# Predicting Vessel Trajectories Using ASTGCN with StemGNN-Derived Correlation Matrix
## 4 Vessel Position Spatio-Temporal Prediction Model
### Description of the Problem
Compared to traditional CNNs that only work on regular data in Euclidean space, the following graph convolution can better capture the interaction and information exchange between nodes in the corresponding graph structure of non-Euclidean data. This is achieved by using a diagonalizable linear operator in the Fourier domain to replace the traditional convolution. The specific convolution formula is as follows:  
\[g_{\theta}\ *_{G}x=g_{\theta}(L)x=g_{\theta}(U\Lambda U^{T})x=Ug_{\theta}( \Lambda)U^{T}x. \tag{9}\]  
\(*_{G}\) here denotes the graph convolution operation. By using Chebyshev polynomials, it is possible to maintain computational accuracy and significantly improve the efficiency of processing large-scale graph data without performing costly feature decomposition:  
\[g_{\theta}\ *_{G}x=g_{\theta}(L)x=\sum\
olimits_{k=0}^{K-1}\theta_{k}T_{k}( \widetilde{L})x. \tag{10}\]  
To dynamically adjust the correlation between points, each term of the Chebyshev polynomial and the spatial attention matrix \(S^{\prime}\) are performed by the Hamada product, where the parameter \(\theta\in R^{K}\) represents the vector of polynomial coefficients and \(\widetilde{L}=\frac{2}{\lambda_{max}^{2}}L-I_{N}\), \(\lambda_{inax}\) is the largest eigenvalue of the Laplacian matrix. The recursion of the Chebyshev polynomial is defined as \(T_{k}(x)=2 xT_{k-1}(x)-T_{k-2}(x)\), \(T_{0}(x)=1\), \(T_{1}(x)=x\). To dynamically adjust the correlation between points, each term of the Chebyshev polynomial and the spatial attention matrix \(S^{\prime}\) are performed by the Hamada product (denoted by \(\odot\)). The graph convolution is transformed as follows:  
\[g_{\theta}\ *_{G}x=g_{\theta}(L)x=\sum\limits_{k=0}^{K-1}\theta_{k}(T_{k}(L) \odot S^{\prime})x. \tag{11}\]  
In the spatial dimension, the graph convolution operation has captured the neighbor information on each node, and then a standard convolutional layer is superimposed on the temporal dimension for calculation, and the node information is updated by merging the information of adjacent time segments:  
\[X_{h}^{(r)}=ReLU\Big{(}\phi*\big{(}ReLU(g_{\theta}*_{G}\widehat{X}_{h}^{(r-1 )})\big{)}\Big{)}R^{C_{r}\times N\times T_{r}}, \tag{12}\]  
where \(*\) represents the standard convolution operation, \(\phi\) is the temporal dimension convolution kernel parameter, and \(ReLU\) is the activation function. In short, the spatio-temporal convolution module can well capture the dependencies in terms of spatio-temporal characteristics of trajectory data. STblock is the core component of the spatio-temporal convolution module, which is composed of a spatio-temporal attention module, spatio-temporal convolution module, and residual module. When stacked, multiple STblocks can further extract a wider range of dynamic spatio-temporal correlations. Finally, a fully connected layer is introduced to map the spatio-temporal features extracted by the STblock module to the dimensional space of the prediction target in order to realize the effective prediction of trajectory data.