  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.
