## 4 Our Proposed ADDGCN System
### Down-Sampling Dynamic Graph Convolution Module Based on Feature Augmentation
#### 4.1.2 Down-Sampling Dynamic Graph Convolution Network  
Figure 4 presents the structure of the down-sampling dynamic graph convolution (DS-DGC) module [36], comprising a down-sampling convolution network and a dynamic graph convolution network (DGCN). The down-sampling convolutional network comprises fundamental down-sampling blocks arranged in a binary tree structure. If necessary, more down-sampling blocks can be stacked to increase the prediction accuracy. At the same time, to augment the model's ability to simultaneously capture spatio-temporal correlation, the DGCN module is incorporated into the down-sampling convolution structure to use the captured spatial information to assist in the training process. Let \(X\in R^{\mathbb{C}\times N\times T}\) serve as the down-sampling convolution module's input. The sequence data are decomposed into odd sequence \(X_{odd}\in R^{\mathbb{C}\times N\times T/2}\) and even sequence \(X_{even}\in R^{\mathbb{C}\times N\times T/2}\) through the sequence decomposition operation, which not only reduces the number of data but also largely preserves the original sequence's information. Next, four \(ID\) convolutional networks _Conv1_, _Conv2_, _Conv3_, _Conv4_ and are used to interactively obtain the two subsequences' respective feature information from the input sequence. Following the initial interactive learning, the output is \(X^{\prime}_{even}\in R^{\mathbb{C}\times N\times T/2}\) and \(X^{\prime}_{odd}\in R^{\mathbb{C}\times N\times T/2}\), the \(X^{\prime}_{even}\) and \(X^{\prime}_{odd}\) continue to learn interactively, and the final output is \(X^{\prime}_{even}\in R^{\mathbb{C}\times N\times T/2}\) and \(X^{\prime}_{odd}\in R^{\mathbb{C}\times N\times T/2}\). The operations in the down-sampling convolution module are characterized as:  
\[ \begin{split}& X_{even},X_{odd}=split(X)\\ & X^{\prime}_{odd}=\tanh(DGCN(Conv1(X_{ever})))\odot X_{odd}\\ & X^{\prime}_{even}=\tanh(DGCN(Conv2(X_{odd})))\odot X_{even}\\ & X^{\prime\prime}_{odd}=X^{\prime}_{odd}-\tanh(DGCN(Conv3(X^{ \prime}_{even})))\\ & X^{\prime\prime}_{even}=X^{\prime}_{even}-\tanh(DGCN(Conv4(X_{odd })))\end{split}  \tag{4}\]  
where \(DGCN\) is the dynamic graph convolution, \(\tanh\) is the stimulation function, and \(\odot\) represents the Hadamard product. The dynamic graph convolution module's design is depicted in Figure 4, where the central component is a graph generator and the diffusion graph convolution network. The initial adjacency matrix \(A\in\mathbb{R}^{N\times N}\) and the hidden feature \(H\in\mathbb{R}^{N\times N\times T}\) are input into the graph generator. The subsequent steps include applying diffusion graph convolution [37] and passing the result through a multi-layer perceptron to generate an adjacency matrix \(A^{\prime}\in R^{N\times N}\) that incorporates spatio-temporal characteristics. The overall procedure can be summarized as follows:  
\[A^{\prime}=SoftMcoc(MLP(GCN(H,A))) \tag{5}\]  
Figure 4: Down-sampling convolution structure diagram and dynamic graph convolution structure diagram. Down-sampling dynamic graph convolutional network structure diagram on the left, and dynamic graph convolution structure graph on the right. where \(GCN\) represents a diffusion graph convolution operation, and \(MLP\) represents a multi-layer perceptron. Since the produced matrix \(A^{\prime}\in R^{N\times N}\) is discontinuous, we must utilize the Gumbel function to reparameterize it in order to verify that the sampling procedure can be derived [38]:  
\[A_{L}=GumbelSoft\max\big{(}A^{\prime}\big{)}=SoftMax\big{(}\big{(}\log\big{(}A^ {\prime}\big{)}-\log(-\log(g))\big{)}/\tau\big{)} \tag{6}\]  
where \(\tau\) is the eigenvalue of the softmax function, which is set to 0.5 here, and \(g\sim Gumbel(0,1)\) is a random variable. Dynamic relationships between nodes may be modeled using \(A_{L}\in R^{N\times N}\). Along with the adjacency matrix \(A_{L}\) created by the graph generator, we also build an adaptive adjacency matrix [39]\(A_{opt}\in R^{N\times N}\).