1. 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. 2 OUTLINE Background Methodology Evaluation Sensitivity Analysis Conclusion

3. 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. 4 Loop detectors Dominant traffic sensors Collect point data: volume, occupancy Can be converted to section travel time Section

5. 5 Section travel time estimate from loop detector data A typical method: – v u and v d : station speed estimates, defined as the weighted average of lane speeds – v i, 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. 6 Probe vehicles Collect area-wide data: travel time Estimation method : 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. 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 (x u and x d ), Used as benchmark section travel time

8. 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. 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. 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. 11 Solution to Kalman Filter Problem State propagation Kalman gain: State estimation

12. 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. 13 Estimation of observation noise Using latest several noise samples An approximation of the observation noise v j : Assuming: noise samples r j can represent v j An unbiased estimator for sample mean and sample variance:

14. 14 Estimation of state noise An approximation of the state noise w j Assuming state noise sample q j can represent w j An unbiased estimator for sample mean and sample variance:

15. 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. 16 Evaluation study Simulation based evaluation: Paramics MOE: MAPE Study site:

17. 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. 18 System error patterns

19. 19 Evaluation scenarios Scenario 1: Recurrent congestion condition Scenario 2: Traffic with an incident Detector Errors Systematic detector ErrorRandom Error (δ) Upstream Detectora = -5%; time-varying error pattern1.0% Downstream Detectora = 8%; constant error pattern1.5% On-ramp Detectora = -5%; constant error pattern0.5% Off-ramp detectora = 10%; constant error pattern2.0%

20. 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. 21 Performance comparisons under the recurrent traffic congestion

22. 22 Performance comparisons under incident scenario

23. 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. 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. 25 Sensitivity Analysis Loop detector errors – Systematic error – Random error – Part of loop detector data missing Performance at other sections

26. 26 Sensitivity to constant loop detector errors

27. 27 Sensitivity to time-dependent loop errors

28. 28 Sensitivity to missing lane data

29. 29 Sensitivity to random errors

30. 30 Performance at other sections Study Section Section between pm 2.35 and 2.99 Section between pm 3.86 and 5.55 Point detector based Method 10.6%16.6%14.0% Probe-based Method (5% probe rate) 10.8%13.8%12.3% AKF Algorithm (5% probe rate) 7.6%9.7%8.8%

31. 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. 32 Thank you! Q & A