Abstract
Estimation of origin-destination (OD) flows in transportation networks is a major step in transportation planning. We are interested in estimating OD flows given data on traffic link volumes over a sequence of days. We propose a dynamic hierarchical Bayesian model for the estimation of day-to-day OD flows. At the first level, we specify a dynamic Gaussian model which describes the evolution of OD flows over time. At the second level, we model the assignment of route flows given OD flows. Route choice probabilities are a function of user-predicted route costs, which depend on past user-experienced costs with a leaning parameter which controls the length of users memory. At the third level we model observed link volumes given route flows. Covariance matrices of OD flows and observed link volumes are modeled through variance functions. We develop a Metropolis-within-Gibbs algorithm to sample from the joint posterior distribution of OD flows, route flows, and parameters of the route choice model and the variance functions. Our model can be applied to both congested and uncongested networks and does not assume network equilibrium. We illustrate application of the model and sampling algorithm through numerical studies on two transportation networks from the literature. In a small test network, results indicate that OD flows and parameters may be identified given uninformative priors, while in a real-scale network OD flows and route choice parameters may be reasonably identified given prior knowledge on OD flows at the beginning of the planning horizon and their temporal variability.
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Notes
Image and data provided by courtesy of Prof. Martin Hazelton, from Massey University, New Zealand.
References
Abareshi M, Zaferanieh M, Keramati B (2017) Path flow estimator in an entropy model using a nonlinear L-shaped algorithm. Netw Spat Econ 17(1):293–315. https://doi.org/10.1007/s11067-016-9327-9
Airoldi E, Haas B (2011) Polytope samplers for inference in ill-posed inverse problems. In: Gordon G, Dunson D, Dudík M (eds) Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, PMLR, Fort Lauderdale, FL, USA, Proceedings of Machine Learning Research, vol 15, pp 110–118
Airoldi EM, Blocker AW (2013) Estimating latent processes on a network from indirect measurements. J Am Stat Assoc 108(501):149–164. https://doi.org/10.1080/01621459.2012.756328
Ashok K, Ben-Akiva ME (2002) Estimation and prediction of time-dependent origin-destination flows with a stochastic mapping to path flows and link flows. Transp Sci 36:184–198. https://doi.org/10.1287/trsc.36.2.184.563
Bell M (1991) The estimation of origin-destination matrices by constrained generalized least squares. Transp Res B 25(1):13–22. https://doi.org/10.1016/0191-2615(91)90010-G
Benameur N, Roberts J (2004) Traffic matrix inference in IP networks. Netw Spat Econ 4(1):103–114. https://doi.org/10.1023/B:NETS.0000015658.75205.ed
Bierlaire M, Frejinger E (2005) Route choice models with subpath components. In: Proceedings 5th Swiss Transport Research Conference, Monte Verita / Ascona
Brenninger-Göthe M, Jörnsten K O (1989) Estimation of origin-destination matrices from traffic counts using multiobjective programming formulations. Transp Res B 23B(4):257–269. https://doi.org/10.1016/0191-2615(89)90028-3
Bureau of Public Roads (1964) Traffic assignment manual for application with a large, high speed computer. Traffic Assignment Manual for Application with a Large, High Speed Computer, U.S. Dept. of Commerce Bureau of Public Roads, Office of Planning, Urban Planning Division
Cao J, Davis D, Wiel SV, Yu B (2000) Time-varying network tomography: Router link data. J Am Stat Assoc 95(452):1063–1075. https://doi.org/10.2307/2669743
Carter CK, Kohn R (1994) On Gibbs sampling for state space models. Biometrika 81(3):541–553. https://doi.org/10.2307/2337125
Cascetta E (1984) Estimation of trip matrices from traffic counts and survey data: a generalized least squares estimator. Transp Res B 16:289–299. https://doi.org/10.1016/0191-2615(84)90012-2
Cascetta E (2009) Transportation Systems Analysis: Models and Applications, 2nd edn. Springer, New York
Cascetta E, Postorino N (2001) Fixed point approaches to the estimation of o/d matrices using traffic counts on congested networks. Transp Sci 35:134–147. https://doi.org/10.1287/trsc.35.2.134.10138
Cascetta E, Inaudi D, Marquis G (1993) Dynamic estimators of origin-destination matrices using traffic counts. Transp Sci 27:363–373. https://doi.org/10.1287/trsc.27.4.363
Cascetta E, Papola A, Marzano V, Simonelli F, Vitiello I (2013) Quasi-dynamic estimation of O–D flows from traffic counts: Formulation, statistical validation and performance analysis on real data. Transp Res B 55:171–187. https://doi.org/10.1016/j.trb.2013.06.007
Chang GL, Wu J (1994) Recursive estimation of time-varying origin-destination flows from traffic counts in freeway corridors. Transp Res B 28 (2):141–160. https://doi.org/10.1016/0191-2615(94)90022-1
Cho HJ, Jou YJ, Lan CL (2009) Time dependent origin-destination estimation from traffic count without prior information. Netw Spat Econ 9(2):145–170. https://doi.org/10.1007/s11067-008-9082-7
Cremer M, Keller H (1987) A new class of dynamic methods for the identification of origin-destination flows. Transp Res B 21:117–132. https://doi.org/10.1016/0191-2615(87)90011-7
Daziano RA, Miranda-Moreno L, Heydari S (2013) Computational bayesian statistics in transportation modeling: From road safety analysis to discrete choice. Transport Rev 33(5):570–592. https://doi.org/10.1080/01441647.2013.829890
Fisk C (1989) Trip matrix estimation from link counts: the congested network case. Transp Res B 23B:331–336. https://doi.org/10.1016/0191-2615(89)90009-X
Friesz TL, Bernstein D, Suo Z, Tobin RL (2001) Dynamic network user equilibrium with state-dependent time lags. Netw Spat Econ 1(3):319–347. https://doi.org/10.1023/A:1012896228490
Frühwirth-Schnatter S (1994) Data augmentation and dynamic linear models. J Time Ser Anal 15(2):183–202. https://doi.org/10.1111/j.1467-9892.1994.tb00184.x
Gelman A, Carlin JB, Stern HS, Rubin DB (2013) Bayesian Data Analysis, 3rd edn. Chapman & Hall, Boca Raton
Gelman A, Simpson D, Betancourt M (2017) The prior can often only be understood in the context of the likelihood. Entropy 19(10). https://doi.org/10.3390/e19100555
Geman S, Geman D (1984) Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans Pattern Anal Mach Intel 6(6):721–741. https://doi.org/10.1109/TPAMI.1984.4767596
de Grange L, González F, Bekhor S (2017) Path flow and trip matrix estimation using link flow density. Netw Spat Econ 17(1):173–195. https://doi.org/10.1007/s11067-016-9322-1
Hastings WK (1970) Monte carlo sampling methods using markov chains and their applications. Biometrika 57(1):97–109. https://doi.org/10.1093/biomet/57.1.97
Hazelton ML (2000) Estimation of origin-destination matrices from link flows on uncongested networks. Transp Res B 34:549–566. https://doi.org/10.1016/S0191-2615(99)00037-5
Hazelton ML (2001a) Estimation of origin–destination trip rates in leicester. J R Stat Soc C-Appl 50(4):423–433. https://doi.org/10.1111/1467-9876.00245
Hazelton ML (2001b) Inference for origin-destination matrices: estimation, prediction and reconstruction. Transp Res B 35:667–676. https://doi.org/10.1016/S0191-2615(00)00009-6
Hazelton ML (2003) Some comments on origin-destination matrix estimation. Transp Res A 37:811–822. https://doi.org/10.1016/S0965-8564(03)00044-2
Hazelton ML (2008) Statistical inference for time varying origin-destination matrices. Transp Res B 42:542–552. https://doi.org/10.1016/j.trb.2007.11.003
Hazelton ML (2010) Bayesian inference for network-based models with a linear inverse structure. Transp Res B 44:674–685. https://doi.org/10.1016/j.trb.2010.01.006
Hazelton ML, Parry K (2016) Statistical methods for comparison of day-to-day traffic models. Transp Res B 92:22–34. https://doi.org/10.1016/j.trb.2015.08.005
Jones E, Oliphant T, Peterson P, et al. (2001) SciPy: Open source scientific tools for Python. http://www.scipy.org. Accessed 26 Apr 2018
Li B (2005) Bayesian inference for origin-destination matrices of transport networks using the em algorithm. Technometrics 47(4):399–408. https://doi.org/10.1198/004017005000000283
Liang G, Taft N, Yu B (2006) A fast lightweight approach to origin-destination IP traffic estimation using partial measurements. IEEE Trans Inf Theory 52(6):2634–2648. https://doi.org/10.1109/TIT.2006.874412
Lo HP, Zhang N, Lam WHK (1996) Estimation of an origin-destination matrix with random link choice proportions: a statistical approach. Transp Res B 30 (4):309–324. https://doi.org/10.1016/0191-2615(95)00036-4
Lu CC, Zhou X, Zhang K (2013) Dynamic origin-destination demand flow estimation under congested traffic conditions. Transp Res C 34:16–37. https://doi.org/10.1016/j.trc.2013.05.006
Lu Z, Rao W, Wu YJ, Guo L, Xia J (2015) A kalman filter approach to dynamic od flow estimation for urban road networks using multi-sensor data. J Adv Transp 49(2):210–227. https://doi.org/10.1002/atr.1292
Maher M (1983) Inferences on trip matrices from observations on link volumes: a Bayesian statistical approach. Transp Res B 17:435–447. https://doi.org/10.1016/0191-2615(83)90030-9
Mahmassani H (2001) Dynamic network traffic assignment and simulation methodology for advanced system management applications. Netw Spat Econ 1 (3):267–292. https://doi.org/10.1023/A:1012831808926
Marzano V, Papola A, Simonelli F (2009) Limits and perspectives of effective O-D matrix correction using traffic counts. Transp Res C 17:120–132. https://doi.org/10.1016/j.trc.2008.09.001
Nie YM, Zhang HM (2010) A relaxation approach for estimating origin–destination trip tables. Netw Spat Econ 10(1):147–172. https://doi.org/10.1007/s11067-007-9059-y
Ortúzar JdD, Willumsen L (2011) Modelling Transport, 4th edn. Wiley, Chichester
Parry K, Hazelton M (2013) Bayesian inference for day-to-day dynamic traffic models. Transp Res B 50:104–115. https://doi.org/10.1016/j.trb.2013.01.003
Peeta S, Ziliaskopoulos AK (2001) Foundations of dynamic traffic assignment: The past, the present and the future. Netw Spat Econ 1(3-4):233–265. https://doi.org/10.1023/A:1012827724856
Pitombeira-Neto AR, Loureiro CFG (2016) A dynamic linear model for the estimation of time-varying origin-destination matrices from link counts. J Adv Transp 50(8):2116–2129. https://doi.org/10.1002/atr.1449
Pitombeira Neto AR, Oliveira Neto FM, Loureiro CFG (2017) Statistical models for the estimation of the origin-destination matrix from traffic counts. Transp 25(4):1–13. https://doi.org/10.14295/transportes.v25i4.1344
Särkkä S (2013) Bayesian Filtering and Smoothing. Cambridge University Press, New York
Shen W, Wynter L (2012) A new one-level convex optimization approach for estimating origin-destination demand. Transp Res B 46:1535–1555. https://doi.org/10.1016/j.trb.2012.07.005
Sherali HD, Park T (2001) Estimation of dynamic origin-destination trip tables for a general network. Transp Res B 35(3):217–235. https://doi.org/10.1016/S0191-2615(99)00048-X
Singhal H, Michailidis G (2007) Identifiability of flow distributions from link measurements with applications to computer networks. Inverse Probl 23(5):1821. https://doi.org/10.1088/0266-5611/23/5/004
Snyder C, Bengtsson T, Bickel P, Anderson J (2008) Obstacles to high-dimensional particle filtering. Mon Weather Rev 136(12):4629–4640. https://doi.org/10.1175/2008MWR2529.1
Tebaldi C, West M (1998) Bayesian inference on network traffic using link count data. J Am Stat Assoc 93(442):557–573. https://doi.org/10.2307/2670105
Van Zuylen HJ, Willumsen LG (1980) The most likely trip matrix estimated from traffic counts. Transp Res B, pp 281–293. https://doi.org/10.1016/0191-2615(80)90008-9
Vardi Y (1996) Network tomography: Estimating source-destination traffic intensities from link data. J Am Stat Assoc 91:365–377. https://doi.org/10.2307/2291416
Wardrop JG (1952) Some theoretical aspects of traffic research. In: Proceedings of the Institution of Civil Engineers part II, Institution of Civil Engineers, vol 1, pp 325–378
Watling D, Hazelton ML (2003) The dynamics and equilibria of day-to-day assignment models. Netw Spat Econ 3(3):349–370. https://doi.org/10.1023/A:1025398302560
Watling DP, Cantarella GE (2013) Modelling sources of variation in transportation systems: theoretical foundations of day-to-day dynamic models. Transportmetrica B 1(1):3–32. https://doi.org/10.1080/21680566.2013.785372
West M, Harrison J (1997) Bayesian Forecasting and Dynamic Models, 2nd edn. Springer, New York
Xie C, Duthie J (2015) An excess-demand dynamic traffic assignment approach for inferring origin-destination trip matrices. Netw Spat Econ 15(4):947–979. https://doi.org/10.1007/s11067-014-9277-z
Yang H (1995) Heuristic algorithms for the bilevel origin-destination matrix estimation problem. Transp Res B 29B(4):231–242. https://doi.org/10.1016/0191-2615(95)00003-V
Yang H, Sasaki T, Iida Y, Asakura Y (1992) Estimation of origin-destination matrices from link counts on congested networks. Transp Res B 26B:417–434. https://doi.org/10.1016/0191-2615(92)90008-K
Yang Y, Fan Y, Wets RJ (2018) Stochastic travel demand estimation: improving network identifiability using multi-day observation sets. Transp Res B 107:192–211. https://doi.org/10.1016/j.trb.2017.10.007
Zhou X, Mahmassani HS (2007) A structural state space model for real-time traffic origin-destination demand estimation and prediction in a day-to-day learning framework. Transp Res B 41(8):823–840. https://doi.org/10.1016/j.trb.2007.02.004
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This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico -CNPq, under grant number 422464/2016-3.
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Pitombeira-Neto, A.R., Loureiro, C.F.G. & Carvalho, L.E. A Dynamic Hierarchical Bayesian Model for the Estimation of day-to-day Origin-destination Flows in Transportation Networks. Netw Spat Econ 20, 499–527 (2020). https://doi.org/10.1007/s11067-019-09490-5
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DOI: https://doi.org/10.1007/s11067-019-09490-5