Bayesian analysis of network flow problems

I 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 class of problems has been of interest in both communication and transportation network studies. The thesis develops the theoretical framework of variants of the origin-destination flow problem, and introduced Bayesian approaches to analysis and inference. As the first and fundamental stage, the so-called fixed routing problems is addressed. 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, I 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. I discuss issues of convergence and related practical matters and illustrate the approach in a network previously studied in Vardi's 1996 article (37). I 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 identification problems. A truly Bayesian approach, imposing partial stochastic constraints through informed prior distributions, offers a way of resolving these problems, and is also perfectly consistent with prevailing trends in updating traffic flow intensities in this field. The second type of problem explored 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, I introduce the concept of a super-network, namely a fixed routing problem in which the stochastic problem may be embedded. This neatly 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. The final part of the thesis is devoted to the analysis of real traffic flows along a highway, adopting a different perspective, since the goal gets shifted to the estimation and prediction of the intensity of traffic along a linear path without distinguishing different origin-destination labels. An hierarchical model is built to estimate the fundamental parameters that rule the evolution of the flow through space and time, and a dynamic linear model is used to update posteriors for out-of-sample data as we move through time. Other possible, future directions of investigation are indicated in both the area touched in this work.