Large-Scale Long-Term Prediction of Ship AIS Tracks via Linear Networks with a Look-Back Window Decomposition Scheme of Time Features

Abstract

Approximating the positions of vessels near underwater devices, such as unmanned underwater vehicles and autonomous underwater vehicles, is crucial for many underwater operations. However, long-term monitoring of vessel trajectories is challenging due to limitations in underwater communications, posing challenges for the execution of underwater exploration missions. Therefore, trajectory prediction based on AIS data is vital in the fusion of underwater detection information. However, traditional models for underwater vessel trajectory prediction typically work well for only small-scale and short-term predictions. In this paper, a novel deep learning method is proposed that leverages a look-back window to decompose the temporal and motion features of ship movement trajectories, enabling long-term vessel prediction in broader sea areas. This research introduces an innovative model structure that enables trajectory features to be simultaneously learned for a larger range of vessels and facilitates long-term prediction. Through this innovative model design, the proposed model can more accurately predict vessel trajectories, providing reliable and comprehensive forecasting results. Our proposed model outperforms the Nlinear model by a 16% improvement in short-term prediction accuracy and an approximately 8% improvement in long-term prediction accuracy. The model also outperforms the Patch model by 5% in accuracy. In summary, the proposed method can produce competitive predictions for the long-term future trajectory trends of ships in large-scale sea areas.

1. Introduction

Predicting ship trajectories using AIS (automatic identification system) data allows for forecasting unknown future motion paths and provides decision support for various maritime activities [1]. This method has wide-ranging potential applications in ship navigation safety, route planning, underwater exploration, and more [2]. For navigation safety [3], long-term trajectory prediction offers valuable support and position information to assist ships in avoiding obstacles at sea [4] and arriving safely at their destinations. Additionally, in terms of route planning and resource allocation, long-term trajectory prediction enables ship routes, docking times, and resource utilization to be more effectively arranged, thereby enhancing logistical efficiency [5].

In underwater exploration, obstacles in water hinder the propagation of electromagnetic signals. As a result, underwater devices are often unable to directly receive real-time position information from surface vessel equipment. When underwater devices cannot obtain real-time location information of surface vessels, they can predict future trajectories by learning from the vessels’ past trajectories to determine their positions. This prediction is crucial in underwater terrain measurement, data collection and transmission, and motion path planning of unmanned underwater vehicles (UUVs) [6]. UUVs in underwater operations complete a wide range of activities and have long diving times. Therefore, they may encounter various complex situations. Having the ability to predict the future trajectories of surface vessels assists UUVs in precise mission planning, conflict avoidance, route planning, and path optimization. Long-term trajectory prediction also enhances the environmental awareness of UUVs during mission execution, accurately assesses potential risks to UUVs, and supports data collection and scientific research.

Large-scale and long-term prediction of ship trajectories can improve the adaptability of UUVs during long-term movement in wide sea areas [7]. It also enables UUVs to better adapt to complex marine environments, ensuring that they operate efficiently despite challenges [8]. In long-term prediction, historical ship AIS data are used in the time series analysis of ship trajectories to predict AIS data values over a wide range of future time steps. The length of the AIS time series refers to the number of observations or the length of the time series, rather than just the size of the time span. The trajectory of a wide-area ship describes the movement of the ship over a large range, usually including multiple areas such as the ocean, straits, ports, and waterways. Analyzing wide-area ship trajectories can reveal the traffic flow modes, route selection, stop preference, and other information of ships in specific areas, providing an important reference for maritime management [8]. In vast sea areas, ships are significantly affected by changes in ocean environmental characteristics such as water flow and wind direction, giving the motion of ships a certain degree of randomness. These differences directly affect the long-term operation statuses and navigation paths of ships. Ships moving at sea for a long time will have some differences in their motion characteristics and patterns. Meanwhile, the differences in factors such as size, speed, shape, payload, and maneuverability of different vessels result in varying motion patterns exhibited by different types of ships at sea. Therefore, the complexity of the marine environment and the different characteristics of ships have made long-term prediction of ship trajectories for multiple ships in vast waters relatively difficult [9].

In this study, to address the aforementioned issues, an improvement to the long-term sequence forecasting model, DLinear [10], is proposed. The improvement allows the DLinear model to simultaneously process tens of thousands of ship trajectories, enhancing its training efficiency and ability to learn more features. The DLinear model combines the decomposition approach used in Autoformer and FEDformer with linear layers [10]. It utilizes a look-back window to decompose a large volume of ship trajectory information in both time and spatial scales, enabling the learning of long-term motion patterns for many vessels over a wide range. The look-back window defines the observable past time steps, allowing the model to perceive dependencies and patterns in the sequence data. The temporal and trend components are decomposed using the same fully connected neural network with shared weights. This enables better learning of motion features over a larger range and longer duration, which aims to provide a solution for long-term ship trajectory prediction in vast maritime areas. By decomposing temporal and trend features, the model can effectively capture historical information to better understand and predict further future data. Additionally, the model structure is modified to enable learning multiple ship trajectory sequences simultaneously. This greatly enhances training efficiency and allows the model to learn the general patterns of long-term vessel motion in vast sea areas. In this study, we collect and process AIS signal trajectory data from a large maritime area to implement the proposed method and compare it with various deep learning models. The results demonstrate that the proposed improved DLinear model outperforms other models in predicting ship motion direction and trends, exhibiting good generalization capability and broad application prospects in long-term sequences. Compared to the Nlinear model, the DLinear model shows a 16% improvement in short-term prediction and a 9% improvement in long-term prediction.

The remainder of this paper is organized as follows. Section 2 introduces the data sources and the process of data cleaning and exploration. Section 3 presents the ship trajectory prediction model, linear networks with time-feature decomposition. Section 4 discusses the experimental setup and presents the results. Finally, Section 5 concludes the paper.

2. Related Work

The uncertainty of the marine environment always poses certain challenges to ship trajectory prediction. Unlike cars moving in road areas, which have fixed tracks and trends [11], ships moving in a wide range of sea areas have a certain degree of randomness. This has brought about the current prediction methods of ship trajectories, including traditional mathematical statistical methods and deep learning methods, such as the LSTM (long short-term memory) and transformer models. Both types of methods can achieve good prediction results in certain areas such as ports. However, making good predictions on a large scale and over a long period with these methods is often a struggle [12].

Traditional mathematical statistical methods include the ship trajectory prediction method using machine learning and the similarity measurement method. This method extracts trajectory features and predicts ship trajectories by using the longest shared path concept and similarity measurement algorithm, respectively [13]. This method is effective and can achieve good results in short-term and small-scale ship trajectory prediction. Using the method of constructing a polynomial Kalman filter [14], the ship motion track can be predicted with longitude and latitude information, which solves the problems of missing information and slow updates in ship track prediction. Accordingly, this method selects a smaller range and has a shorter prediction duration than other methods.

As a deep learning method, the LSTM network algorithm can be used to accurately predict the next several moments according to the AIS signal of the ship track near a port, where the ship has strong regularity. Another deep learning model uses the BP neural network [15]. This model can achieve good results in the short-term prediction of ship trajectories in river channels [16]. By utilizing the transformer network structure of AIS data depicted by enhanced, sparse, and high-dimensional representations, complex patterns in ship trajectories can be learned. The performance of this method is significantly superior to that of previous methods, and this method can predict the trajectory of specific ships over a large range [17]. Although the aforementioned ship trajectory prediction methods achieve good results in some cases, predicting large-scale and long-term ship trajectories using these methods is still difficult. The prediction duration of these methods is limited to short-term predictions, and the range of ship motion is often limited to small areas such as rivers and ports. Moreover, the models are often restricted to certain types of vessels.

Compared to the aforementioned issues, our improved DLinear model has significant advantages in ship trajectory prediction. The proposed model, which utilizes a long-time prediction method, can better handle the uncertainties and challenges posed by marine environments. Long-time predictions can provide more complete information since they can observe and consider more historical data and environmental changes. This allows the model to better understand and adapt to the various complex factors of the marine environment, accurately capturing the impact of these factors on ship trajectories during the prediction process. Therefore, the proposed model is better suited for handling the uncertainties and challenges of marine environments and can provide more reliable results in predicting ship trajectories. Moreover, it possesses the capability to forecast over long time ranges, outperforming traditional mathematical statistical methods and deep learning approaches in terms of accuracy for large-scale and long-term predictions. Additionally, our model exhibits good scalability and adaptability, enabling it to accommodate different types of data and real-world application scenarios, thereby providing valuable information for ship trajectory prediction across various ship types and diverse marine environments. In conclusion, our model excels in handling complex environments, long-term forecasting, and scalability, enabling more accurate ship trajectory predictions.

3. The Proposed Method

3.1. Definition

In this section, we offer precise definitions for the concepts employed in formulating this study. These definitions play a crucial role in enhancing the understanding and clarification of the diverse concepts discussed within the context of this paper.

• $ℝ$ is the dataset used in this experiment, which includes all the AIS ship trajectory sequences.
• $T$ refers to a segment of future time steps in a sequence, while $t$ represents a specific point within that future segment.
• $L$ represents the length of the sequence.
• $X$ represents a given historical sequence $X={\{X_1^{\mathrm{t}},\dots ,X_{\mathrm{C}}^{\mathrm{t}}\}}_{\mathrm{t}=1}^L$. The task of time series forecasting is to predict the values $\widehat{X}={\{{\widehat{X}}_1^{\mathrm{t}},\dots ,{\widehat{X}}_{\mathrm{C}}^{\mathrm{t}}\}}_{\mathrm{t}=L+1}^{L+T}$ at $T$ future time steps, where t ranges from $L+1$ to $L+T$.
• $C$ represents the number of variables contained within a time series.
• $H$ represents the output obtained after passing the input data through a single-layer network during the training process.
• $S$ represents trajectory prediction for a specific time step $S$ in the future.
• Channel refers to the different features in the input AIS data. By creating independent channels for each feature, the model can better learn and express the dynamic changes and relationships between different features, thereby improving its ability to predict, analyze, and make decisions.
• Individual means creating a separate channel for each feature so that the model can independently learn and represent each feature.
• Kernel size represents the sliding window size used in the model to decompose temporal and spatial features.

3.2. Linear Networks with Time-Feature Decomposition

Due to the challenges described in the Section 1, a large portion of samples in the AIS ship trajectory dataset are random and unpredictable. Therefore, it is hard to effectively predict ship trajectories based on AIS signals using conventional long-term sequence prediction models. In our experiments, the Autoformer, Informer, and transformer models all performed poorly in predicting AIS trajectories.

In contrast, the DLinear model [10] is a linear model specifically designed for long-term time series prediction, challenging the effectiveness of transformer models in long-term time series prediction tasks. The DLinear model is more suitable than other models for large-scale ship trajectory prediction because it has a linear structure and thus is capable of capturing linear relationships. It also incorporates deep feature representation to extract high-level features and exhibits long-term prediction ability as well as good generalization capability.

The core idea behind linear networks with time-feature decomposition is directly regressing historical time series data to predict future time series. This differs from traditional methods such as the sliding windows and ARIMA methods, which treat time series data as random processes and require multiple iterations to obtain useful prediction results.

The proposed model is based on decomposing time features and motion features using the look-back window, and it is applicable in long-term ship prediction in wider sea areas. The look-back window refers to a data window that looks back a certain specific time duration from the current moment when predicting ship trajectories. Through the look-back window, we can obtain historical trajectory information of ships over a period of time. The selection of this time window depends on the requirements of the prediction task, and the appropriate window size can be set according to specific circumstances. Using the look-back window, we can decompose time features and motion features. Time features refer to time-related information such as hours and weeks. These features can be used to capture the periodicity and trend of time factors affecting ship behavior. Motion features describe the ship’s motion behavior, such as speed, acceleration, steering angle, etc., within the look-back window. By decomposing time features and motion features, we can model and predict the dynamic changes in ship behavior.

Furthermore, the look-back window provides a continuous data stream that enables models to better capture the time series relationship of ship trajectories. Ship behavior often exhibits time correlations, where past and future behaviors may have certain associations and regularities. By leveraging the look-back window, models can learn time series patterns and trends, which contributes to more accurate predictions of future ship behavior.

Ship trajectory prediction tasks require considering various factors, such as time series properties, historical behavior, and environmental factors. Using the look-back window and decomposition strategy allows us to obtain information more comprehensively and perform modeling and prediction.

When dealing with AIS data, the DLinear model decomposes the original data into trend components and remainder (temporal) components using a moving average kernel. It applies a linear layer to each of these components separately and then combines their features to obtain the final prediction result. This approach can capture the periodicity and trends in AIS data better than other approaches, thus improving the prediction accuracy. The structure of the basic linear model is shown in Figure 1.

The DLinear model encompasses three variants: Vanilla Linear, DLinear, and NLinear. Vanilla Linear is the most fundamental version of these three and employs only linear functions for regressing historical time series data. DLinear enhances performance by combining position encoding strategies with linear layers, while NLinear introduces addition and subtraction operations for normalization, enabling the model to better adapt to diverse time series data [10].

Specifically, the DLinear model transforms historical time series data into vector representations and employs linear layers to regressively predict these vectors. Compared to the deconstruction of Autoformer and Fedformer, DLinear further utilizes certain position encoding strategies to enhance model performance. These position encodings consist of absolute position encodings that combine time steps with absolute positional information and relative position encodings arranged periodically. We can understand the basic structure of the model in Figure 2.

3.3. Improved Linear Networks with Multiple Feature Processing Capabilities

The proposed model is used to learn the patterns of individual ship trajectories. By analyzing the motion data of a single ship, it extracts patterns and features to predict the ship’s future trajectory. This model is highly effective in studying ship movements within a local range, but it is inadequate for handling numerous ship trajectories in a large-scale area.

When there are many ship trajectories in a wide area, individually modeling and predicting each ship becomes extremely difficult. Furthermore, the movement trajectory of a single ship can be influenced by various factors, such as other ships, ocean currents, and wind direction, making the prediction more complex.

We have conducted a reconstruction of the model in order to enhance its training efficiency and improve its ability to learn multiple features. The original model could only extract and learn the trajectory features of one ship at a time, resulting in low training efficiency and limited generalization ability. However, in this experiment, we have employed a method of sequentially concatenating the extracted input features and feeding them into the model.

For individual trajectory inputs, the approach remains consistent with the previously mentioned model methods. The key lies in concatenating the extracted features and uniformly inputting them into the model for learning. Through this approach, the information of multiple ships is integrated into a single input sample, enabling the model to simultaneously learn patterns and features from multiple ship trajectories and capture more comprehensive and diverse ship behaviors and relationships. As a result, the performance of the model is significantly enhanced. With the reconstructed model, we are able to train the model more efficiently and improve its generalization ability, thus extracting and learning a wider range of ship features.

This improvement significantly enhanced the training efficiency of the model. The current linear networks with a time-feature decomposition model can handle a larger amount of data simultaneously than the previous model, which can learn only one trajectory at a time. The improved linear networks with a time-feature decomposition model can effectively leverage the correlations and shared feature information among different trajectories during the learning process.

The proposed model integrates multiple ship trajectories to learn patterns and features, enabling the prediction of future trajectories for individual ships based on analysis of their motion data. It is designed to adapt to large-scale and long-duration marine environments that are complex and constantly changing. Although there is some randomness in the short-term trajectories of vessels, overall trends can still be clearly observed. Training the model on tens of thousands of vessel trajectories enhances its understanding of vessel motion and captures universal patterns. This innovative approach endows linear networks with a time-feature decomposition model with improved generalizability, enabling vessel motion trends to be accurately predicted in extensive and prolonged scenarios. This approach also provides strong support for applications such as marine environment monitoring and maritime safety.

Through this enhancement, we can train linear networks with a time-feature decomposition model faster, while maintaining accuracy, improving the overall training efficiency. This is crucial for processing large-scale AIS data and for real-time predictions, providing better support for shipping operations and underwater research needs. We can see that the improvement of the model structure is shown in Figure 3.

In NLinear, addition and subtraction operations are introduced for normalization. Specifically, NLinear divides historical time series data into multiple regions and calculates an average value for addition and subtraction in each region. These averaged values serve as normalization factors to better adapt the model to various time series data. In addition, to improve performance in long-term prediction problems with data distribution shifts, the NLinear model adopts a novel normalization method. It subtracts the last value of the sequence from the input, processes it through linear layers, and then adds back the previously subtracted portion before making the final prediction, completing a simple normalization of the input sequence. This subtraction and addition process scales and shifts the input sequence, effectively eliminating distribution shifts in the sequence.

The linear networks with the time-feature decomposition model exhibit superior performance to the NLinear model in long-term prediction. The proposed model is a machine learning-based approach for time series forecasting that is applicable to the time series prediction tasks of various industries. By incorporating strategies such as time feature advance and positional encoding, linear networks with a time-feature decomposition model can effectively handle long-term time series data. The vector representation transforms time series data into continuous vector forms, enabling the model to better capture the structure and trends of the sequence. Meanwhile, positional encoding assigns specific positional information to each element in the sequence, assisting the model in capturing changes at different time points. With these strategies, the proposed model can effectively handle long-term time series and overcome challenges faced by traditional methods in long-term prediction problems. The model accurately captures long-term dependencies and trends within the sequences, improving prediction accuracy and stability.

In summary, the proposed model is a deep learning-based approach for time series prediction that utilizes time feature advancements and positional encoding strategies to effectively handle long-term time series and achieve efficient forecasting. The proposed model demonstrates superior performance to the NLinear model in long-term prediction. The proposed model is a worthwhile and promising choice for time series prediction problems regardless of the industry. From Algorithm 1, we can see the execution process of the improved linear networks with the time-feature decomposition model.

Algorithm 1 Pseudocode of the improved multi-input multi-output linear networks with time-feature decomposition mode.

1:  Input: A series of multiple sets of ship trajectory sequences $X={\{X_1^{\mathrm{t}},\dots ,X_{\mathrm{C}}^{\mathrm{t}}\}}_{\mathrm{t}=1}^L$, $t$ 2:  represents the t-th time step in the future. 3:  Outupt: Prediction of trajectories within a specific time step $S$ in the future 4:  Initialization: $t$ = 0, $k$ = 0, $i$ = 0, window_size denotes the length of the sliding 5:  window. 6:  For $t$ in 0, …, $L$ do: 7:    Initialize temporal and spatial weight features 8:    Input initial sequence set $X$, window_size 9:    Window feature = [] 10:   If channel individual is False: 11:     num_windows = len($X^k$) − window_size + 1 12:     For $i$ to num_windows: 13:       window = time_series [i: i + window_size] 14:       Decomposing time features through moving windows 15:       Decomposing motion features through moving windows 16:       End For 17:     Else: 18:     For sequence in each channel: 19:       For $k$ in 1, …, $C$ do: 20:         num_windows = len($X_{{}^c}^k$) − window_size + 1 21:         For $i$ to num_windows: 22:           window = time_series [i: i + window_size] 23:           Decompose time and motion trends in channels and update network 24:           layer parameters 25:           Learning sequence changes over time and motion through moving 26:           windows 27:         End for 28:       End for 29:     End for 30:   End If 31: End for 32: Restructure the obtained features as the input of the improved DLinear model 33: Train the DLinear model until the training iterations are finished 34: Apply a trained DLinear model to predict the ship trajectories of $S$ step in the 35: future sequence $X$ 36: Output: Prediction of trajectories within a specific time step $S$ in the future

3.4. AIS Data Pre-Processing

Several issues arise when processing and analyzing AIS information, such as missing data and erroneous data. Data cleaning and pre-processing are necessary for several reasons [18]:

First, AIS data may be incomplete or missing. These problems are often caused by equipment malfunctions, network issues, signal interference, etc. Before analyzing AIS data, the data must be cleaned to ensure their integrity and accuracy. AIS data may also have duplicated data records containing the same vessel information, which can lead to errors or distortions in the data analysis results. Therefore, deduplicating and addressing duplicate data rows is needed. AIS data also suffer from inconsistent time intervals, nonstandard formats, and misclassified vessel types. To facilitate efficient data processing, these issues must be addressed by standardizing and normalizing the data format and using interpolation to ensure consistent data time intervals.

Second, due to the wide range of obtained data and the significant differences between ships, we must filter the ship motion trajectories. We selected the corresponding vessels based on the longitude and latitude ranges within the Gulf of Mexico. To ensure modeling accuracy, we deleted vessels with a length less than 3 m and a width less than 2 m during the data preprocessing stage. We adopted this strategy because small vessels are often more influenced by environmental factors such as ocean currents and waves, making determining their trajectories more difficult and significantly impacting subsequent analysis and modeling. Next, we extracted the MMSI (Maritime Mobile Service Identity) number for each ship on each day and removed isolated data rows, as well as repetitive data with the same MMSI number caused by duplicated data. When multiple data rows of the same ship were present, we sorted them in chronological order to determine the movement trajectory of the ship.

Finally, as tracking the long-term motion of ships in vast waters is a complex process, more features must be added for training. To identify ship features, we extracted turning features based on the difference in COC (course over ground) values between each ship. This enables us to represent the distance and angle at which each ship initiates a turn, considering its current navigational initial speed. Because AIS data come from different time points with different time intervals, these data are asynchronous. To further improve the modeling accuracy, in the data preprocessing stage, we processed the time difference between the previous and next points of the same MMSI and added the resulting time interval feature Δ time as a one-dimensional feature to the model input feature. With this processing method, we can more accurately describe the changes in ship motion status, providing a more reliable data basis for subsequent modeling. In addition, we also calculated and interpolated the longitude, latitude, velocity, heading, and trajectory angle characteristics of each ship’s route trajectory.

After processing the data in the process shown in Figure 4, we need to input the preprocessed time series into the model for learning. By feeding data into the model, the model can extract valuable information and update model weights by learning the relationships and patterns between the data. The input of a time series is divided into seq_len, pred_len, and label_len. These parameters are used in time series prediction models to describe the length of the time series data and the predicted target.

Table 1. Time series parameters.

Parameter Name	Meaning of Parameter
seq_len	The length of the time series data.
pred_len	The length of the prediction result that the model outputs.
label_len	The length of the label data used to train or evaluate the model.
predict_len	The length of the predicted trajectory.

provides a summary of the meaning and significance of the parameters seq_len, pred_len, label_len, and predict_len. These parameters are crucial for describing the time series data lengths, prediction results, and target sequences used for training and evaluation purposes. The partitions of these parameters in the input sequence are shown in Figure 5.

When referring to various series data, we found that some time series are actually unpredictable. The predictability of a time series generally depends on its nature and trend characteristics. The future trends of relatively stable and regular time series, such as sales revenue or temperature, can often be predicted based on historical data. These time series thus exhibit a certain level of predictability. However, the future trends of irregular and nonlinear time series, such as stock prices in financial markets or traffic congestion indices, are often influenced by multiple complex factors. As a result, these series are difficult to predict using simple models and thus have lower predictability. Additionally, some time series that exhibit obvious temporal and periodic patterns can be more accurately predicted by considering historical data when assessing the current time point. These series thus demonstrate higher predictability.

On the other hand, the prediction of ship AIS trajectories is feasible but not easy, as they exhibit certain regularity and predictability in their behavior. This is because AIS trajectories are influenced by various factors, such as routes, ship speeds, weather conditions, and sea states, which can be controlled and predicted to some extent.

However, in practical applications, AIS trajectories may also exhibit random behavior to some degree. For instance, when ships encounter sudden weather or sea condition anomalies their navigation paths may experience significant disturbances; it is difficult to predict the ship trajectory in this situation.

AIS trajectories under irregular vessel behaviors, such as illegal fishing or theft, may exhibit random and unpredictable characteristics. In this scenario, as shown in Figure 6, it is difficult to predict the AIS trajectory of ships. Therefore, filtering and cleaning the data before training is necessary for the model to better identify temporal and spatial motion patterns within these irregular trajectories. The AIS trajectories of ships in stable motion in the ocean are often clear and regular, as shown in Figure 7, which is relatively easy to predict.

4. Experiment and Discussion

4.1. AIS Dataset

In this study, we utilized the AIS dataset, which is a publicly available dataset obtained from the official website of Marine Cadastre [19], the national marine observation viewer of the United States of America. The dataset contains maritime and coastal geospatial information data for the United States and its territorial waters. The downloaded AIS dataset encompasses ship data for the entire year of 2022. The dataset is publicly available, ensuring research integrity and compliance with relevant laws and regulations regarding data usage. Prior to usage, we performed preliminary data cleaning and processing, including deduplication and handling missing values, to ensure data quality and reliability. By analyzing data, we found that the downloaded AIS data cover a significant portion of the Northern Hemisphere. For experimental convenience, we applied cluster analysis to identify the largest region with the highest number of ship voyages.

To identify the areas with the densest ship trajectories, we used an algorithm called clustering analysis to process all ship trajectory data [20]. Cluster analysis is an important algorithm for analyzing vessel trajectories. We applied the K-means algorithm to the latitude and longitude position data from ships’ AIS signals. By clustering and analyzing the trajectories of these ships, we partitioned the ships into different clusters, where each cluster represented a dense area. This allowed us to determine which areas have closer and more concentrated ship trajectories. These areas indicate regions where ships pass through more frequently. The process involves applying the K-means algorithm to ship position data to partition the ships into different clusters, with each cluster representing a dense area. Through iterative refinement until convergence, we ultimately identified the most densely populated regions of ships. This process helped us identify hotspots of ship activities and provided a foundation for further analysis and planning [21].

The principle of cluster analysis is based on the similarity or distance calculation between data points. For AIS data, we can use the longitude and latitude information of ships as features to measure the distance between them. We extracted the ship position information from the AIS data and calculated the distance between ships based on their locations. Then, by selecting the K-means clustering algorithm, we partitioned the ships into several clusters, with each cluster representing a dense area. Subsequently, we iteratively assigned ships to different clusters using the algorithm that minimizes the intra-cluster distance, updating cluster centroids until convergence was achieved. Once the central clusters were determined, we assigned ship data points to the nearest cluster, resulting in the identification of areas with the highest ship trajectory density.

Therefore, we identified a rectangular region covering the entire Gulf of Mexico with a longitude range of −98 to −77 and a latitude range of 30 to 16 and selected the ship information within this region for training. The range selection is shown in Figure 8 below.

The Gulf of Mexico usually encompasses a vast expanse and a considerable number of vessels. The wide-area AIS trajectory prediction and small-area prediction have clear differences in data coverage, data density, prediction methods, and target applications. Wide-area AIS data include vessel information from various marine regions. As a result, this dataset exhibits a massive data volume and uneven spatial distribution, as well as sparsity. In contrast, small-area datasets are limited to specific marine areas or to near individual ports, resulting in higher data density.

Wide-area AIS trajectory prediction is more complex than small-area prediction. Due to its data sparsity and tremendous coverage, data gaps and uncertainties must be addressed, and sophisticated models must be employed for trajectory prediction.

In this study, long-term prediction not only refers to the length of time being predicted but also involves predicting multiple time steps into the future. Long-term prediction is challenged by increasing uncertainty as the prediction horizon extends. This makes long-term forecasting difficult, especially for vessels operating at sea [22]. Due to differing time intervals, the predictions for future durations may vary. Extracting AIS data with different time intervals from the interpolated dataset allows data of various durations to be obtained.

Through this approach, we can make predictions at different time scales according to our needs. If we are concerned about accurate and detailed predictions, we can choose smaller time intervals, such as a ten-minute interval. If we are more concerned about overall trends and long-term changes, we can choose larger time intervals, such as a daily interval. In this way, we can adjust the duration of predictions based on actual requirements and obtain predictions with different granularities. In this experiment, a time interval of 90 s was selected.

With this method, we can flexibly predict future durations and select appropriate time intervals based on specific application scenarios. This research design can provide more comprehensive and accurate prediction results, meeting the needs of different usage scenarios. The interval selection is shown in Figure 9 below.

4.2. Experimental Settings

All models utilized in this experiment were specifically designed for time series forecasting of ship trajectories. Seven different models were used: Autoformer [23], Informer [24], Transformer [25], Linear, NLinear, DLinear, and PatchTST [20]. The input consisted of a time series of length seq_len, which contained both multivariate and univariate feature data. The models predicted the values for the next pred_len time steps, with a starting token length of label_len.

In the data loading phase, the model imported batched ship trajectory data for each day, with a sampling frequency of 90 s. During the training process, a moving average kernel was applied to smooth the data. The moving average kernel computes a weighted average of the current time step value and the historical values within a certain time range, capturing trends or periodic patterns in the data. By applying the moving average kernel, noise and fluctuations in the time series data can be reduced, and trend information can be extracted. Four feature encoding methods were available for this experiment: value embedding with time embedding and position embedding, value embedding with time embedding, value embedding with position embedding, and value embedding. In this experiment, the value embedding with time embedding and position embedding method [10] was chosen. This approach provides a richer and more comprehensive representation of features. By incorporating value, time, and position information, the performance and accuracy of the model in sequence tasks can be improved.

The optimization techniques employed in this study included gradient descent and early stopping. The model was trained for 500 iterations using a batch size of 128. A learning rate of 0.005 was applied, along with the Adam optimizer and the mean squared error (MSE) loss function. Furthermore, GPU acceleration and multi-GPU parallel training were utilized. For the kernel size, it was set to 25, while the channels were configured to be individual of each other.

4.3. Evaluation Criteria

The evaluation metric for the experiment is defined as follows: The prediction error at time steps was measured using the Haversine distance [26] between the true position and the predicted position. The Haversine distance [26], a mathematical formula that calculates the great-circle distance between two points on the Earth’s surface, is a particularly useful metric for evaluating the accuracy of models in long-term prediction tasks. By measuring the direct distance between two points on the globe, without considering the curvature of the Earth, the Haversine distance provides a reliable measure of the overall predictive performance over an extended period [27]. The calculation method of the Haversine distance is shown in Formula (1).

On the other hand, regarding analyzing the navigation trends of ships in various sea areas, the track deviation angle [28] emerges as a more suitable metric. This angle represents the angular difference between the intended ship’s path and its actual route. By quantifying the extent to which a ship deviates from its desired course, the track deviation angle offers insights into the effectiveness and efficiency of ship navigation in different maritime regions. The calculation method of the track deviation angle is shown in Formula (2).

In summary, while the Haversine distance serves as an accurate measure of model accuracy for long-term predictions, the track deviation angle is better suited to capture ship navigation trends across diverse sea areas. These two metrics complement each other and together provide a more comprehensive assessment of ship performance and navigation capabilities.

$$d=2r\arcsin \left(\sqrt{{\sin }^2\left(\frac{La{t}_2-La{t}_1}{2}\right)+\cos \left(La{t}_1\right)\cos \left(La{t}_2\right){\sin }^2\left(\frac{Lo{n}_2-Lo{n}_1}{2}\right)}\right)$$

(1)

Calculating the track deviation angle based on latitude and longitude requires more mathematical and geographical knowledge. Typically, the conversion of latitude and longitude coordinates into a planar rectangular coordinate system is required prior to the calculation of track deviation angle.

This method aims to more simply and effectively calculate the track deviation angle [28]. By using this method, we can quickly and accurately estimate track deviation angles to evaluate the prediction results of ship motion trajectories. The latitude and longitude coordinates of the starting point and the ending point are converted into coordinates in a planar rectangular coordinate system. Assuming that the coordinates of the starting and ending points are $\left(x_1,{ \mathrm{y}}_1\right)$ and $\left(x_2,{ \mathrm{y}}_2\right)$, respectively, $\sigma$ represents the angle between the ship’s heading and the expected heading (i.e., the deviation angle). The calculation method is shown in Formula (2).

$$\alpha =a\mathrm{rctant}\frac{\left(y_2-y_1\right)}{\left(x_2-{ \mathrm{x}}_1\right)}-\sigma$$

(2)

The Haversine distance is more suitable for expressing the model accuracy in long-term prediction, while the track deviation angle is better able to express the navigation trends of ships in different sea areas. In this article, we can more intuitively understand the prediction of AIS trajectories for each ship through the Haversine distance and understand the changes in ship motion trends through the track deviation angles.

4.4. Experimental Results

This experiment focuses on studying ship AIS trajectories in the dataset downloaded. Most ships’ navigation data are within one day, and each data file contains one day’s navigation data. Some long sailing data may exceed one day, which is reflected in multiple data files. Analysis shows that the sailing time of ships varies widely, from hours to days. Cross-ocean navigation between the West Coast and East Coast of the United States can take several days, while sailing from the East Coast to coastal areas such as the Caribbean Sea may take only a few hours. In summary, to study the prediction of future sailing time based on our actual data situation, different prediction step lengths (50, 100, 150, 200, 250, 300) were chosen. The time interval between steps is 90 s, with a maximum predicted time of 7.5 h. Through prior data analysis, it can be determined that a predicted time of 7.5 h covers approximately one-fourth to one-half of the total time that most ships spend at sea, possibly exceeding one-half in some cases, such as fishing boats and short-distance passenger ships. However, longer prediction step sizes lead to increased prediction error and reduced accuracy, potentially overfitting the model. Therefore, a maximum prediction step size of 300 (7.5 h) was chosen for this experiment, striking a balance between prediction accuracy and generalization ability to generate effective results.

For our experimental results graph, we used a logarithmic coordinate since the maximum value of the experimental results is above 100, while the minimum value is only below 10. This type of numerical difference is significant, making it difficult to clearly display the changes between smaller values using ordinary linear ordinates. The experimental results are shown in Figure 10 and Figure 11 below. Analyzing the experimental results demonstrates that the improved linear networks with time-feature decomposition have the best predictive performance among all compared models.

As finding the overall operation trends in the prediction results of multiple transformer models is difficult, the prediction results of multiple transformer models are not displayed in Figure 11.

The trends of these line curves in the graph demonstrate that as the prediction step size increases, the prediction error of different models also increases. However, our improved model always maintains a lower prediction error than the other models. Notably, in short-term prediction, the transformer model performs better than the transformer model, as shown by the intersecting line changes in the graph. However, as the prediction step size increases, the transformer model gradually outperforms the Autoformer and Informer model, especially in long-term prediction. This indicates that some models may be good at short-term prediction but unsatisfactory in long-term prediction.

The quantitative information of the experimental results can be referred to in

Table 2. Experimental results of the Haversine distance criterion.

The Haversine Distance on Different Predict Lengths	Pred Length 24	Pred Length 48	Pred Length 96	Pred Length 128	Pred Length 192	Pred Length 256	Pred Length 300
Transformer	32.67	46.45	55.47	76.93	96.77	104.35	123.34
Autoformer	23.64	30.93	56.64	70.31	83.74	98.34	103.49
Informer	47.98	59.74	73.36	89.26	97.64	103.64	113.31
Linear	5.34	13.76	26.49	30.33	47.36	52.23	58.64
NLinear	4.30	4.87	8.68	13.98	15.27	23.66	27.87
PatchTST	2.47	5.73	8.75	10.6	13.34	15.37	23.76
Improved DLinear	1.26	3.73	4.23	7.65	8.98	10.73	12.23

and

Table 3. Experimental results of the track deviation angle criterion.

The Track Deviation Angle on Different Predict Lengths	Pred Length 24	Pred Length 48	Pred Length 96	Pred Length 128	Pred Length 192	Pred Length 256	Pred Length 300
Linear	0.67	3.64	7.89	10.37	17.38	29.34	31.96
NLinear	0.36	1.37	3.64	6.31	9.87	13.21	16.97
PatchTST	1.63	2.31	6.23	8.69	12.64	24.56	46.98
Improved DLinear	0.27	0.73	1.46	2.67	2.93	3.56	5.14

. These tables include various measurement results under different experimental conditions used in the study. By analyzing these tables, we can obtain more comprehensive and specific data, from which we can see that our improved model achieved the best results in both evaluation criteria.

Thus, the above results clearly illustrates that the improved linear networks with the time-feature decomposition model have the best prediction performance in terms of the Haversine distance and track deviation angle in both long- and short-term prediction. This means that among many models, our improved model can achieve the best results in both long-term and large-scale AIS trajectory predictions.

4.5. Parameter Selection

In order to find the most suitable parameters for deep learning, this section will conduct Taguchi experiments [29] on three parameters that have a significant impact on the model. The Taguchi experiment is a quality management experimental design method based on statistical principles and quality engineering strategies. Its objective is to determine the optimal parameter settings with as few experiments as possible, in order to achieve the best performance and stability of the model. Taguchi experiments focus on identifying the key parameters that have the greatest impact on the process or product performance, and optimizing the results by adjusting these parameters. Through Taguchi experiments, we can systematically search for the optimal parameter configuration to improve the performance of the deep learning model.

Through preliminary experiments, we found that the three parameters that have the greatest impact on the model performance are channel independence, loss function configuration, and kernel size (which determines the window size used by the model to extract temporal and spatial features). These parameters play important roles in deep learning models, and their different settings may significantly affect the model’s performance. The loss functions used include MSE (mean squared error), MAE (mean absolute error), and MBE (mean bias error), which can all measure different aspects of model performance and help optimize the model training process. Therefore, we used the Taguchi experimental method to determine the optimal parameter configuration in order to achieve the best model performance. By systematically adjusting and optimizing these three key parameters, we were able to enhance the performance and stability of the model.

After determining the parameters, we used an orthogonal table to design the Taguchi experiment. An orthogonal table is a design table used to select different combinations of factor levels, aiming to cover possible parameter combinations with minimal number of experiments. By utilizing an orthogonal table, we can systematically select the values for key parameters and obtain information about the impact of these parameters on the results. Additionally, the use of an orthogonal table helps to reduce the influence of random errors, thereby improving the reliability and interpretability of experimental data.

Table 4. Parameter settings in orthogonal tables.

Is Individual	Loss Function	Kernel Size
True	MSE	25
True	MAE	75
True	MBE	55
False	MSE	75
False	MAE	55
False	MBE	25

below displays the settings of different parameters in the orthogonal table.

We applied these parameters to the prediction of future 64-step DLinear model experiments with the evaluation function of the Haversine distance. The experimental results are shown in

Table 5. Orthogonal experimental results.

Is Individual	Loss Function	Kernel Size	Experimental ResultHaversine Distance (m)
True	MSE	25	11.572
True	MAE	75	5.770
True	MBE	55	8.653
False	MSE	75	6.971
False	MAE	55	7.856
False	MBE	25	12.579

.

From the above experimental results, it can be seen that when each channel is independent, the loss function used is MAE, and the Kernel size is 75, the model achieves the best performance.

4.6. Discussion

When conducting learning and prediction experiments with Transformer, Autoformer, and Informer, we found that these three models performed particularly poorly. In general, these three models have certain deficiencies in time series forecasting tasks and require further research and improvement. The results of Autoformer are shown in Figure 12 and Figure 13.

From Figure 12, we can learn that these results demonstrate that the unmodified models of the former series perform poorly in ship trajectory prediction. Figure 13 also illustrates that the improved DLinear model performs well in predicting the ship trajectory. Moreover, even if the trajectory has many twists and turns, the DLinear model can still predict the future movement direction.

Figure 14 shows that the PatchTST model performs relatively well on some simple trajectories. However, when the trajectories become more complex, the prediction results deteriorate significantly, and in some cases even become opposite to the actual trajectory.

Through a series of comparative experiments, we found that the improved DLinear model excels in ship trajectory prediction tasks, outperforming models such as Transformer, Autoformer, Informer, and PatchTST. Specifically, the improved model, utilizing a look-back approach, accurately predicts ship trajectories due to its superior memories of past trajectories and historical dependency capabilities. Finally, compared to the Nlinear and Linear models, the improved model demonstrates superior prediction performance to the Nlinear and Linear models by learning temporal features and providing better predictive results under different lengths of time steps.

Taking everything into account, our improved model demonstrates significant enhancements in ship trajectory sequence prediction. Compared with the traditional DLinear model, our model extracts features from large datasets by concatenation, which greatly improves the efficiency of the model and its generalization ability.

In addition, our model has a wider prediction range. Based on comparative results from references [11,12,13], our research shows a broader prediction range in ship trajectory sequence prediction. Moreover, compared to references [12,14], our model can predict trajectories with longer time steps.

Significantly, our model has achieved remarkable accuracy in prediction. Through extensive feature extraction and concatenation learning from large data sets, the model captures key features more accurately. This method not only improves the learning efficiency of the model but also enhances the model’s generalization ability, making it more reliable when dealing with complex situations. Furthermore, in comparison to reference [16], our model can predict trajectories of more types of ships.

Finally, our improved model has strong scalability and adaptability. It can handle large-scale data sets and different types of trajectory sequences with high performance in real-world applications. This provides broad prospects for our model and satisfies high requirements for future time prediction.

5. Conclusions

We propose a methodology to tackle the challenge of large-scale and long-term prediction of ship trajectories based on AIS data. Our approach employs an improved DLinear model that decomposes the spatiotemporal movement patterns of numerous ship trajectories using a retrospective window. The model is established following these steps: Firstly, it extracts and concatenates features from multiple ship trajectories as inputs utilizing the retrospective window. Next, it employs a shared-weight neural network to learn the underlying movement patterns embedded in these features. By learning from the information extracted from multiple trajectories, the proposed model can capture the regularities in ship movements, enabling accurate long-term predictions across diverse scenarios. To evaluate the model’s performance, we utilized a public AIS dataset in the Gulf of Mexico for experimentation. Moreover, we conducted a Taguchi experiment to determine the optimal parameter configuration specific to our model. The experimental results demonstrate that the proposed model outperforms other contrast models such as Transformer, Autoformer, Informer, PatchTST, and so on, in terms of prediction accuracy of ship trajectories. Furthermore, our model also exhibits robustness in handling a wider range of scenarios, longer prediction intervals, and various types of vessels.

5.1. Limitations

The current methodology has a few limitations. First of all, the process of extracting and concatenating multiple ship features is time-consuming, resulting in a long period of training for each iteration. Secondly, although the model achieves long-term predictions at the hourly level, there is still a limitation for an even longer prediction.

5.2. Future Work

In future work, we aim to incorporate multimodal data such as weather conditions and oceanographic hydrological data into our experiments to assist in ship trajectory prediction. Moreover, it would be valuable to explore the scalability of our model on the datasets that cover even larger geographic areas. To achieve this, we will optimize our model architecture and the training process, as well as leverage distributed computing techniques to speed up training times.