## 3 Methods
### Dual-Channel Deep Counterfactual Prediction Model for OD Matrix Prediction
#### 3.3.3 Loss Function and Prediction Accuracy Evaluation Metrics
In addition, Deep Twin Networks for counterfactuals estimation [11] which incorporates counterfactual structure but uses a different data matching approach, was included for comparison. The results, presented in Table 4, reveal that the baseline models like LSTM+GCN and LSTM+GAT exhibit higher errors across all passenger flow conditions compared to DC-DCF, particularly in high passenger flow scenarios. LSTM-based models (LSTM+GCN, LSTM+GAT) show the poorest performance. This is attributable to their reliance on consistent temporal patterns, which are disrupted during incidents, causing significant prediction bias. Although the Transformer model improves over the baseline models due to its encoding of incident features, it still underperforms compared to DC-DCF. The Deep Twins and DC-DCF models, both of which integrate factual and counterfactual data, perform significantly better. Among these, the DC-DCF model, which leverages KL divergence-based propensity matching, outperforms Deep Twins, demonstrating the effectiveness of the fact-counterfactual data matching mechanism developed in this study. The ablation experiments further support the importance of the components within the DC-DCF framework. The results shown in Table 5 validate the necessity of several key components, such as the fact-counterfactual passenger flow tensor matching, graph-based contrastive learning for station semantics, the GAT layer for spatial knowledge updating, the temporal convolutional layers for time-based learning, the shared layer for collaborative learning of factual and counterfactual information, and the incident type embedding feature. The experiments emphasize that the inclusion of incident type knowledge embedding, fact-counterfactual tensor matching, and the shared layer are the most critical elements for optimal model performance. The removal of fact-counterfactual propensity matching produces the largest performance drop. Without matched factual reference samples, the counterfactual channel lacks a reliable baseline for estimating passenger flow deviations and results in greater prediction errors. To assess the statistical significance of the differences \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Time Scale**} & \multicolumn{3}{c}{**High Passenger Flow Case**} & \multicolumn{3}{c}{**Medium Passenger Flow Case**} & \multicolumn{3}{c}{**Low Passenger Flow Case**} \\ \cline{2-10}  & **MAE** & **MAPE** & **RMSE** & **MAE** & **MAPE** & **RMSE** & **MAE** & **MAPE** & **RMSE** \\ \hline \multirow{2}{*}{10 min} & 0.773 & 6.437 & 4.518 & 0.210 & 2.105 & 1.887 & 0.176 & 1.850 & 1.504 \\  & ** & ** & ** & ** & ** & ** & ** & ** & ** \\ \hline \multirow{2}{*}{20 min} & 1.559 & 8.905 & 8.981 & 0.421 & 3.167 & 3.522 & 0.358 & 2.801 & 2.811 \\  & ** & ** & ** & ** & ** & ** & ** & ** & ** \\ \hline \multirow{2}{*}{30 min} & 2.347 & 10.587 & 13.368 & 0.648 & 4.013 & 5.347 & 0.526 & 3.995 & 3.437 \\  & ** & ** & ** & ** & ** & ** & ** & ** & ** \\ \hline \multirow{2}{*}{60 min} & 4.655 & 13.849 & 25.642 & 1.268 & 5.654 & 9.821 & 1.036 & 4.907 & 7.201 \\  & ** & ** & ** & ** & ** & ** & ** & ** & ** \\ \hline \multirow{4}{*}{60 min} & 4.655 & 13.849 & 25.642 & 1.268 & 5.654 & 9.821 & 1.036 & 4.907 & 7.201 \\  & ** & ** & ** & ** & ** & ** & ** & ** & ** \\ \hline \multirow{4}{*}{To evaluate the concentration and statistical significance of the prediction results, we assess the proportion of predicted samples that fall within three times the standard deviation (\(\pm\)3\(\sigma\)) of the prediction errors for each evaluation metric, namely Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE). Based on the coverage ratio within this interval, significance levels are categorized as follows: a coverage of 99% or higher is marked with ***, 95% to 99% with **, and 90% to 95% with *.** \\ \cline{1-1} \cline{2-10}  & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}