Bayesian inference of time-varying origin–destination matrices from boarding and alighting counts for transit services

Origin–destination (OD) demand matrices are crucial for transit agencies to design and operate transit systems. This paper presents a novel temporal Bayesian model designed to estimate transit OD matrices at the individual bus-journey level from boarding and alighting counts at bus stops. Our approach begins by modeling the number of alighting passengers at subsequent bus stops, given a boarding stop, through a multinomial distribution parameterized by alighting probabilities. Given the large scale of the problem, we generate alighting probabilities with a latent variable matrix and factorize it into a mapping matrix and a temporal matrix, thereby substantially reducing the number of parameters. To further encode a temporally-smooth structure in the parameters, we impose a Gaussian process prior on the columns of the temporal factor matrix. For model inference, we develop a two-stage algorithm with the Markov chain Monte Carlo (MCMC) method. In the first stage, latent OD matrices are sampled conditional on model parameters using a Metropolis–Hastings sampling algorithm with a Markov model-based proposal distribution. In the second stage, we sample model parameters conditional on latent OD matrices using slice and elliptical slice sampling algorithms. We assess the proposed model using real-world data collected from three bus routes with varying numbers of stops, and the results demonstrate that our model achieves accurate posterior mean estimation and outperforms the widely used iterative proportional fitting (IPF) method. Additionally, our model can provide uncertainty quantification for the OD demand matrices, thus benefiting many downstream planning/operational tasks that require robust decisions.