Probabilistic modeling of delays for train journeys with transfers

Reliability plays a key role in the experience of a rail traveler. The reliability of journeys involving transfers is affected by the reliability of the transfers and the consequences of missing a transfer, as well as the possible delay of the train used to reach the destination. In this paper, we propose a flexible method to model the reliability of train journeys with any number of transfers. The method combines a transfer reliability model based on gradient boosting responsible for predicting the reliability of transfers between trains and a delay model based on probabilistic Bayesian regression, which is used to model train arrival delays. The models are trained on delay data from four Swedish train stations and evaluated on delay data from another two stations, in order to evaluate the generalization performance of the models. We show that the probabilistic delay model, which models train delays following a mixture distribution with two lognormal components, allows to much more realistically model the distribution of actual train delays compared to a standard lognormal model. Finally, we show how these models can be used together to sample the arrival delay at the final destination of the entire journey. The results indicate that the method accurately predicts the reliability for nine out of ten tested journeys. The method could be used to improve journey planners by providing reliability information to travelers. Further applications include timetable planning and transport modeling.