Particle Methods for High- Dimensional Traffic Estimation Problems Mila Mihaylova 1 1 Lancaster University, United Kingdom Collaborative work with Rene.

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

Outline

I. Motivation

II. Parallelised Particle Filters for

Traffic Flow Estimation

III. Performance Evaluation

IV. Conclusions and Open Issues for

Future Research

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).

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

* Distributed estimation over space and time

* Develop efficient traffic control methods

The Fundamental Diagram

0 20 40 60 80 100 120 140 160 180500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500F

low

, [ve

h/h]

Density, [veh/km]

crit

jam

Traffic Flow and Measurements

Li,k

Segment i

i,k, vi,ki-1,k, vi-1,k i+1,k, vi+1,k

qi-1,k qi,k

Sensor measurements in tk ts

1 2 i-1 i i+1 n-1 n

qk in, vk

in qk out, vk

out

n+1

z1,szj,s zm,s

L vmax t

Two types of states:- inside segments (speed and density of vehicles)- inflow/ outflow (boundary conditions)

Results from Modelling. Comparison with Real Data

Real data from the video cameras Results from the developed compositional model

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

kk zzzZ ,...,, 21

11111 )/()/()/( kkkkkkk dxZxpxxpZxp

)/(

)/()/(/

1

1

kk

kkkkkk Zzp

ZxpxzpZxp

)/( kk Zxp

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.

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.

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

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

Centralised Approach

• Global states and weights

• Communications only for measurements

Approach I: Shared Particles

• Functionally equivalent to the centralised PF, but calculations are distributed over several processing units.

• Communication of states over boundaries

• Communication of weights to a central unit when resampling is necessary.

Approach II: Separate Particles

• Neighbour combination: based on weights• Communication of neighbouring states over the

boundaries,• No need of central unit for resampling.

Centralised Particle Filter

The posterior density at k is approximated as:

Centralised Particle Filter

• Typically

The state and measurement vectors are partitioned into S subvectors

Partitioning the Traffic Network into Subnetworks

The vector collects all neighbouring state variables that act as an input to subnetwork s.

Assumptions:

• Not all states of the neighbouring networks are communicated, only the variables that serve as an input to subnetwork s.

• Measurements in a subnetwork depend only on the state in that subnetwork.

• Independent state noises between the subnetworks

• Independent measurement noises between the networks

Partitioning the Traffic Network into Subnetworks

Boundary states

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)

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

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.

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

• 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.

Approach II

Applying Monte Carlo sampling to the product

with a proposal distribution results in the approximation

: state variables at the boundaries

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

Approach II

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.

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

Scenario

METANET Traffic Model

Law of conservation of vehicles

Results: Accuracy

Scenario with the shock wave, 500 particles in the PFs

CPU Time vs Number of ParticlesApproach 1

Approach 2

Results: Communications

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

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

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

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

Thank you for your

attention !