Statistical inference of boarding and alighting counts in transit systems with incomplete data

• A Bayesian framework estimates missing boarding and alighting counts in transit systems. • Boarding demand is modeled with a time-varying Poisson process informed by headways. • A softmax-based low-rank factorization with Gaussian processes models alighting probabilities. • An elliptical slice sampling-based MCMC algorithm ensures efficient inference. Automatic passenger counting (APC) systems have been widely used in public transit systems to collect boarding and alighting counts, which are essential for understanding travel demand, optimizing transit operations, and improving transit service quality. However, missing boarding and alighting counts remain a pervasive problem due to APC deployment, hardware malfunctions, or operational disruptions. The reconstruction of these missing data is particularly challenging because boarding and alighting counts must satisfy real-world constraints, such as balance conditions and onboard passenger limits. To address this issue, we propose a probabilistic framework that integrates passenger travel behavior and operational constraints to estimate missing boarding and alighting counts. The framework builds a time-varying Poisson model to estimate boarding demand and employs a method to infer time-varying alighting probabilities. Further, the alighting counts are derived by assigning estimated boarding counts to downstream stops with time-varying alighting probabilities, ensuring that the reconstructed data meet operational constraints. We validate the proposed framework using real-world transit data. The results demonstrate the method’s accuracy and robustness in estimating missing APC data, while also providing valuable insights into time-varying passenger travel behaviors, including arrival rates and alighting probabilities. This framework offers a practical and interpretable solution for reconstructing incomplete boarding and alighting data, with significant implications for improving transit planning and operational decision-making.