Bayesian inference for link travel time correlation of a bus route

Estimation of link travel time correlation of a bus route is essential to many bus operation applications, such as timetable scheduling, travel time forecasting and transit service assessment/improvement. Most previous studies rely on either independent assumptions or simplified local spatial correlation structures. In the real world, however, link travel time on a bus route could exhibit complex correlation structures, such as long-range correlations, negative correlations, and time-varying correlations. Therefore, before introducing strong assumptions, it is essential to empirically quantify and examine the correlation structure of link travel time from real-world bus operation data. To this end, this paper develops a Bayesian Gaussian model to estimate the link travel time correlation matrix of a bus route using smart-card-like data. Our method overcomes the small-sample-size problem in correlation matrix estimation by borrowing/integrating those incomplete observations (i.e., with missing/ragged values and overlapped link segments) from other bus routes. Next, we propose an efficient Gibbs sampling framework to marginalize over the missing and ragged values and obtain the posterior distribution of the correlation matrix. Three numerical experiments are conducted to evaluate model performance. We first conduct a synthetic experiment and our results show that the proposed method produces an accurate estimation for travel time correlations with credible intervals. Next, we perform experiments on a real-world bus route with smart card data; our results show that both local and long-range correlations exist on this bus route. Finally, we demonstrate an application of using the estimated covariance matrix to make probabilistic forecasting of link and trip travel time.