Due to the complex maritime environment in real scenarios, the sensor equipment carried by fishing vessels often has problems such as signal loss and equipment failure, which will lead to problems such as incorrect reporting coordinates, reported data loss, and even crazy reporting of some equipment. Therefore, we need to preprocess the collected data and clean up the unwanted data. The original GPS trajectory data contains rich types of information, including ship ID, longitude (LON), latitude (LAT), heading, speed, time, ship type, tonnage, etc. For the ship trajectory prediction task, we want to carry out, only part of the information data is used, so we need to extract the trajectory of the ship. The extraction process of our required ship trajectory is as follows:

Suppose the characteristics of the ship’s navigation trajectory at a certain time t can be expressed as:

The target ship’s longitude, latitude, speed over the ground, and course over the ground are represented by the variables x_t , y_t ,v_t ,a_t respectively, at time t.

M-STCP (Matrix-Based Ship trajectory Cleaning and Prediction) is a data cleaning and trajectory prediction method based on matrix neural networks. It leverages the advantages of matrix neural networks to efficiently clean ship trajectory data and accurately predict future ship trajectories. By applying matrix neural networks to the fields of data cleaning and trajectory prediction, M-STCP demonstrates capabilities in enhancing data processing efficiency and accuracy. The flow chart is shown in Figure 3 . The core method of this method is described as follows.

Each ship trajectory has its initial position ( ( a 0 f , b 0 f ) ) , even if the trajectory shape of the ship trajectory is different, it will be interpreted as a different trajectory due to the difference in the initial position. This also shows that there is a spatial bias in the trajectory dataset because the ship coordinates are described by latitude and longitude coordinates. To eliminate the spatial deviation, in the ship trajectory prediction, we can perform a phase shift reconstruction of the coordinates of all samples, and after the reconstruction, all the trajectory points are moved to the (0,0) point relative to the starting position of the trajectory, thereby eliminating the spatial deviation. For each time step t and each sample f from the original position ( a f , b f ) ( f > = 0 ) reconstructed to the same starting point (0,0), the process is as in Eq (5):

where ( a 0 f , b 0 f ) represents the latitude and longitude coordinates of the fth sample in the first time step.

In this way, we can unify the starting point of all trajectories to the same location, eliminating the spatial bias caused by different starting point locations, to make trajectory prediction more efficient.

In ship trajectory prediction, we use matrix neural networks for autoregressive or sliding window prediction frameworks. Data embedding reconstruction for time series is also referred to as Taken’s theorem. According to Taken’s theorem, the size is chosen to be α and a regular interval of β to reconstruct the time series, thus the original time series’ properties are preserved in the reconstructed time series. The given ship trajectories are binary sequences, and each trajectory data is embedded as a matrix of size ( T f − α ) × ( α × 2 ) , and each data point contains two feature information of latitude and longitude (a_t,b_t ). The time series data takes the location coordinate information as an input point in chronological order, and then forms an input matrix m t f . Each row of the input matrix contains two eigenvalues, latitude and longitude. Starting from the α + 1st position coordinate to the last position coordinate as a data point, the output matrix is formed. Each row of the output matrix also contains two eigenvalues. The input matrix and output matrix form are shown in Eq(6): (t ≥ 0)

In this way, we can convert the trajectory data into matrix form and use it to train a matrix neural network model to achieve predictions of ship trajectories.

When dealing with problems involving spatial correlation such as ship trajectory coordinates, if the traditional neural network input vector sum is used, information will be lost. As a result, matrix neural networks are better suited to handle these issues. The mapping between MNN layers and layers is defined in Eq(7).

The expanded form is as follows:

Among them, X^l is the matrix variable of the L layer, X^{l +1} is the output in the L layer, W, X, V, B are matrices with compatible dimensions, B^l is the offset of the current layer, and σ()is the activation function. The matrix form of the data can be maintained by Eq(8). Through the changes of W(l), X(l), and V (l), the matrix form of the hidden layer can be reconstructed. The specific matrix change process is shown in Eq(9):

The matrix variation formula of a matrix neural network is shown the matrix transformation process from input matrix X^l to output matrix X^{l +1}, which is different from traditional vector-based input. Also shown is the matrix transformation for the ith window from time t to t + α, which gives the latitude and longitude of the ship’s trajectory. Among them, the matrix dimension is changed from a × b to p × q by the dimension dot product of the input matrix.

The definition of the ship trajectory indicates that the input data m ^f of the matrix neural network is a three-dimensional tensor of size, that is, the number of sliding windows for each input data is T_f - α), each The size of the input unit is α × 2. The number of each output data sliding window of the output data is also, and the size of each output unit is 2. That is, we need to find the mapping relationship that changes the size of the matrix from α × 2 to 2, that is, f: ℝ ^{a ×2} → ℝ². To lessen the discrepancy between the supplied output n ^f and the network output n ^ f , we adopt Eq(10) for learning.

The ship’s track in space is some time series data. Because our track data is measured by various sensors, there will be some unavoidable errors. To ensure the accuracy of trajectory prediction, we need to smooth the data. Rudolph E. Kalman put forward the Kalman filter in 1960 and gave a new method to solve the prediction problem. Kalman filter takes into account the possible uncertainty in the prediction process and estimates the optimal result through incomplete accurate prediction models and incomplete accurate measurement results. The prediction equation includes the system’s status right now and the uncertainty of the system. The specific process of KF is to evaluate the prediction equation and correction equation through recursion. The prediction equation estimates the current state and uncertainty of the system. The correction equation uses the measured current state quantity of the system to update the estimation. It should be noted that in KF, we assume that both system uncertainty and measurement uncertainty follow a normal distribution.

The prediction equation of Kalman filter is:

In Eq(11) X ^ k − represents a prior estimate at time k, A is the state matrix, X ^ k − 1 represents the estimated value at k − 1 time, B represents the control matrix, u _{k −1} represents the control variable at the moment. In Eq(12) P k − represents the prior error covariance, A represents the state matrix, P k − 1 represents the error covariance at k − 1, and Q represents the covariance of process noise. The correction equation of the

Kalman filter is:

In Eq(13), K _k represents the Kalman ga in, H represents the observation matrix, and R represents the error covariance of the measurement process. In Equation (14), X ^ k represents a posterior estimate, that is, the optimal estimate we need, Z _k represents the measured value at time k. In Eq(15), P _k represents the update error covariance at time k. The covariance matrices of process noise and measurement noise are represented by the matrices Q and R in the Kalman filter, respectively. The larger the value of Q, the more trust we have in the measured results, the greater the value of R will be, and the more trust we have in the estimated results.

When the M-STCP method is combined with the Kalman filter for ship trajectory prediction, the Kalman filter needs to be set including the state transition equation, measurement equation, initial state, and covariance matrix. First, a dynamic trajectory forecast model is determined by the vessel’s historical trajectory. Then, the observation values such as position and speed information are input into the matrix neural network, and the state estimation is carried out to obtain the predicted value of the ship’s state at that moment. The predicted value is then corrected using a Kalman filter to obtain a ship condition estimate that is closer to the real value. The specific implementation is shown in algorithm 3 .

We can define the state of a ship as a four-dimensional vector X(k) ={x_k, y_k, v_xk, v_xyk}, where x_k and y_k represent the ship’s position on a planar coordinate system, and v_xk and v_yk represent the ship’s velocity in the x-axis and y-axis directions. This state variable evolves between adjacent temporal steps, so we can use a transition matrix A to describe the evolution of state variables in each temporal step. In particular, under the assumption that the time step is dt, the transition matrix can be defined as:

The Q process noise covariance matrix represents the uncertainty or noise of the state variable during the forecast. Ships can be affected by factors such as wind and currents while sailing, so we can build a random walking pattern. Then, depending on the location of the ship, we can define the Q corresponding to where t is the sampling interval.

Observe the value of the noise covariance matrix R We measure the position and speed data of the ship, record the measurement data and observe the error to obtain the value of the R matrix, and you can set it to:

To ensure the validity of trajectory prediction, we utilize real trajectory data. The ship tracking data used in this article is from the Wenzhou Fishing Vessel Safety and Rescue Center, which contains about 5 million tracks in Jiangsu and Zhejiang in 2016. The GPS data collection cycle is 15-30 minutes, the data set size is approximately 70GB, and it contains about 40 million pieces of trajectory data. Oracle Database is used for data storage. In Table 1 , a sample of the GPS data are displayed. To obtain the density/frequency feature of the trajectory data, we plot ship trajectories every four months in the same heatmap respectively. The space heat map is shown in Figure 4 .

The data distribution after extraction and processing of the original ship trajectory and anomaly point cleaning is shown in Figure 5 . The trajectory points set in this paper have a longitude range of [119.6,131.0], a latitude range of [22.0,33.8], a direction range of [0.0,360.0], and a velocity range of [0.0,25.0].

Due to the complex ship motion pattern in the real data, the ship’s trajectory is not single linear, we cluster it using the DBSCAN technique based on density. According to the collected large number of ship trajectory data, the corresponding characteristics such as ship number and ship steering angle are selected, and the ship trajectories with similar patterns are regarded as the same cluster through DBSCAN. The key to the clustering algorithm is to adjust the two parameters of the Eps radius and the minimum number of MinPts samples. Following experimental tests, our Epsilon is set to 0.4 and MinPts to 8, optimizing the clustering effect. The results of the ship trajectory clustering are displayed in Figure 6 , including the number of the four clusters and the approximate direction of the trajectory. Although there are some anomalous trajectories in the clustering results, we can see significant differences between each cluster.

For neural networks, the value range of the input features is too large to cause the gradient descent algorithm to converge slowly or difficult to converge, so we normalize the input features, scale the data to the range of [0,1], and limit the value range to a certain range, to eliminate the adverse effects caused by the data, accelerate the model convergence speed, and enhance the model’s generalizability and stability. The normalization formula is shown in Eq(16):

where x is the original data, x _min and x _max are the minimum and maximum values of the data in the sample, respectively, and x_norm is the normalized data.

At present, the mainstream methods in the field of ship trajectory prediction include recurrent neural networks, deep learning, LSTM, GRU, etc. We compare these methods, as shown in Table 2 . We can see that MNN is a new method of predicting trajectory, which has greater efficiency and data processing capacity than the traditional neural network method. Comparison of mainstream methods for ship trajectory prediction.

Parameter selection is an important factor influencing prediction outcomes. A number of neurons and a greater number of layers can extract more information about the vessel’s track characteristics, but excessive complexity can also lead to overflow. The experiments use the RMSProp optimizer, a method to optimize adaptive learning rate and stochastic gradient descent. It can efficiently address oscillation and instability issues in traditional gradient descent algorithms. After experimental verification and parameter comparison, the M-STCP architecture comprises the following elements: input layer, matrix multiplication layer, non-linear transformation layer, loop layer and output layer. The input layer is used to input ship trajectory data into the neural network; The matrix multiplication layer converts the input trajectory data into matrix form for input to the next layer; The nonlinear change layer applies nonlinear functions to each element of the matrix to enhance the expressiveness of the model; The recurrent layer will apply the recurrent neural network to process the matrix to better capture the time series information in the trajectory data. The output layer produces the last line of the matrix subsequent to the prediction. Xavier initialization and ReLU activation are used to initialize the parameters. The loss function selects the average square error to reduce the error between the expected outcome and the actual value. In the case that the default learning rate is 0.001, in order to find the optimal batch size and the number of neurons, we conduct experimental comparisons and determine that the batch size is 32 and the number of neurons is 128. Supplementary parameter selections for the remaining two methods are presented in Table 3 . The details of experimental hardware configuration used in this paper are shown in Table 4 .

For additional evidence that the suggested model is workable and successful, we use 70% of the experimental dataset as the training set and 30% as the test set. The results of the M-STCP method and Kalman filter processing were compared and analyzed with the prediction results of the LSTM model and the GRU (Han et al., 2021) model, and the experimental results were verified by the test set. In the experiment, we selected two sets of data sets for comparison to compare the prediction effects of the three methods on simple and complex ship trajectories. The first set of datasets contains relatively complex, curved ship trajectory data; The second set of data includes simpler ship trajectory data. In Figure 7 , the experimental outcomes are displayed. Figure 7 ’s a and c show that at the beginning of the ship trajectory, only the prediction results of the M-STCP method processed by KF are more accurate, the M-STCP method has the same movement trend as the ship trajectory, but due to the influence of noise, there is a deviation in latitude and longitude, and the performance of the LSTM model is the worst. Although the prediction results of the three models are roughly the same as those of the real ship trajectory, M-STCP is better than LSTM and GRU in the trajectory prediction performance of the ship with large changes in the course of the ship. In Figure 7 , it is visible from b that for simple trajectories, although the prediction results of LSTM and GRU have the same motion trend as the real trajectory, the position information has a large deviation, while the M-STCP method and the real trajectory not only have the same motion trend but also similar position information. This is because although LSTM and GRU have the strong fitting ability, they are also prone to overfitting. The matrix neural network adopts the matrix decomposition method to reduce the number of parameters, effectively reduces the risk of overfitting, and retains the spatiotemporal correlation in the trajectory data when predicting the trajectory of the ship, which also makes its prediction effect better than that of the traditional neural network.

To provide a more precise demonstration of the deviation between predicted and actual results, we have chosen to examine 100 sets of ship trajectories through three different methods, ultimately presenting their longitude and latitude error scatter plots in Figure 8 . The M-STCP method’s error in latitude and longitude can be found to be lower than those of the other two approaches, and the error of the M-STCP method after the Kalman filter treatment is further reduced, and the GRU model has the lowest prediction accuracy. This also proves the good performance of M-STCP in ship trajectory prediction.

For the prediction results of the experiment, We employ evaluation metrics including root mean square error RMSE, mean absolute error MAE, mean absolute percentage error MAPE. All four evaluation methods are that smaller values indicate more accurate predictions from the model. where y_i represents the true trajectory value and y ^ i represents the predicted value of the trajectory. As stated in Table 5 , the specific values are tabulated, and the histogram is shown in Figure 9 . Taking MAPE and MSE as indicators to evaluate the prediction results, it can be seen that the M-STCP value is the smallest and the prediction effect is the best, followed by LSTM and GRU have the worst performance. Using MAE and RMSE as indicators to evaluate the prediction results, M-STCP has the best prediction effect, GRU and LSTM values were similar and higher than those of M-STCP method. In summary, the value of the M-STCP method is smaller than that of the other two models under the three evaluation criteria, which indicates that M-STCP has a high precision than the LSTM and GRU models.