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Estimation of Parameters of Network Equilibrium Models: A Maximum Likelihood Method and Statistical Properties of Network Flow

  • Shoichiro Nakayama
  • Richard D.
Chapter

Abstract

Estimation of the parameters in network equilibrium models, including OD matrix elements, is essential when applying the models to real-world networks. Link flow data are convenient for estimating parameters because it is relatively easy for us to obtain them. In this study, we propose a maximum likelihood method for estimating parameters of network equilibrium models using link flow data, and derive first and second derivatives of the likelihood function under the equilibrium constraint. Using the likelihood function and its derivatives, t-values and other statistical indices are provided to examine the confidence interval of estimated parameters and the model’s goodness-of-fit. Also, we examine which conditions are needed for consistency, asymptotic efficiency, and asymptotic normality for the maximum likelihood estimators with non-I.I.D. link flow data. In order to investigate the validity and applicability, the proposed ML method is applied to a simple network and the road network in Kanazawa City, Japan.

Keywords

Central Limit Theorem Maximum Likelihood Method Route Choice Transportation Research Part Asymptotic Efficiency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgments

I would like to express my gratitude to Prof. Jun-ichi Takayama (Kanazawa University) for providing valuable comments and advices. I am also grateful to Mr. Tomonari Anaguchi (Kanazawa University) for his computational support.

References

  1. Anas, A. and Kim, I. (1990). Network loading versus equilibrium estimation of the stochastic route choice model: maximum likelihood and least squares revisited. Journal of Regional Science, 30, 89-103.CrossRefGoogle Scholar
  2. Bell, M.G.H. (1991). The estimation of origin-destination matrices by constrained generalised least squares. Transportation Research Part B, 25, 13-22.CrossRefGoogle Scholar
  3. 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.CrossRefGoogle Scholar
  4. Chen, M. and Alfa, A. (1991). A network design algorithm using a stochastic incremental traffic assignment approach. Transportation Science, 25, 215-224.CrossRefGoogle Scholar
  5. Clark, S.D and Watling, D.P. (2002). Sensitivity analysis of the probit-based stochastic user equilibrium assignment model. Transportation Research Part B, 36, 617-635.CrossRefGoogle Scholar
  6. Clark, S.D. and Watling, D.P. (2005). Modelling network travel time reliability under stochastic demand. Transportation Research Part B, 39, 119-140.CrossRefGoogle Scholar
  7. Daganzo, C.F. (1977). Some statistical problems in connection with traffic assignment. Transportation Research Part B, 11, 385-389.Google Scholar
  8. Davis, G.A. (1994). Exact local solution of the continuous network design problem via stochastic user equilibrium assignment. Transportation Research Part B, 28, 61-75.CrossRefGoogle Scholar
  9. Durrett, R. (1991). Probability: Theory and Examples. Pacific Grove, CA.: Wadsworth and Brooks/Cole.Google Scholar
  10. Fiacco, A. (1983). Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. New York: Academic Press.Google Scholar
  11. Fisk, C. (1977). Note on the maximum likelihood calibration on dial’s assignment method. Transportation Research Part B, 11, 67-68.Google Scholar
  12. Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, Vol. 1, Distribution Theory. 4th edition, London: Charles Griffin.Google Scholar
  13. Hazelton, M.L. (2000). Estimation of origin-destination matrices from link flows under uncongested networks. Transportation Research Part B, 34, 549-566.CrossRefGoogle Scholar
  14. Lehmann, E.L. and Casella, G. (1998). Theory of Point Estimation. Berlin: Springer.Google Scholar
  15. Liu, S.S. and Fricker, J.D. (1996). Estimation of a trip table and the θ Parameter in a stochastic network. Transportation Research Part A, 30, 287-305.Google Scholar
  16. Lo, H.P. and Chan, C.P. (2003). Simultaneous estimation of an origin-destination matrix and link choice proportions using traffic counts. Transportation Research Part B, 37, 771-788.Google Scholar
  17. Nie, Y., Zhang, H.M. and Recker W.W. (2005). Inferring origin-destination trip matrices with a decoupled GLS path flow estimator. Transportation Research Part B, 39, 497-518.CrossRefGoogle Scholar
  18. Robillard, P. (1974). Calibration of dial’s assignment method. Transportation Science, 8, 117-125.CrossRefGoogle Scholar
  19. Stuart, A., Ord, J.K. and Arnold, S. (1999). Kendall’s Advanced Theory of Statistics, Vol. 2A, 6th edition, London: Arnold Publisher.Google Scholar
  20. Yai, T., Iwakura, S. and Morichi, S. (1997). Multinomial robit with structured covariance for route choice behavior. Transportation Research Part B, 31, 195-207.CrossRefGoogle Scholar
  21. Yang, H., Sasaki, T. and Iida, Y. (1992). Estimation of origin-destination matrices from link traffic counts on congested networks, Transportation Research Part B, 26, 417-434.CrossRefGoogle Scholar
  22. Yang, H., Meng, Q. and Bell, M.G.H (2001). Simultaneous estimation of the origin-destination matrices and travel-cost coefficient for congested networks in a stochastic user equilibrium. Transportation Science, 35, 107-123.CrossRefGoogle Scholar
  23. Ying, J.Q. and Yang, H. (2005). Sensitivity analysis of stochastic user equilibrium flows in a Bi-Modal network with application to optimal pricing. Transportation Research Part B, 39, 769-795.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  • Shoichiro Nakayama
    • 1
  • Richard D.
    • 2
  1. 1.Kanazawa University, Japan and University of LeedsBritainU.K
  2. 2.Connors and David Watling, University of LeedsBritainU.K

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