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).  doi: 10.1142/S0218202512300049.  Google Scholar

[3]

M. Boltes and A. Seyfried, Collecting pedestrian trajectories,, Neurocomputing, 100 (2013), 127.  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, (2014), 937.   Google Scholar

[5]

A. Corbetta, L. Bruno, A. Muntean and F. Toschi, High statistics measurements of pedestrian dynamics,, Transportation Research Procedia, 2 (2014), 96.  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.  doi: 10.1119/1.1990764.  Google Scholar

[8]

R. Cox, Algebra of Probable Inference,, Algebra of Probable Inference, (1961).   Google Scholar

[9]

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

[10]

C. De Boor, A Practical Guide to Splines,, Vol. 27, (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.  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.  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.  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, 2062 (2013), 271.  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).  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, 368 (2010), 4497.  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.  doi: 10.1002/cav.1511.  Google Scholar

[19]

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

[20]

E. T. Jaynes, The well-posed problem,, Foundations of Physics, 3 (1973), 477.  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.  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.  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.  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.  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.  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, (2012), 657.  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.  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.  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).  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).  doi: 10.1142/S0218202512300049.  Google Scholar

[3]

M. Boltes and A. Seyfried, Collecting pedestrian trajectories,, Neurocomputing, 100 (2013), 127.  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, (2014), 937.   Google Scholar

[5]

A. Corbetta, L. Bruno, A. Muntean and F. Toschi, High statistics measurements of pedestrian dynamics,, Transportation Research Procedia, 2 (2014), 96.  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.  doi: 10.1119/1.1990764.  Google Scholar

[8]

R. Cox, Algebra of Probable Inference,, Algebra of Probable Inference, (1961).   Google Scholar

[9]

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

[10]

C. De Boor, A Practical Guide to Splines,, Vol. 27, (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.  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.  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.  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, 2062 (2013), 271.  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).  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, 368 (2010), 4497.  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.  doi: 10.1002/cav.1511.  Google Scholar

[19]

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

[20]

E. T. Jaynes, The well-posed problem,, Foundations of Physics, 3 (1973), 477.  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.  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.  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.  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.  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.  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, (2012), 657.  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.  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.  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).  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

[1]

Sebastien Motsch, Mehdi Moussaïd, Elsa G. Guillot, Mathieu Moreau, Julien Pettré, Guy Theraulaz, Cécile Appert-Rolland, Pierre Degond. Modeling crowd dynamics through coarse-grained data analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1271-1290. doi: 10.3934/mbe.2018059
[2]

Robert Azencott, Yutheeka Gadhyan. Accurate parameter estimation for coupled stochastic dynamics. Conference Publications, 2009, 2009 (Special) : 44-53. doi: 10.3934/proc.2009.2009.44
[3]

Jiang Xie, Junfu Xu, Celine Nie, Qing Nie. Machine learning of swimming data via wisdom of crowd and regression analysis. Mathematical Biosciences & Engineering, 2017, 14 (2) : 511-527. doi: 10.3934/mbe.2017031
[4]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521
[5]

Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001
[6]

Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601
[7]

Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems & Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053
[8]

Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1
[9]

Nicola Bellomo, Abdelghani Bellouquid. On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Networks & Heterogeneous Media, 2011, 6 (3) : 383-399. doi: 10.3934/nhm.2011.6.383
[10]

Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221
[11]

Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems & Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007
[12]

Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
[13]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027
[14]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719
[15]

Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic & Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025
[16]

Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393
[17]

Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487
[18]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
[19]

Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289
[20]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences