Peng Cheng, Member, IEEE, Zhijun Qiu, and Bin Ran Presented By: Guru Prasanna Gopalakrishnan

Overview Background- Where it fits? Problem Formulation Traffic Models First Order Traffic Model Second Order Traffic Model Particle Filter Design Experimental Results Conclusion

Introduction-I Traffic time and congestion information valued by road users and road system managers 1 Applications- Incident detection, Traffic management, Traveler information, Performance monitoring Two approaches to collect real-time traffic data - Fixed Sensors - Mobile Sensors

Introduction-II Fixed Sensor System - Inductive loops, Radar, etc - Real-time information collection - Dense Sampling technique Mobile Sensor System - Handset Based Solutions - Network Based Solutions - Sparse Sampling Technique

Problem Formulation- I Key Points: - Microcells of similar size - Randomization of Handoff points

Problem Formulation II Definitions: - H=(ID cell phone, t handoff, Cell from, Cell to ) - Handoff pair

Traffic Model Traffic flow modeled as stochastic dynamic system with discrete- time states State Variable: - x i,k = {N i,k, i,k } T Generic model of system state evolution - x k +1 =f k (x k, w k ) - f k is system transition function and w k is the system noise - y k =h k (x k,  k ) - h k is measurement function and  k measurement error

Important Terminologies N i,k -Number of vehicles in section I at sampling time t k i,k -Average speed of the vehicles Q, i,k - number of vehicles crossing the cell boundary from link i to link i+1 during the time interval k i,k,  - Constant and scale co-efficient respectively i,t, e - Intermediate speed and equilibrium speed  i,t,  crit - Anticipated traffic density and critical density S i,t, R i+1,t - Sending and Receiving functions respectively

First-order Traffic Model Traffic speed is the only state variable System State Equation:  i,k+1 =  i,k i-1,k +  i,k i,k +  i,k i+1,k + w i,k i=1,2,3,….n Measurement Equation:  y i,k = avg i,k +  k i=1,2,3,….n - avg i,k = L i /(t j + -t j - ) - For stable road-traffic,  i,k+  i,k +  i,k =1

Second Order Traffic Model-I Traffic volume is the second state variable Macroscopic level- System State Equation:  Q i,k+1 = U i,t + W 1 i,k  V i,k+1 = (1/ Ʈ k ) i,t + w 2 i,k i=1,2…n; k=1,2,….K Macroscopic level- Measurement Equation:  Y 1 i,k = (1/ i,k ) Q i,k. e -  L i / v i,k +  1 i,k  Y 2 i,k = V i,k +  2 i,k i=1,2…n; k=1,2,….K Note:  N i,k+1 =N i,k + Q i-1,k -Q i,k

Second Order Traffic Model-II Microscopic System State Equation:  N i,t+1 =N i,t + U i-1,t -U i,t  i,t+1 =  i,t+1 + (1-  ) e (  i,t+1 ) + w 3 i,t   [0,1] Where - i,t+1 = { ( i-1,t Q i-1,t + i,t (N i,t - Q i,t ))/N i,t+1 N i,t+1 K 0 { free o.w -  i,t+1 =   i,t+1 + (1-  )  i+1,t+1   [0,1] - N i,t =  i,t.L i i=1,2,…n

Second Order Traffic Model-III Microsocopic System State Equation Contd… - e (  )={ free.e -(0.5)(  /  crit) 3.5 if  <=  crit { free.e -(0.5)(  -  crit) Otherwise - U i,t = min(S i,t, R i,t+1 ) - S i,t = max(N i,t.( i,t  t)/L i + W 4 i,t, N i,t (V out,min  t)/L i ) - R i+1,t= (L i+1.l/A l )+U i+1,t - N i+1,t

State Transition and Reconstruction Y 1 i,k Y 1 i,k+1 Y 2 i,k Y 2 i,k+1 State Transition Macroscopic Level Microscopic level State Reconstruction Q i,k V i,k,t Q i,k+1 V i,k+1,t U i,1 i,1 U i,3 i,3 U i,2 i,2 U i,k i, Ʈ k

Particle Filter- Why? I Bayesian estimation to construct conditional PDF of the current state x k given all available information Y k = {y j j=1,2,…..k} Two steps used in construction of p(x k /Y k ) 1) Prediction p(x k /Y k-1 )= f p(x k /X k-1 ) p(x k-1 /Y k-1 ) dx k-1 and 2) Updation p(x k /Y k )= p(Y k /X k ) p(x k-1 /Y k-1 ) / p(Y k /Y k-1 )

Particle Filter-Why? II p( Y k /Y k-1 ) – A normalized constant p(x k / X k-1 )= fc (x k -f k-1 ( X k-1, W k-1 ))p( W k-1 ) dw k-1 - p(W k ) is PDF of noise term in system equation p(Y k / X k )= fc (Y k -h k ( X k,  k ))p(  k ) d  k - p(  k ) is PDF of noise term in measurement equation - c (.) is dirac delta function

Particle Filter- Why? III No Simple analytical solution for p(x k /Y k ) Particle filter is used to find an approximate solution by empirical histogram corresponding to a collection of M particles

Particle Filter Implementation-I Step 1: Initialization For l=1,2,….M, Sample x 0 (l) ~ p(x 0 ) q 0 = 1/M set K=1 Step 2: Prediction For l=1,2,….M, Sample x k (l) ~ p(x k / x k-1) Step 3: Importance Evaluation For l=1,2,….M, q k = (p(y k / x k ) q (l) k-1 / ( p(y k / x (j) k ) q (j) k-1

Particle Filter Implementation-II Selection - Multiple/suppress M particles {x k (l) } according to their importance weights and obtain new M unweighted particles. Output P(x k /Y k )= q k (l).c (x k -x k (l) ) Posterior mean, x k =E(x k /Y k )=(1/M ) x k (l) Posterior Co-Variance, V(x k /Y k )=(1/(M-1) ) (x k (l) -x k ) (x k (l) -x k ) T Last Step - Let k=k+1 and Goto Step-2

Experimental Results-I

Experimental Results-II

Experimental Results-III

Experimental Results-IV

Conclusion Implemented using an existing infrastructure Some Critiques Interference due to parallel freeways 2 Cannot differentiate between pedestrians and moving vehicles Some Unrealistic assumptions

References 1. G. Rose, Mobile phones as traffic probes, Technical Report, Institute of Transportation Studies, Monash University, L. Mihaylova and R. Boel, “A Particle Filter for Freeway Traffic Estimation,” Proc. of 43rd IEEE Conference on Decision and Control, Vol. 2, pp , 2004.