September  2013, 8(3): 825-842. doi: 10.3934/nhm.2013.8.825

Probability hypothesis density filtering for real-time traffic state estimation and prediction

1. 

Université de Lyon, F-69000, Lyon, France, France, France

2. 

Department of Automatic Control and Systems Engineering, Mappin Street, University of Sheffield, Sheffield S1 3JD, United Kingdom

Received  April 2012 Revised  June 2013 Published  October 2013

The probability hypothesis density (PHD) methodology is widely used by the research community for the purposes of multiple object tracking. This problem consists in the recursive state estimation of several targets by using the information coming from an observation process. The purpose of this paper is to investigate the potential of the PHD filters for real-time traffic state estimation. This investigation is based on a Cell Transmission Model (CTM) coupled with the PHD filter. It brings a novel tool to the state estimation problem and allows to estimate the densities in traffic networks in the presence of measurement origin uncertainty, detection uncertainty and noises. In this work, we compare the PHD filter performance with a particle filter (PF), both taking into account the measurement origin uncertainty and show that they can provide high accuracy in a traffic setting and real-time computational costs. The PHD filtering framework opens new research avenues and has the abilities to solve challenging problems of vehicular networks.
Citation: Matthieu Canaud, Lyudmila Mihaylova, Jacques Sau, Nour-Eddin El Faouzi. Probability hypothesis density filtering for real-time traffic state estimation and prediction. Networks & Heterogeneous Media, 2013, 8 (3) : 825-842. doi: 10.3934/nhm.2013.8.825
References:
[1]

M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 50 (2002), 174-188. doi: 10.1109/78.978374.  Google Scholar

[2]

G. Battistelli, L. Chisci, S. Morrocchi, F. Papi, A. Benavoli, A. Di Lallo, A. Farina and A. Graziano, Traffic intensity estimation via PHD filtering, In "Proc. 5th European Radar Conf.," Amsterdam, The Netherlands, (2008), 340-343. Google Scholar

[3]

A. Ben Aissa, J. Sau, N-E. El Faouzi and O. De Mouzon, Sequential Monte Carlo traffic estimation for intelligent transportation system: Motorway travel time prediction application, "In Proc. Of the 2nd ISTS," Lausanne, Switzerland (2006). Google Scholar

[4]

R. Billot, N-E. El Faouzi, J. Sau and F. De Vuyst, Integrating the impact of rain into traffic management: Online traffic state estimation using sequential Monte Carlo techniques, Transportation Research Record: Journal of the Transportation Research Board, 2169 (2010), 141-149. doi: 10.3141/2169-15.  Google Scholar

[5]

Z. Chen, Bayesian filtering: From Kalman filters to particle filters, and beyond, Adaptive Systems Lab., Technical Report, McMaster University, ON, Canada (2003). Google Scholar

[6]

M. Canaud, N-E. El Faouzi and J. Sau, Reservoir-based urban traffic modeling for travel time estimation: Sensitivity analysis and case study, In "Proc. Of the 91th Transportation Research Board Annual Meeting," Washington D. C., USA, (2012). Google Scholar

[7]

C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[8]

A. Doucet, "Monte Carlo Methods for Bayesian Estimation of Hidden Markov Models. Application to Radiation Signals," PhD thesis, Université Paris-Sud, Orsay (1997). Google Scholar

[9]

A. Doucet, On sequential simulation-based methods for Bayesian filtering, Departement of Engineering, Technical report CUED/F-INFENG/TR.310, Cambridge University (1998). Google Scholar

[10]

A. Doucet, N. De Freitas and N. Gordon, "Sequential Monte Carlo Methods in Practice," Statistics for Engineering and Information Science. Springer-Verlag, New York, 2001. xxviii+581 pp.  Google Scholar

[11]

N-E. El Faouzi, Research needs for real time monitoring, surveillance and control of road networks under adverse weather conditions, Research Agenda for the European Cooperation in the field of scientific and technical research (COST), (2007) - www.COST-TU0702.org. Google Scholar

[12]

O. Erdinc, P. Willett and Y. Bar-Shalom, Probability hypothesis density filter for multitarget multisensor tracking, In "Proc. Of the 8th Int. Conf. On Information Fusion," 146-153, Philadelphia, PA, USA (2005). doi: 10.1109/ICIF.2005.1591848.  Google Scholar

[13]

K. Gilholm, S. Godsill, S. Maskell and D. Salmond, Poisson models for extended target and group tracking, In "Proc. SPIE: Signal and Data Processing of Small Targets 5913," {230-241}, San. Diego, CA, USA (2005). doi: 10.1117/12.618730.  Google Scholar

[14]

K. Gilholm and D. Salmond, Spatial distribution model for tracking extended objects, In "Proc. IEEE on Radar, Sonar and Navigation,'' 152 (2005), 364-371. doi: 10.1049/ip-rsn:20045114.  Google Scholar

[15]

A. Gning, L. Mihaylova and F. Abdallah, Mixture of uniform probability density functions for non linear state estimation using interval analysis, In "Proc. Of the 13th Int. Conf. On Information Fusion," Edinburgh, UK (2010). Google Scholar

[16]

A. Gning, B. Ristic and L. Mihaylova, A box particle filter for stochastic set-theoretic measurements with association uncertainty, In "Proc. Of the 14th Int. Conf. On Information on Fusion," Chicago, IL, USA (2011). Google Scholar

[17]

A. Gning, B. Ristic and L. Mihaylova, Bernouilli particle/box particle filters for detection and tracking in the presence of triple uncertainty, IEEE Trans. Signal Processing, 60 (2012), 2138-2151. doi: 10.1109/TSP.2012.2184538.  Google Scholar

[18]

A. Hegyi, D. Girimonte, R. Babuska and B. De Schutter, A comparison of filter configurations for freeway traffic state estimation, In "Proc. Of the 2006 IEEE Intelligent Transportation Systems Conference (ITSC 2006)," 1029-1034, Toronto, Canada (2006). doi: 10.1109/ITSC.2006.1707357.  Google Scholar

[19]

R. Juang and P. Burlina, Comparative performance evaluation of GM-PHD filter in clutter, In "Proc. Of the 12t Internatiinal Conf. On Information Fusion," (2009), 1195-1202. Google Scholar

[20]

S. Julier and J. Uhlmann, A new extension of the Kalman filter to nonlinear systems, In "International Symposium on Aerospace/Defense Sensing, Simulation and Controls," 182-193, Orlando, FL, USA (1997). doi: 10.1117/12.280797.  Google Scholar

[21]

R. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552.  Google Scholar

[22]

J. Lebacque, The Godunov scheme and what it means for first order traffic flow models, In "Proc. Of the 13th nternaional symposium on transportation and traffic theory (ISTTT)," (1995), 647-677. Google Scholar

[23]

R. Mahler, Multitarget bayes filtering via first-order multitarget moments, IEEE Transactions on Aerospace and Electronic Systems, 39 (2003), 1152-1178. doi: 10.1109/TAES.2003.1261119.  Google Scholar

[24]

R. Mahler, Statistical multisources multitarget information fusion, Artech House, 2007. Google Scholar

[25]

R. Mahler, PHD filters for nonstandard targets, I: Extended targets, In "Proc. Of the 12th International Conference on Information Fusion," 914-921, Seattle, WA, USA (2009). Google Scholar

[26]

R. Mahler, B-T. Vo and B-N. Vo, CPHD filtering with unknown clutter rate and detection profile, IEEE Transactions on Signal Processing, 59 (2011), 3497-3513. doi: 10.1109/TSP.2011.2128316.  Google Scholar

[27]

L. Mihaylova and R. Boel, A particle filter for freeway traffic estimation, In "Proc. Of the 43rd IEEE Conference on Decision and Control," Atlantis, Paradise Island, Bahamas, 2 (2004), 2106-2111. doi: 10.1109/CDC.2004.1430359.  Google Scholar

[28]

L. Mihaylova, R. Boel and A. Hegyi, Freeway traffic estimation within recursive bayesian framework, Automatica J. IFAC, 43 (2007), 290-300. doi: 10.1016/j.automatica.2006.08.023.  Google Scholar

[29]

{L. Mihaylova, A. Hegyi, A. Gning and R. Boel, Parallelized Particle and Gaussian Sum Particle Filters for Large Scale Traffic Systems,}, IEEE transactions on intelligent transportation systems. Special issue on emergent cooperative technologies in intelligent transp. Systems, 13 (2012), 36-48. Google Scholar

[30]

K. Panta, B. Vo, S. Singh and A. Doucet, Probability hypothesis density filter versus multiple hypothesis tracking, In "Proceedings of SPIE," 5429 (2004), 284-295. doi: 10.1117/12.543357.  Google Scholar

[31]

B. Ristic, M. Arulampalam and N. Gordon, "Beyond the Kalman Filter: Particle Filters for Tracking Applications," Artech House, Boston, 2004. Google Scholar

[32]

B. Ristic, D. Clark and B. Vo, Improved SMC implementation of the PHD filter, In "Proc. Of the 13th International Conference on Information Fusion," Edinburgh, UK (2010). Google Scholar

[33]

J. Sau, N-E. El Faouzi and O. De Mouzon, Particle-filter traffic state estimation and sequential test for real-time traffic sensor diagnosis, In "Proc. Of ISTS'08 Symposium," Queensland (2008). Google Scholar

[34]

M. Schikora, A. Gning, L. Mihaylova, D. Cremers and W. Koch, Box-particle PHD filter for multi-target tracking, IEEE Trans. on Aerospace and Electronic Systems, to appear (2013). Google Scholar

[35]

H. Sidenbladh, Multi-target particle filtering for the probability hypothesis density, In "Proc. 6th Int'l Conf. on Information Fusion," Cairns, Australia (2003). Google Scholar

[36]

A. Sumalee, R. X. Zhong, T. L. Pan and W. Y. Szeto, Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment, Transportation Research Part B, 45 (2011), 507-533. doi: 10.1016/j.trb.2010.09.006.  Google Scholar

[37]

X. Sun, L. Munoz and R. Horowitz, Highway traffic state estimation using improved mixture Kalman filters for effective ramp metering control, In "Proc. Of th 42nd IEEE Conf. On Decision and Control," 6333-6338, Maui, Hawaii, USA (2003). Google Scholar

[38]

J. Sussman, Introduction to transportation problems, Artech House, Norwood, Masachussets, 2000. Google Scholar

[39]

B. Vo, S. Singh and A. Doucet, Sequential Monte Carlo methods for multi-target filtering with random finite sets, IEEE Trans. Aerospace and Electronic Systems, 41 (2005), 1224-1245. Google Scholar

[40]

B. Vo and W. Ma, The gaussian mixture probability hypothesis density filter, IEEE Trans. Signal Processing, 54 (2006), 4091-4104. doi: 10.1109/TSP.2006.881190.  Google Scholar

[41]

B-T. Vo, B-N. Vo and A. Cantoni, Analytic implementations of the cardinalized probability hypothesis density filter, IEEE Trans. Signal. Processing, 55 (2007), 3553-3567. doi: 10.1109/TSP.2007.894241.  Google Scholar

[42]

N.-N. Vo, B.-T. Vo and D. Clark, Bayesian multiple target tracking using random finite sets, ch. 3 in "Integrated Tracking, Classification, and Sensor Management: Theory and Applications," Eds. M. Mallick, V. Krishnamurthy, and B.-N. Vo, 75-125. John Wiley & Sons, 2012. Google Scholar

[43]

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), 167-181. doi: 10.1287/trsc.1070.0194.  Google Scholar

[44]

Y. Wang, M. Papageorgiou, A. Messmer, P. Coppola, A. Tzimitsi and A. Nuzzolo, An adaptive freeway traffic state estimator, Automatica, 45 (2009), 10-24. doi: 10.1016/j.automatica.2008.05.019.  Google Scholar

[45]

N. Whiteley, S. Singh and S. Godsill, Auxiliary particle implementation of the probability hypothesis density filter, IEEE Trans. on Aerospace and Electronic Systems, 46 (2010), 1437-1454. Google Scholar

[46]

D. Work, S. Blandin, O-P. Tossavainen, B. Piccoli and A. Bayen, A Traffic Model for Velocity Data Assimilation, Applied Mathematics Research eXpress, 2010 (2010), 1-35.  Google Scholar

[47]

D. Work, O.-P. Tossavainen, S. Blandin, A. M. Bayen, T. Iwuchukwu and K. Tracton, An ensemble Kalman filtering approach to highway traffic estimation using GPS enabled mobile devices, Proceedings of CDC, (2008), 5062-5068. doi: 10.1109/CDC.2008.4739016.  Google Scholar

[48]

T. Zajic and R. Mahler, Particle-systems implementation of the PHD multitarget tracking filter, In "Proceedings of SPIE," Signal Processing, Sensor Fusion, and Target Recognition, XII, 5096 (2003), 291-299, Bellingham. doi: 10.1117/12.488533.  Google Scholar

show all references

References:
[1]

M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 50 (2002), 174-188. doi: 10.1109/78.978374.  Google Scholar

[2]

G. Battistelli, L. Chisci, S. Morrocchi, F. Papi, A. Benavoli, A. Di Lallo, A. Farina and A. Graziano, Traffic intensity estimation via PHD filtering, In "Proc. 5th European Radar Conf.," Amsterdam, The Netherlands, (2008), 340-343. Google Scholar

[3]

A. Ben Aissa, J. Sau, N-E. El Faouzi and O. De Mouzon, Sequential Monte Carlo traffic estimation for intelligent transportation system: Motorway travel time prediction application, "In Proc. Of the 2nd ISTS," Lausanne, Switzerland (2006). Google Scholar

[4]

R. Billot, N-E. El Faouzi, J. Sau and F. De Vuyst, Integrating the impact of rain into traffic management: Online traffic state estimation using sequential Monte Carlo techniques, Transportation Research Record: Journal of the Transportation Research Board, 2169 (2010), 141-149. doi: 10.3141/2169-15.  Google Scholar

[5]

Z. Chen, Bayesian filtering: From Kalman filters to particle filters, and beyond, Adaptive Systems Lab., Technical Report, McMaster University, ON, Canada (2003). Google Scholar

[6]

M. Canaud, N-E. El Faouzi and J. Sau, Reservoir-based urban traffic modeling for travel time estimation: Sensitivity analysis and case study, In "Proc. Of the 91th Transportation Research Board Annual Meeting," Washington D. C., USA, (2012). Google Scholar

[7]

C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[8]

A. Doucet, "Monte Carlo Methods for Bayesian Estimation of Hidden Markov Models. Application to Radiation Signals," PhD thesis, Université Paris-Sud, Orsay (1997). Google Scholar

[9]

A. Doucet, On sequential simulation-based methods for Bayesian filtering, Departement of Engineering, Technical report CUED/F-INFENG/TR.310, Cambridge University (1998). Google Scholar

[10]

A. Doucet, N. De Freitas and N. Gordon, "Sequential Monte Carlo Methods in Practice," Statistics for Engineering and Information Science. Springer-Verlag, New York, 2001. xxviii+581 pp.  Google Scholar

[11]

N-E. El Faouzi, Research needs for real time monitoring, surveillance and control of road networks under adverse weather conditions, Research Agenda for the European Cooperation in the field of scientific and technical research (COST), (2007) - www.COST-TU0702.org. Google Scholar

[12]

O. Erdinc, P. Willett and Y. Bar-Shalom, Probability hypothesis density filter for multitarget multisensor tracking, In "Proc. Of the 8th Int. Conf. On Information Fusion," 146-153, Philadelphia, PA, USA (2005). doi: 10.1109/ICIF.2005.1591848.  Google Scholar

[13]

K. Gilholm, S. Godsill, S. Maskell and D. Salmond, Poisson models for extended target and group tracking, In "Proc. SPIE: Signal and Data Processing of Small Targets 5913," {230-241}, San. Diego, CA, USA (2005). doi: 10.1117/12.618730.  Google Scholar

[14]

K. Gilholm and D. Salmond, Spatial distribution model for tracking extended objects, In "Proc. IEEE on Radar, Sonar and Navigation,'' 152 (2005), 364-371. doi: 10.1049/ip-rsn:20045114.  Google Scholar

[15]

A. Gning, L. Mihaylova and F. Abdallah, Mixture of uniform probability density functions for non linear state estimation using interval analysis, In "Proc. Of the 13th Int. Conf. On Information Fusion," Edinburgh, UK (2010). Google Scholar

[16]

A. Gning, B. Ristic and L. Mihaylova, A box particle filter for stochastic set-theoretic measurements with association uncertainty, In "Proc. Of the 14th Int. Conf. On Information on Fusion," Chicago, IL, USA (2011). Google Scholar

[17]

A. Gning, B. Ristic and L. Mihaylova, Bernouilli particle/box particle filters for detection and tracking in the presence of triple uncertainty, IEEE Trans. Signal Processing, 60 (2012), 2138-2151. doi: 10.1109/TSP.2012.2184538.  Google Scholar

[18]

A. Hegyi, D. Girimonte, R. Babuska and B. De Schutter, A comparison of filter configurations for freeway traffic state estimation, In "Proc. Of the 2006 IEEE Intelligent Transportation Systems Conference (ITSC 2006)," 1029-1034, Toronto, Canada (2006). doi: 10.1109/ITSC.2006.1707357.  Google Scholar

[19]

R. Juang and P. Burlina, Comparative performance evaluation of GM-PHD filter in clutter, In "Proc. Of the 12t Internatiinal Conf. On Information Fusion," (2009), 1195-1202. Google Scholar

[20]

S. Julier and J. Uhlmann, A new extension of the Kalman filter to nonlinear systems, In "International Symposium on Aerospace/Defense Sensing, Simulation and Controls," 182-193, Orlando, FL, USA (1997). doi: 10.1117/12.280797.  Google Scholar

[21]

R. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552.  Google Scholar

[22]

J. Lebacque, The Godunov scheme and what it means for first order traffic flow models, In "Proc. Of the 13th nternaional symposium on transportation and traffic theory (ISTTT)," (1995), 647-677. Google Scholar

[23]

R. Mahler, Multitarget bayes filtering via first-order multitarget moments, IEEE Transactions on Aerospace and Electronic Systems, 39 (2003), 1152-1178. doi: 10.1109/TAES.2003.1261119.  Google Scholar

[24]

R. Mahler, Statistical multisources multitarget information fusion, Artech House, 2007. Google Scholar

[25]

R. Mahler, PHD filters for nonstandard targets, I: Extended targets, In "Proc. Of the 12th International Conference on Information Fusion," 914-921, Seattle, WA, USA (2009). Google Scholar

[26]

R. Mahler, B-T. Vo and B-N. Vo, CPHD filtering with unknown clutter rate and detection profile, IEEE Transactions on Signal Processing, 59 (2011), 3497-3513. doi: 10.1109/TSP.2011.2128316.  Google Scholar

[27]

L. Mihaylova and R. Boel, A particle filter for freeway traffic estimation, In "Proc. Of the 43rd IEEE Conference on Decision and Control," Atlantis, Paradise Island, Bahamas, 2 (2004), 2106-2111. doi: 10.1109/CDC.2004.1430359.  Google Scholar

[28]

L. Mihaylova, R. Boel and A. Hegyi, Freeway traffic estimation within recursive bayesian framework, Automatica J. IFAC, 43 (2007), 290-300. doi: 10.1016/j.automatica.2006.08.023.  Google Scholar

[29]

{L. Mihaylova, A. Hegyi, A. Gning and R. Boel, Parallelized Particle and Gaussian Sum Particle Filters for Large Scale Traffic Systems,}, IEEE transactions on intelligent transportation systems. Special issue on emergent cooperative technologies in intelligent transp. Systems, 13 (2012), 36-48. Google Scholar

[30]

K. Panta, B. Vo, S. Singh and A. Doucet, Probability hypothesis density filter versus multiple hypothesis tracking, In "Proceedings of SPIE," 5429 (2004), 284-295. doi: 10.1117/12.543357.  Google Scholar

[31]

B. Ristic, M. Arulampalam and N. Gordon, "Beyond the Kalman Filter: Particle Filters for Tracking Applications," Artech House, Boston, 2004. Google Scholar

[32]

B. Ristic, D. Clark and B. Vo, Improved SMC implementation of the PHD filter, In "Proc. Of the 13th International Conference on Information Fusion," Edinburgh, UK (2010). Google Scholar

[33]

J. Sau, N-E. El Faouzi and O. De Mouzon, Particle-filter traffic state estimation and sequential test for real-time traffic sensor diagnosis, In "Proc. Of ISTS'08 Symposium," Queensland (2008). Google Scholar

[34]

M. Schikora, A. Gning, L. Mihaylova, D. Cremers and W. Koch, Box-particle PHD filter for multi-target tracking, IEEE Trans. on Aerospace and Electronic Systems, to appear (2013). Google Scholar

[35]

H. Sidenbladh, Multi-target particle filtering for the probability hypothesis density, In "Proc. 6th Int'l Conf. on Information Fusion," Cairns, Australia (2003). Google Scholar

[36]

A. Sumalee, R. X. Zhong, T. L. Pan and W. Y. Szeto, Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment, Transportation Research Part B, 45 (2011), 507-533. doi: 10.1016/j.trb.2010.09.006.  Google Scholar

[37]

X. Sun, L. Munoz and R. Horowitz, Highway traffic state estimation using improved mixture Kalman filters for effective ramp metering control, In "Proc. Of th 42nd IEEE Conf. On Decision and Control," 6333-6338, Maui, Hawaii, USA (2003). Google Scholar

[38]

J. Sussman, Introduction to transportation problems, Artech House, Norwood, Masachussets, 2000. Google Scholar

[39]

B. Vo, S. Singh and A. Doucet, Sequential Monte Carlo methods for multi-target filtering with random finite sets, IEEE Trans. Aerospace and Electronic Systems, 41 (2005), 1224-1245. Google Scholar

[40]

B. Vo and W. Ma, The gaussian mixture probability hypothesis density filter, IEEE Trans. Signal Processing, 54 (2006), 4091-4104. doi: 10.1109/TSP.2006.881190.  Google Scholar

[41]

B-T. Vo, B-N. Vo and A. Cantoni, Analytic implementations of the cardinalized probability hypothesis density filter, IEEE Trans. Signal. Processing, 55 (2007), 3553-3567. doi: 10.1109/TSP.2007.894241.  Google Scholar

[42]

N.-N. Vo, B.-T. Vo and D. Clark, Bayesian multiple target tracking using random finite sets, ch. 3 in "Integrated Tracking, Classification, and Sensor Management: Theory and Applications," Eds. M. Mallick, V. Krishnamurthy, and B.-N. Vo, 75-125. John Wiley & Sons, 2012. Google Scholar

[43]

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), 167-181. doi: 10.1287/trsc.1070.0194.  Google Scholar

[44]

Y. Wang, M. Papageorgiou, A. Messmer, P. Coppola, A. Tzimitsi and A. Nuzzolo, An adaptive freeway traffic state estimator, Automatica, 45 (2009), 10-24. doi: 10.1016/j.automatica.2008.05.019.  Google Scholar

[45]

N. Whiteley, S. Singh and S. Godsill, Auxiliary particle implementation of the probability hypothesis density filter, IEEE Trans. on Aerospace and Electronic Systems, 46 (2010), 1437-1454. Google Scholar

[46]

D. Work, S. Blandin, O-P. Tossavainen, B. Piccoli and A. Bayen, A Traffic Model for Velocity Data Assimilation, Applied Mathematics Research eXpress, 2010 (2010), 1-35.  Google Scholar

[47]

D. Work, O.-P. Tossavainen, S. Blandin, A. M. Bayen, T. Iwuchukwu and K. Tracton, An ensemble Kalman filtering approach to highway traffic estimation using GPS enabled mobile devices, Proceedings of CDC, (2008), 5062-5068. doi: 10.1109/CDC.2008.4739016.  Google Scholar

[48]

T. Zajic and R. Mahler, Particle-systems implementation of the PHD multitarget tracking filter, In "Proceedings of SPIE," Signal Processing, Sensor Fusion, and Target Recognition, XII, 5096 (2003), 291-299, Bellingham. doi: 10.1117/12.488533.  Google Scholar

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