# An Attention-Based Deep Convolution Network for Mining Airport Delay Propagation Causality
## 3 Methodology
### Delay Prediction Model Based on Deep Convolutional Network
#### 3.2.3 Depthwise Separable Convolution  
The next question is how to implement the convolution of time series. In this paper, with the help of the depthwise separable convolution idea proposed in reference [47], the convolution process is divided into channelwise depthwise convolution and pointwise convolution.  
Channel-by-channel depthwise convolution is a convolution kernel responsible for a delay time series, that is, a delay time series is convolved by a convolution kernel. After channel-by-channel convolution processing, the number of output feature sequences is the same as the number of input time series. As shown in Figure 5, \(N\) delay time series are convolved with different convolution kernels, respectively. The depth convolution is to use different convolution kernels to convolve each input channel, so as to separate the input \(N\) airport delay time series and study the impact of each airport delay on the target airport delay time series. As shown in Figure 5, for each input channel, the contribution parameter is multiplied by the input sequence to obtain a new input sequence, and the new input sequence is subjected to hole convolution with different convolution kernels to obtain the time series. Residual connections obtain the output sequence. Compared with ordinary convolution kernels, atrous convolution greatly increases the receptive field by reducing the number of layers and reducing computational complexity. Residual connections make it easier to optimize networks with multiple convolutional layers during backpropagation, reducing network errors and improving model performance. Figure 6 shows a schematic diagram of a three-layer atrous convolution module for the channel \(j\) (airport \(j\)). The bottom layer is the input layer, the middle contains two hidden convolution layers, and the top layer is the output layer. The size of the convolution kernel is \(k=3\), the expansion coefficient of the convolution kernel is \(2\), and the stride size of the convolution kernel is \(d\), from top to bottom, from the output layer to the input layer. To predict the output layer, the atrous convolution obtains the first \(15\) receptive fields of the input layer by inserting \"0\" into the convolution kernel. Compared with the traditional convolution, which can only obtain the first seven receptive fields of the input layer, the atrous convolution expands the receptive field without increasing the number of layers, and obtains more information from the input layer, thereby improving the prediction accuracy.  
The convolution budget of pointwise convolution is similar to that of regular convolution. Layer-by-layer convolution is mainly to linearly combine the output information of channelwise convolution. The information from different channels is fused together by merging the outputs of the channelwise convolution layers. As shown in Figure 7, by performing ordinary \(1\times 1\) convolution on the output of the time series convolution, the \(N\) outputs are combined into one output, and the influence of the delay characteristics of the \(N\) airports on the target airport is also considered. Figure 7 shows that the time series \(\mathbf{X}_{i}^{\prime}\) undergoes a depthwise convolution and then carries out point-to-point convolution and outputs \(\overline{\mathbf{X}}_{j}\), that is, the predicted delay time series of the target airport.  
Figure 6: Single-channel depthwise convolution.  
Figure 7: Depthwise separable convolution.