## 3 Construction of PIs Using the Bootstrap Method  
### Mathematical Description of the Problem  
Due to the limitation of the bus departure time interval (10-30 min), only one data record of the travel time can be extracted during each time cycle from departure to arrival. Therefore, the travel time data of buses have typical characteristics of small samples. It is reasonable to use the bootstrap method to predict bus travel time PIs. Referring to reference [16; 40], assuming the existence of a set of data \(D=\{(x_{i},y_{i}),i=1,\ 2\ldots N\}\), there is a non-linear mapping relationship \(y(x_{i})\) between the target value \(y\) and the input variable \(x\). Taking into account the measurement error, the travel time can be expressed as:  
\[y_{i}^{*}=y(x_{i})+e(x_{i}) \tag{1}\]  
where \(y_{i}^{*}\) indicates the measured value, \(y(x_{i})\) represents the true value of the \(i\)-th sampling, and \(e(x_{i})\) represents noise. Assuming that the output of the prediction model is \(\hat{y}_{i}(x_{i})\), the error of the model can be expressed as:  
\[y_{i}^{*}-\hat{y}_{i}=[y(x_{i})-\hat{y}_{i}(x_{i})]+e(x_{i}) \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Paper** & **Data Type** & **Method** & **Spatial** & **Temporal** & **Cluster** & **Target** \\ \hline
[24] & AVL & ANN, LR, HA & 2 routes & 6 months & No & Point Prediction \\
[3] & Surveys data & SVM, ANN, K-NN, LR & 8 routes & 3 days & No & Point Prediction \\
[26] & Unknown & SVM, ANN & 1 route & 1 month & No & Point Prediction \\
[27] & GPS & SVM, ANN, K-NN & 3 routes & 1 month & Yes & Point Prediction \\
[28] & GPS & HA, ANN, KF & 1 route & 6 months & Yes & Point Prediction \\
[6] & GPS & RFNN & 2 routes & 3 days & No & Point Prediction \\
[29] & Unknown & RF, PPR, SVM & 6 routes & 3 months & No & Point Prediction \\
[30] & AVL, APC & KF & 1 route & 5 days & No & Point Prediction \\
[31] & GPS & KF & 1 route & 30 days & No & Point Prediction \\
[38] & APC & ANN + KF & 1 route & 12 months & No & Point Prediction \\
[39] & On-board survey & SVM + KF & 1 route & 1 month & No & Point Prediction \\
[4] & GPS & LSTM & 30 routes & 2 months & Yes & Point Prediction \\
[21] & GPS & CNN & 5 routes & 6 months & Yes & Point Prediction \\
[32] & GPS & RNN & 47 routes & 1 month & No & Point Prediction \\
[33] & GPS & ConvLSTM & 1 route & 5 months & No & Point Prediction \\
[34] & GPS & Geo-convLSTM & 2 routes & 3 months & No & Point Prediction \\
[14] & GPS & Delta & 1 route & 6 months & No & PI Prediction \\
[15] & GPS & Delta, Bayesian & 1 route & 6 months & No & PI Prediction \\
[13] & GPS & Bootstrap + ANN & 1 route & 6 months & No & PI Prediction \\
[8] & GPS & Bootstrap + ANN & 1 route & 1 month & Yes & PI Prediction \\ Ours & GPS & Bootstrap + ANN & 3 routes & 1 month & Yes & PI Prediction \\ \hline \hline \end{tabular} Note: AVL—automatic vehicle location, APC—automatic passenger counter, HA—Historical Average.  
\end{table}
Table 1: Overview of bus travel time point prediction and PI prediction.  
where \(y(x_{i})-\hat{y}_{i}(x_{i})\) represents the error between the predicted value of the model and the real value, \(\epsilon(x_{i})\) represents the noise of the data, the predicted variance can be expressed as:  
\[\sigma_{y}^{2}(x_{i})=\sigma_{y}^{2}(x_{i})+\sigma_{\epsilon}^{2}(x_{i}) \tag{3}\]  
If \(\sigma_{y}^{2}(x_{i})\) and \(\sigma_{\epsilon}^{2}(x_{i})\) are estimated, then \(\sigma_{y}^{2}(x_{i})\) can be calculated, and then the prediction intervals can be estimated based on the variance value. Next, in Section 3.2, we will introduce the process of using the bootstrap method to estimate \(\sigma_{y}^{2}(x_{i})\) and \(\sigma_{\epsilon}^{2}(x_{i})\).