd1982; Likas et al., Reference Likas, Vlassis and Verbeek2003). Most common approaches are based on the k-means clustering algorithm. With the k-means algorithm, vectors in a given dataset are partitioned into $K$ clusters and a code is assigned to each cluster. The code and centroid pair of each cluster (reference information) is stored as a codebook. Accordingly, each given vector can be expressed by the code of the cluster to which it belongs. For decoding, the code can be replaced by its centroid in the codebook. As a result, one can use a code number instead of a vector with many elements and there are only a finite number of codes for a specific application.

However, when using the k-means clustering algorithm for quantising large-scale vector datasets, the problems of vast memory consumption and prohibitive runtime costs become remarkable. Recently, a new approach that improved this problem significantly was introduced. The idea of product-quantisation (PQ) was originally developed for source coding (Gray, Reference Gray1984) and further applied to approximated nearest neighbour search (Jégou et al., Reference Jégou, Douze and Schmid2011). In the later work, each vector was compressed into a short code, called PQ-code, and the search was conducted over the PQ codes using lookup tables. More precisely, an intermediate code is introduced to compress the original vector dataset to a lower dimension and short-length dataset. Based on this concept, a billion-scale memory-efficient clustering algorithm has been developed, called the PQk-means algorithm (Matsui et al., Reference Matsui, Ogaki, Yamasaki and Aizawa2017).

Assuming a dataset of D-dimensional vectors $\mathbf {p}_n = \{x_1,\, x_2,\, \ldots,\, x_D\}$ is given, where $n \in \{1,\,2,\, \ldots,\, N\}$, first, split the dataset into M disjoined subvectors $\mathbf {p}_n = \{s_1,\,s_2,\, \ldots,\, s_M \}$. Then for each D/M-demensional vector, the closest codeword from pre-trained L codewords is determined. Finally, vector $\mathbf {p}_n$ can be expressed with a code vector, whose element is the code determined by its subvectors, as follows:

(3.1)\begin{equation} \mathbf{c}_n = \{c^{s1}, c^{s2}, \ldots, c^{sM} \} \end{equation}

For two given code vectors, the squared Euclidian distance is given by

(3.2)\begin{equation} d(\mathbf{c}_i, \mathbf{c}_j) = \sum_{m=1}^{M}(c^{im}-c^{jm})^2 = \sum_{m=1}^{M} \sum_{l=1}^{L} (\bar{\mathbf{p}}_{l}^{im} - \bar{\mathbf{p}}_{l}^{jm})^2 \end{equation}

and then cluster the resultant intermediate codes with the k-means algorithm, that is, the cluster to which each vector belongs is determined by assigning the vector to $K$ clusters to minimise the following cost function:

(3.3)\begin{equation} W(C)=\sum_{k=1}^{K}N_{k} \sum_{C(i)=k} d(\mathbf{c}_i, \mathbf{c}_k ) \end{equation}

where $N_k=\sum _{i=1}^N I(C(i)=k)$ is the number of vectors of the $k$th cluster, and $d(\mathbf {c}_i,\, \mathbf {c}_k$ is the distance between the $i$th code vector and $k$th code vector. With this algorithm, clustering a one billion dataset with 100,000 clusters is achieved in 14 h on a 32GB RAM personal computer (Matsui et al., Reference Matsui, Ogaki, Yamasaki and Aizawa2017). Unlike other existing large-scale clustering methods, such as Bk-means (Gong et al., Reference Gong, Pawlowski, Yang, Brandy, Bourdev and Fergus2015) and IQk-means (Avrithis et al., Reference Avrithis, Kalantidis, Anagnostopoulos and Emiris2015), the original vector sample can also be reconstructed approximately from a given code, this being particularly efficient for large AIS trajectory data.

3.3 Topic model

The topic model provides an effective approach for extracting conceptually coherent topics from a massive document dataset towards a considerably small set of latent variables. To the best of our knowledge, one of the most useful implementations of the topic model is the latent Dirichlet allocation (LDA) algorithm (Blei et al., Reference Blei, Ng and Jordan2003), which is an unsupervised machine learning technique to identify latent topic information from a massive document collection.

Let us give a brief description of the LDA algorithm. In a topic model, a document is described as a bag of words, i.e. attention is not given to the order of these words, just care about the appearance and frequency of words. In this way, the topic model can focus on solving problems with a probability distribution of words. Furthermore, the topic model assumes that a document is composed of several topics. This assumption means that the distribution of words depends on the topics selected for an article. It is also assumed that all documents in a data collection can be composed of a limited number of topics.

From this standpoint, the probability distribution of words in a given document collection $\mathbf {W}$ under the condition of topic proportion matrix $\theta$ can be expressed as follows:

(3.4)\begin{equation} P(\mathbf{W} \,| \,\theta) = \prod_{n=1}^{N} \sum_{z} P(w_{n} \,|\, z) P(z \,|\, \theta) \end{equation}

where $P(z \,|\, \theta )$ is the topic distribution on $\theta$ and $P(w_{n} \,|\, z)$ is the word distribution on topic $z$. In an LDA implementation, the topic proportion is assumed as a Dirichlet prior distribution, so that Equation (3.4) is reduced to the following equation:

(3.5)\begin{equation} P(\mathbf{W}) = \int P(\mathbf{W} \,|\, \theta) P(\theta) \,d \theta = \int P(\theta)\prod_{n=1}^{N} \sum_{z} P(w_{n} \,|\, z) P(z \,|\, \theta) \,d \theta \end{equation}

where $P(\theta )$ is a Dirichlet prior distribution. Based on Equation (3.5), the LDA model can be schematised and depicted by a graphical representation as shown in Figure 1. In Figure 1, $\theta$ is a topic distribution of documents and $\phi$ is a word distribution of topics. Here, $\alpha$ is an hyper parameter for generating $\theta$, $\beta$ is an hyper parameter for generating $\phi$, $z$ is a topic matrix, and $w$ is a word count matrix for each document and each word. Furthermore, $K$ stands for the number of topics, $M$ stands for the number of documents and $N$ stands for the number of unique words in the given document collection.

Figure 1. Graphical model representation of LDA

According to this graphical model representation, a document collection can be generated by the following procedures.

1. (i) For each topic $k=1,\, 2,\, \ldots,\, K$:

   1. (a) draw word distribution,

      (3.6)\begin{equation} \phi_{k} \sim Dir(\beta) \end{equation}

2. (ii) For each document $d=1,\, 2,\, \ldots,\, M$:

   1. (a) draw topic distribution,

      (3.7)\begin{equation} \theta_{d} \sim Dir(\alpha) \end{equation}

3. (iii) For each word $n=1,\, 2,\, \ldots,\, N_{d}$:

   1. (a) draw topic,

      (3.8)\begin{equation} z_{dn} \sim Mult(\theta_{d} ) \end{equation}

   2. (b) draw word,

      (3.9)\begin{equation} w_{nm} \sim Mult(\phi_{z_{dn}} ) \end{equation}

where $\phi _{k}$ denotes the distribution of codes in topic $k$, and $\theta _{d}$ denotes the topic proportion in document $d$. Here, $Dir(\cdot )$ denotes the Dirichlet prior distribution and $Mult(\cdot )$ denotes the multinomial distribution. After the inference of LDA to fit the given document collection, the results are represented as a distribution over codes in which top probability codes form a semantically coherent concept, and each document can be represented as a distribution over the discovered topics. As a result, the LDA provides two matrices, one is $\mathbf {\Theta }$, called a topic distribution of documents, and the other is $\mathbf {\Phi }$, called a word distribution of topics, as shown in Equations (3.10) and (3.11):

(3.10)\begin{align} & \mathbf{\Theta} =\begin{pmatrix} \mathbf{\theta}_{1} \\ \mathbf{\theta}_{2} \\ \vdots \\ \mathbf{\theta}_{M} \end{pmatrix} =\begin{pmatrix} \theta_{11} & \theta_{12} & \ldots & \theta_{1K} \\ \theta_{21} & \theta_{22} & \ldots & \theta_{2K} \\ \vdots & \vdots & \vdots & \vdots & \\ \theta_{M1} & \theta_{M2} & \ldots & \theta_{MK} \end{pmatrix} \end{align}

(3.11)\begin{align} & \mathbf{\Phi} =\begin{pmatrix} \mathbf{\phi}_{1} \\ \mathbf{\phi}_{2} \\ \vdots \\ \mathbf{\phi}_{K} \end{pmatrix} =\begin{pmatrix} \phi_{11} & \phi_{12} & \ldots & \phi_{1N} \\ \phi_{21} & \phi_{22} & \ldots & \phi_{2N} \\ \vdots & \vdots & \vdots & \vdots & \\ \phi_{K1} & \phi_{K2} & \ldots & \phi_{KN} \end{pmatrix} \end{align}

The matrix $\mathbf {\Theta }$ of Equation (3.10) gives a topic distribution for each document, which means a document can be represented by a combination of topics and the elements of the $\theta$ matrix are the weights for each topic here. However, the number of topics $K$ is quite small, as is the number of words $N$, so the document can be converted to a very small dimension vector by the $\theta$ matrix. The matrix $\mathbf {\Phi }$ of Equation (3.11) gives a word distribution for each topic, which means that each word has a specific appearance probability. In other words, some words with high probability may appear very frequently for a specific topic, this yields that the top-ranked words in the distribution are a kind of representative words for the topic.

3.4 Performance measure for vector quantisation

To evaluate the result of vector quantisation, a performance measure is introduced. For each element $x$ of a given vector, its relative mean root squared error (RRMSE) is defined as follows:

(3.12)\begin{equation} {\rm RRMSE}=\frac{\sqrt{\frac{1}{N} \sum_{n=1}^{N}(x_{n}-\tilde{x}_{n})^{2} }} {\sqrt{\frac{1}{N}\sum_{n=1}^{N} x_{n}^{2} }} =\sqrt{\frac{\sum_{n=1}^{N}(x_{n}-\tilde{x}_{n})^{2} } {\sum_{n=1}^{N} x_{n}^{2} }} \end{equation}

where $x$ is the original value of the element and $\tilde {x}$ is the reconstructed value of the element from its code.

3.5 Performance measure of the topic model

Because the topic model intends to separate the latent factor into its basic components, a valuable topic model should reflect sufficient differences between any two topics. Considering the Jessen–Shannon divergence gives the distance or unsimilarity between two probability distributions, an average of the Jessen–Shannon divergence between any two different topics is chosen as the total performance measure for a specific topic model, whose definition is given as follows:

(3.13)\begin{equation} \overline{D_{JS}} = \frac{1}{K(K-1)}\sum_{p \neq q} D_{JS} (\phi_{p}, \phi_{q}) \end{equation}

where $D_{JS} (\phi _{p},\, \phi _{q})$ is the Jessen–Shannon divergence between topic $p$ and $q$. Also, $\phi _{p}$ and $\phi _{q}$ are the code distribution of topic $p$ and $q$, respectively.

4.1 Preliminary experiment: moving vessels dataset

The original AIS dataset used for our experiments is the one distributed by the 2nd Maritime Big Data Workshop (MBDW) (Ray et al., Reference Ray, Dréo, Camossi and Jousselme2018; MBDW, 2020), collected from Brest Bay in North West France from October 1, 2015 to March 31, 2016. The dataset attributes are sourcemmsi, navigationalstatus, rateofturn, speedoverground, courseoverground, trueheading, lot(longitude), lat(latitude) and t(timestamp). The original dataset contains 19,035,630 AIS records, which come from 3,928 ships. However, there are 8,203,254 records whose speed equals 0 knot/s in this dataset.

For a preliminary exploration of the AIS trajectory of the moving vessels, a moving vessels’ dataset is first prepared by the following procedures.

1. (i) A region of interest is defined in Brest Bay and its vicinity, which gives a rectangle area of (${\rm longitude}=-7\,{\cdot }\,0$, ${\rm latitude}=49\,{\cdot }\,5$) and (${\rm longitude}=-2\,{\cdot }\,0$, ${\rm latitude}=46\,{\cdot }\,5$). Data records out of this coverage are ignored.

2. (ii) To clean the dataset, records without valid heading values are removed.

3. (iii) To select the moving vessels, records with a speed value equal to 0 knot/s are removed.

4. (iv) Only retain the attributes of sourcemmsi, lon, lat, speedoverground, trueheading and t, others are removed.

After these procedures, the dataset shrinks to 4,980,363 records of 3,928 ships. Then AIS data records are categorised according to vessels’ MMSI and segmented to sails when the interval of its timestamp is longer than one hour. Furthermore, based on the following two assumptions, a minimum length of data records is introduced for making this dataset.

1. (i) Because navigation patterns are composed of a series of AIS data records, longer trajectories are more likely to reflect representative patterns.

2. (ii) Because computational costs for vector quantisation depend on the total number of data records, short-distance sails are not retained.

Without loss of generality, a minimum length of 3,000 is experimentally chosen. This yields a subset of the original dataset, so-called as the MovingVessels dataset in the rest of the paper. The descriptions of basic items for the original dataset and MovingVessels dataset are shown in Table 1.

Table 1. Descriptions of the original dataset and two derived moving vessels’ dataset

As a whole image of the dataset, the sail trajectories of the MovingVessels dataset are shown in Figure 2(a). Moreover, by dividing the coverage into $33\times 34$ grids, where each grid area is approximately $10\ {\rm km}\times 10\ {\rm km}$, distribution of all data records is generated by counting the data records in each grid. The result of the experiment for the MovingVessels dataset is shown as a heatmap in Figure 2(b).

Figure 2. Trajectory and distribution of the MovingVessels dataset. (a) All trajectory in the MovingVessels dataset. (b) Distribution of the MovingVessels dataset

It is clear that data records occur intensively only in a few central grids, whereas only a very small number of data records occur in the other grids. Actually, this leads us to consider a four-dimension vector for our research, as scalar quantisation generates a huge number of data points and leads to expensive computation time and resources. This experimental result further motivates the introduction of vector quantisation when processing AIS trajectory data.

4.2 Experiments for vector quantisation

4.2.1 Experiment 1: vector quantisation via k-means algorithm

Vector quantisation is applied to these four-dimension data records obtained from previous procedures, with the codebook length $L$ assigned in a style of $L=B^{4}$, although each dimension is not equally divided into $B$ intervals, where the number $B$ denotes the precision of a vector quantisation setting. For applying the k-means algorithm of vector quantisation, experiments with $B=8,\, 12,\, 16$ have been conducted. As an example of the results, a trajectory of original data and three reconstructed trajectories from different codebook lengths are shown in Figure 3. From these results, it can be confirmed that a bigger value of $B$ gives better precision of reconstruction. However, there still remain some sparse data away from the trajectory. These sparse data arise because the encoding method is not a completely reversible process. Therefore, when reconstructing the initial data, that is, longitude, latitude, heading and speeds, the centroid of each code is used, leading to a scattering of the data as compared to the initial data. This trend is improved by the choice of longer codebooks as revealed by Figure 3. For different codebook lengths, the RRMSE of each attribute and the elapsed time are shown in Table 2. These results show that a good precision of reconstructed data can be achieved as the codebook length is efficiently long, but it takes unaffordable computation time.

Figure 3. Trajectory of AIS data reconstructed from codes of k-means vector quantisation. (a) Original data, (b) vector quantisation with $B=8$, (c) vector quantisation with $B=12$ and (d) vector quantisation with $B=16$

Table 2. Performance and elapsed time of k-means algorithm for different codebook lengths

4.2.2 Experiment 2: vector quantisation via PQk-means algorithm

For applying the PQk-means algorithm of vector quantisation, experiments with $B=8,\, 12,\, 16,\, 20$ and $24$ are conducted. As an example of the results, four reconstructed trajectories when $B=12,\, 16,\, 20$ and $24$ are shown in Figure 4, whereas their original trajectory is the same one as in Figure 3(a). From these results, it can be confirmed that a bigger value of $B$ gives better precision of reconstruction, similar to Figure 3. Furthermore, the case of $B=24$ in Figure 4 provided a better reconstruction precision and less sparse data than the case of $B=16$ in Figure 3.

Figure 4. Trajectory of AIS data reconstructed from codes of PQk-means vector quantisation. (a) Vector quantisation $B=12$, (b) vector quantisation $B=16$, (c) vector quantisation $B=20$ and (d) vector quantisation $B=24$

The RRMSE of each attribute and the elapsed time are shown in Table 3. These results show that a good precision of reconstructed data can be achieved as the codebook length is efficiently long, but its computation time becomes longer according to the codebook length. However, compared with the k-means algorithm, this result shows a great improvement of approximately 127 times in computation speed when $B=16$. This provides a significant benefit when processing large amounts of data in a limited time. On the other side, suppose $B=16$, when assigning the minimum length to 2,000 and 1,000, the total number of codes in the MovingVessels dataset becomes 1,898,973 and 3,590,309. The elapsed time for each case is 0:13:56 and 0:26:35, respectively. This implies that the computational time increases almost linearly when the minimum length decreases.

Table 3. Performance and elapsed time of the PQk-means algorithm for different codebook lengths

4.3 Experiments for validating the topic model

4.3.1 Experiment 3: clustering moving dataset – DBSCAN algorithm

The DBSCAN algorithm has been reported to extract waypoints from AIS data (Birant and Kut, Reference Birant and Kut2007). To make a comparison with the topic model, a trajectory clustering experiment with the DBSCAN algorithm is conducted. As the DBSCAN algorithm directly clusters data records, data records of all sails are gathered together at first. In this experiment, the parameter of maximum distance is assigned to 0$\,{\cdot}\,$2 and the number of samples in a neighbourhood is assigned to 5. With this setting, all the 1,146,372 trajectory data records in the MovingVessels dataset are clustered in 227 clusters, whereas 1,625 data records are judged as noise, so do not belong to any cluster. From the results, the clustering results appear as considerably imbalanced, the maximum number of data records is 1,105,352, whereas the number of data records in most clusters is less than 200. Because DBSCAN is a density-based algorithm, this result can be explained easily by the distribution of data records shown in Figure 2(b). To provide the typical trajectories of the data records, these clusters with length $\geq 200$ are shown in Figure 5.

Figure 5. Navigation pattern results extracted by DBSCAN algorithm (selected cluster (length of cluster $\geq 200$))

4.3.2 Experiment 4: clustering moving dataset – Gaussian mixture model

Gaussian mixture model is reported to detect an anomaly in sea traffic (Laxhammer, Reference Laxhammer2008). To compare with the topic model, a trajectory clustering experiment with the Gaussian mixture model is conducted. As the Gaussian mixture model cluster data records directly, the data records of all sails are first gathered together. Without loss of generality, the number of components is assigned to 20. With this setting, all the 1,146,372 trajectory data records in the MovingVessels dataset are categorised into each one of 20 clusters with different belonging probabilities. To provide the typical trajectories of the data records, the top 300 records for each cluster are shown in Figure 6.

Figure 6. Navigation pattern results extracted by Gaussian mixture model (the number of components $=$ 20, constructed from the top 300 records)

4.3.3 Experiment 5: topic model – vessels’ navigation patterns extraction

After vector quantisation to the MovingVessels dataset with parameter $B=24$, 209 code documents are obtained. Then, according to the previous procedure, the LDA algorithm of the topic model is applied to these code documents.

The results of the experiment yield two matrices, document-topic matrix $\mathbf {\Theta }$ and topic-code matrix $\mathbf {\Phi }$. The topic-code matrix $\mathbf {\Phi }$ can be used to extract the navigation patterns. Because each row of the $\mathbf {\Phi }$ matrix gives a probability distribution on all the codes in the codebook, the top-ranked codes are used to represent the topic. However, a topic denotes a vessel navigation pattern. Therefore, these top-ranked codes can be selected as a representative of a navigation pattern. Furthermore, by referencing the codebook of vector quantisation, these representative codes can be reconstructed to their corresponding approximate data of longitude, latitude, heading and speed. Then, these data are depicted on the map of Brest Bay, so that the navigation patterns of vessels can be clearly visualised.

According to these considerations, for all rows of the $\mathbf {\Phi }$ matrix, these reconstructed codes give us all the navigation patterns for a specific topic set. Moreover, to investigate the effect of the number of topics, this work is applied to all $\mathbf {\Phi }$ matrices of experiments with $K=20$, 40, 60, 80, 100, 120, 140, 180, 200, 220 and $240$. For an overview of the results of this extraction, the navigation patterns when $K=20$ are shown in Figure 7. Although $K=20$ is not the best performance case among all of these experiments, considering space constraints, the case of $K=20$ is chosen here. Additional details are presented in Section 4.4.3.

Figure 7. Navigation pattern results extracted by the topic model (the number of topics $=$ 20, reconstructed from the top 100 codes)

To provide a better understanding of vessel navigation patterns, an example of comparing the results between different $K$ is shown in Figure 8. Figure 8(a) illustrates the reconstructed trajectory of topic #16 from the experiment results of the number of topics $K=20$. For comparison, Figures 8(b), 8(c) and 8(d) illustrate the reconstructed trajectory of topics #9, #19 and #115 in the experiment results for the number of topics $K=120$. By observation, it appears clearly that the pattern in Figure 8(a) is decomposed to the patterns in Figures 8(b), 8(c) and 8(d). Therefore, this confirmed that higher resolution of vessel navigation patterns can be achieved by assigning higher topic numbers.

Figure 8. Comparison of navigation pattern between different numbers of topics. (a) Topic #16 when $K=20$, (b) topic #9 when $K=120$, (c) topic #19 when $K=120$ and (d) topic #115 when $K=120$

4.4 Further discussions on the topic model

4.4.2 Experiment 6: topic model – find similar sails

The experiment results give document-topic matrix $\mathbf {\Theta }$ and topic-code matrix $\mathbf {\Phi }$. The usages of topic-code matrix $\mathbf {\Phi }$ have been described in Section 3.3 In this subsection, the usages of document-topic matrix $\mathbf {\Theta }$ will be described.

As mentioned in Section 3.3, document-topic matrix $\mathbf {\Theta }$ gives a set of proportions on topics for all sails. this means that similar sails can be derived by calculating the cosine similarity between their topic proportion vector. For illustration purposes, examples of similar sails regarding the topic proportion are shown in Figure 9. By comparing these results, it can be easily confirmed that proportions have top values for the same topic number from Figures 9(a) and 9(b), also the trajectory of the pair sails are very similar to each other from Figures 9(c) and 9(d). The significance of this fact can be stated as follows: if a set of topics have been extracted by the topic model, a sail can be represented with a topic proportion vector of the same length as $K$, instead of its original AIS trajectory data which have very long and uncertain lengths. This will lead to high performance for identifying a specific vessel or finding similar vessels.

Figure 9. (a) Topic proportion of sail #58, (b) topic proportion of sail #198, (c) trajectory of sail #58 and (d) trajectory of sail #198. Cosine similarity between panels (a) and (b) is 0$\,{\cdot}\,$9677

4.4.3 Experiment 7: topic model – the number of topics

To evaluate the performance of the topic extraction, a set of topic model experiments with different numbers of topics has been conducted, where the number of topics $K$ has been set to 10, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220 and 240. From the topic distribution results of each experiment, the average of Jessen–Shannon divergence between any two different topics for each setting of $K$ has been calculated. Results of this calculation are shown in Figure 10(a), also the elapsed times for each topic model setting are shown in Figure 10(b) together. From Figure 10(a), as the number of topics increases, the average performance makes clear increases at the lower range from 10 to 80, but almost makes no difference at the higher range from 80 to 240. Because Jessen–Shannon divergence represents the differences between topics, a higher topic difference means a better model. The results of Figure 10(a) denote that the performance of the topic model is improved between $K=10$ and $80$, while no more improvements are observed between $K=80$ and $240$. The computation times of the topic model for different $K$ are shown in Figure 10(b). From this result, one can understand that the computation time is proportional to the number of topics. Therefore, considering the computation cost, the best choice of the number of topics should be $K=80$ for this MovingVessels dataset.

Figure 10. Performance of topic model according to the number of topics. (a) Average of Jessen–Shannon divergences between any two different topics. (b) Elapsed time for calculating topic model