Interpretable machine learning for identifying the determinants of bus delays

Abstract

The study proposes an interpretable machine-learning approach for predicting bus arrival delays based on the Automatic Vehicle Location (AVL) system. The data describe vehicle movements and schedule adherence at successive stops, enabling delay estimation both along the route and across different time periods. Delays are defined as deviations between observed arrival times and scheduled times at individual stops, allowing both spatial and temporal delay patterns to be examined. The approach integrates ensemble machine-learning models with nested, temporally aware cross-validation and hyperparameter optimization. This validation strategy preserves the temporal order of observations, ensuring that model evaluation reflects realistic forecasting conditions and avoids information leakage. In particular, training and test sets are separated along the time axis, preventing future observations from influencing model training. Three ensemble models—Random Forest, XGBoost, and CatBoost—achieved comparable accuracy (RMSE ≈ 5.4 min; MAE ≈ 3.6 min; R² ≈ 0.44). Model performance was evaluated at the level of individual stop arrivals, reflecting short-term delay prediction under operational conditions. Comparable performance across models indicates that the observed patterns are not specific to a single modelling technique. This consistency suggests that the identified delay patterns are robust with respect to model choice rather than driven by algorithm-specific assumptions. The results show that the proposed approach enables the identification of combined temporal and spatial feature interactions associated with bus delay magnitudes within the analysed case study. These interactions mainly involve time-of-day effects and stop sequence, illustrating how delays accumulate as vehicles progress along the route. This pattern reflects the cumulative nature of delays in mixed-traffic operations, where early disturbances propagate toward downstream stops. SHAP values were used to provide a quantitative interpretation of these interactions and to identify the most influential predictors in the analysed models. The analytical structure, designed around weekly data partitions, supports delay monitoring and operational analysis. The results indicate that the proposed approach can support the analysis and interpretation of bus delay patterns within similar operational contexts.