## 2 Methods
### Sequence-to-Sequence Learning for Temporal Dynamics Modeling
### Comparison with Alternative Models

To demonstrate the advantages of the proposed DCRNN model, we will compare it with the other five classic models. The comparative models include two classic RNN \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Baseline Parameter** & **Hyper Parameter** & **MAPE** & **MAE** & **RMSE** & **Number of Parameters** \\ \hline \multirow{4}{*}{1} & RNN Layers = 2, & RNN Units = 32 & 34.80\% & 6.47 & 9.56 & 94,017 \\  & Diffusion steps = 2, & RNN Units = 64 & 34.47\% & 6.38 & 9.43 & 372,353 \\  & Dropout = 0 & RNN Units = 128 & 33.32\% & 6.36 & 9.50 & 1,481,985 \\  & RNN Units = 256 & 39.81\% & 7.10 & 10.56 & 5,913,089 \\ \hline \multirow{4}{*}{2} & RNN Units = 128, & RNN Layers = 1 & 33.36\% & 6.36 & 9.38 & 498,177 \\  & Diffusion steps = 2, & RNN Layers = 2 & 33.32\% & 6.36 & 9.50 & 1,481,985 \\  & Dropout = 0 & RNN Layers = 3 & 35.46\% & 6.74 & 10.01 & 2,465,793 \\ \hline \multirow{4}{*}{3} & RNN Units = 128, & Diffusion Steps = 1 & 34.24\% & 6.49 & 9.74 & 889,857 \\  & RNN Layers = 2, & Diffusion Steps = 2 & 33.32\% & 6.36 & 9.50 & 1,481,985 \\  & Dropout = 0 & Diffusion Steps = 3 & 46.58\% & 7.27 & 10.79 & 2,074,113 \\ \hline \multirow{4}{*}{4} & RNN Units = 128, & Dropout = 0 & 33.32\% & 6.36 & 9.50 & 1,481,985 \\  & RNN Layers = 2, & Dropout = 0.1 & 34.67\% & 6.66 & 10.02 & 1,481,985 \\ \cline{1-1}  & Diffusion steps = 2 & Dropout = 0.2 & 39.08\% & 6.77 & 9.88 & 1,481,985 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Four sets of experiments on hyperparameters. models, LSTM and GRU [40], and three statistical models--the static model, the historical average (HA) model, and the vector auto-regression (VAR) model. Specifically, the static model simply assumes that \(X^{t+1}=X^{t}\), meaning the data for the next time period is the same as the previous cycle. The HA captures the evolution of bus travel demand as a seasonal process. It predicts future values by using a weighted average from the past seasons. In this work, we use a period of one week and aggregate the data from previous weeks for predictions. For instance, to predict the bus passenger flow on this Friday, we take the average bus passenger flow from past Fridays. As the HA method does not rely on short-term data, its performance remains stable in the presence of slight disturbances and noise. The VAR models the relationship between multiple time series variables. It has been widely used in transportation research to predict bus passenger flows by considering the interdependencies among multiple transit-related variables. The lag value is set to three in this paper. Due to the fact that LSTM and GRU are both deep neural network models, we also performed hyperparameter selection for fine-tuning these two models. As shown in Table 2, the hyperparameters were chosen based on the same criteria as for DCRNN. We select the set of hyperparameters with the smallest MAPE. For both models, the set of hyperparameters is [the number of units of each RNN layer = 256, the number of network layers = 2, dropout = 0.2]. The datasets for training and prediction data are kept consistent for all models. The model comparison results are given in Table 3 and visualized in Figure 7. It is evident that all three neural network models outperform the other three traditional models. Compared with the LSTM and GRU, the DCRNN presents a significant advantage of about 4-5% improvement in terms of MAPE with a comparable performance of MAE and RMSE.