Bayesian inference on network traffic using link count data
We study Bayesian models and methods for analysing network traffic counts in problems of inference about the traffic intensity between directed pairs of origins and destinations in networks. This is a class of problems very recently discussed by Vardi in a 1996 JASA article and is of interest in both communication and transportation network studies. The current article develops the theoretical framework of variants of the origin-destination flow problem and introduces Bayesian approaches to analysis and inference. In the first, the so-called fixed routing problem, traffic or messages pass between nodes in a network, with each message originating at a specific source node, and ultimately moving through the network to a predetermined destination node. All nodes are candidate origin and destination points. The framework assumes no travel time complications, considering only the number of messages passing between pairs of nodes in a specified time interval. The route count, or route flow, problem is to infer the set of actual number of messages passed between each directed origin-destination pair in the time interval, based on the observed counts flowing between all directed pairs of adjacent nodes. Based on some development of the theoretical structure of the problem and assumptions about prior distributional forms, we develop posterior distributions for inference on actual origin-destination counts and associated flow rates. This involves iterative simulation methods, or Markov chain Monte Carlo (MCMC), that combine Metropolis–Hastings steps within an overall Gibbs sampling framework. We discuss issues of convergence and related practical matters, and illustrate the approach in a network previously studied in Vardi's article. We explore both methodological and applied aspects much further in a concrete problem of a road network in North Carolina, studied in transportation flow assessment contexts by civil engineers. This investigation generates critical insight into limitations of statistical analysis, and particularly of non-Bayesian approaches, due to inherent structural features of the problem. A truly Bayesian approach, imposing partial stochastic constraints through informed prior distributions, offers a way of resolving these problems and is consistent with prevailing trends in updating traffic flow intensities in this field. Following this, we explore a second version of the problem that introduces elements of uncertainty about routes taken by individual messages in terms of Markov selection of outgoing links for messages at any given node. For specified route choice probabilities, we introduce the concept of a super-network—namely, a fixed routing problem in which the stochastic problem may be embedded. This leads to solution of the stochastic version of the problem using the methods developed for the original formulation of the fixed routing problem. This is also illustrated. Finally, we discuss various related issues and model extensions, including inference on stochastic route choice selection probabilities, questions of missing data and partially observed link counts, and relationships with current research on road traffic network problems in which travel times within links are nonnegligible and may be estimated from additional data. © 1998 Taylor & Francis Group, LLC.
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