3.1 Douglas–Peucker algorithm

The Douglas–Peucker algorithm, also called the DP algorithm, was proposed for simplifying two-dimensional curves (Douglas and Peucker, Reference Douglas and Peucker1973). The DP algorithm has also been used to compress vessel trajectory data (Liu et al., Reference Liu, Li, Yang, Wu, Liu and Liu2019, Reference Liu, Shi and Zhu2020). A schematic diagram of the DP algorithm is shown in Figure 2. The vessel trajectory compression procedure based on the DP algorithm is as follows.

• • Step 1. Set a threshold according the acceptable margin of error for practical application. Then, the first point ${p_1}$ and last point ${p_n}$ of the trajectory are respectively selected as the initial anchor point and initial floating point to create a baseline $\overline {{p_1}{p_n}}$. Next, the distances of the other points to the baseline are calculated one by one to find the point with the maximum distance. If the maximum distance is smaller than the setting threshold, the points except the first point ${p_1}$ and last point ${p_n}$ should be cleared, which means that the first point ${p_1}$ and last point ${p_n}$are the characteristic points and the compressed trajectory data are the set of $P^{\prime}= ({{p_1},{p_n}} )$. Otherwise, the next step is taken.

• • Step 2. The point with the maximum distance greater than the setting threshold is a reference characteristic point, acting as both the floating point corresponding to the previous anchor point and the anchor point corresponding to the previous floating point in Step 1. Suppose the reference characteristic point is ${p_k}$. The $\overline {{p_1}{p_k}}$ is the new baseline of points of $({{p_1},{p_2}, \cdots ,{p_i}, \cdots ,{p_k}} )$, and the $\overline {{p_k}{p_n}}$ is the new baseline of points of $({{p_k},{p_{k + 1}}, \cdots ,{p_j}, \cdots ,{p_n}} )$. Then, the distances of the points of $({{p_2}, \cdots ,{p_i}, \cdots ,{p_{k - 1}}} )$ to the baseline of $\overline {{p_1}{p_k}}$ are calculated one by one to find the point with the maximum distance, and the existence of a new reference characteristic point in $({{p_1},{p_2}, \cdots ,{p_i}, \cdots ,{p_k}} )$ is checked by comparing the value of the maximum distance and the setting threshold. Additionally, the points of $({{p_k},{p_{k + 1}}, \cdots ,{p_j}, \cdots ,{p_n}} )$ and the baseline of $\overline {{p_k}{p_n}}$ are also handled for checking the existence of a new reference characteristic point. If there are no new reference characteristic points, the compressed trajectory data are the set of $P^{\prime} = ({{p_1},{p_k},{p_n}} )$. Otherwise, the next step is taken.

• • Step 3. The points corresponding to the baseline with the existing new reference characteristic points continue to be processed like Step 2 until there is no unique reference characteristic point in each subset of points. The first and last point as well as all reference characteristic points are retained as the compressed trajectory data.

Figure 2. Schematic diagram of DP algorithm

The threshold is the basis of the DP algorithm. A more significant threshold may cause that compressed trajectory to have low similarity with the original trajectory. Meanwhile, a smaller threshold may cause a low compression ratio to result in failure of achieving a satisfactory outcome. In addition, the DP algorithm is time-consuming due to a large amount of distance calculation.

3.2 Uniform sampling algorithm

The US algorithm is used to compresses the trajectory data by uniformly selecting the sample data in the time sequence. Suppose the interval of retained trajectory samples is m, then the compressed data of trajectory P is $({{p_1},{p_{m + 1}}, \cdots ,{p_{im + 1}}, \cdots } )$. However, the US algorithm can't effectively compress the trajectory data with large fluctuation.

3.3 Spatial quality simplification heuristic algorithm

The SQUISH algorithm is proposed for offset data compression by deleting the points that cause an acceptable margin of error (van Emde Boas, Reference van Emde Boas1975; Muckell et al., Reference Muckell, Hwang, Patil, Lawson, Ping and Ravi2011, Reference Muckell, Olsen, Hwang, Lawson and Ravi2014). The schematic diagram of the SQUISH algorithm is shown in Figure 3. The effect of data compression depends on the size of buffer based on the SQUISH algorithm. The vessel trajectory compression procedure based on the SQUISH algorithm is as follows.

• • Step 1. Set a buffer size, which determines the number of the trajectory points after compression according to the compression ratio. Suppose there are six points of $({{p_1},{p_2}, \cdots ,{p_i}, \cdots ,{p_6}} )$ in the original vessel trajectory dataset and four points in the buffer.

• • Step 2. Input a new point of vessel trajectory into the buffer. When the next point is recorded in the buffer, the point among the previous points with the smallest error will be deleted. Then, a new set in the buffer will be obtained. Continually, input the points by the timestamp, which are not in the buffer, and delete the points with the smallest error until there are no points out of the buffer.

• • Step 3. Finally, the complete compressed vessel trajectory data with a set of m points will be obtained. In Figure 2, the points of $({{p_2},{p_3}} )$ have the smallest error in turn, so the points of $({{p_1},{p_4},{p_5},{p_6}} )$ are the compression trajectory.

Figure 3. Schematic diagram of SQUISH algorithm

3.4 Dead reckoning algorithm

The DR algorithm is a compression algorithm that uses the integral vector technique of estimating or measuring displacement to the fixed position (Zhao et al., Reference Zhao, Ochieng, Quddus and Noland2003; Trajcevski et al., Reference Trajcevski, Cao, Scheuermanny, Wolfsonz and Vaccaro2006). The schematic diagram of the DR algorithm is shown in Figure 4. Predicting ship position from ship speed and previous position is the basis for ship trajectory compression based on the DR algorithm. The vessel trajectory compression procedure based on the DR algorithm is as follows.

• • Step 1. Set an error threshold, which determines the retained points of vessel trajectory data. Suppose there are seven points of $({{p_1},{p_2}, \cdots ,{p_i}, \cdots ,{p_7}} )$ in the original vessel trajectory dataset and the error threshold is ${d_0}$.

• • Step 2. The ship's trajectory is predicted based on the first point, and the speed and the distance between the original and the corresponding trajectory points are calculated in the time series. Furthermore, the previous point of the point with a distance more significant than the error threshold is a compressed trajectory point, which is the subsequent base point to predict the vessel trajectory. This process is constantly implemented until the last point of the compressed trajectory is determined.

• • Step 3. Determine the compression trajectory data. In Figure 3, ${d_5}\mathrm{\ > }{d_0}$ and ${d_7}\mathrm{\ > }{d_0}$, so the points of $({{p_1},{p_4},{p_6}} )$ are the compression trajectory.

Figure 4. Schematic diagram of DR algorithm

3.5 OPW algorithm

The OPW algorithm is a classical online compression algorithm (Keogh et al., Reference Keogh, Chu, Hart and Pazzani2001), which compresses the trajectory data based on the window line of the connection between the two points. The vessel trajectory compression procedure based on the OPW algorithm is as follows.

• • Step 1. Set a threshold, which determines the retained points of vessel trajectory data. Suppose there are n points of $({{p_1},{p_2}, \cdots ,{p_i}, \cdots ,{p_n}} )$ in the original vessel trajectory dataset.

• • Step 2. Determine a window line by connecting the first point ${p_1}$ and the third point ${p_3}$ in the time series, and calculate the distance from the second point ${p_2}$ to the window line. If the distance is smaller than the threshold, the second point should not be retained in the compressed dataset. Otherwise, the second point should be retained in the compressed dataset. Next, a new window line should be replaced as the connection between the second point and the fourth point, and afterwards, the distance from the third point to the new window line is calculated to determine whether the third point should be in the compressed dataset or not. The compression will continue until the last point of the trajectory.

• • Step 3. Determine the compression trajectory data. Meanwhile, there is some improved OPW algorithm used for compression trajectory. The before open window (BOPW) algorithm is used to compress the trajectory data by changing the window line, and there are two or more points between the former point and the latter point. Moreover, when the sum of errors in the window is greater than a threshold, the second-to-last trajectory point in the window is used as the end point of the approximate line segment of that window. Another improved OPW algorithm is called the normal open window (NOPW) algorithm, which changes the window line like BOPW, but it generates a data point in the window that exceeds the error threshold as the end point of the approximate line segment of the window (Muckell et al., Reference Muckell, Hwang, Lawson and Ravi2010). In addition, there is an improved OPW algorithm that uses SED to replace PED (Meratnia and Rolf, Reference Meratnia and Rolf2004).

4.1 Step 1: AIS vessel trajectory data preprocessing

The information from AIS consists of the MMSI number, timestamp, position, speed, course and other vessel attributes (Wang et al., Reference Wang, Liu, Yan, Wang and Zhang2022). However, the vessel trajectory data always have abnormal data, due to the equipment. The abnormal data, which is caused by data loss and other phenomena, will be deleted to improve the dataset quality by data preprocessing (Zhao et al., Reference Zhao, Shi and Yang2018a, Reference Zhao, Shang, Wang, Zheng, Nguyen and Zheng2018b).

For the vessel AIS information in the relevant area, the vessel position information collected by the AIS equipment at times ${T_1}$ and ${T_2}$ are $({{\lambda_1},{\varphi_1}} )$ and $({{\lambda_2},{\varphi_2}} )$. If ${T_1} < T < {T_2}$, then the position information of vessel AIS at time T can be calculated by the following formulae:

(1)\begin{align}{\lambda _T}& = {\lambda _1}\times \frac{{{T_2} - T}}{{{T_2} - {T_1}}}+ {\lambda _2}\times \frac{{T - {T_1}}}{{{T_2} - {T_1}}}\end{align}

(2)\begin{align}{\varphi _T}& = {\varphi _1}\times \frac{{{T_2} - T}}{{{T_2} - {T_1}}}+ {\varphi _2}\times \frac{{T - {T_1}}}{{{T_2} - {T_1}}}\end{align}

For large and medium-sized vessels, their course and speed will not be significantly changed in a short time, nor will their bow direction change significantly, which can be approximated to uniform linear motion. However, for small vessels, their manoeuvrability is better. When the speed and course change rate is significant, some errors will be caused. At this point, the vessel classification as a small vessel is derived from the size data obtained through the AIS. Subsequently, the rate of change of the vessel's speed and course is computed, leading to a refinement of the previously presented formulae.

4.2 Step 2: AIS vessel trajectory compression based on improved open window algorithm

An improved algorithm to compress the AIS trajectory data generated by inland and coastal vessels is proposed to explore whether online compression technology can better compress vessel trajectory data. The algorithm based on an existing online compression technology called the OPW (Keogh et al., Reference Keogh, Chu, Hart and Pazzani2001) is an IOPW algorithm. In addition, various performance indicators are used in this context. The OPW algorithm first sets a window for data compression, containing some trajectory points. Unlike other window algorithms, however, the OPW algorithm then calculates the PED for all the points in the window. Based on the relative error between the sum of the errors within the window and a given threshold, the endpoints of the approximate line segments can be selected.

The OPW algorithm relies on the distance generated by the combination of PED. Although PED can retain the contour information of the vessel trajectory, its time information is almost random. SED is introduced to limit the trajectory's offset further to achieve better information retention. Compared with the OPW algorithm, which only considers the spatial features of the trajectory points, the IOPW algorithm also finds the time characteristics of the trajectory. This algorithm employs iterative methods to measure the amount of information loss incurred by trajectory simplification, but no longer iterates over the entire trajectory. The OPW algorithm uses the established window to iterate and continuously update the window information until simplification is completed.

The improved OPW algorithm is illustrated in Figure 6. The original OPW algorithm uses the PED as a measure to calculate the error, for example, $|\overrightarrow {{\textbf{t}_4}{\textbf{t}_{4}^{\ast}}}|$ is the PED of data point ${{\textbf{t}}_4}$ to line segment $\overrightarrow {{{\textbf{t}}_0}{{\textbf{t}}_{10}}}$ in Figure 7(a). The trajectory data compression algorithm using the PED has a good compression ratio (Cao et al., Reference Cao, Wolfson and Trajcevski2006; Shi and Cheung, Reference Shi and Cheung2006; Liu et al., Reference Liu, Zhao, Sommer, Shang, Kusy and Jurdak2015). However, the time information is ignored in the PED. Therefore, when the PED line segment simplification algorithm is applied in the spatio-temporal query of a compressed trajectory to determine ‘the position of the object at time m’, and it returns an approximate perpendicular point $\textbf{t}_{m}^{\ast}$. However, the distance from that point to the actual position t of the moving object at time m is not limited. In the present study, the coordinates of the ${\textrm{t}_{\boldsymbol{i}}}$,${\textrm{t}_{\boldsymbol{j}}}$ and ${{\textbf{t}}_{\boldsymbol{m}}}(\boldsymbol{i} < \boldsymbol{m} < \boldsymbol{j})$ trajectory points in the original trajectory sequence are $({{x_i},\;{y_i},\;{z_i}} ),\;({{x_j},{y_j},{z_j}} ),({{x_m},{y_m},{z_m}} )$ (the z coordinates represent time information). Thus, the PED of data point ${{\textbf{t}}_{\boldsymbol{m}}}$ to point $\textbf{t}_{m}^{\ast}$ (assuming its coordinates are $(x_{m}^{\ast},y_{m}^{\ast},z_{m}^{\ast})$ and the window line formed by${{\textbf{t}}_{\boldsymbol{i}}}$,${{\textbf{t}}_{\boldsymbol{j}}}$) can be obtained as

(3)\begin{equation}\textrm{PED}({{{\textbf{t}}_{\boldsymbol{m}}}} )= \left|{\frac{{({{y_j} - {y_i}} ){x_m}+ ({{x_i} - {x_j}} ){y_m}+ {x_j}{y_i} - {x_i}{y_j}}}{{\sqrt {{{({{y_j} - {y_i}} )}^2}+ {{({{x_j} - {x_i}} )}^2}} }}} \right|\end{equation}

Figure 6. Flow chart of IOPW algorithm

Figure 7. (a) PED and (b) SED illustrate by using $T[{{{\textbf{t}}_0}, \ldots ,{{\textbf{t}}_{10}}} ]$ and $T^{\prime}[{{{\textbf{t}}_0},{{\textbf{t}}_4},{{\textbf{t}}_{10}}} ]$

Figures 7(a) and 7(b) show the trajectory $T[{{{\textbf{t}}_0}, \ldots ,{{\textbf{t}}_{10}}} ]$ with a total of 11 points, represented by two and four continuous segments (black) and compressed by the DP algorithm using PED and SED, respectively. In the two figures, the DP algorithm first creates a line segment$\overrightarrow {{{\textbf{t}}_0}{{\textbf{t}}_{10}}}$. Then, it calculates the distance from the other points on the trajectory to the $\overrightarrow {{{\textbf{t}}_0}{{\textbf{t}}_{10}}}$ in panel (a). The DP algorithm finds that point ${{\textbf{t}}_4}$ has the maximum distance to$\overrightarrow {{{\textbf{t}}_0}{{\textbf{t}}_{10}}}$, being greater than the specified error threshold. Therefore, the trajectory is divided into two segments $[{{{\textbf{t}}_0}, \ldots ,{{\textbf{t}}_4}} ]$ and$[{{{\textbf{t}}_4}, \ldots ,{{\textbf{t}}_{10}}} ]$.

The SED is the Euclidean distance between the data points and the approximate spatio-temporal data points on the corresponding line segment (van Emde Boas, Reference van Emde Boas1975). The approximate spatio-temporal data point ${\textbf{t}^{\prime}}$ is obtained according to the point ${\textbf{t}}$ on the original trajectory (${\textbf{t}^{\prime}}$ can be obtained from the first and last points of the trajectory according to interpolation in time), and just calculates the distance between ${\textbf{t}^{\prime}}$ and ${\textbf{t}}$. After the original text is modified, SED should be calculated by trajectory points and their approximate spatio-temporal data points. In Figure 7(b),${{\textbf{t}^{\prime}}_2}\textrm{,}{{\textbf{t}^{\prime}}_4}$ and ${{\textbf{t}^{\prime}}_7}$ are approximate spatio-temporal data points of ${{\textbf{t}}_2},{{\textbf{t}}_4},$ and ${{\textbf{t}}_7}$, and more trajectory points can be preserved using the SED compression algorithm. Meanwhile, the SED can be used to limit the error threshold between trajectory points and their approximate spatio-temporal data points (e.g. ${{\textbf{t}^{\prime}}_4}$ is the point where ${{\textbf{t}}_0}$ and ${{\textbf{t}}_{10}}$ interpolate in proportion to time). The SED represents the Euclidean distance between the point on the original path and the point on the simplified path in proportion to time in a general way. Similarly, if it is assumed that the trajectories from ${{\textbf{t}}_{\boldsymbol{i}}}$ to ${{\textbf{t}}_{\boldsymbol{j}}}$ in the original trajectory sequence are compressed into ${{\textbf{t}}_{\boldsymbol{i}}}$ and${{\textbf{t}}_{\boldsymbol{j}}}$, then t_m is located in the middle of ${{\textbf{t}}_{\boldsymbol{i}}}$ and ${{\textbf{t}}_{\boldsymbol{j}}}$ assuming the coordinates are$({{x_m},\;{y_m},\;{z_m}} )$. The time scale projection position ${{\textbf{t}^{\prime}}_{\boldsymbol{m}}}$ on the trajectory segment is$({x_m^{\prime},\;y_m^{\prime},\;z_m^{\prime}} )$:

(4)\begin{align}{x^{\prime}_m}& = {x_i}+ \frac{{{z_m} - {z_i}}}{{{z_j} - {z_i}}}({{x_j} - {x_i}} ),\;{z_i}\mathrm{\ < }{z_m}\mathrm{\ < }{z_j}\end{align}

(5)\begin{align}{y^{\prime}_m}& = {x_i}+ \frac{{{z_m} - {z_i}}}{{{z_j} - {z_i}}}({{y_j} - {y_i}} ),\;{z_i}\mathrm{\ < }{z_m}\mathrm{\ < }{z_j}\end{align}

When the position of ${{\textbf{t}^{\prime}}_{\boldsymbol{m}}}$ is computed according to ${{\textbf{t}}_{\boldsymbol{i}}}$ and$\;{{\textbf{t}}_{\boldsymbol{j}}}$ by interpolation in time, the SED distance of ${{\textbf{t}}_{\boldsymbol{m}}}$ to$\; {{\textbf{t}^{\prime}}_{\boldsymbol{m}}}$ can be calculated:

(6)\begin{equation}\textrm{SED}({{{\textbf{t}}_{\boldsymbol{m}}}} )= \sqrt {{{({{x_m} - {{x^{\prime}}_m}} )}^2} + {{({{y_m} - {{y^{\prime}}_m}} )}^2}} \end{equation}

Therefore, with the IOPW algorithm using the fusion distance between SED and PED as a parameter to compare with the error threshold, $\textrm{D}({{{\textbf{t}}_{{\textbf{m}}}}} )$ is the newly defined fusion distance, based on the original algorithm using the PED distance as the measurement error threshold. Since there is an order of magnitude difference between SED and PED, $\mathrm{\alpha }$ is multiplied by the formula to unify the order of magnitude of SED and PED. The new parameter SED is continuously introduced. Combining the two distances and the resulting distance D, the value of $\textrm{D}({{{\textbf{t}}_{{\textbf{m}}}}} )$ is determined through the application of Equations (3) and (6):

(7)\begin{equation}D({{\boldsymbol{t}_{\boldsymbol{m}}}} )= \lambda \times \textrm{PED}({{\boldsymbol{t}_{\boldsymbol{m}}}} )+ ({1 - \lambda } )\times \alpha \times\textrm{SED}({{\boldsymbol{t}_{\boldsymbol{m}}}} )\end{equation}

After multiple experiments, it was found that when $\mathrm{\alpha =\ }0.01,\;\lambda = 0.87,$ better compression results can be obtained. As described in Section 5, the above parameters were used for the experiments.

In the present study, the IOPW algorithm is employed with the OPW method to determine the end point of the approximate linear segment. The trajectory of a vessel can be represented by a set of points $T= \{ {{\textbf{t}}_{{\textbf{i}}}}|{{\textbf{t}}_{{\textbf{i}}}},{{\textbf{i}}} = 1,2,3,4 \ldots ,n\}$, where $\;{{\textbf{t}}_{{\textbf{i}}}}$ represents an AIS information point on the vessel trajectory and n is the number of points. Window lines were constructed from ${{\textbf{t}}_1}$ to $\;{{\textbf{t}}_{{\textbf{i}}}}\;({{{\textbf{i}}} \ge 3} )$. The PED(T) is first calculated from the point between t₁ and ${{\textbf{t}}_{{\textbf{i}}}}$ to the window line. Second, the synchronisation point ${{\textbf{t}^{\prime}}_{{\textbf{i}}}}$ of the ${{\textbf{t}}_{{\textbf{i}}}}$ point within the error range is determined, and the SED from ${{\textbf{t}}_{{\textbf{i}}}}$ to ${{\textbf{t}^{\prime}}_{{\textbf{i}}}}$ is calculated. Here, ${\textbf{D}}\;({{{\textbf{t}}_{{\textbf{i}}}}} )$ is calculated by the relation $\textrm{D}({{{\textbf{t}}_{{\textbf{m}}}}} )= \lambda \mathrm{\times PED}({{{\textbf{t}}_{{\textbf{m}}}}} )+ ({1 - \mathrm{\lambda }} )\mathrm{\times \alpha \times SED}({{{\textbf{t}}_{\boldsymbol{m}}}} )$. If $\;{\textbf{D}}\;({{{\textbf{t}}_{{\textbf{i}}}}} )> \textrm{epsilon}$, ${{\textbf{t}}_{{\textbf{i}}}}$ is regarded as the end point of the approximate trajectory of the previous segment and is also used as the starting point for the subsequent compression simplification. This point is also saved as a compressed point to the point set $\;F= \{ f(k )|k= 1,2 \ldots ,m\}$, where k is the number corresponding to the specific point and m is the number of points. If the point between ${{\textbf{t}}_1}$ and ${{\textbf{t}}_{{\textbf{i}}}}$ does not have a vertical Euclidean distance exceeding the threshold, the point ${{\textbf{t}}_{{{\textbf{i}}} + 1}}$ after ${{\textbf{t}}_{{\textbf{i}}}}$ is introduced to the window. The operation process is then repeated. The symbols and terminology used in the algorithm pseudo code and the pseudo code itself are given in the Nomenclature section and Table 1, respectively.

Table 1. IOPW algorithm

Figure 7 shows that the AIS data points exceeding the threshold are segmentation points. The data sequence is interrupted at data points t₄, t₈, t₁₂ and t₁₆. There are multiple termination conditions for each iteration of the window algorithm, e.g. based on the number of points, the SED and PED, or the maximum error in the window. Once the threshold for any of these conditions is not satisfied, the current iteration is terminated and the next iteration is initiated. The fusion distance formed by SED and PED at t₄ point exceeds the error threshold. So t₄ is retained in the compressed trajectory. Meanwhile, the previous iteration ends and t₄ is taken as the initial trajectory point of the new iteration to continue the compression. Finally, the points retained on the compressed trajectory are t₁, t₄, t₈, t₁₂ and t₁₆, and after the trajectory point t₁₉, the compression will continue until the final point of the trajectory. Therefore, countermeasures to address this problem are required. As the algorithm only considers the position of the AIS trajectory point in the window, it can achieve the best performance in each provincial simplification. However, it cannot balance the overall trend.

Although the computational cost of the OPW algorithm is high, its application in road traffic trajectory compression is widespread (Cao et al., Reference Cao, Cong and Jensen2010). Most of this is attributable to the fact that the algorithm is less affected by noise when computing relatively short data sequences. The IOPW algorithm can perform dynamic data calculation. As the input, only the start point of the trajectory is required. The algorithm does n't need to determine the trajectory endpoint.

4.3 Step 3: Performance assessment indices

The performance indices examined in the compression experiments are delineated below.

4.3.1 Trajectory compression ratio

The compression ratio measures how much storage capacity has been saved by compressing original vessel trajectories (Muckell et al., Reference Muckell, Hwang, Patil, Lawson, Ping and Ravi2011). The higher compression ratio is related to effectively reducing the storage cost while preserving the utility of trajectories. The trajectory compression ratio (TCR) adopted in this study is mathematically defined as follows:

(8)\begin{equation}\textrm{TCR = }\left( {1 - \frac{{{N_c}}}{{{N_o}}}} \right) \times 100 \% \end{equation}

where ${N_c}$ and ${N_o}$ denote the numbers of timestamped points in compressed and original vessel trajectories, respectively.

4.3.3 Rate of length loss

The rate of length loss (RLL) essentially indicates the rate of the length loss and the total length of one original vessel trajectory (Huang et al., Reference Huang, Li, Zhang and Liu2020). Theoretically, the low value of RLL is naturally related to high-quality trajectory compression, which could preserve the main geometrical features in the compressed trajectories. In this study, the variables $\{{{S_1},{S_2}, \cdots ,{S_M}} \}$ and $\{{{{S^{\prime}}_1},{{S^{\prime}}_2}, \cdots ,{{S^{\prime}}_M}} \}$ are designated respectively denoting the original and compressed vessel trajectories. Here, the symbol M denotes the total count of vessel trajectories employed in the experiments. The mathematical definition of RLL is given by

(9)\begin{equation}\textrm{RLL = }\frac{{\mathop \sum \nolimits_{m = 1}^M |{{S_m}} |- \mathop \sum \nolimits_{m = 1}^M |{S_m^{\prime}} |}}{{\mathop \sum \nolimits_{m = 1}^M |{{S_m}} |}}\end{equation}

where $\sum\nolimits_{m = 1}^M {|{{S_m}} |- \sum\nolimits_{m = 1}^M {|{{{S^{\prime}}_m}} |} }$ is the length loss after trajectory compression, $|{{S_m}} |$ and $|{{{S^{\prime}}_m}} |$ denote the lengths of the $m$th original trajectory and its compressed version. In particular, the trajectory length can be regarded as the sum of the geometrical distance between any two adjacent timestamped points in one trajectory.

4.3.5 Contrastive analysis

The compression algorithm consists of an offline compression algorithm (also called an offline algorithm), e.g. DP algorithm and US algorithm, and an online compression algorithm, e.g. SQUISH algorithm, DR algorithm, OPW algorithm and IOPW algorithm. The DP algorithm is a popular batch-processing algorithm for line segment simplification. The US algorithm is a highly efficient compression method for batch data. The online compression algorithm is more efficient and conducive to real-time maritime monitoring than offline compression. Additionally, the time complexity of an offline compression algorithm is generally higher than that of an online compression algorithm. The SQUISH algorithm is an online compression algorithm with superior performance that is achieved by simplifying the original trajectory using local optimisation. The OPW algorithm is an online compression algorithm that can perform local optimisation according to window compression processing. The DR algorithm cannot estimate the cumulative error and offset for online trajectory compression of motion state changes. The details of the traditional compression algorithms are listed in Table 2.

Table 2. Details of the traditional compression algorithms

a m indicates the queue size and N indicates the number of variables.

5.1 Data sources

The MMSI number and time stamp were used to obtain AIS data for all the vessels in a specific area. For the three water areas considered herein, i.e. MarineCadastre, WSYR and SCYRE, 596, 401 and 1,476 vessel trajectories that could represent traffic patterns were chosen to form the experiment data sources. The details of the dataset are given in Table 3.

Table 3. Statistical information related to three different water areas

5.2 Experimental results

In this study, 0 ⋅ 001, 0 ⋅ 003, 0 ⋅ 005 and 0 ⋅ 04 were selected as the algorithm error threshold, and these step sizes were applied in the DP, DR, OPW and uniform sampling algorithms. For the SQUISH algorithm, the compression rate was taken as the algorithm reciprocal of the compression ratio with values of 0 ⋅ 01, 0 ⋅ 04, 0 ⋅ 07 and 0 ⋅ 20. Parameters of 3, 7, 10 and 60 were selected for the uniform sampling algorithm. To verify the advantages of IOPW, parallel experiments were performed involving nine different parameters for each of the five algorithms. The experimental results are listed in Tables 4–6.

Table 4. Comparative studies for the Wuhan section of the Yangtze river

Table 4 displays the compression rate, DTW matching distance, RLL and running time on the application of the algorithms to the AIS data of vessels on the Wuhan Section of the Yangtze River. The results demonstrate that the compression rates of the DR, uniform sampling, DP, OPW and IOPW increase with the parameter values. The DP algorithm has a higher running rate than IOPW, whereas the compression rate of the two algorithms is very close when the threshold is significant. However, the average DTW matching distance and RLL of the IOPW algorithm are generally smaller than those of the traditional DP algorithm at the same threshold. The compression performance of the DR algorithm is closer to that of IOPW. However, the information retention of the compressed AIS trajectory is lower than that of IOPW. The SQUISH algorithm has the best preservation effect on the trajectory information when the compression rate is 19 · 83%. When the compression rate is significant, the SQUISH compression performance is poor, but the trajectory information retention is better.

Further, when the parameter is changed from 0 · 01 to 0 · 04, the RLL is changed from 19 · 958% to 5 · 027%. Generally, the AIS data transmission frequency is within 2–120 s. Therefore, the AIS emission frequency and terrain are considered to be the causes of large fluctuations in indicators such as distance loss. Little change was observed for other datasets due to terrain factors. Tables 5 and 6 present the examined algorithms' results when applied based on vessel AIS data from the MarineCadastre database and the South Channel of the Yangtze River, respectively.

Table 5. Comparative studies using the MarineCadastre database

Table 6. Comparative studies for the south channel of the Yangtze river estuary

A comparison of the experimental results presented in Tables 4–6 reveals that the DP algorithm is better than IOPW in terms of compression performance and the information integrity of the compressed trajectory. The three tables show that the compression effect of IOPW is close to that of the DP algorithm. In Table 5, when the compression rates of the IOPW and DP algorithms are 23 · 96% and 6 · 8%, respectively, the DTW matching distance of the two algorithms is very close. Table 6 shows the same results.

The experimental results show that a smaller TCR and smaller RLL result in a better compression effect of the algorithm. Another observation depicted in Figure 8 is that the compression effect of the IOPW algorithm is stronger than the OPW algorithm when the ratio of the linear trajectory and the curved trajectory is standard.

Figure 8. Values of RLL and TCR under different PED thresholds