• Joint probabilistic forecasting of link travel times for all running buses on a route. • A time-varying multivariate Gaussian mixture distribution captures intra-bus and inter-bus correlations. • Kronecker-structured covariance enables scalable high-dimensional parameter estimation. • Real-time conditional forecasting achieves competitive accuracy on three real-world bus routes. Recent advancements in statistical and machine learning models have substantially enhanced bus travel time forecasting accuracy. However, these studies primarily rely on deterministic models and fail to quantify forecasting uncertainty, which is crucial for travelers and transit operators. Moreover, existing probabilistic methods are either computationally prohibitive or restricted to local dependencies, preventing them from modeling the joint travel time correlation among multiple running buses and links along the route. To address these issues, this paper introduces a probabilistic deep structured Gaussian model that performs joint forecasting of link travel times for all running buses on a route by explicitly capturing both intra-bus and inter-bus correlations. We model the travel time of each bus on every link as a random variable, whose joint correlations are captured using a time-varying multivariate Gaussian mixture distribution. Efficient high-dimensional parameter estimation is achieved through a novel deep neural network that incorporates a Kronecker product-based structure for the covariance matrix, with the multivariate Gaussian mixture likelihood as the loss function. This specialized architecture enables the network to effectively learn dynamic intra-bus and inter-bus correlations by fusing spatiotemporal features encoded from the travel times of the preceding buses. Probabilistic forecasting is then conducted by computing the conditional distribution of downstream bus link travel times based on partially observed upstream travel times, facilitating real-time predictions for all remaining links of all running buses. We evaluate the proposed model with three routes from two bus systems. Compared with other baseline models, results show that our approach achieves an average improvement of 0.44% in MAPE, 1.12 in RMSE, 0.36 in CRPS, 0.20 in 0.5-risk, and 0.24 in 0.9-risk. Furthermore, the model provides interpretable operational insights, capturing time-varying inter-bus correlations with complex short-range and long-range intra-bus dependencies.
Zhao et al. (Wed,) studied this question.