# An Urban Traffic Flow Fusion Network Based on a Causal Spatiotemporal Graph Convolution Network
## 3 Model Framework
### LPF Convolution Module  
Usually, a traffic network will produce an irregular graph; even if junctions are deemed as the vertex, roads as sides, and the road network as a graph of points and sides, it cannot be deemed as a grid. This is because the distances between junctions are different and the spatial relationship between the junctions would be ignored if the road network were deemed as a grid. The spatial domain information captured by such convolution is not reasonable. Therefore, we introduced graph convolution and defined convolution directly on the graph data to capture the features that impacted different weights on junctions. The graph convolution operations in Formula (1) require the decomposition of Laplacian matrix features. The calculation complexity is \(O\big{(}n^{2}\big{)}\), and great challenges can be encountered in cases of large-scale graph data caused by large numbers of nodes. Therefore, we used Chebyshev polynomials as the convolutional kernel and effectively solved this problem. Using Chebyshev polynomial approximation, the kernel \(g_{\theta}\) can be limited to the polynomial of \(\Lambda\) in order to localize the filter and reduce the number of parameters. For example \(g_{\theta}(\Lambda)=\sum_{k=0}^{k=k-1}\theta_{k}\Lambda^{k}\), where the kernel is replaced with Chebyshev polynomial approximation, can be approximately considered as \(g_{\theta}(\Lambda)=\sum_{k=0}^{k=k-1}\theta_{k}T_{k}(\hat{\Lambda})\); after Chebyshev polynomials are introduced, the graph convolution is changed into:  
\[x*_{G}g_{\theta}=Ug_{\theta}(\Lambda)U^{T}x=U\sum_{k=0}^{k=k-1}\theta_{k}T_{k }(\hat{\Lambda})U^{T}x=\sum_{k=0}^{k=k-1}\theta_{k}T_{k}(\hat{L})x \tag{2}\]  
where \(\theta\in R^{k}\) is the polynomial coefficient, k is the size of the graph convolution kernel which determines the number of node orders covered by the target node convolution \(\hat{L}=\frac{2}{\lambda_{max}-2}L-l_{n}\), and \(\lambda_{max}\) is the maximum feature value of the Laplacian matrix. Chebyshev graph convolution is a filter in which the weight of high-frequency signal increases with the rising \(k\) value. \[g\theta(\Lambda)=\sum_{k=0}^{K-1}\theta_{k}\Lambda^{k} \tag{3}\]  
As the weight of high-frequency features increases, it interferes with researchers' ability to capture low-frequency features, which in turn affects the prediction accuracy. To solve this problem, we propose a LPF to replace it, which can increase the weight of low-frequency signals and suppress high-frequency signals. The data processed by LPF convolution are smoother, and the interference of high-frequency noise can be avoided. From the signal \(x\) on the graph, we can obtain:  
\[x=\alpha_{1}\overline{u_{1}}+,\ldots,+\alpha_{n}\overline{u_{n}} \tag{4}\]  
Because the Laplacian matrix is a symmetric positive semi-definite matrix, we can obtain:  
\[\overline{u_{i}u_{i}}^{T}=1\,\ \overline{u_{i}u_{j}}^{T}=0,\ \ i\
eq j \tag{5}\]  
In formula (1) \(Ug_{\theta}(\Lambda)U^{T}x=\big{(}\theta_{1}\overline{u_{1}u_{1}}^{T}+,\ldots,+\theta_{n}\overline{u_{n}u_{n}}^{T}\big{)}x\), and we can obtain:  
\[\overline{u_{n}u_{n}}^{T}x=\alpha_{1}\overline{u_{1}}\overline{u_{1}}^{T} \overline{u_{1}}+,\ldots,+\alpha_{n}\overline{u_{n}u_{n}}^{T}\overline{u_{n}}= \alpha_{n}\overline{u_{n}} \tag{6}\]  
Therefore, \(\left\{\overline{u_{n}u_{n}}^{T}\right\}_{n=1}^{n}\) can be considered a set of basic filters in graph convolution, and allowing only the passage of feature values associated with \(\overline{u_{n}}\). \(\alpha_{n}\) is equivalent to the weight assigned to the filters. Polynomial approximation graph convolution includes a set of combined filters \(\left\{\hat{L}^{k}\right\}_{k=0}^{k-1}\). So, for the basic filters of polynomial approximation graph convolution, the basic filter of high-frequency signals has a greater weight, \(\alpha_{n}=\lambda_{n}^{k}\). We therefore considered that an LPF could be designed to replace the previous filters.