Adaptive Kalman Filter Based Freeway Travel time Estimation

1. Adaptive Kalman Filter Based Freeway Travel time Estimation Lianyu Chu CCIT, University of California Berkeley Jun-Seok Oh Western Michigan University Will Recker University of California Irvine

2. OUTLINE • Background • Methodology • Evaluation • Sensitivity Analysis • Conclusion

3. Issues in Travel Time Estimation • Source of information • Point detection • Loop detectors • Video, microwave, etc • Probe vehicle • GPS-based • AVI • Cellular phone positioning • Vehicle re-identification • However, still need to estimate section travel times • Involves errors in estimation

4. Section Loop detectors • Dominant traffic sensors • Collect point data: volume, occupancy • Can be converted to section travel time

5. Section travel time estimate from loop detector data • A typical method: • vu and vd: station speed estimates, defined as the weighted average of lane speeds • vi , speed of lane i, can be estimated from single loop, or obtained from double loops directly • two kinds of estimation errors: • (1) speed estimation from loop detector data and • (2) travel time conversion from speed

6. Probe vehicles • Collect area-wide data: travel time • Estimationmethod: Arrival-based • Estimation error: • Low probe rate: a biased estimate with high variance • Vehicles arrived during (t-1, t) may enter the section during (t-2, t-1)

7. Representative section travel time • Travel time: preferable for some ATMIS applications • e.g. traffic information and route guidance • Representative section travel time: • mean travel time within the closed area defined by the time (t and t+1) and space (xu and xd), • Used as benchmark section travel time

8. Objective & Approach • Improve travel time estimation using • both point detection and probe vehicle data • Kalman filtering • Key issue • covariance matrices of the state and observation noises • many traffic studies • noise statistics was assumed constant • Our method: • Adaptive Kalman Filtering (AKF) • Dynamic estimation of noise statistics • On-line applications

9. Travel Time Estimation based on Section Density • Fundamental equation of traffic flow: • q=u*k • Assuming: traffic inside the section is homogeneous • Section travel time:

10. Kalman Filter for Data Fusion • State equation: • Observation equation: • Two data sources: • Traffic volume from loops • Travel time from probes • State noise, w(t): • a Gaussian noise: mean: q(t), variance: Q(t) • Observation noise, v(t): • a Gaussian noise: mean: r(t), variance: R(t)

11. Solution to Kalman Filter Problem • State propagation • Kalman gain: • State estimation

12. Adaptive Kalman Filter (AKF) problem • Limitation in applying KF: • statistics of the state and observation errors are assumed to be known • noise statistics may change with time • due to the nature of the traffic system and detection errors • {q, Q, r, R} needs to be simultaneously estimated • an empirical estimation method for AKF • proposed by Myers K.A. • simple • handle both systematic and random errors • On-line applications: a limited memory algorithm

13. Estimation of observation noise • Using latest several noise samples • An approximation of the observation noise vj: • Assuming: noise samples rj can represent vj • An unbiased estimator for sample mean and sample variance:

14. Estimation of state noise • An approximation of the state noise wj • Assuming state noise sample qj can represent wj • An unbiased estimator for sample mean and sample variance:

15. Summary of the proposed algorithm • Calculating model parameters • u(t) and H(t) for state and observation equations • State propagation • calculating a priori estimate of section density and estimation covariance • Estimating observation noise (r, R) • Updating Kalman gain • State estimation • calculating a posteriori estimate of section density and estimation covariance • Calculating the section travel time

16. Evaluation study • Simulation based evaluation: Paramics • MOE: MAPE • Study site:

17. Modeling detector errors • Detection errors of loops: • inductance may change with temperature, moisture, corrosion, and mechanical deformation • Traffic controllers and communication devices may also malfunction • Considers such errors: • α(t) represents the systematic • constant or a time-dependent value • β(t) represents random error • varies randomly between measurements • a Gaussian white noise (0, δ) • 95% is within (-2δ, 2δ)

18. System error patterns

19. Evaluation scenarios • Scenario 1: Recurrent congestion condition • Scenario 2: Traffic with an incident • Detector Errors

20. Simulation study • Implement algorithms in Paramics using API • Estimation from loop data only • Estimation from probe data only • Proposed AKF algorithm with both data • Simulation runs • from 6:30 AM to 9:00 AM • First 30 min: warm-up • Compared with the benchmark travel time in terms of MAPE

21. Performance comparisons under the recurrent traffic congestion

22. Performance comparisons under incident scenario

23. Performance comparison • Point-detector-based algorithm • not robust, showing strong fluctuations during the congestion period • Probe based algorithm • over-estimate section travel time during a certain time period after traffic congestion • AKF algorithm outperforms the other two methods • Especially, during the congestion period

24. On-line estimation of noise mean and variances • q(t), r(t) • capture the systematic errors in the state equation and observation equation. • Q and R: • capture random errors in the state equation and observation equation .

25. Sensitivity Analysis • Loop detector errors • Systematic error • Random error • Part of loop detector data missing • Performance at other sections

26. Sensitivity to constant loop detector errors

27. Sensitivity to time-dependent loop errors

28. Sensitivity to missing lane data

29. Sensitivity to random errors

30. Performance at other sections

31. Conclusion • Developed an AKF-based travel time estimation method. • Advantages: • Dynamic • Work with detection errors • Robust • Useful tool until reaching enough probe rate. • Future task • Longer section with multiple loop detectors

32. Thank you! Q & A