# An Attention-Based Deep Convolution Network for Mining Airport Delay Propagation Causality
## 3 Methodology
### Causality Verification
#### 3.3.2 True Causality Verification Based on \(t\)-Test  
All the candidate causal relationships screened according to the attention score have satisfied the time sequence of the causal relationship, and the attention score can reflect the degree of influence of the \"cause\" airport on the \"effect\" airport. In order to verify that each candidate causal airport pair is a true direct causal relationship, we control the candidate \"cause\" airports in the candidate causal relationship pair and observe the changes of the candidate \"effect\" airports. If a change in the candidate \"cause\" airport affects the candidate \"effect\" airport, then there is a true causal relationship between the pair of airports; otherwise, there is no causal relationship.  
Since the airport network is a dynamic and highly complex network, it is impossible to judge the impact on another airport by actually controlling the delays at an airport. Therefore, in a random way, the delay time series of the \"cause\" airport is rearranged, so that the new sequence is not time-sequential, and the rearranged sequence is further input into the trained model to re-predict the \"effect\" airport delay. A potential causal relationship pair is considered a true causal relationship if the prediction error varies widely. We employ permutation importance and \(t\)-tests to determine whether candidate causality is true causality.  
For any candidate causality \(X_{i}\to X_{j}\), we assume that the original delay time series dataset of the \"cause\" airport \(i\) is rearranged as \(D_{i}^{I}\). The datasets \(D^{O}=\{D_{1},\cdots,D_{i-1},D_{i},D_{i+1},\cdots D_{N}\}\) and \(D^{I}=\{D_{1},\cdots,D_{i-1},D_{i}^{I},D_{i+1},\cdots D_{N}\}\) are respectively input into the prediction model, resulting in two delay time prediction error sets \(E_{O}\) and \(E_{I}\) about the \"effect\" airport \(j\). If it is a true causal relationship of \(X_{i}\to X_{j}\), then the value of \(E_{I}\) is significantly greater than that of \(E_{O}\). Conversely, if the value of \(E_{I}\) is not significantly greater than \(E_{O}\), then there is no true causality because the timing of the airport \(i\) is not exploited when predicting delays at the airport \(j\). Assuming that \(K\) days of data are utilized, the dataset contains \(K\) delay time series. For a model trained with this data, computing the forecast error for each time series can be viewed as a different sample from the same distribution. Through the statistics of delay prediction error, it is found that it obeys the \(t\) distribution with \(K\)-\(I\) degrees of freedom. Therefore, we judged the significance of the increase in delay prediction error by a \(t\)-test.  
For each delay time series, calculate the forecast errors \(E_{O}\) and \(E_{I}\). Let \(error(E_{O  
In order to judge the significance level of \(E_{O}\) and \(E_{I}\), the significance level \(\sigma\) is selected. Since each value in \(E_{I}\) is greater than or equal to the value in \(E_{O}\), it belongs to a class of one-tailed \(t\)-distributions (see Figure 8). From the \(t\)-distribution table, the critical value \(z\) corresponding to the degree of freedom \(K\)-\(I\) and the significance level \(O\) are found. If \(t>z\), the value of t falls in the rejection domain, meaning that the hypothesis that the mean values of \(E_{O}\) and \(E_{I}\) are the same can be rejected. It can thus be asserted that the prediction error \(E_{O}\) is significantly larger than the prediction error \(E_{I}\).