Vessel Trajectory Prediction Based on Automatic Identification System Data: Multi-Gated Attention Encoder Decoder Network

Abstract

Utilizing time-series data from ship trajectories to forecast their subsequent movement is crucial for enhancing the safety within maritime traffic environments. The application of deep learning techniques, leveraging Automatic Identification System (AIS) data, has emerged as a pivotal area in maritime traffic studies. Within this domain, the precise forecasting of ship trajectories stands as a central challenge. In this study, we propose the multi-gated attention encoder decoder (MGAED) network, a model based on an encoder–decoder structure specialized for predicting ship trajectories in canals. The model employs a long short-term memory network (LSTM) as an encoder, combined with multiple Gated Recurrent Units (GRUs) and an attention mechanism for the decoder. Long-term dependencies in time-series data are captured through GRUs, while the attention mechanism is used to strengthen the model’s ability to capture key information, and a soft threshold residual structure is introduced to handle sparse features, thus enhancing the model’s generalization ability and robustness. The efficacy of our model is substantiated by an extensive evaluation against current deep learning benchmarks. Through comprehensive comparison experiments with existing deep learning methods, our model shows significant improvements in prediction accuracy, with an at least 9.63% reduction in the mean error (MAE) and an at least 20.0% reduction in the mean square error (MSE), providing a new solution to improve the accuracy and efficiency of ship trajectory prediction.

1. Introduction

In recent years, shipping has become increasingly important as a key component of the global transportation system [1,2]. Inland waterway vessels play an important role in cargo transportation, tourism and resource development. However, the environment in inland waterways is complex and changeable, and the safe operation of ships faces many challenges [3,4]. In order to improve navigation safety and efficiency, research into inland waterway ship trajectory prediction technology has become a hot spot.

The Automatic Identification System (AIS) is a sophisticated digital navigation aid that plays a pivotal role in maritime traffic management [5,6]. When integrated with the Global Positioning System (GPS), the AIS is capable of transmitting both static and dynamic information about vessels over Very High Frequency (VHF) communication channels. This technology enhances the safety and efficiency of maritime operations by providing real-time data exchange among ships and coastal stations. AIS as the main source of data for ship trajectory prediction [7], providing a wealth of real-time information on the ship’s position, speed, and heading, etc [8]. By utilizing AIS data, the dynamic behavior of ships can be better captured, providing a solid foundation for trajectory prediction. The application of ship trajectory prediction as a technology can not only help ship navigation systems optimize sailing paths, reduce fuel consumption and sailing time, but also plays an important role in ship traffic management, accident prevention and rescue work [9,10].

Traditional trajectory prediction methods mainly rely on physical models and statistical analyses such as Kalman filtering [11] and particle filtering [12], but these methods have certain limitations when dealing with the complex and variable inland waterway environments. Physical models and statistical analyses are usually based on certain assumptions, such as linear systems and Gaussian noise, which are often difficult to satisfy. As big data and Artificial intelligence technologies advance, trajectory forecasting techniques that leverage machine learning and deep learning are increasingly capturing the interest of researchers and practitioners alike. The support vector machine (SVM) [13,14], the clustering algorithm [15], multilayer perceptron (MLP) [16,17], backpropagation neural network (BPNN) [18], and long short-term memory (LSTM) [19] have been used in the prediction of ship trajectory and are widely applied. These deep learning-based methods are able to more accurately grasp the patterns of ship movements by training massive historical data and show excellent performance in real-time prediction. Currently, deep learning-based trajectory prediction is mainly applied to spatio-temporal data for ocean-going vessels, but in the field of inland shipping, the application of this method faces some challenges due to the limitations of waterway characteristics such as river flow and river shape. To address these challenges, we propose a multi-gated attention encoder–decoder network (MGAED), and our main contributions are as follows:

• We propose a decoder with an information extraction structure made of multiple GRU units combined with an attention mechanism to efficiently capture the key information in the input sequence. The output sequences generated by each GRU unit are weighted and combined by the attention mechanism, which significantly improves the model’s ability to model sequences.
• We introduce a soft thresholding residual operation on the model output, a mechanism that not only effectively reduces the influence of noise, but also preserves important feature information. By learning the thresholds and applying them to the model output, the accuracy and stability of the predictions are further improved.
• We benchmark our method against state-of-the-art schemes on real AIS datasets and report significantly better prediction errors than previous work. This demonstrates the high effectiveness and practicality of our proposed method in real-world applications.

The subsequent sections of this paper are organized as follows: Section 2 provides an overview of the pertinent literature and identifies the existing constraints within AIS-driven ship trajectory prediction methodologies. Section 3 elaborates on the trajectory prediction model and the data preprocessing techniques employed in this study. The experimental comparative analysis is thoroughly examined in Section 4, while Section 5 presents a synthesis of the findings and concludes the paper.

2. Related Works

2.1. Machine Learning Methods

Machine learning has a wide range of applications in predicting ship trajectories. The Kalman Filter [20,21] is a recursive filter suitable for estimating the state of dynamic systems. It predicts ship trajectories by processing noise and other inaccuracies in a series of measurements. The method excels in handling linear systems. The support vector machine (SVM) [13] is a supervised learning model widely used in classification and regression analysis. It distinguishes between different classes of data points by finding the optimal hyperplane, and can be used to process nonlinear trajectory data. A Markov Chain [22,23] predicts the future state of a system by assuming that the future state of the system depends only on the current state and not on the past state. Using Markov Chains makes it possible to model the probability of a ship transferring between different locations and thus predict future locations. The Clustering [15] algorithm simplifies complex trajectory patterns by grouping similar trajectories into the same class. The Back Propagation Neural Network (BP) [18] minimizes prediction error by adjusting weights and biases. It is powerful in handling nonlinear and complex relationships and is suitable for ship trajectory prediction.

The advantage of machine learning methods is that they usually have a high interpretability and low computational complexity, but they can be ineffective in dealing with complex and nonlinear problems and tend to produce large prediction errors.

2.2. Deep Learning Methods

In the past decade, research on deep learning-based ship trajectory prediction has increased dramatically [24], with Recurrent Neural Networks (RNNs) [25,26] being particularly popular, Long Short-Term Memory (LSTM) [19] is a special kind of RNN that specializes in capturing long-term dependencies in time-series data. It efficiently processes and predicts the ship’s trajectory through memory cells and gating mechanisms. The Gated Recurrent Unit (GRU) [27] is a simplified version of LSTM with similar functionality but higher computational efficiency. It captures and maintains important time series information by resetting gates and updating them. Bi-directional LSTM (Bi-LSTM) [28] enhances the model’s prediction of trajectories by taking into account both forward and backward time series data. Bidirectional GRU (Bi-GRU) [29], which is similar to Bi-LSTM, combines the advantages of both forward and backward GRUs, improves the ability to capture time-series data, and is suitable for high-precision trajectory prediction. Sequence to Sequence (Seq2Seq) [30,31] includes two parts, an encoder and a decoder, and is good at dealing with sequences with different input and output lengths. It has significant advantages in handling variable length trajectory prediction tasks. Transformer [32] utilizes the self-attention mechanism and is able to process the entire sequence data in parallel, which significantly improves the training efficiency and prediction performance of the model.

In contrast to prior research, this study introduces an encoder-decoder based model that incorporates an information extraction framework featuring multiple Gated Recurrent Unit components. This model is enhanced with an attention mechanism, which is instrumental in accurately identifying and extracting pivotal information from the decoder’s input sequence. Additionally, the model employs an adaptive soft thresholding residual operation. This operation is designed to mitigate the impact of noise while preserving and effectively capturing the underlying potential information and correlations within the ship’s positional sequence.

3. Methodology

3.1. Problem Formulation

The main task in ship trajectory prediction is to predict the position of a ship at a future moment by analyzing historical AIS data. The attribute vectors of the past N trajectory points, $P_{past}=\left\{p_i\in {\mathbb{R}}^d|i=1,2,\dots ,N\right\}$, can be used to predict the attribute vectors for the M future trajectory points, $P_{future}=\left\{p_j\in {\mathbb{R}}^d|j=N+1,N+2,\dots ,N+M\right\}$.

The state of the ship at time t can be represented by the following equation.

$$p_t=[LO{N}_t,LA{T}_t,SO{G}_t,CO{G}_t]$$

(1)

where $LON$ and $LAT$ and $SOG$ and $COG$ denote longitude, latitude, speed over ground and course over ground, respectively. We wish to find a mapping function f such that $f\left(P_{past}\right)=P_{future}$. The exact form can be expressed by the following equation.

$$P_{N+1},P_{N+2},\dots ,P_{N+M}=f\{P_1,P_2,\dots ,P_N\}$$

(2)

If a deep learning model is used to represent this mapping function f, it can be expressed by the following equation.

$$P_{future}=f_{\theta }\left(P_{past}\right)$$

(3)

where $P_{future}$ is a vector of attributes of future trajectory points predicted by the model and $\theta$ is a parameter of the model.

3.2. Multi-Gated Attention Encoder Decoder Network

The encoder-decoder framework, initially developed for Neural Machine Translation (NMT) [33], has demonstrated remarkable efficacy in time-series data forecasting. This architecture comprises two principal components: an encoder and a decoder. The encoder’s role is to transform the input sequence into a fixed-length context vector, which encapsulates the temporal dynamics and latent characteristics inherent in the sequence. The decoder, on the other hand, constructs the forecasted sequence by utilizing this context vector, sequentially producing the predicted value for each time step. A visual representation of the proposed model’s architecture is depicted in Figure 1.

The encoder in our proposed model utilizes a long short-term memory neural network to capture the temporal dependencies within the sequence data. Conversely, the decoder employs feature fusion and multi-layer Gated Recurrent Units to distill potential correlations between future trajectory information and the historical and current states. It generates a representation of feature correlations at each future prediction step by integrating the hidden state with contextual information through an attention mechanism. This innovative design empowers the model to effectively capture and forecast intricate time series data.

3.2.1. LSTM Based Encoder

LSTM is a special type of RNN, which effectively solves the problem of gradient explosion or disappearance encountered by traditional RNNs when dealing with long term dependencies through a gating mechanism. The LSTM structural unit is shown in Figure 2.

The updated equations are as follows.

$$\begin{array}{c}{f}_t=\sigma (W_f·[h_{t-1},x_t]+b_f)\end{array}$$

(4)

$$\begin{array}{c}{i}_t=\sigma (W_i·[h_{t-1},x_t]+b_i)\end{array}$$

(5)

$$\begin{array}{c}{\tilde{C}}_t=tanh(W_C·[h_{t-1},x_t]+b_C)\end{array}$$

(6)

$$\begin{array}{c}{C}_t=f_t\odot C_{t-1}+i_t\odot {\tilde{C}}_t\end{array}$$

(7)

$$\begin{array}{c}{O}_t=\sigma (W_O·[h_{t-1},x_t]+b_O)\end{array}$$

(8)

$$\begin{array}{c}{h}_t=O_t\odot tanh\left(C_t\right)\end{array}$$

(9)

where $f_t$ is the forgetting gate, $i_t$ is the input gate, ${\tilde{C}}_t$ is the candidate cell state and $O_t$ is the output gate; $\sigma$ denotes the sigmoid function; tanh denotes the tanh function; ⊙ denotes the element-by-element multiplication; $h_t$ denotes the output gate; and $W_f,W_i,W_C,W_O$ and $b_f,b_i,b_C,b_O$ are the weight matrix and bias terms, respectively. This structure allows the LSTM to maintain and update information over a long period of time, effectively solving the long-term dependency problem in RNNs.

The model proposed in this paper uses the hidden state $C_n$ of the last time step of the LSTM encoder for the subsequent decoder. The $C_n$ contains contextual information about the input sequence, which can help the decoder better understand the long and short-term dependencies in the sequence, thus improving the accuracy of the prediction.

3.2.2. GRU and Attention Mechanism-Based Decoder

The decoder layer employs a multi-layer Gated Recurrent Unit alongside an attention mechanism to identify potential correlations among future trajectory states, utilizing insights from both past and present state information. The GRU structural unit is shown in Figure 3.

The updated equations are as follows.

$$\begin{array}{c}{r}_t=\sigma (W_r·[h_{t-1},x_t]+b_r)\end{array}$$

(10)

$$\begin{array}{c}{z}_t=\sigma (W_z·[h_{t-1},x_t]+b_z)\end{array}$$

(11)

$$\begin{array}{c}{\tilde{h}}_t=tanh(W_h·[r_t\odot h_{t-1},x_t]+b_h)\end{array}$$

(12)

$$\begin{array}{c}{h}_t=(1-z_t)\odot h_{t-1}+z_t\odot {\tilde{h}}_t\end{array}$$

(13)

where $z_t$ is the update gate and $r_t$ is the reset gate, and ${\tilde{h}}_t$ is the candidate hidden state; ⊙ denotes element-by-element multiplication, and $\sigma$ denotes the sigmoid function; $h_t$ is the hidden state of the current GRU unit; $W_r,W_z,W_h$ and $b_r,b_z,b_h$ are the weight matrix and bias term, respectively.

The computational procedure of the proposed decoder algorithm based on a GRU and attention mechanism is shown in Equations (14)–(18). For the ith GRU block, the input is X and the hidden state is $H_{i-1}$, and the output $Y_i$ and the new hidden state is $H_i$. The algorithm is based on the following equation.

$$(Y_i,H_i)=GR{U}_i(X,H_{i-1})$$

(14)

For the output $Y_i$ of each GRU block, the attention score $A_i$ is obtained by linear transformation ($score$) and averaged over the time dimension and softmax normalized to obtain the attention weights. As shown in the following equations.

$$\begin{array}{c}{A}_i=score\left(Y_i\right)\end{array}$$

(15)

$$\begin{array}{c}{\alpha }_i={\displaystyle \frac{1}{T}}\sum _{t=1}^T{A}_{i,t}\end{array}$$

(16)

$$\begin{array}{c}{\alpha }_i={\displaystyle \frac{exp\left({\alpha }_i\right)}{{\sum }_{j=1}^{K}exp\left({\alpha }_j\right)}}\end{array}$$

(17)

where K is the number of GRU blocks. The outputs of all the GRU blocks are weighted and summed according to the attentional weights to obtain the final output Y. The outputs of all the GRU blocks are weighted and summed according to the attentional weights. As shown in the following equation.

$$Y=\sum _{i=1}^K{\alpha }_i{Y}_i$$

(18)

By parallelizing multiple GRU blocks, each processing the same input data independently, multiple feature representations are provided, improving the expressive power of the model. By weighting and summing the outputs of multiple GRU blocks through the attention mechanism, the weight of each GRU block can be adaptively adjusted according to the features of the input data, which improves the flexibility and generalization ability of the model.

3.2.3. Adaptive Soft Thresholding Residual Layer

Soft thresholding is usually used in noise removal [34], in this paper, we implement an adaptive soft thresholding activation function to generate thresholds by global average pooling ($GAP$) and fully connected layer ($MLP$), and process the input data. The calculation process is shown in the following equations.

$$\begin{array}{c}T=GAP\left(\right|x\left|\right)Sigmoid\left(MLP\right(GAP\left(\right|x\left|\right)\left)\right)\end{array}$$

(19)

$$\begin{array}{c}soft(x,T)=sign\left(x\right)max\left(\right|x|-T,0)\end{array}$$

(20)

$$\begin{array}{c}{x}_{out}=soft(x,T)+x\end{array}$$

(21)

This layer dynamically generates the threshold value according to the statistical characteristics of the input signal, which enhances the flexibility and adaptability of the model. And the residual operation is performed after the thresholding operation, which maintains the integrity and continuity of the signal.

3.2.4. Output Layer

Following the soft thresholding layer, we incorporate a fully connected layer. This layer’s primary function is to facilitate a transformation from one feature space to another. Its role is to distill the correlations within the features extracted from preceding layers, after they have undergone linear and nonlinear transformations, ultimately projecting them into the output space.

3.3. Preprocessing of AIS Data

The uploading and recording of AIS data relies on GPS signals. Interruption or blockage of GPS signals can lead to interruption of data collection, which results in anomalies that can have an impact on the training of the model. The following processing sterps were performed for the acquired raw AIS data.

• Deleted data where the navigational status was anchored, moored, or aground.
• Deleted data that did not meet the parameter specifications such as missing data sub-items, abnormal data values, etc., and deleted duplicate data by comparing timestamps, vessel IDs, and positions.
• Interpolated and filled in missing values based on data from surrounding time points.
• For each MMSI, the corresponding track data were sorted to split track segments in chronological order, and the track points were divided into different consecutive track segments according to the set time intervals and distance thresholds, and a unique ID was assigned to each segmented track segment for subsequent analysis.

The processed segmented AIS data are shown in

Table 1. Partial navigational data of ship’s AIS information.

ID	Time	MMSI	Lat	Lon	Sog	Cog
0	2019-12-31T22:00:06	367315790	45.65590	−121.01632	7.4	283.1
0	2019-12-31T22:01:07	367315790	45.65638	−121.01921	7.4	283.7
0	2019-12-31T22:02:16	367315790	45.65695	−121.02245	7.3	284.2
0	2019-12-31T22:03:26	367315790	45.65755	−121.02567	7.3	284.7
0	2019-12-31T22:04:35	367315790	45.65808	−121.02893	7.3	281.0
0	2019-12-31T22:05:36	367315790	45.65840	−121.03176	7.3	275.6

.

For example, for trajectory ID 0, on 31 December 2019, at 22:00:06, the ship with MMSI number 367315790 was positioned at 121.01632 degrees west and 45.65590 degrees north. The vessel had a ground speed of 7.4 knots and a heading of 283.1 degrees.

4. Experiments and Results

4.1. Experimental Data Description

We used real AIS data that are publicly available on the marinecadastre.gov website, jointly managed by the Bureau of Ocean Energy Management (BOEM) and National Oceanic and Atmospheric Administration (NOAA) [35], to select historical AIS data for the Columbia River and the Hudson River. The Hudson River channel is relatively straight and ships navigate smoothly here, with fewer changes in heading. However, in the Columbia River, the channel bends sharply. Model predictions were made in different geographic environments to better test the applicability of the model. The trajectory of the treated river after Section 3.3 is shown in Figure 4.

(1)

Columbia River Waterway Dataset

The trajectory data are AIS data from 1 January 2020 to 31 December 2020 for vessels in the Columbia River channel. The geographic range is from 120.92° to 121.46° W longitude and 45.60° to 45.70° N latitude. The dataset includes 1869 tracks with 103,732 track points.

(2)

Hudson River Track Dataset

The trajectory data are AIS data from 1 January 2020 to 31 December 2020 for vessels in the Hudson River channel. The geographic range is from 73.93° to 73.96° W longitude and 41.57° to 41.87° N latitude. The dataset includes 1444 tracks totaling 84,435 track points.

The statistics for trajectories of different lengths, sampling time intervals and sampling distance intervals are shown in Figure 5.

It can be seen that the processed datasets have trajectory lengths of 40 or more, sampling times between 1 and 2 min and sampling distances of about 300 meters. In the experiments, the trajectory sequences were sampled sequentially with a sliding window length of step 1, as shown in Figure 6. Then, the dataset after sliding window sampling was randomly divided into a training set, a validation set and a test set according to the ratio of 80%, 10% and 10%.

4.2. Experimental Setup

4.2.1. Data Normalization

Data normalization is a very important step in the data preprocessing process, it can transform data of different scales and magnitudes to the same scale, thus improving the training effect and stability of the model. This experiment uses min–max normalization, and the equations are shown below.

$$\begin{array}{c}{x}_{std}={\displaystyle \frac{x-x_{min}}{x_{max}-x_{min}}}\end{array}$$

(22)

$$\begin{array}{c}x{=x}_{std}(x_{max}-x_{min})+x_{min}\end{array}$$

(23)

where x is the original trajectory data, and $x_{max}$ and $x_{min}$ denote the minimum and maximum values in the original trajectory data, respectively. $x_{std}$ denotes the normalized data.

4.2.2. Model and Hyperparameter Settings

In order to evaluate and compare the proposed ship trajectory prediction models, seven baseline models are used as follows:

• BP: Back Propagation Neural Networks are multi-layer feed-forward architectures that optimize prediction accuracy by iteratively adjusting network weights through a backpropagation algorithm, which minimizes prediction error.
• RNN: Recurrent Neural Networks are adept at handling sequential data, preserving time-series information by feeding the output of one time step as input to the subsequent step, facilitated by recurrent connections.
• LSTM: Long Short-Term Memory networks are a specialized form of RNNs that manage information flow through the use of memory cells, input gates, forget gates, and output gates. This design enables them to capture long-term dependencies effectively.
• GRU: Gated Recurrent Units offer a streamlined approach compared to LSTMs, addressing the gradient vanishing issue in RNNs. GRUs regulate the flow of state information with update and reset gates, and due to their reduced parameter count, they can offer faster training times in certain scenarios.
• BiLSTM: The bidirectional long and short term memory network combines two LSTMs, forward and backward, which can take into account both past and future information to improve prediction accuracy. This method performs well in tasks that require contextual information.
• BiGRU: The Bidirectional GRU is similar to Bi-LSTM in that it improves the prediction performance by processing both forward and reverse sequence information. Compared with Bi-LSTM, Bi-GRU is more computationally efficient and is suitable for scenarios with limited computational resources.
• Seq2Seq: The Sequence to Sequence model, a cornerstone in tasks like machine translation and text generation, typically comprises an encoder and a decoder. The encoder’s role is to convert the input sequence into a fixed-length, high-dimensional vector that encapsulates the input’s semantic content. This vector then serves as the context for the decoder, which generates the output sequence step by step, ensuring that the generated text is coherent and relevant to the input.

Since both the proposed and baseline models are based on Recurrent Neural Networks and Multilayer Perceptrons, we ensured consistency between the two in terms of initial conditions and parameter settings to fairly evaluate their performance.The optimizer in this experiment uses Adam, the loss function uses MSELoss, the learning rate is adjusted using cosine annealing and the remaining hyperparameters are shown in

Table 2. Related hyperparameter combination.

Learning Rate	Train Epochs	Batch Size	Hidden Size	Num Layers	Seed
0.001	300	1024	512	2	2024

.

Table 3. Time complexity of different models.

Model	Time Complexity	Model	Time Complexity
BP	$O(L·H)$	RNN	$O(L·H^2)$
LSTM	$O(L·4H^2)$	Bi-LSTM	$O(L·8H^2)$
GRU	$O(L·3H^2)$	Bi-GRU	$O(L·6H^2)$
Seq2Seq	$O(L·8H^2)$	MGAED	$O(L·(4+3k)H^2)$

shows the time complexity of the different models, where L denotes the length of the sequence, H denotes the size of the hidden layer and k denotes the number of GRU blocks in the decoder for the MAGED model.

As can be seen from Figure 7, the loss function curves for the model proposed in this study converge relatively fast and there is no overfitting compared to other models; the following hardware configuration was used: the operating system was Ubuntu 20.04 (Canonical Ltd., London, UK), the CPU was an Intel Xeon Platinum 8255C @ 2.50 GHz (Intel Corporation, Santa Clara, CA, USA) and the GPU was an NVIDIA RTX 3080 (10 GB) (NVIDIA Corporation, Santa Clara, CA, USA). The proposed method was implemented in a programming environment based on Python 3.8, using the PyTorch 2.0.0 deep learning framework and CUDA version 11.8.

4.3. Experimental Evaluation Index

When evaluating the precision of forecasting models, it is crucial to consider the discrepancies between predictions and real-world outcomes that can arise from numerous influences. This research utilized five key metrics to gauge the efficacy of the models: mean absolute error (MAE), mean square error (MSE), mean absolute percentage error (MAPE), mean squared percentage error (MSPE) and mean Haversine angular distance error (MHADE).

MSE indicates the mean value of the squared differences between the projected and actual figures. MAE quantifies the deviation between the forecasted and actual data, which is the mean of the absolute deviations. MAPE, expressed as a percentage of the actual data, serves as a measure of forecasting inaccuracy and is frequently employed to assess the reliability of predictions. MSPE is akin to MAPE but imposes a more stringent penalty on errors, as it incorporates the squared magnitude of the discrepancies. Lastly, MHADE calculates the shortest distance between two geographical points and provides a visual estimation of the actual discrepancy (in meters) between the forecasted and true values. The equations are shown below.

$$mae={\displaystyle \frac{1}{n}}\sum _{i=1}^n|tru{e}_i-{\widetilde{pred}}_i|$$

(24)

$$mse={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(tru{e}_i-{\widetilde{pred}}_i)}^2$$

(25)

$$mape={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{tru{e}_i-{\widetilde{pred}}_i}{tru{e}_i}}\right|$$

(26)

$$mspe={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left({\displaystyle \frac{tru{e}_i-{\widetilde{pred}}_i}{tru{e}_i}}\right)}^2$$

(27)

$$\begin{array}{c}{d}_i=2r·arcsin\sqrt{{sin}^2{\displaystyle \frac{{l\tilde{a}t}_i-la{t}_i}{2}}+cos{l\tilde{a}t}_i·cosla{t}_i·{sin}^2{\displaystyle \frac{{l\tilde{o}n}_i-lo{n}_i}{2}}}\times 1000\\ mhade={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{d}_i\end{array}$$

(28)

The actual position of the ship is represented by $tru{e}_i$, while ${\widetilde{pred}}_i$ signifies the forecasted trajectory data. The predicted geographic coordinates are given by ${l\tilde{o}n}_i$ for longitude and ${l\tilde{a}t}_i$ for latitude. Conversely, $lo{n}_i$ and $la{t}_i$ represent the actual longitude and latitude, respectively. The value r, approximately 6371 km, corresponds to the Earth’s radius.

4.4. Results and Analysis

In the experiment, the initial 10 data points from each trajectory served as the starting input for the neural network, aiming to forecast the vessel’s path for the subsequent 20 temporal intervals, which equates to approximately 30 min. A comparative analysis was conducted on the prediction inaccuracies of various models, including BP, RNN, LSTM, GRU, Bi-LSTM, Bi-GRU and Seq2Seq.

4.4.1. Visualization and Analysis of Prediction Results

In order to provide a detailed comparative analysis of the prediction performance of the eight models, one representative trajectory was selected from both the Columbia River test set and the Hudson River test set. Figure 8 shows the comparison of the trajectory prediction of a vessel traveling in a curved channel for the Columbia River test set with an MMSI of 368121540, and Figure 9 shows the comparison of the trajectory prediction of a vessel traveling in a straight channel for the Hudson River test set with an MMSI of 367728850.

Our model’s trajectory predictions exhibit the highest degree of resemblance to the actual paths, affirming its superiority as the most precise forecasting tool among the eight models evaluated.

4.4.2. Assessment of Methodological Performance through Quantitative Metrics

Table 4. Quantitative assessment of different methods in two regions (output 20).

Model	Columbia River					Hudson River
	MAE	MSE	MAPE	MSPE	MHADE	MAE	MSE	MAPE	MSPE	MHADE
BP	$9.20\times {10}^{-4}$	$2.41\times {10}^{-6}$	$1.17\times {10}^{-5}$	$3.32\times {10}^{-10}$	147.63	$1.48\times {10}^{-3}$	$9.70\times {10}^{-6}$	$3.23\times {10}^{-5}$	$5.37\times {10}^{-9}$	112.10
RNN	$1.13\times {10}^{-3}$	$4.12\times {10}^{-6}$	$1.44\times {10}^{-5}$	$5.18\times {10}^{-10}$	184.20	$1.50\times {10}^{-3}$	$1.21\times {10}^{-5}$	$3.27\times {10}^{-5}$	$6.77\times {10}^{-9}$	113.91
LSTM	$9.79\times {10}^{-4}$	$2.72\times {10}^{-6}$	$1.22\times {10}^{-5}$	$3.44\times {10}^{-10}$	160.31	$1.47\times {10}^{-3}$	$9.47\times {10}^{-6}$	$3.23\times {10}^{-5}$	$5.27\times {10}^{-9}$	107.51
GRU	$9.83\times {10}^{-4}$	$2.62\times {10}^{-6}$	$1.23\times {10}^{-5}$	$3.49\times {10}^{-10}$	159.54	$1.46\times {10}^{-3}$	$9.47\times {10}^{-6}$	$3.21\times {10}^{-5}$	$5.27\times {10}^{-9}$	107.41
BiLSTM	$8.65\times {10}^{-4}$	$2.15\times {10}^{-6}$	$1.08\times {10}^{-5}$	$2.80\times {10}^{-10}$	140.27	$1.35\times {10}^{-3}$	$7.95\times {10}^{-6}$	$2.96\times {10}^{-5}$	$4.43\times {10}^{-9}$	98.73
BiGRU	$8.85\times {10}^{-4}$	$2.22\times {10}^{-6}$	$1.12\times {10}^{-5}$	$2.98\times {10}^{-10}$	142.39	$1.39\times {10}^{-3}$	$8.66\times {10}^{-6}$	$3.05\times {10}^{-5}$	$4.82\times {10}^{-9}$	102.27
Seq2Seq	$9.85\times {10}^{-4}$	$2.73\times {10}^{-6}$	$1.21\times {10}^{-5}$	$3.33\times {10}^{-10}$	163.06	$1.41\times {10}^{-3}$	$7.83\times {10}^{-6}$	$3.10\times {10}^{-5}$	$4.35\times {10}^{-9}$	101.66
MGAED	$7.64\times {10}^{-4}$	$1.72\times {10}^{-6}$	$9.57\times {10}^{-6}$	$2.21\times {10}^{-10}$	124.65	$1.22\times {10}^{-3}$	$6.09\times {10}^{-6}$	$2.67\times {10}^{-5}$	$3.38\times {10}^{-9}$	89.70

shows the longitude and latitude mean prediction errors for the various models used for ship trajectory prediction using the MAE, MSE, MAPE, MSPE and MHADE after inverse normalization of the evaluation metrics for the Columbia River test set and the Hudson River test set.

From the statistics of the seven comparative datasets, the average percentage of improvement in the three indicators can be obtained, as shown in

Table 5. Average percentage improvement in different methods in two regions.

MGAED VS	Columbia River					Hudson River
	MAE	MSE	MAPE	MSPE	MHADE	MAE	MSE	MAPE	MSPE	MHADE
BP	16.96%	28.63%	18.21%	33.43%	15.57%	17.57%	37.22%	17.34%	37.06%	19.98%
RNN	32.92%	58.25%	33.54%	57.34%	32.33%	18.67%	49.84%	18.35%	50.07%	21.25%
LSTM	21.96%	36.76%	21.56%	35.76%	22.24%	17.01%	35.69%	17.34%	5.86%	16.57%
GRU	22.28%	34.35%	22.20%	36.68%	21.87%	16.44%	35.69%	16.82%	35.86%	16.49%
BiLSTM	11.68%	20.00%	11.39%	21.07%	11.14%	9.63%	23.40%	9.80%	23.70%	9.15%
BiGRU	13.67%	22.52%	14.55%	25.84%	12.46%	12.23%	29.68%	12.46%	29.88%	12.29%
Seq2Seq	22.44%	37.00%	20.91%	33.63%	23.56%	13.48%	22.22%	13.87%	22.30%	11.76%

. For the rates of improvement in the 3 indicators, the highest was 58.25%, and the lowest was 9.15%. Improvements were more significant in the meandering Columbia River than in the gently sloping Hudson River. Therefore, the results for the three indicators and the improvement in the accuracy of the Columbia watershed further validate that the present model is superior to the other seven models.

4.4.3. Comparison of Models with Different Output Feature Lengths

The length of the model’s output features is directly related to the model’s complexity, training cost, and ultimately its predictive performance. Shorter feature vectors may mean that the model is more concise, but may not capture all the important information. Conversely, longer feature vectors may contain more detail, but may also result in a more complex model with higher time costs for training and inference. In order to verify the performance of the proposed model with different output lengths,

Table 6. Quantitative assessment of different methods in two regions (output 15).

Model	Columbia River					Hudson River
	MAE	MSE	MAPE	MSPE	MHADE	MAE	MSE	MAPE	MSPE	MHADE
BP	$7.81\times {10}^{-4}$	$1.74\times {10}^{-6}$	$9.93\times {10}^{-6}$	$2.42\times {10}^{-10}$	125.44	$1.18\times {10}^{-3}$	$5.54\times {10}^{-6}$	$2.55\times {10}^{-5}$	$3.04\times {10}^{-9}$	92.36
RNN	$1.01\times {10}^{-3}$	$2.63\times {10}^{-6}$	$1.29\times {10}^{-5}$	$3.71\times {10}^{-10}$	162.16	$1.19\times {10}^{-3}$	$7.31\times {10}^{-6}$	$2.58\times {10}^{-5}$	$4.06\times {10}^{-9}$	92.53
LSTM	$8.50\times {10}^{-4}$	$2.06\times {10}^{-6}$	$1.05\times {10}^{-5}$	$2.63\times {10}^{-10}$	139.25	$1.13\times {10}^{-3}$	$5.15\times {10}^{-6}$	$2.46\times {10}^{-5}$	$2.84\times {10}^{-9}$	85.99
GRU	$8.23\times {10}^{-4}$	$1.89\times {10}^{-6}$	$1.03\times {10}^{-5}$	$2.52\times {10}^{-10}$	133.52	$1.15\times {10}^{-3}$	$5.69\times {10}^{-6}$	$2.49\times {10}^{-5}$	$3.15\times {10}^{-9}$	87.37
BiLSTM	$7.26\times {10}^{-4}$	$1.55\times {10}^{-6}$	$9.21\times {10}^{-6}$	$2.09\times {10}^{-10}$	116.86	$1.08\times {10}^{-3}$	$4.44\times {10}^{-6}$	$2.35\times {10}^{-5}$	$2.45\times {10}^{-9}$	81.69
BiGRU	$7.68\times {10}^{-4}$	$1.71\times {10}^{-6}$	$9.83\times {10}^{-6}$	$2.36\times {10}^{-10}$	122.66	$1.11\times {10}^{-3}$	$4.95\times {10}^{-6}$	$2.41\times {10}^{-5}$	$2.73\times {10}^{-9}$	83.47
Seq2Seq	$9.11\times {10}^{-4}$	$2.34\times {10}^{-6}$	$1.12\times {10}^{-5}$	$2.86\times {10}^{-10}$	150.24	$1.16\times {10}^{-3}$	$5.22\times {10}^{-6}$	$2.54\times {10}^{-5}$	$2.89\times {10}^{-9}$	85.99
MGAED	$6.43\times {10}^{-4}$	$1.29\times {10}^{-6}$	$8.12\times {10}^{-6}$	$1.69\times {10}^{-10}$	103.87	$9.87\times {10}^{-4}$	$3.77\times {10}^{-6}$	$2.14\times {10}^{-5}$	$2.08\times {10}^{-9}$	75.33

and

Table 7. Quantitative assessment of different methods in two regions (output 10).

Model	Columbia River					Hudson River
	MAE	MSE	MAPE	MSPE	MHADE	MAE	MSE	MAPE	MSPE	MHADE
BP	$6.21\times {10}^{-4}$	$1.15\times {10}^{-6}$	$7.84\times {10}^{-6}$	$1.52\times {10}^{-10}$	100.38	$8.56\times {10}^{-4}$	$2.68\times {10}^{-6}$	$1.82\times {10}^{-5}$	$1.45\times {10}^{-9}$	70.09
RNN	$8.32\times {10}^{-4}$	$1.76\times {10}^{-6}$	$1.06\times {10}^{-5}$	$2.56\times {10}^{-10}$	133.41	$8.47\times {10}^{-4}$	$3.22\times {10}^{-6}$	$1.79\times {10}^{-5}$	$1.75\times {10}^{-9}$	70.65
LSTM	$6.83\times {10}^{-4}$	$1.32\times {10}^{-6}$	$8.45\times {10}^{-6}$	$1.65\times {10}^{-10}$	112.45	$8.18\times {10}^{-4}$	$2.48\times {10}^{-6}$	$1.75\times {10}^{-5}$	$1.35\times {10}^{-9}$	65.77
GRU	$6.42\times {10}^{-4}$	$1.18\times {10}^{-6}$	$8.06\times {10}^{-6}$	$1.53\times {10}^{-10}$	104.23	$8.14\times {10}^{-4}$	$2.58\times {10}^{-6}$	$1.74\times {10}^{-5}$	$1.40\times {10}^{-9}$	65.77
BiLSTM	$5.90\times {10}^{-4}$	$1.02\times {10}^{-6}$	$7.48\times {10}^{-6}$	$1.35\times {10}^{-10}$	94.90	$7.91\times {10}^{-4}$	$2.33\times {10}^{-6}$	$1.69\times {10}^{-5}$	$1.27\times {10}^{-9}$	62.92
BiGRU	$6.13\times {10}^{-4}$	$1.07\times {10}^{-6}$	$7.81\times {10}^{-6}$	$1.44\times {10}^{-10}$	98.50	$8.20\times {10}^{-4}$	$2.59\times {10}^{-6}$	$1.76\times {10}^{-5}$	$1.42\times {10}^{-9}$	65.10
Seq2Seq	$7.33\times {10}^{-4}$	$1.48\times {10}^{-6}$	$9.02\times {10}^{-6}$	$1.80\times {10}^{-10}$	121.22	$8.64\times {10}^{-4}$	$2.65\times {10}^{-6}$	$1.86\times {10}^{-5}$	$1.45\times {10}^{-9}$	66.84
MGAED	$5.23\times {10}^{-4}$	$8.43\times {10}^{-7}$	$6.59\times {10}^{-6}$	$1.10\times {10}^{-10}$	84.78	$7.45\times {10}^{-4}$	$1.96\times {10}^{-6}$	$1.60\times {10}^{-5}$	$1.07\times {10}^{-9}$	59.40

show the quantitative evaluation with an input step size of 10 and output step sizes of 15 and 10, respectively.

In the comprehensive evaluation of the time series prediction tasks for the Columbia River and Hudson River regions, the MGAED model has demonstrated a superior performance across all key metrics, including MAE, MSE, MAPE, MSPE, and MHADE. The model’s lowest values in error metrics underscore its superior predictive accuracy and stability. In the Columbia River region, the MGAED model’s MAE of $5.23\times {10}^{-4}$ and MSE of $8.43\times {10}^{-7}$ are significantly lower than other models, indicating a substantial reduction in prediction errors, especially when compared with traditional models such as BP, RNN, and Seq2Seq. The model also exhibits the lowest MHADE of 84.78, highlighting its advantage in managing complex fluctuations. Similarly, in the Hudson River region, the MGAED model’s MAE of $7.45\times {10}^{-4}$ and MSE of $1.96\times {10}^{-6}$ outperform other models, suggesting that it is more effective in reducing prediction errors compared to traditional RNN and Seq2Seq models.

The quantitative assessment results reveal that bidirectional models, such as BiLSTM and BiGRU, generally perform better than unidirectional models like RNN, LSTM and GRU in both regions. This advantage is attributed to the bidirectional structure’s ability to capture both forward and backward dependencies in data, which is particularly beneficial when dealing with complex time series data. On the other hand, traditional models like BP and RNN show limitations in several evaluation metrics, especially in MAPE and MHADE, indicating their struggles with complex nonlinear time series data.

In conclusion, the MGAED model’s exceptional performance in time series prediction tasks, particularly in modeling complex hydrological data, is evident. The model’s powerful predictive capabilities are fully demonstrated, while bidirectional neural network models also show clear advantages. Traditional models, however, appear somewhat lacking in accuracy, providing valuable insights for future research and application in hydrological data modeling and time series prediction.

4.4.4. Ablation Experiment

To assess the individual contributions of each component within the model, we conducted ablation studies, as outlined below. In doing so, we systematically eliminated various modules and developed four distinct model configurations.

A1

Remove the decoder attention mechanism and average the output of each GRU block;

A2

Remove the adaptive soft threshold residual module;

A3

Use LSTM as a decoder backbone network;

A4

Remove the decoder and use the encoder to directly feed the output to the adaptive soft threshold residual module.

The model configuration and dataset are the same as above, and the experimental results are shown in

Table 8. Quantitative assessment of different methods in two regions (output 20).

Model	Columbia River					Hudson River
	MAE	MSE	MAPE	MSPE	MHADE	MAE	MSE	MAPE	MSPE	MHADE
A1	$7.91\times {10}^{-4}$	$1.78\times {10}^{-6}$	$9.89\times {10}^{-6}$	$2.30\times {10}^{-10}$	128.97	$1.24\times {10}^{-3}$	$6.23\times {10}^{-6}$	$2.72\times {10}^{-5}$	$3.46\times {10}^{-9}$	90.98
A2	$8.89\times {10}^{-4}$	$2.21\times {10}^{-6}$	$1.10\times {10}^{-5}$	$2.81\times {10}^{-10}$	145.68	$1.36\times {10}^{-3}$	$8.92\times {10}^{-6}$	$2.99\times {10}^{-5}$	$4.98\times {10}^{-9}$	99.21
A3	$7.98\times {10}^{-4}$	$1.83\times {10}^{-6}$	$9.94\times {10}^{-6}$	$2.34\times {10}^{-10}$	130.40	$1.25\times {10}^{-3}$	$6.62\times {10}^{-6}$	$3.64\times {10}^{-5}$	$5.37\times {10}^{-9}$	92.23
A4	$8.63\times {10}^{-4}$	$2.15\times {10}^{-6}$	$1.07\times {10}^{-5}$	$2.83\times {10}^{-10}$	140.57	$1.29\times {10}^{-3}$	$6.92\times {10}^{-6}$	$2.83\times {10}^{-5}$	$3.84\times {10}^{-9}$	94.90
MGAED	$7.64\times {10}^{-4}$	$1.72\times {10}^{-6}$	$9.57\times {10}^{-6}$	$2.21\times {10}^{-10}$	124.65	$1.22\times {10}^{-3}$	$6.09\times {10}^{-6}$	$2.67\times {10}^{-5}$	$3.38\times {10}^{-9}$	89.70

.

The table clearly demonstrates that the model’s precision diminishes upon the removal of any component. Specifically, excluding the adaptive soft threshold residual module results in a performance level that is on par with the baseline model, underscoring the module’s critical role in enhancing model accuracy. Similarly, the omission of either the attention or GRU module leads to a marked decline in performance.

The findings from the ablation study confirm the indispensability of each module within the model, highlighting their collective contribution to the model’s predictive accuracy.

5. Conclusions

In this study, we introduce a novel model known as MGAED, which employs an encoder–decoder framework to predict ship trajectories within canals with precision. This model is distinguished by its integration of a long short-term memory encoder, multiple Gated Recurrent Unit components, and a decoder that is enhanced with an attention mechanism. This architecture is designed to capture the intricacies of long-term dependencies within time series data. The model’s focus on critical information is facilitated by the attention mechanism, while a soft threshold residual architecture adeptly handles sparse features, enhancing the model’s generalization and robustness. Our model’s performance is rigorously validated through a comprehensive comparative analysis against seven other state-of-the-art deep learning methods across five different evaluation metrics. The results demonstrate a marked improvement in prediction accuracy, with a minimum reduction of 9.63% in mean absolute error and an at least a 20.0% decrease in mean squared error. These empirical findings confirm that our model surpasses traditional approaches in terms of predictive accuracy, presenting a pioneering solution for enhancing the precision and efficiency of ship trajectory forecasting.

However, it is also crucial to recognize the limitations of our work. The performance of the model may be affected by data variability, and its ability to predict in real-time under dynamic and unpredictable conditions requires further validation. Scalability of the model to accommodate larger and more complex marine networks is another area that needs to be further explored. In addition, integrating MGAED with existing marine traffic management systems to improve operational efficiency is a potential avenue for future research. Addressing these limitations will help refine the model and ensure its continued relevance and effectiveness in the field of marine traffic prediction.