The Annals of Applied Statistics

Network tomography for integer-valued traffic

Martin L. Hazelton

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A classic network tomography problem is estimation of properties of the distribution of route traffic volumes based on counts taken on the network links. We consider inference for a general class of models for integer-valued traffic. Model identifiability is examined. We investigate both maximum likelihood and Bayesian methods of estimation. In practice, these must be implemented using stochastic EM and MCMC approaches. This requires a methodology for sampling latent route flows conditional on the observed link counts. We show that existing algorithms for doing so can fail entirely, because inflexibility in the choice of sampling directions can leave the sampler trapped at a vertex of the convex polytope that describes the feasible set of route flows. We prove that so long as the network’s link-path incidence matrix is totally unimodular, it is always possible to select a coordinate system representation of the polytope for which sampling parallel to the axes is adequate. This motivates a modified sampler in which the representation of the polytope adapts to provide good mixing behavior. This methodology is applied to three road traffic data sets. We conclude with a discussion of the ramifications of the unimodularity requirements for the routing matrix.

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Ann. Appl. Stat., Volume 9, Number 1 (2015), 474-506.

First available in Project Euclid: 28 April 2015

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EM algorithm MCMC origin–destination polytope transport unimodular matrix


Hazelton, Martin L. Network tomography for integer-valued traffic. Ann. Appl. Stat. 9 (2015), no. 1, 474--506. doi:10.1214/15-AOAS805.

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  • Airoldi, E. M. and Blocker, A. W. (2013). Estimating latent processes on a network from indirect measurements. J. Amer. Statist. Assoc. 108 149–164.
  • Airoldi, E. M. and Haas, B. (2011). Polytope samplers for inference in ill-posed inverse problems. In International Conference on Artificial Intelligence and Statistics 15. Ft. Lauderdale, FL.
  • Bell, M. G. H. (1991). The estimation of origin-destination matrices by constrained generalised least squares. Transp. Res. Part B 25 13–22.
  • Ben-Akiva, M. and Lerman, S. R. (1985). Discrete Choice Analysis. MIT Press, Cambridge, MA.
  • Blocker, A. W., Koullick, P. and Airoldi, E. (2012). networkTomography: Tools for network tomography. R package version 0.2.
  • Caffo, B. S., Jank, W. and Jones, G. L. (2005). Ascent-based Monte Carlo expectation-maximization. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 235–251.
  • Cao, J., Davis, D., Vander Wiel, S. and Yu, B. (2000). Time-varying network tomography: Router link data. J. Amer. Statist. Assoc. 95 1063–1075.
  • Cascetta, E. (1984). Estimation of trip matrices from traffic counts and survey data: A generalized least squares estimator. Transportation Research Part B 18 289–299.
  • Cascetta, E. (1989). A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transportation Research Part B 23 1–17.
  • Cascetta, E., Nuzzolo, A., Russo, F. and Vitetta, A. (1996). A modified logit route choice model overcoming path overlapping problems: Specification and some calibration results for interurban networks. In Proceedings of the 13th International Symposium on Transportation and Traffic Theory 697–711. Elsevier Science, Lyon, France.
  • Castro, R., Coates, M., Liang, G., Nowak, R. and Yu, B. (2004). Network tomography: Recent developments. Statist. Sci. 19 499–517.
  • Daganzo, C. F. and Sheffi, Y. (1977). On stochastic models of traffic assignment. Transp. Sci. 11 253–274.
  • Denby, L., Landwehr, J. M., Mallows, C. L., Meloche, J., Tuck, J., Xi, B., Michailidis, G. and Nair, V. N. (2007). Statistical aspects of the analysis of data networks. Technometrics 49 318–334.
  • Hazelton, M. L. (2001). Estimation of origin-destination trip rates in Leicester. J. Roy. Statist. Soc. Ser. C 50 423–433.
  • Hazelton, M. L. (2010). Statistical inference for transit system origin-destination matrices. Technometrics 52 221–230.
  • Hazelton, M. L. (2015). Supplement to “Network tomography for integer-valued traffic.” DOI:10.1214/15-AOAS805SUPP.
  • Heaton, L., Obara, B., Grau, V., Jones, N., Nakagaki, T., Boddy, L. and Fricker, M. D. (2012). Analysis of fungal networks. Fungal Biology Reviews 26 12–29.
  • Hoffman, A. J. and Kruskal, J. B. (1956). Integral boundary points of convex polyhedra. In Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.) 223–246. Princeton Univ. Press, Princeton, NJ.
  • Kolaczyk, E. D. (2009). Statistical Analysis of Network Data: Methods and Models. Springer, New York.
  • Koppelman, F. S. and Wen, C. H. (2000). The paired combinatorial logit model: Properties, estimation and application. Transp. Res., Part B: Methodol. 34 75–89.
  • Lawrence, E., Michailidis, G. and Nair, V. N. (2006). Network delay tomography using flexicast experiments. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 785–813.
  • Li, B. (2005). Bayesian inference for origin-destination matrices of transport networks using the EM algorithm. Technometrics 47 399–408.
  • Liang, G. and Yu, B. (2003). Maximum pseudo-likelihood estimation in network tomography. IEEE Trans. Signal Process. 51 2043–2053.
  • Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226–233.
  • Maher, M. J. (1983). Inferences on trip matrices from observations on link volumes: A Bayesian statistical approach. Transp. Res., Part B: Methodol. 17 435–447.
  • Parry, K. and Hazelton, M. L. (2013). Bayesian inference for day-to-day dynamic traffic models. Transp. Res., Part B: Methodol. 50 104–115.
  • R Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
  • Singhal, H. and Michailidis, G. (2007). Identifiability of flow distributions from link measurements with applications to computer networks. Inverse Problems 23 1821–1849.
  • Tanner, M. A. (1996). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd ed. Springer, New York.
  • Tebaldi, C. and West, M. (1998). Bayesian inference on network traffic using link count data. J. Amer. Statist. Assoc. 93 557–576.
  • Vanderbei, R. J. and Iannone, J. (1994). An EM approach to OD matrix estimation. Technical Report SOR 94-04, Princeton Univ., Princeton, NJ.
  • Vardi, Y. (1996). Network tomography: Estimating source-destination traffic intensities from link data. J. Amer. Statist. Assoc. 91 365–377.
  • Veinott, A. F. Jr. and Dantzig, G. B. (1968). Integral extreme points. SIAM Rev. 10 371–372.
  • Yai, T., Iwakura, S. and Morichi, S. (1997). Multinomial probit with structured covariance for route choice behavior. Transp. Res., Part B: Methodol. 31 195–207.
  • Ziegler, G. M. (1995). Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer, New York.
  • Zuylen, H. J. V. and Willumsen, L. G. (1980). The most likely trip matrix estimated from traffic counts. Transp. Res. Part B 14 281–293.

Supplemental materials

  • Supplement to "Network tomography for integer-valued traffic".: The supplementary materials, stored as a zip archive, include data and additional numerical results for the applications in Section 5. The data comprise link-path incidence matrices, observed traffic counts and prior pseudo counts for Bayesian analyses. The additional results include effective sample sizes for MCMC output, computing times and summaries of the slack for the route flow samplers considered.