## 4 Experiment and Results
### Performance Metrics  
The Root Mean Squared Error (RMSE) is a widely used metric in bus arrival time forecasting that provides the forecast error in actual time units (seconds or minutes), allowing for an intuitive interpretation of the forecast performance. We chose RMSE as the main performance-evaluation metric because it is sensitive to large errors between predicted and actual values, making it suitable for conservatively evaluating the prediction accuracy of a model, and in time series data such as bus arrival time prediction, RMSE is very useful for comparing the overall prediction performance of a model. In this Formula (1), \(T_{i}\) represents the actual bus travel time, \(P_{i}\) represents the predicted bus travel time, and \(n\) represents the sample size. \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(T_{i}-P_{i})^{2}} \tag{1}\]  
The mean absolute percentage error (MAPE) expresses the accuracy of a prediction model as a percentage, which is useful for comparing the performance of models across different datasets and scales, especially when the values in the data span a wide range, as MAPE shows the relative prediction error, making it intuitively easier to understand the accuracy of the model. In this study, MAPE was used to evaluate the predictive accuracy of the model and to see how consistently the model performed overall. In this formula, \(T_{i}\) represents the true value and \(P_{i}\) represents the predicted value. \[\text{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{T_{i}-P_{i}}{T_{i}}\right| \times 100 \tag{2}\]  
The coefficient of determination (R\({}^{2}\)) is a metric that evaluates how well a model explains the variability in the data, indicating how much of the variance in the true value can be explained by the model, and is a widely used metric in regression analysis, quantifying the proportion of the variance in the dependent variable that can be predicted from the independent variable. A higher R\({}^{2}\) value indicates that the model is explaining the data well, so in this study, we used the R\({}^{2}\) metric to see how well the model explains the variability in bus arrival times. A value closer to 1 indicates that the model explains a higher percentage of the variance, indicating a better fit. As expressed in Formula (3), \(T_{i}\) represents the actual bus travel time, Pi represents the predicted bus travel time, \(\bar{T}\) represents the average of the actual values, and \(n\) represents the sample size. \[R^{2}=1-\frac{\sum_{i=1}^{n}(T_{i}-P_{i})^{2}}{\sum_{i=1}^{n}(T_{i}-\bar{T})^{2}} \tag{3}\]  
The Time-weighted error (TWE) is a performance metric that incorporates the importance of forecast error at different points in time by applying weights. In bus arrival prediction, forecast accuracy may be more important at certain times of the day, so TWE is used to reflect the importance of these times. TWE can be weighted to evaluate forecast errors at different times of the day, providing a more precise measure of the accuracy of the forecast over time. Since forecast accuracy at specific times of day has a direct impact on congestion management and transit user satisfaction, it was important to use TWE to highlight forecast performance at these critical points in time. As expressed in Formula (4), where \(T_{i}\) represents the actual bus travel time, \(P_{i}\) represents the predicted bus travel time, \(w_{i}\) represents the weight associated with the i-th data point, and \(n\) represents the total number of data points. \[\text{TWE}=\frac{\sum_{i=1}^{n}w_{i}\cdot|T_{i}-P_{i}|}{\sum_{i=1}^{n}w_{i}} \tag{4}\]  
In this study, we used a combination of these performance metrics to evaluate the model's performance from multiple angles. RMSE intuitively shows the overall error size of the model, while MAPE helps to understand the relative error. R\({}^{2}\) indicates the explanatory power of the model, and TWE reflects the importance of prediction error over time, allowing us to evaluate the performance of the model from different perspectives.