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

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.
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, CRSC-TR12-21, North Carolina State University, Raleigh, NC.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

Microsoft Corp., Kinect for Xbox 360, Redmond, WA, USA.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

The OpenPTV Consortium, OpenPTV: Open source particle tracking velocimetry, 2012.

[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.

[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.

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, CRSC-TR12-21, North Carolina State University, Raleigh, NC.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

Microsoft Corp., Kinect for Xbox 360, Redmond, WA, USA.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

The OpenPTV Consortium, OpenPTV: Open source particle tracking velocimetry, 2012.

[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.

[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.

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