Parameter estimation for pedestrian dynamics models Susana Gomes Mathematics Institute, University of Warwick Joint work with A.M. Stuart (Caltech) and M.-T. Wolfram (Warwick) Crowds: Models and Control CIRM, Marseille, 5 th of June 2019
Motivation We use a simple model exploring a coupling between - a microscopic model for the trajectories of pedestrians (data from experiments 1 ), and - a macroscopic model for the density of pedestrians to identify and estimate quantities of interest for the dynamics of the crowd. hello world 1BaSiGo experiments (Forschungszentrum Jülich) 2Figure from Chattaraj, Seyfried, Chakroborty, Advances in Complex Systems, 2009. 1 / 17
Motivation We use a simple model exploring a coupling between - a microscopic model for the trajectories of pedestrians (data from experiments 1 ), and - a macroscopic model for the density of pedestrians to identify and estimate quantities of interest for the dynamics of the crowd. hello world hello world hello world In particular, we would like to hello world - estimate parameters appearing in the fundamental diagram 2 , or - learn the functional form of the fundamental diagram, and - explore whether we can use the fundamental diagram for the time dependent evolution of crowds. 1BaSiGo experiments (Forschungszentrum Jülich) 2Figure from Chattaraj, Seyfried, Chakroborty, Advances in Complex Systems, 2009. 1 / 17
Our model: The microscopic equation Individual trajectories are realizations of a generalized Γ N McKean–Vlasov equation ( 0 , ℓ ) ( L , ℓ ) √ d X t = F ( ρ ( X t , t )) dt + 2 Σ d W t . F ( ρ ( x , t )) Γ in Γ out - W ( · ) is a unit Brownian motion - Σ is a diffusion tensor ( 0 , − ℓ ) ( L , − ℓ ) Γ N - ρ is the density of the process � � 1 − ρ ( x , t ) - F ( ρ ( x , t )) = v max e 1 (from ρ max fundamental diagram). The microscopic equation depends on the density of the process. 2 / 17
Our model: The macroscopic equation Γ N The corresponding Fokker–Planck equation is ( 0 , ℓ ) ( L , ℓ ) � � � � ρ ρ t = ∇ · Σ ∇ ρ − v max 1 − ρ , F ( ρ ( x , t )) ρ max Γ in Γ out ( 0 , − ℓ ) ( L , − ℓ ) Γ N with boundary conditions: ( − Σ ∇ ρ + F ( ρ ) ρ ) · n = − a ( ρ max − ρ ) , for x ∈ Γ in , ( − Σ ∇ ρ + F ( ρ ) ρ ) · n for x ∈ Γ out , = b ρ, ( − Σ ∇ ρ + F ( ρ ) ρ ) · n = 0 , for x ∈ Γ N . where n is the unit normal, a is the inflow rate and b the outflow rate . 3 / 17
The Fokker-Planck equation rescaled ρ By defining ˜ ρ = ρ max , we obtain a rescaled SDE-PDE system: √ d X t = v max ( 1 − ˜ ρ ) dt + 2 Σ d W t , ˜ ρ t = ∇ · (Σ ∇ ˜ ρ − v max ˜ ρ ( 1 − ˜ ρ )) , with the boundary conditions ˜ j · n = − a ( 1 − ˜ ρ ) , for x ∈ Γ in , ˜ j · n = b ˜ for x ∈ Γ out , ρ, ˜ j · n = 0 , for x ∈ Γ N . and where ˜ j = ( − Σ ∇ ˜ ρ + v max ( 1 − ˜ ρ )˜ ρ ) . 4 / 17
The Fokker-Planck equation rescaled ρ By defining ˜ ρ = ρ max , we obtain a rescaled SDE-PDE system: √ d X t = v max ( 1 − ˜ ρ ) dt + 2 Σ d W t , ˜ ρ t = ∇ · (Σ ∇ ˜ ρ − v max ˜ ρ ( 1 − ˜ ρ )) , with the boundary conditions ˜ j · n = − a ( 1 − ˜ ρ ) , for x ∈ Γ in , ˜ j · n = b ˜ for x ∈ Γ out , ρ, ˜ j · n = 0 , for x ∈ Γ N . and where ˜ j = ( − Σ ∇ ˜ ρ + v max ( 1 − ˜ ρ )˜ ρ ) . ρ max does not influence the dynamics in the rescaled SDE-PDE system! ⇒ we cannot estimate ρ max using this model. 4 / 17
Rich behaviour of solutions Steady solutions of the FP equation are influenced by a and b . 3 3Burger and Pietschmann, Nonlinearity, 2016. 5 / 17
Rich behaviour of solutions Steady solutions of the FP equation are influenced by a and b . 3 Properties of solution Solution in 1D ⋆ Maximal current if min( a , b ) ≥ v max 2 ; ρ = 1 asymptotic density ˜ 2 and boundary layers at both entrance and exit. 3Burger and Pietschmann, Nonlinearity, 2016. 5 / 17
Rich behaviour of solutions Steady solutions of the FP equation are influenced by a and b . 3 Properties of solution Solution in 1D ⋆ Maximal current if min( a , b ) ≥ v max 2 ; ρ = 1 asymptotic density ˜ 2 and boundary layers at both entrance and exit. ⋆ Influx limited if a < b and a < v max 2 ; ρ < 1 asymptotic low ˜ 2 and boundary layer at the exit. 3Burger and Pietschmann, Nonlinearity, 2016. 5 / 17
Rich behaviour of solutions Steady solutions of the FP equation are influenced by a and b . 3 Properties of solution Solution in 1D ⋆ Maximal current if min( a , b ) ≥ v max 2 ; ρ = 1 asymptotic density ˜ 2 and boundary layers at both entrance and exit. ⋆ Influx limited if a < b and a < v max 2 ; ρ < 1 asymptotic low ˜ 2 and boundary layer at the exit. ⋆ Outflux limited if b < a and b < v max 2 . ρ > 1 asymptotic high ˜ 2 and boundary layer at the entrance. 3Burger and Pietschmann, Nonlinearity, 2016. 5 / 17
❘ The time dependent PDE The Fokker–Planck equation has a gradient flow structure with respect to the entropy � E ( ρ ) = ( ρ log ρ + ( 1 − ρ ) log( 1 − ρ ) − ρ V dx , Ω with V = x . 6 / 17
❘ The time dependent PDE The Fokker–Planck equation has a gradient flow structure with respect to the entropy � E ( ρ ) = ( ρ log ρ + ( 1 − ρ ) log( 1 − ρ ) − ρ V dx , Ω with V = x . Using this gradient structure we can - Deduce the necessary a-priori estimates to prove global-in-time existence of weak solutions. - These solutions verify ρ ∈ L 2 ( 0 , T ; H 1 (Ω)) . - Using bootstrapping arguments, we can extend the proof to obtain solutions ρ ∈ C (¯ Ω × [ 0 , T ]) . 6 / 17
The time dependent PDE The Fokker–Planck equation has a gradient flow structure with respect to the entropy � E ( ρ ) = ( ρ log ρ + ( 1 − ρ ) log( 1 − ρ ) − ρ V dx , Ω with V = x . Using this gradient structure we can - Deduce the necessary a-priori estimates to prove global-in-time existence of weak solutions. - These solutions verify ρ ∈ L 2 ( 0 , T ; H 1 (Ω)) . - Using bootstrapping arguments, we can extend the proof to obtain solutions ρ ∈ C (¯ Ω × [ 0 , T ]) . Further extensions on these results would allow us to prove existence and uniqueness of strong solutions for the SDE if it was posed in ❘ 2 . 6 / 17
Parameter estimation methodology Goal: Estimate v max given a set of trajectories. 7 / 17
Parameter estimation methodology Goal: Estimate v max given a set of trajectories. Finding the best value of v given an observation of a trajectory X corresponds to minimizing the function � T Φ( v ; X ) = 1 | ˙ X − F ( ρ ( X ( t ) , t ); v ) | 2 Σ , 4 0 over all values of v . 7 / 17
Parameter estimation methodology Goal: Estimate v max given a set of trajectories. Finding the best value of v given an observation of a trajectory X corresponds to minimizing the function � T Φ( v ; X ) = 1 | ˙ X − F ( ρ ( X ( t ) , t ); v ) | 2 Σ , 4 0 over all values of v . However, Φ( v ; · ) is almost surely infinite. 7 / 17
Parameter estimation methodology Goal: Estimate v max given a set of trajectories. Finding the best value of v given an observation of a trajectory X corresponds to minimizing the function � T Φ( v ; X ) = 1 | ˙ X − F ( ρ ( X ( t ) , t ); v ) | 2 Σ , 4 0 over all values of v . However, Φ( v ; · ) is almost surely infinite. But we can write � T Φ( v ; X ) = Ψ( v ; X ) + 1 X | 2 dt , | ˙ 4 0 where the function Ψ is defined as � T Ψ( v ; X ) = 1 � � | F ( ρ ( X ( t ) , t ); v ) | 2 Σ dt − 2 � F ( ρ ( X ( t ) , t ); v ) , dX ( t ) � Σ . 4 0 Ψ( v , X ) is the log-likelihood function of this process... ... which we can obtain it due to the PDE-SDE coupling in our model. 7 / 17
✶ P P Parameter estimation methodology We will include prior information and consider instead the functional J ( v ; X t ) = Ψ( v ; X t ) + 1 2 | v − m | 2 c . 8 / 17
P Parameter estimation methodology We will include prior information and consider instead the functional J ( v ; X t ) = Ψ( v ; X t ) + 1 2 | v − m | 2 c . Using Bayes’ and Girsanov’s theorems, we can show that the function e −J ( v ; X t ) ✶ ( v > 0 ) , appropriately normalised, is the posterior distribution P ( v | X ) of v given the observation X . 8 / 17
Parameter estimation methodology We will include prior information and consider instead the functional J ( v ; X t ) = Ψ( v ; X t ) + 1 2 | v − m | 2 c . Using Bayes’ and Girsanov’s theorems, we can show that the function e −J ( v ; X t ) ✶ ( v > 0 ) , appropriately normalised, is the posterior distribution P ( v | X ) of v given the observation X . Note that - Minimising Ψ gives the maximum likelihood estimator (MLE). - Minimising J gives the maximum a posteriori estimator (MAP). - Ψ measures the misfit between the data and the model. - Sampling from P ( v | X ) corresponds to the Bayesian methodology. 8 / 17
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