Nonparametric Bayesian Estimation of Freeway Capacity Distribution from Censored Observations

Previous studies have been made of the usefulness and effectiveness of survival analysis in transportation and traffic engineering studies with incomplete data in which the Kaplan–Meier estimate is proposed for determining traffic capacity distribution. However, well-known estimators like Kaplan–Meier and Nelson–Aalen have several disadvantages that make it difficult to obtain the traffic capacity distribution. First, neither estimator is defined for all values of traffic flows possible. That is, the maximum flow followed by a breakdown defines the final point of the estimated distribution curve. Therefore, parametric fitting tools have to be applied to obtain the remaining portion of the curve. Moreover, the discontinuity and nonsmoothness of the Kaplan–Meier and Nelson–Aalen estimates make it difficult to ensure the robustness of the estimation. In this paper the Kaplan–Meier and Nelson–Aalen nonparametric estimators are used to obtain the traffic capacity function of four freeway sections. Then a Bayesian nonparametric estimator, which is shown to be a Bayesian extension of the Kaplan–Meier estimator, is introduced for estimating the capacity distribution. This estimator assumes a Dirichlet process prior for the survival function under the minimization of a squared-error loss function. The results indicate that the curves obtained by using the Bayesian estimation method are smoother than those obtained with the other estimator. This smoothness also ensures the continuity in the vicinity of censored observations. Furthermore, the Bayesian estimates can be obtained for any traffic flow value regardless of the availability of data only for certain ranges of observations (including censored data).