# Integrated Spatio-Temporal Graph Neural Network for Traffic Forecasting
## 3 Methodology
### Proposed Architecture: Integrated Spatio-Temporal Graph Neural Network  
The integrated spatio-temporal graph neural network (ISTGNN) is designed to improve traffic forecasting by simultaneously modeling spatial and temporal dependencies in traffic data. It combines graph convolutional layers to capture spatial relationships among traffic nodes, represented through an adjacency matrix, with a temporal component that processes sequential data across time steps. The model constructs a combined spatio-temporal adjacency matrix to integrate current and historical data, enabling it to capture complex interactions within the traffic network. Node embeddings are updated iteratively through a series of graph convolutions and non-linear transformations, which aggregate information from neighboring nodes and time steps. These embeddings are then used to predict future traffic conditions by minimizing a loss function, such as mean squared error (MSE), which measures the discrepancy between predicted and actual traffic states. This architecture allows ISTGNN to learn intricate patterns in traffic dynamics, making it well suited for applications in intelligent transportation systems and smart city infrastructure, where accurate short-term predictions are crucial for effective traffic management and planning. An integrated spatio-temporal graph convolutional network (STGCN) is used to capture the complex spatial and temporal dependencies inherent in the dataset. The model is designed to handle graph-structured data over time, where nodes represent entities (e.g., sensors, individuals) and edges represent the relationships between them, evolving across multiple time steps. The architecture incorporates both graph convolutional networks (GCNs) for spatial information and temporal convolutional networks (TCNs) to model sequential dependencies over time. Integrated spatio-temporal data are represented as a graph \(G=(V,E)\), where \(V\) is the set of nodes and \(E\) is the set of edges. Each node \(\
u\in\
u\) represents an entity with features at a specific time step. The edges \(e\in E\) define the relationships between nodes and are encoded in an adjacency matrix \(A\). The adjacency matrix \(A_{ij}\) is defined as follows:  
\[A_{ij}=\left\{ \begin{array}{ll}1,&if\ node\ i\ connected\ to\ node\ j\\ 0,&otherwise\end{array} \right.\]  
Self-loops are added to the adjacency matrix \(A_{\text{st}}\), allowing each node to aggregate its own information during the convolution operation. The degree matrix \(F_{\text{st}}\) is then used to normalize the adjacency matrix, ensuring that nodes with different degrees are processed equally. Using the spatio-temporal adjacency matrix to capture both spatial and temporal dependencies, we construct a block-diagonal spatio-temporal adjacency matrix \(A_{\text{ij}}\), where each block corresponds to the adjacency matrix at a particular time step. This allows the model to handle spatial relationships at each time point while also incorporating temporal dependencies by connecting nodes across different time steps. The graph convolutional network (GCN) is used to aggregate spatial information from neighboring nodes. Each node's feature matrix \(x_{s|f}^{l}\) at layer \(l\) is updated by considering the features of its neighbors, weighted by the adjacency matrix. The propagation rule for the spatial convolution at layer \(l\) is given by:  
\[X_{Self}^{l+1}=\sigma\left(F_{st}^{-\frac{1}{2}}A_{st}F_{st}^{-\frac{1}{2}}X_{ self}^{l}W_{spatial}^{l}\right) \tag{3}\]  
where \(A_{st}\) is the adjacency matrix representing the spatial relationships between nodes. \(F_{st}\) is the degree matrix used for normalization. \(X_{self}^{l}\) is the node feature matrix at layer \(l\), \(W_{spatial}^{l}\) is a learnable weight matrix for the spatial transformation, and \(\sigma\) is a non-linear activation function. This equation aggregates information from neighboring nodes in the graph, allowing each node to update its feature representation by incorporating the features of spatially adjacent nodes.