American Institute of Mathematical Sciences

Advanced
2015, 12(2): 337-356. doi: 10.3934/mbe.2015.12.337

Parameter estimation of social forces in pedestrian dynamics models via a probabilistic method

Alessandro Corbetta 1, , Adrian Muntean 2, and  Kiamars Vafayi 1,

1. 

CASA- Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands, Netherlands
2. 

CASA- Centre for Analysis, Scientific computing and Applications, ICMS - Institute for Complex Molecular Systems, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands

Received  April 2014 Revised  October 2014 Published  December 2014

Focusing on a specific crowd dynamics situation, including real life experiments and measurements, our paper targets a twofold aim: (1) we present a Bayesian probabilistic method to estimate the value and the uncertainty (in the form of a probability density function) of parameters in crowd dynamic models from the experimental data; and (2) we introduce a fitness measure for the models to classify a couple of model structures (forces) according to their fitness to the experimental data, preparing the stage for a more general model-selection and validation strategy inspired by probabilistic data analysis. Finally, we review the essential aspects of our experimental setup and measurement technique.
Keywords: Bayes theorem, Crowd dynamics, models classification, data analysis., parameter estimation.
Mathematics Subject Classification: Primary: 35R30, 91D10; Secondary: 62F15, 91C9.
Citation: Alessandro Corbetta, Adrian Muntean, Kiamars Vafayi. Parameter estimation of social forces in pedestrian dynamics models via a probabilistic method. Mathematical Biosciences & Engineering, 2015, 12 (2) : 337-356. doi: 10.3934/mbe.2015.12.337
References:
[1]

H. T. Banks and W. C. Thompson, Least Squares Estimation of Probability Measures in the Prohorov Metric Framework,, Center for Research in Scientific Computation Tech Rep, (): 12.   Google Scholar

[2]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1230004, 29 pp. doi: 10.1142/S0218202512300049.  Google Scholar

[3]

M. Boltes and A. Seyfried, Collecting pedestrian trajectories, Neurocomputing, 100 (2013), 127-133. doi: 10.1016/j.neucom.2012.01.036.  Google Scholar

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L. Bruno and A. Corbetta, Multiscale probabilistic evaluation of the footbridge crowding. Part 2: Crossing pedestrian position, in Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 (eds. A. Cunha, E. Caetano, P. Ribeiro and G. Müller), 2014, 937-944. Google Scholar

[5]

A. Corbetta, L. Bruno, A. Muntean and F. Toschi, High statistics measurements of pedestrian dynamics, Transportation Research Procedia, 2 (2014), 96-104; The Conference on Pedestrian and Evacuation Dynamics 2014 (PED 2014), Delft, The Netherlands, October 2014, 22-24. doi: 10.1016/j.trpro.2014.09.013.  Google Scholar

[6]

A. Corbetta, A. Muntean, F. Toschi and K. Vafayi, Structural identification of interaction terms in a Langevin-like model for crowd dynamics, in preparation, 2014. Google Scholar

[7]

R. T. Cox, Probability, frequency and reasonable expectation, American Journal of Physics, 14 (1946), 1-13. doi: 10.1119/1.1990764.  Google Scholar

[8]

R. Cox, Algebra of Probable Inference, Algebra of Probable Inference, Johns Hopkins University Press, 1961.  Google Scholar

[9]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

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C. De Boor, A Practical Guide to Splines, Vol. 27, Springer-Verlag, New York, 1978. Google Scholar

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P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, Journal of Statistical Physics, 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.  Google Scholar

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R. O. Duda, P. E. Hart and D. G. Stork, Pattern Classification, John Wiley & Sons, 2012. Google Scholar

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D. C. Duives, W. Daamen and S. P. Hoogendoorn, State-of-the-art crowd motion simulation models, Transportation Research Part C: Emerging Technologies, 37 (2013), 193-209. doi: 10.1016/j.trc.2013.02.005.  Google Scholar

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J. Evers, S. C. Hille and A. Muntean, Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions, Comptes Rendus Mathematique, 352 (2014), 51-54. doi: 10.1016/j.crma.2013.11.012.  Google Scholar

[15]

D. Helbing and A. Johansson, On the controversy around Daganzo's requiem for an Aw-Rascle's resurrection of second-order traffic flow models, Modelling and Optimisation of Flows on Networks, Lecture Notes in Mathematics, 2062 (2013), 271-302. doi: 10.1007/978-3-642-32160-3_4.  Google Scholar

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D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), pp. 4282. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

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S. Hoogendoorn and R. Hoogendoorn, Calibration of microscopic traffic-flow models using multiple data sources, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 4497-4517. doi: 10.1098/rsta.2010.0189.  Google Scholar

[18]

N. Jaklin, A. Cook and R. Geraerts, Real-time path planning in heterogeneous environments, Computer Animation and Virtual Worlds, 24 (2013), 285-295. doi: 10.1002/cav.1511.  Google Scholar

[19]

E. T. Jaynes, Prior probabilities, Systems Science and Cybernetics, IEEE Transactions on, 4 (1968), 227-241. doi: 10.1109/TSSC.1968.300117.  Google Scholar

[20]

E. T. Jaynes, The well-posed problem, Foundations of Physics, 3 (1973), 477-492. doi: 10.1007/BF00709116.  Google Scholar

[21]

E. T. Jaynes, Probability Theory: the Logic of Science, Cambridge University Press, 2003. doi: 10.1017/CBO9780511790423.  Google Scholar

[22]

S. Kirkpatrick and M. Vecchi et al., Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.  Google Scholar

[23]

X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing, Physica A: Statistical Mechanics and its Applications, 388 (2009), 2717-2726. doi: 10.1016/j.physa.2009.03.017.  Google Scholar

[24]

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21 (1953), 1087-1092. doi: 10.1063/1.1699114.  Google Scholar

[25]

Microsoft Corp., Kinect for Xbox 360,, Redmond, ().   Google Scholar

[26]

M. Moussaid, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proceedings of the National Academy of Sciences, 108 (2011), 6884-6888. doi: 10.1073/pnas.1016507108.  Google Scholar

[27]

P. Romanczuk, M. Bär, W. Ebeling, B. Lindner and L. Schimansky-Geier, Active Brownian particles, The European Physical Journal Special Topics, 202 (2012), 1-162. doi: 10.1140/epjst/e2012-01529-y.  Google Scholar

[28]

C. Rudloff, T. Matyus and S. Seer, Comparison of different calibration techniques on simulated data, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, 2014, 657-672. doi: 10.1007/978-3-319-02447-9_55.  Google Scholar

[29]

A. Schadschneider, D. Chowdhury and K. Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier, 2010. Google Scholar

[30]

S. Seer, N. Brändle and C. Ratti, Kinects and human kinetics: A new approach for studying pedestrian behavior, Transportation Research Part C: Emerging Technologies, 48 (2014), 212-228. doi: 10.1016/j.trc.2014.08.012.  Google Scholar

[31]

D. S. Sivia, Data Analysis: A Bayesian Tutorial, Oxford University Press, 1996.  Google Scholar

[32]

J. Skilling, Probabilistic data analysis: An introductory guide, Journal of Microscopy, 190 (1998), 28-36. doi: 10.1046/j.1365-2818.1998.2780835.x.  Google Scholar

[33]

The OpenPTV Consortium, OpenPTV: Open source particle tracking velocimetry,, 2012., ().   Google Scholar

[34]

A. U. K. Wagoum, A. Seyfried and S. Holl, Modeling the dynamic route choice of pedestrians to assess the criticality of building evacuation, Advances in Complex Systems, 15 (2012), 1250029, 22 pp. doi: 10.1142/S0219525912500294.  Google Scholar

[35]

J. Willneff, A Spatio-Temporal Matching Algorithm for 3D Particle Tracking Velocimetry, Mitteilungen-Institut für Geodäsie und Photogrammetrie an der Eidgenossischen Technischen Hochschule Zürich, 2003. Google Scholar

show all references

References:
[1]

H. T. Banks and W. C. Thompson, Least Squares Estimation of Probability Measures in the Prohorov Metric Framework,, Center for Research in Scientific Computation Tech Rep, (): 12.   Google Scholar

[2]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1230004, 29 pp. doi: 10.1142/S0218202512300049.  Google Scholar

[3]

M. Boltes and A. Seyfried, Collecting pedestrian trajectories, Neurocomputing, 100 (2013), 127-133. doi: 10.1016/j.neucom.2012.01.036.  Google Scholar

[4]

L. Bruno and A. Corbetta, Multiscale probabilistic evaluation of the footbridge crowding. Part 2: Crossing pedestrian position, in Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 (eds. A. Cunha, E. Caetano, P. Ribeiro and G. Müller), 2014, 937-944. Google Scholar

[5]

A. Corbetta, L. Bruno, A. Muntean and F. Toschi, High statistics measurements of pedestrian dynamics, Transportation Research Procedia, 2 (2014), 96-104; The Conference on Pedestrian and Evacuation Dynamics 2014 (PED 2014), Delft, The Netherlands, October 2014, 22-24. doi: 10.1016/j.trpro.2014.09.013.  Google Scholar

[6]

A. Corbetta, A. Muntean, F. Toschi and K. Vafayi, Structural identification of interaction terms in a Langevin-like model for crowd dynamics, in preparation, 2014. Google Scholar

[7]

R. T. Cox, Probability, frequency and reasonable expectation, American Journal of Physics, 14 (1946), 1-13. doi: 10.1119/1.1990764.  Google Scholar

[8]

R. Cox, Algebra of Probable Inference, Algebra of Probable Inference, Johns Hopkins University Press, 1961.  Google Scholar

[9]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[10]

C. De Boor, A Practical Guide to Splines, Vol. 27, Springer-Verlag, New York, 1978. Google Scholar

[11]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, Journal of Statistical Physics, 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.  Google Scholar

[12]

R. O. Duda, P. E. Hart and D. G. Stork, Pattern Classification, John Wiley & Sons, 2012. Google Scholar

[13]

D. C. Duives, W. Daamen and S. P. Hoogendoorn, State-of-the-art crowd motion simulation models, Transportation Research Part C: Emerging Technologies, 37 (2013), 193-209. doi: 10.1016/j.trc.2013.02.005.  Google Scholar

[14]

J. Evers, S. C. Hille and A. Muntean, Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions, Comptes Rendus Mathematique, 352 (2014), 51-54. doi: 10.1016/j.crma.2013.11.012.  Google Scholar

[15]

D. Helbing and A. Johansson, On the controversy around Daganzo's requiem for an Aw-Rascle's resurrection of second-order traffic flow models, Modelling and Optimisation of Flows on Networks, Lecture Notes in Mathematics, 2062 (2013), 271-302. doi: 10.1007/978-3-642-32160-3_4.  Google Scholar

[16]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), pp. 4282. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[17]

S. Hoogendoorn and R. Hoogendoorn, Calibration of microscopic traffic-flow models using multiple data sources, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368 (2010), 4497-4517. doi: 10.1098/rsta.2010.0189.  Google Scholar

[18]

N. Jaklin, A. Cook and R. Geraerts, Real-time path planning in heterogeneous environments, Computer Animation and Virtual Worlds, 24 (2013), 285-295. doi: 10.1002/cav.1511.  Google Scholar

[19]

E. T. Jaynes, Prior probabilities, Systems Science and Cybernetics, IEEE Transactions on, 4 (1968), 227-241. doi: 10.1109/TSSC.1968.300117.  Google Scholar

[20]

E. T. Jaynes, The well-posed problem, Foundations of Physics, 3 (1973), 477-492. doi: 10.1007/BF00709116.  Google Scholar

[21]

E. T. Jaynes, Probability Theory: the Logic of Science, Cambridge University Press, 2003. doi: 10.1017/CBO9780511790423.  Google Scholar

[22]

S. Kirkpatrick and M. Vecchi et al., Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.  Google Scholar

[23]

X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing, Physica A: Statistical Mechanics and its Applications, 388 (2009), 2717-2726. doi: 10.1016/j.physa.2009.03.017.  Google Scholar

[24]

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21 (1953), 1087-1092. doi: 10.1063/1.1699114.  Google Scholar

[25]

Microsoft Corp., Kinect for Xbox 360,, Redmond, ().   Google Scholar

[26]

M. Moussaid, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proceedings of the National Academy of Sciences, 108 (2011), 6884-6888. doi: 10.1073/pnas.1016507108.  Google Scholar

[27]

P. Romanczuk, M. Bär, W. Ebeling, B. Lindner and L. Schimansky-Geier, Active Brownian particles, The European Physical Journal Special Topics, 202 (2012), 1-162. doi: 10.1140/epjst/e2012-01529-y.  Google Scholar

[28]

C. Rudloff, T. Matyus and S. Seer, Comparison of different calibration techniques on simulated data, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, 2014, 657-672. doi: 10.1007/978-3-319-02447-9_55.  Google Scholar

[29]

A. Schadschneider, D. Chowdhury and K. Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier, 2010. Google Scholar

[30]

S. Seer, N. Brändle and C. Ratti, Kinects and human kinetics: A new approach for studying pedestrian behavior, Transportation Research Part C: Emerging Technologies, 48 (2014), 212-228. doi: 10.1016/j.trc.2014.08.012.  Google Scholar

[31]

D. S. Sivia, Data Analysis: A Bayesian Tutorial, Oxford University Press, 1996.  Google Scholar

[32]

J. Skilling, Probabilistic data analysis: An introductory guide, Journal of Microscopy, 190 (1998), 28-36. doi: 10.1046/j.1365-2818.1998.2780835.x.  Google Scholar

[33]

The OpenPTV Consortium, OpenPTV: Open source particle tracking velocimetry,, 2012., ().   Google Scholar

[34]

A. U. K. Wagoum, A. Seyfried and S. Holl, Modeling the dynamic route choice of pedestrians to assess the criticality of building evacuation, Advances in Complex Systems, 15 (2012), 1250029, 22 pp. doi: 10.1142/S0219525912500294.  Google Scholar

[35]

J. Willneff, A Spatio-Temporal Matching Algorithm for 3D Particle Tracking Velocimetry, Mitteilungen-Institut für Geodäsie und Photogrammetrie an der Eidgenossischen Technischen Hochschule Zürich, 2003. Google Scholar

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