## 3 The VTP-GCN Model
### Graph Representation Based on Raw Vehicle Motion Data  
The complex spatial and temporal dependencies among vehicles in driving scenes are crucial to vehicle trajectory prediction. In order to effectively extract the vehicle spatial  
In previous research on vehicle trajectory prediction based on graph neural networks, in order to simplify calculations, researchers have assumed that each edge has the same weight when constructing the spatial graph, that is, the intensity of spatial interactions between the vehicle and its neighboring vehicles are the same. However, this assumption completely violates the natural driving situation, and the different motion states of different neighboring vehicles have completely different effects on the vehicle. For example, when a vehicle is driving on a lane at a certain speed, a neighboring vehicle located 10 m ahead will interact more strongly with said vehicle than a neighboring vehicle 60 m ahead. At the same time, neighboring vehicles that are separated from the vehicle in question by the same distance but travel at different velocities will also have different effects on the vehicle. Therefore, to mark the intensity of spatial interactions among vehicles, we propose the concept of spatial interaction coefficient (SIC). Its numerical value is inversely proportional to the Euclidean distance between two vehicles and directly proportional to the velocity difference, something which is in line with people's conventional cognition. Subsequent ablation experiments have proven that constructing the spatial interaction graph based on the SIC can help improve the accuracy of vehicle trajectory prediction. We will elaborate on the specific content in Section 4.4. SIC is calculated with the following formula:  
\[SIC_{ij}=\left\{ \begin{array}{ll}\frac{|v_{i}-v_{j}|}{D_{ij}},\ i\
eq j\text { and }d_{ij}\leq L\\ 0\qquad,\text{ Otherwise}\end{array} \right. \tag{5}\]  
where \(v_{i}\) is the velocity of vehicle \(i\), \(D_{ij}\) is the Euclidean distance between vehicle \(i\) and vehicle \(j\), and \(d_{ij}\) is the longitudinal distance between the two vehicles. Based on the SIC, we constructed a weighted adjacency matrix \(A_{spa\_t}\in\mathbb{R}^{N\times N}\) to represent the spatial interdependencies of all vehicles in the scene. Thus, \(A_{spa\_t}\) can be obtained by the following equation:  
\[A_{spa\_t}=\left( \begin{array}{cccc}0&SIC_{1,2}&SIC_{1,3}&\cdots&SIC_{1,N}\\ SIC_{2,1}&0&SIC_{2,3}&\cdots&SIC_{2,N}\\ SIC_{3,1}&SIC_{3,2}&0&\cdots&SIC_{3,N}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ SIC_{N,1}&SIC_{N,2}&SIC_{N,3}&\cdots&0\end{array} \right) \tag{6}\]  
We can obtain the adjacency matrix \(A_{spa}\in\mathbb{R}^{N\times N\times l_{h}}\) of the spatial graph \(G_{s}\) by stacking \(A_{spa\_t}\) at each historical time moment, and obtain the vertex feature matrix \(V_{spa}\in\mathbb{R}^{l_{h}\times N\times C_{g}}\) of the spatial graph \(G_{s}\) by stacking the \(V_{spa\_t}\), where \(C_{g}=2\) represents the position coordinates \((x,y)\). (2) Temporal Graph Representation Learning  
Most of the existing studies on GCN-based vehicle trajectory prediction [21; 22] use a single spatial graph to extract spatial and temporal interaction features successively, and this method based on the extracted spatial interaction information may lead to insufficient extraction of temporal interaction features. To overcome this problem, we propose the  
Figure 2: An illustration of the spatial interaction range between two vehicles. construction of the temporal graph based on the raw vehicle data to extract the interaction relations of vehicles at different historical time instants and, successively, its fusion with the spatial interaction feature to obtain the spatial-temporal interaction feature.