# Multivariate Time Series Deep Spatiotemporal Forecasting with Graph Neural Network
## 3 Methods
### Down-Sampling Convolution Module  
The down-sampling convolution module in this paper is a multilayer module. Multiple resolution temporal features are captured by down-sampling convolutions to improve model performance. The down-sampling convolution module consists of basic down-sampling blocks, each D-s Block contains a structure on the right of the figure. The last layer is connected via Concat. FC is a fully connected layer, which maps the output to the specified dimension, as shown on the left of Figure 2. The output of the previous module is used as the input of the binary tree structure, and multiple spatiotemporal short-sequence information is obtained after passing through L layers. The short-sequence information is rearranged into new deep spatiotemporal sequence information. Through residual connection, the original input is added to the newly extracted deep spatiotemporal information sequence to ensure the integrity of the information. A basic down-sampling block of a binary tree structure consists of sequence segmentation, convolution filtering, and information crossing. The output data of the previous module is divided into two subsequences in the down-sampling component as shown in Figure 2 right. The output \(D_{out}\) of the graph convolution module is used as input, and then each layer takes the output of the upper layer as input. \(D_{even}\) and \(D_{odd}\) represent sequences with odd and even subscripts, respectively. Conv1, Conv2, Conv3, and Conv4 are four identical 1D convolutional networks, respectively. The exp function is used to highlight peaks of information. Hadamard product is represented by \(\odot\), which is the multiplication between the elements. Finally, the output of the current layer is obtained by subtracting the two sequences. Different convolutional filters are used to extract new sequence information from each sequence. In order to avoid the loss of information caused by dividing the sequence multiple times, the important information of multiple subsequences is preserved through information crossing. The input sequence is decomposed into odd and even sequences by the sequence decomposition operation in the down-sampling block of Figure 2. The odd sequences take data at positions 1, 3, 5, etc., in the input sequence. The even sequences take data at positions 0, 2, 4, etc., in the input sequence. The original sequence \(D_{out}\) is decomposed into two subsequences \(D_{even}\) and \(D_{odd}\). Odd and even sequences reduce the amount of data while retaining most of the information of the original sequence. After that, different convolutional networks are employed to extract the feature information of the input sequence  
Figure 2: Down-sampling Convolution Module and Down-sampling block (D-s Block). from the two sequences. The feature information obtained by different convolutional networks has stronger representation ability after fusion:  
\[D_{even},D_{odd}=split(D_{out}) \tag{16}\]  
\[D^{{}^{\prime}}_{even}=D_{even}\odot\exp(conv_{1}(D_{odd})),\ \ \ D^{{}^{\prime}}_{odd}=D_{odd}\odot\exp( conv_{2}(D_{even})) \tag{17}\]  
\[D^{{}^{\prime}}_{even}=D^{{}^{\prime}}_{even}-conv_{3}(D^{{}^{\prime}}_{odd}), \ \ D^{{}^{\prime\prime}}_{odd}=D^{{}^{\prime}}_{odd}-conv_{4}(D^{{}^{\prime}}_{ even}) \tag{18}\]  
In Formula (16), \(D_{out}\) is split into \(D_{even}\) and \(D_{odd}\). In Formula (17), \(D_{even}\) and \(D_{odd}\) are mapped to the hidden state through two convolutional networks and converted to \(\exp\) format. In Formula (18), two other convolutional networks map \(D^{{}^{\prime}}_{even}\) and \(D^{{}^{\prime}}_{odd}\) to new hidden states, which are subtracted to obtain the final result. The mapping process of the hidden state can be viewed as a scaling transformation of the two subsequences. The scaling factor is learned during network training. The down-sampling convolution module enlarges the receptive field compared to the dilated convolution used in the WaveNet architecture.