Predicting the probability distribution of bus travel time to measure the reliability of public transport services

An important aspect of the quality of a public transport service is its reliability, which is defined as the invariability of the service attributes. In order to measure reliability during the service planning phase, a key piece of information is the long-term prediction of the density of the travel time, which conveys the uncertainty of travel times. This work empirically compares probabilistic models for the prediction of the conditional probability density function (PDF) of the travel time and proposes a simulation framework taking as input the latter distributions to approximate the expected secondary delays, a measure of the reliability of public transport services. Two types of probabilistic models, namely similarity-based density estimation models and a smoothed logistic regression for probabilistic classification model, are compared on a dataset of more than 41,000 trips and 50 bus routes of the city of Montréal. A similarity-based density estimation model using a k-nearest neighbors method and a log-logistic distribution predicted the best estimate of the true conditional PDF of the travel time and generated the most accurate approximations of the expected secondary delays on this dataset. This model reduced the mean squared error of the expected secondary delay by approximately 9% compared to the benchmark model, namely a random forest. This result highlights the added value of modeling the conditional PDF of the travel time with probabilistic models.