Particle Methods for High-Dimensional Traffic Estimation Problems

1. Particle Methods for High-Dimensional Traffic Estimation Problems Mila Mihaylova1 1 Lancaster University, United Kingdom Collaborative work with Rene Boel 2, Andreas Hegiy 3 2University of Ghent, Belgium 3 Delft University, the Netherlands

2. Outline • Motivation • Parallelised Particle Filters for Traffic Flow Estimation • Performance Evaluation • Conclusions and Open Issues for Future Research

3. Motivation • Traffic: complex nonstationary, nonlinear behaviour, with different modes such as: free flow motion, congestions, stop-and-go waves. • Changes are due to the traffic dynamics, or external events (e.g. accidents, road works, weather conditions).

4. Traffic Flow Problems of Interest * Analysis of the accuracy of sensor data(from video cameras and magnetic detectors) * Build up traffic and sensor models of traffic on motorways and in urban environment * Develop traffic models for adversary weather conditions *Distributedestimation overspace and time * Develop efficient traffic control methods

5. 5500 5000 Flow, [veh/h] 4500 4000 3500 3000 2500 r r crit jam 2000 1500 Density, [veh/km] 1000 500 0 20 40 60 80 100 120 140 160 180 The Fundamental Diagram

6. Sensor measurements in tk ts qkin, vkin qk out, vk out 1 2 n-1 n n+1 i-1 i+1 i zj,s zm,s z1,s Segment i qi-1,k qi,k i-1,k, vi-1,k i,k, vi,k i+1,k, vi+1,k Li,k Traffic Flow and Measurements L vmax t Two types of states: - inside segments (speed and density of vehicles) - inflow/ outflow (boundary conditions)

7. Results from Modelling. Comparison with Real Data Results from the developed compositional model Real data from the video cameras

8. Traffic State Estimation Within Bayesian Framework • The posterior state probability density function (PDF) is estimated given a data set • The sensor information updates recursively the state distribution. Prediction : Update : The conditional state PDF is represented as a set of random samples which are updated and propagated by a particle filter

9. Parallelised Particle Filtering for Freeway Traffic State Estimation A. Hegyi, L. Mihaylova, R. Boel and Z. Lendek, Parallelized Particle Filtering for Freeway Traffic State Tracking, Proc. of the European Control Conf., Greece, 2007, TuD15.3, pp. 2442-2449 L. Mihaylova, R. Boel, A. Hegyi, Freeway Traffic Estimation within Recursive Bayesian Framework, Automatica, 2007, Vol. 43, No. 2, pp. 290-300, February.

10. Parallelised Particle Filters for Freeway Traffic State Estimation Aims: • Cope with the high computational demands. • For traffic state estimation the required number of particles grows exponentially with network size. • Achieve: • high accuracy • deal with nonlinearities and non-Gaussian processes Approach: Parallelise the traffic network • Why parallelisation is possible: • A traffic network can be simulated in parallel (limited interaction at subnetwork boundaries), • Measurements are related to local states.

11. Related Works • M. Bolic, P.M. Djuric, and S. Hong, Resampling Algorithms and Architectures for Distributed Particle Filters, IEEE Trans. Signal Processing, 53:2442-2450, 2005. • C. Coates. Distributed Particle Filtering for Sensor Networks, Proc. of the Int. Symp. Information Processing in Sensor Networks, Berkeley, California, April 2004. • S. Maskell, K. Weekes, and M. Briers, Distributed tracking of stealthy targets using particle Filters, Proc. of IEE Seminar on Target Tracking: Algorithms and Applications, pages 13-20. IEE, Birmingham, UK, March 2006. • X. Sheng, Y. H. Hu, and P. Ramanathan. Distributed particle Filter with GMM Approximation for Multiple Targets Localization and Tracking in Wireless Sensor Network, Proc. of the 4th Intl. Conf. on Information Processing in Sensor Networks (IPSN), pages 181-188, 2005. • A. S. Bashi, V. P. Jilkov, X. R. Li, H. Chen, “Distributed implementations of particle filters,” Proc. of the 2003 International Conf. Information Fusion, Australia, 2003. • Algorithms transmitting: • particles and their weights between processing units (PUs) • communicating a parametric approximation

12. Main Idea: Partition the Traffic Network into Subnetworks • Applicable: when the whole state vector can be partitioned into subsets of states and most interactions are within the subsets • A traffic network can be simulated in parallel • Divide the traffic network into several sub-networks where each PU is responsible for one sub-network and the relevant variables of the neighbouring segments are communicated

13. Centralised Approach • Global states and weights • Communications only for measurements

14. Approach I: Shared Particles • The same as the centralised particle filter, but calculations are distributed over several processing units. • Communication of states over boundaries • Communication of weights to a central unit when resampling is necessary.

15. Approach II: Separate Particles • Neighbour combination: based on weights • Communication of neighbouring states over the boundaries, • No need of central unit for resampling. • Approximates joint probabilities of neighboring states by assuming independence.

16. Centralised Particle Filter The posterior density at k is approximated as:

17. Centralised Particle Filter • Typically

18. Partitioning the Traffic Network into Subnetworks The state and measurement vectors are partitioned into S subvectors The vector collects all neighbouring state variables that act as an input to subnetwork s.

19. Partitioning the Traffic Network into Subnetworks Assumptions: • Not all states of the neighbouring networks are communicated, only the variables that serve as an input to subnetwork s. • The measurements taken in a subnetwork depend only on the state in that subnetwork. • Independent state noises between the subnetworks • Independent measurement noises between the networks Boundary states

20. Approach I: Shared Particles • PUs of different subnetworks share the same particles . Particles are partitioned into subparticles for each subnetwork s. • The PU of subnetwork s is responsible for the calculation of subparticles • Approach I: equivalent to the centralised approach if the conditions of independence (for the noises) hold • In the state update step, the subparticles are drawn from a distribution which is based on local information only (including the neighbour states)

21. Approach I: Shared Particles • Choosing the proposal distribution such that using the independence conditions and the fact that the weight update equation can be written as

22. Approach I: Shared Particles • State and measurement update: performed locally (divided over S processors) • The weights can be calculated locally and only the result is communicated to the central PU to determine • The centrally calculated weights are normalised and sent back to the local PUs (after resampling) • Resampling: • for the residual and systematic resampling: not need to communicate particles, only weights since these methods use only weights as inputs. After resampling, only indices are communicated back to the PUs.

23. Approach II • There is no central PU • Communications only between the neighbouring PUs: statistics of neighbouring states is exchanged Advantages of Approach II over Approach I • Requires less particles: the dimension of the state space is reduced by a factor S (if all subnetworks have the same number of states). • For each subnetwork a different number of particles can be used Disadvantage of Approach II • An approximation is introduced in the interaction (joint pdf) of the local states with the states in neighbouring subnetworks.

24. Approach II Applying Monte Carlo sampling to the product with a proposal distribution results in the approximation : state variables at the boundaries

25. Approach II • By assumption the pdf of the communicated state variables is independent on and then • Taking one sample from for each i and choosing

28. Experimental Setup • Motorway with a traffic jam • Research questions: • Compare the centralised filter and approaches 1 and 2 for several numbers of particles • Tracking accuracy • Computational complexity (CPU time) • Communication • Each test executed 10 times.

29. Experimental Setup • Two links, two lanes, 10 segments in each link; • Measurements: at segments 1 and 10 every minute • State update step: 10 seconds • Boundary conditions estimated as part of the state vector • Gaussian noises • State vector = [ states, boundary states] • METANET model for state update

30. Scenario

31. METANET Traffic Model Law of conservation of vehicles

32. Results: Accuracy Scenario with the shock wave, 500 particles in the PFs

33. CPU Time vs Number of Particles Approach 1 Approach 2

34. Results: Communications Number of communicated doubles (real numbers) for each approach as a function of the number of particles I:

35. Conclusions • Two parallelised particle filters are developed for traffic state estimation • The centralised and the parallelised approaches compared for: • estimation accuracy • computational complexity • communication needs • Performance of Approach I : similar to the centralised approach w.r.t accuracy, slightly less computational load • Approach II is less computationally complex than Approach I • Approach II: gives more accurate results than the centralised PF, less CPU time • Approach II is superior than the other PFs • Approaches I and II: need more communications than the centralised

36. Conclusions and Future Work • The presented approach for parallelisation is in general applicable to systems where it is possible: • to partition the overall state into subsets of states, • such that most of the interactions take place within the subsets. • Fusion of sensor data from different modalities (e.g., from radars and video cameras) • Open issues: • distributed estimation • algorithms robust to missing data and sensor failures • what is the optimal configuration of the detectors (optimal sensor placement) • Modelling traffic to reflect different weather conditions

37. Related Works • L. Mihaylova, R. Boel, A. Hegyi, Freeway Traffic Estimation within Recursive Bayesian Framework, Automatica, 2007, Vol. 43, No. 2, pp. 290-300. • L. Mihaylova, R. Boel, A Particle Filter for Freeway Traffic Estimation, Proc. 43rd IEEE Conf. on Decision & Control, 2004, pp. 2106-2111. • L. Mihaylova, R. Boel, A. Hegyi, An Unscented Kalman Filter for Freeway Traffic Estimation, Proc. of the 11th IFAC Symposium on Control in Transportation Systems, The Netherlands, pp. 31-36, 2006 • A. Hegyi, L. Mihaylova, R. Boel, Z. Lendek, Parallelized Particle Filtering for Freeway Traffic State Tracking, Proc. of the European Control Conference, Kos, Greece, 2-5 July 2007, TuD15.3, pp. 2442-2449

38. Thank you for your attention  !