BAYESIAN INFERENCE ON NETWORK TRAFFIC USING LINK COUNT DATA
Claudia Tebaldi & Mike West
Original version: June 1996
Final revision: September 1997
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 of interest in both
communication and transportation network studies. The current paper 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 1996 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 non-negligible and may be estimated from additional data.
Claudia Tebaldi, PhD, is postdoctoral fellow in the
Climate Analysis Section/Geophysical Statistics Project, National
Center for Atmospheric Research, Boulder CO 80307. The research reported here was partially supported by NSF grant DMS-9313013
to the National Institute of Statistical Sciences. This paper appeared (with invited discussion) in the
June 1998 issue of the Journal of the American Statistical Association.
The manuscript is available in either
postscript or pdf