Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data

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September 2013, 8(3): 803-824. doi: 10.3934/nhm.2013.8.803

Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data

Olli-Pekka Tossavainen ^1, and Daniel B. Work ^2,

1.	Microsoft, 1065 La Avenida St, Mountain View, CA 94043, United States
2.	University of Illinois at Urbana-Champaign, 1203 Newmark Civil Engineering Laboratory, 205 N. Mathews Ave, Urbana, IL 61801, United States

Received October 2012 Revised June 2013 Published October 2013

Abstract
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In large scale deployments of traffic flow models, estimation of the model parameters is a critical but cumbersome task. A poorly calibrated model leads to erroneous estimates in data--poor environments, and limited forecasting ability. In this article we present a method for calibrating flow model parameters for a discretized scalar conservation law using only velocity measurements. The method is based on a Markov Chain Monte Carlo technique, which is used to approximate statistics of the posterior distribution of the model parameters. Numerical experiments highlight the difficulty in estimating jam densities and provide a new approach to improve performance of the sampling through re-parameterization of the model. Supporting source code for the numerical experiments is available for download at https://github.com/dbwork/MCMC-based-inverse-modeling-of-traffic.

Keywords: Markov Chain, Monte Carlo, cell transmission model., Parameter estimation, fundamental diagram, traffic flow.

Mathematics Subject Classification: 90B20, 65C05, 65C40, 35L65, 65N2.

Citation: Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803

References:

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[3]	S. Blandin, A. Couque, A. Bayen and D. Work, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D: Nonlinear Phenomena, 241 (2012), 1421. Google Scholar
[4]	G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510. doi: 10.1137/070697768. Google Scholar
[5]	R. M. Colombo and A. Marson, A Hölder continuous ode related to traffic flow,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759. doi: 10.1017/S0308210500002663. Google Scholar
[6]	M. Cremer and M. Papageorgiou, Parameter identification for a traffic flow model,, Automatica J. IFAC, 17 (1981), 837. doi: 10.1016/0005-1098(81)90071-6. Google Scholar
[7]	E. Cristiani, C. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensor data,, Communications in Applied and Industrial Mathematics, 1 (2010), 54. Google Scholar
[8]	G. Dervisoglu, G. Gomes, J. Kwon, R. Horowitz and P. Varaiya, Automatic calibration of the fundamental diagram and empirical observations on capacity,, in, (2009). Google Scholar
[9]	M. Garavello and B. Piccoli, "Traffic Flow on Networks,", Conservation laws models. AIMS Series on Applied Mathematics, (2006). Google Scholar
[10]	W. Gilks, S. Richardson and D. Spegelhalter, "Markov Chain Monte Carlo in Practice,", Interdisciplinary Statistics. Chapman & Hall, (1996). Google Scholar
[11]	S. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations,, (Russian) Mat. Sb. (N. S.), 47 (1959), 271. Google Scholar
[12]	A. Hegyi, D. Girimonte, R. Babŭska and B. D. Schutter, A comparison of filter configurations for freeway traffic state estimation,, in, (2006), 1029. doi: 10.1109/ITSC.2006.1707357. Google Scholar
[13]	J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar
[14]	J. Lebacque, The Godunov scheme and what it means for first order traffic flow models,, in, (1996), 647. Google Scholar
[15]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, (1992). doi: 10.1007/978-3-0348-8629-1. Google Scholar
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[17]	J. V. Lint, S. Hoogendoorn and A. Hegyi, Dual EKF state and parameter estimation in multi-class first-order traffic flow models,, in, (2008). Google Scholar
[18]	X.-Y. Lu, P. Varaiya and R. Horowitz, Fundamental diagram modeling and analysis based on NGSIM data,, in, (2009). Google Scholar
[19]	L. Mihaylova, R. Boel and A. Hegyi, Freeway traffic estimation within particle filtering framework,, Automatica J. IFAC, 43 (2007), 290. doi: 10.1016/j.automatica.2006.08.023. Google Scholar
[20]	L. Munoz, X. Sun, D. Sun, G. Gomez and R. Horowitz, Methodological calibration of the cell transmission model,, in, 1 (2004), 798. Google Scholar
[21]	A. Muralidharan, G. Dervisoglu and R. Horowitz, Probabilistic graphical models of fundamental diagram parameters for freeway traffic simulations,, in, (2011). doi: 10.3141/2249-10. Google Scholar
[22]	P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar
[23]	S. Smulders, Control of freeway traffic flow by variable speed signs,, Transportation Research Part B: Methodological, 24 (1990), 111. doi: 10.1016/0191-2615(90)90023-R. Google Scholar
[24]	Transportation Research Board, "HCM 2010: Highway Capacity Manual,", (2010)., (2010). Google Scholar
[25]	, , (). Google Scholar
[26]	Y. Wang, M. Papageorgiou and A. Messmer, RENAISSANCE - A unified macroscopic model-based approach to real-time freeway network traffic surveillance,, Transportation Research Part C: Emerging Technologies, 14 (2006), 190. doi: 10.1016/j.trc.2006.06.001. Google Scholar
[27]	Y. Wang, M. Papageorgiou and A. Messmer, Real-time freeway traffic state estimation based on extended kalman filter: A case study,, Transportation Science, 41 (2007), 167. doi: 10.1287/trsc.1070.0194. Google Scholar
[28]	D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express. AMRX, 2010 (2010), 1. Google Scholar
[29]	J. Yan, Parameter identification of freeway traffic flow model and adaptive ramp metering,, in, (2009), 235. doi: 10.1109/ISECS.2009.39. Google Scholar

show all references

References:

[1]	T. Bellemans, B. D. Schutter and B. D. Moor, Model predictive control with repeated model fitting for ramp metering,, in, (2002), 236. doi: 10.1109/ITSC.2002.1041221. Google Scholar
[2]	T. Bellemans, B. D. Schutter, G. Wets and B. D. Moor, Model predictive control for ramp metering combined with extended Kalman filter-based traffic state estimation,, in, (2006), 406. doi: 10.1109/ITSC.2006.1706775. Google Scholar
[3]	S. Blandin, A. Couque, A. Bayen and D. Work, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D: Nonlinear Phenomena, 241 (2012), 1421. Google Scholar
[4]	G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510. doi: 10.1137/070697768. Google Scholar
[5]	R. M. Colombo and A. Marson, A Hölder continuous ode related to traffic flow,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759. doi: 10.1017/S0308210500002663. Google Scholar
[6]	M. Cremer and M. Papageorgiou, Parameter identification for a traffic flow model,, Automatica J. IFAC, 17 (1981), 837. doi: 10.1016/0005-1098(81)90071-6. Google Scholar
[7]	E. Cristiani, C. de Fabritiis and B. Piccoli, A fluid dynamic approach for traffic forecast from mobile sensor data,, Communications in Applied and Industrial Mathematics, 1 (2010), 54. Google Scholar
[8]	G. Dervisoglu, G. Gomes, J. Kwon, R. Horowitz and P. Varaiya, Automatic calibration of the fundamental diagram and empirical observations on capacity,, in, (2009). Google Scholar
[9]	M. Garavello and B. Piccoli, "Traffic Flow on Networks,", Conservation laws models. AIMS Series on Applied Mathematics, (2006). Google Scholar
[10]	W. Gilks, S. Richardson and D. Spegelhalter, "Markov Chain Monte Carlo in Practice,", Interdisciplinary Statistics. Chapman & Hall, (1996). Google Scholar
[11]	S. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations,, (Russian) Mat. Sb. (N. S.), 47 (1959), 271. Google Scholar
[12]	A. Hegyi, D. Girimonte, R. Babŭska and B. D. Schutter, A comparison of filter configurations for freeway traffic state estimation,, in, (2006), 1029. doi: 10.1109/ITSC.2006.1707357. Google Scholar
[13]	J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar
[14]	J. Lebacque, The Godunov scheme and what it means for first order traffic flow models,, in, (1996), 647. Google Scholar
[15]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, (1992). doi: 10.1007/978-3-0348-8629-1. Google Scholar
[16]	M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A. 229* (1955), 229* (1955), 317. doi: 10.1098/rspa.1955.0089. Google Scholar
[17]	J. V. Lint, S. Hoogendoorn and A. Hegyi, Dual EKF state and parameter estimation in multi-class first-order traffic flow models,, in, (2008). Google Scholar
[18]	X.-Y. Lu, P. Varaiya and R. Horowitz, Fundamental diagram modeling and analysis based on NGSIM data,, in, (2009). Google Scholar
[19]	L. Mihaylova, R. Boel and A. Hegyi, Freeway traffic estimation within particle filtering framework,, Automatica J. IFAC, 43 (2007), 290. doi: 10.1016/j.automatica.2006.08.023. Google Scholar
[20]	L. Munoz, X. Sun, D. Sun, G. Gomez and R. Horowitz, Methodological calibration of the cell transmission model,, in, 1 (2004), 798. Google Scholar
[21]	A. Muralidharan, G. Dervisoglu and R. Horowitz, Probabilistic graphical models of fundamental diagram parameters for freeway traffic simulations,, in, (2011). doi: 10.3141/2249-10. Google Scholar
[22]	P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar
[23]	S. Smulders, Control of freeway traffic flow by variable speed signs,, Transportation Research Part B: Methodological, 24 (1990), 111. doi: 10.1016/0191-2615(90)90023-R. Google Scholar
[24]	Transportation Research Board, "HCM 2010: Highway Capacity Manual,", (2010)., (2010). Google Scholar
[25]	, , (). Google Scholar
[26]	Y. Wang, M. Papageorgiou and A. Messmer, RENAISSANCE - A unified macroscopic model-based approach to real-time freeway network traffic surveillance,, Transportation Research Part C: Emerging Technologies, 14 (2006), 190. doi: 10.1016/j.trc.2006.06.001. Google Scholar
[27]	Y. Wang, M. Papageorgiou and A. Messmer, Real-time freeway traffic state estimation based on extended kalman filter: A case study,, Transportation Science, 41 (2007), 167. doi: 10.1287/trsc.1070.0194. Google Scholar
[28]	D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express. AMRX, 2010 (2010), 1. Google Scholar
[29]	J. Yan, Parameter identification of freeway traffic flow model and adaptive ramp metering,, in, (2009), 235. doi: 10.1109/ISECS.2009.39. Google Scholar

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