## 3 Methodology
### Model Implementation
We deployed this experimental program in Keras, a neural network application programming interface of TensorFlow, to build a completed time series processing model. As in other studies [46; 47], we used the Adam optimizer and the loss function Mean Squared Error (MSE) in the Joint-IF model. Figures 4 and 5 display the convergence of Joint-IF for different numbers of training samples. From Figures 4 and 5, we can clearly see that the loss value of Joint-IF decreases sharply as the number of iterations and Batch_size increase, and Joint-IF eventually stabilizes. Thus, as shown in Table 2, we set the number of iterations and epochs to 75 and 512, respectively. In addition, as shown in Table 3, we used special settings for the structure and parameters of LSTM. Of these, to prevent the Joint-IF from over-fitting, we set the Dropout layer between the two layers. Figure 4: Convergence of the Joint-IF model using MSE with 75 iterations. (**a**) Batch_size = 64; (**b**) Batch_size = 128. Figure 5: Convergence of the Joint-IF model using MSE with 75 iterations. (**a**) Batch_size = 256; (**b**) Batch_size = 512. ### Experimental Results and Analysis

#### 4.5.1 Overall Analysis

To validate the performance of Joint-IF in predicting missing transfer passenger flow, we tested the forecasting results of Joint-IF for transfer passenger flow with the combinations of RM and NM and missing ratios of 10%, 30%, 50% and 70%, respectively, and then made comparisons with those baseline models. Figures 6 and 7 show four evaluation metrics of MAE, MAPE, RMSE and R\({}^{2}\) of Joint-IF under various missing combinations for the 1, 6, 12 and 18 h forecasting tasks of transfer passenger flow at different metro stations. Comparing the prediction results of Joint-IF for transfer flow under different missing modes in Figures 6 and 7, we can obtain several interesting findings:

* Under the RM and NM scenarios, the prediction errors were similar for transfer passenger flow with the same missing rates;
* The prediction errors gradually increased with the increase of missing rates, indicating the corresponding decrease of the forecasting performance. In the cases of the missing rates of 10% and 70%, it is obvious that the evaluation indexes (including MAE, MAPE and RMSE) were larger and the R\({}^{2}\) was smaller when the missing rate was 70%;
* The differences in the forecasting results were smaller when the missing rates were small, as illustrated in Figure 6. At the missing ratios of 10% and 30%, the differences among the assessment metrics were not significant and the predicted values were relatively close. Overall, our Joint-IF model consistently achieved favorable performance (except for the first step), although its errors increased with time interval and such tendency is accelerated. We can also observe that Joint-IF has a low forecasting error when dealing with transfer passenger flow in different missing cases, and all of the metrics show good stability and robustness. #### 4.5.2 Performance Comparison with Baseline Models

To further verify the superiority of the Joint-IF in predicting transfer passenger flow, we compared it against those of advanced baseline methods (i.e., MLR, GPR, GWR, RF and LSTM) for the 1 h forecasting task without loss. In Table 4, we present the performance metrics of these baselines and the Joint-IF model. It can be seen that Joint-IF can accurately estimate the transfer passenger flow with lower error and exhibit better prediction performance than those multivariate baseline models with the complete transfer passenger flow. In particular, the MAE, MAPE and RMSE of the Joint-IF model were 1.38-76.19%, 21.51-69.50% and 0.86-78.81% lower than baselines and its R\({}^{2}\) was 2.36-65.04% higher than baselines; the GWR had the most similarity and Joint-IF the least difference. Comparing the forecasting results of each statistical method, it can be found that the evaluation metrics of MAE, MAPE and RMSE of GWR were significantly smaller than MLR and GPR, while R\({}^{2}\) was larger.