1. Bayesian Travel Time Reliability
Feng Guo, Associate Professor
Department of Statistics, Virginia Tech
Virginia Tech Transportation Institute
Dengfeng Zhang, Department of Statistics, Virginia Tech

2. Travel Time Reliability
Travel time is random in nature.
Effects to quantify the uncertainty
Percentage Variation
Misery index
Buffer time index
Distribution
Normal
Log-normal distribution …

3. Multi-State Travel Time Reliability Models
Better fitting for the data
Easy for interpretation and prediction, similar to weather forecasting:
The probability of encountering congestion
The estimated travel time IF congestion
Guo et al 2010

4. Multi-State Travel Time Reliability Models
Direct link with underline traffic condition and fundamental diagram
Can be extended to skewed component distributions such as log-normal
Park et al 2010; Guo et al 2012

5. Model Specification 𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 1 + 1−𝜆 𝑓 2 (𝑦| 𝜇 2 , 𝜎 2 )
𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 −𝜆 𝑓 2 (𝑦| 𝜇 2 , 𝜎 2 )
𝑓 1 : distribution under free-flow condition
𝑓 2 : distribution under congested condition
𝜆 is the proportion of the free-flow component.

6. Model Parameters Vary by Time of Day
Mean
Variance
90th Percentile
Probability in congested state
What is the root cause of this fluctuation?

7. Bayesian Multi-State Travel Time Models
The fluctuation by time-of-day most like due to traffic volume
How to incorporate this into the model?

8. Model Specification: Model 1
𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 −𝜆 𝑓 2 (𝑦| 𝜇 2 , 𝜎 2 )
𝑓 1 : distribution under free-flow condition
𝑓 2 : distribution under congested condition
𝜆 is the proportion of the free-flow component.
𝜇 2 = 𝜃 0 + 𝜃 1 ∗𝑥
Φ −1 1−𝜆 = 𝛽 0 + 𝛽 1 ∗𝑥
Link mean travel time of congested state with covariates
Link probability of travel time state with covariates

9. Bayesian Model Setup Inference based on posterior distribution
Using non-informative priors: let data dominate results.
Developed Markov China Monte Carlo (MCMC) to simulate posterior distributions

10. Issues with Model 1
When traffic volume is low, the two component distribution can be very similar to each other
The mixture proportion estimation is not stable

11. Model Specification: Model 2
𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 −𝜆 𝑓 2 (𝑦| 𝜇 2 , 𝜎 2 )
𝑓 1 : distribution under free-flow condition
𝑓 2 : distribution under congested condition
𝜆 is the proportion of the free-flow component.
𝜇 2 = 𝜽 𝒔 ∗ 𝝁 𝟏 + 𝜃 1 ∗𝑥
Φ −1 1−𝜆 = 𝛽 0 + 𝛽 1 ∗𝑥
where 𝜽 𝒔 is a predefined scale parameter: How large the minimum value of 𝝁 𝟐 comparing to 𝝁 𝟏

12. Comparing Model 1 and 2 𝜇 2 = 𝜽 𝒔 ∗ 𝝁 𝟏 + 𝜃 1 ∗𝑥
𝜽 𝒔 =1: the minimum value of congested state is the same as free flow
𝜽 𝒔 =1.5: congested state is at least 50% higher than free flow
𝜽 𝒔 -1

13. Simulation Study To evaluate the performance of models
Based on two metrics
Average of posterior mean
Coverage probability
Set n=Number of simulations we plan to run.
For (i in 1:n) {
Generate data Do
Markov Chain Monte Carlo
}While convergence
Record if the 95% credible intervals cover the true values }

14. Simulation Study: Data Generation
𝑋 𝑖𝑗 : Observed traffic volume at time interval i on day j
𝜇 𝑖 : Average Traffic volume at time interval i (e.g. 8:00-9:00)
𝜇 𝑖 = 𝑘 𝑋 𝑖𝑘 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠
𝑌 𝑖𝑗 : Simulated traffic volume at time interval i on day j 𝑌 𝑖𝑗 = 𝑐∗ 𝜇 𝑖 + 𝜖 𝑖𝑗 + , 𝜖 𝑖𝑗 ~𝑁(0, 𝑑 2 )

15. Model 1 VS Model 2: Posterior Means

16. Model 1 VS Model 2: Coverage Probability
Setting 3: when both 𝜃 0 and 𝜃 1 are small

17. Robustness What if… the true value of 𝜃 𝑠 is unknown
The two components are too close
We showed that the overall estimation are quite stable, even if the tuning parameter is misspecified
When the two components are too close, by selecting a misspecified tuning parameter could improve the coverage probabilities of some parameters

18. Robustness

19. Robustness

20. Robustness: Coverage Probabilities

21. Data The data set contains 4306 observations
A section of the I-35 freeway in San Antonio, Texas.
Vehicles were tagged by a radio frequency device
High precision

22. Study Corridor

23. Modeling Results

24. Real Data Analysis

25. Probability of Congested State as A Function of Traffic Volume

26. Next Step… Apply the model to a large dataset
Any available data are welcome!
Hidden Markov Model

27. HMM
The models discussed are based on the assumption that all the observations are independent. Is it realistic?

28. Hidden Markov Model
Hidden Markov model is able to incorporate the dependency structure of the data.
Markov chain is a sequence satisfies:
𝑃 𝑥 𝑡+1 =𝑥 𝑥 1 , 𝑥 2 ,… 𝑥 𝑡 =𝑃 𝑥 𝑡+1 =𝑥 𝑥 𝑡
In hidden Markov Chain, the state 𝑥 𝑡 is not visible (i.e. latent) but the output 𝑦 𝑡 is determined by 𝑥 𝑡

29. Hidden Markov Model Latent state: 𝑤 𝑡 = 1:𝑓𝑟𝑒𝑒−𝑓𝑙𝑜𝑤 2:𝑐𝑜𝑛𝑔𝑒𝑠𝑡𝑒𝑑
𝑤 𝑡 = 1:𝑓𝑟𝑒𝑒−𝑓𝑙𝑜𝑤 2:𝑐𝑜𝑛𝑔𝑒𝑠𝑡𝑒𝑑
Distribution of travel time:
𝑦 𝑡 | 𝑤 𝑡 ~ 𝑓 1 , 𝑖𝑓 𝑤 𝑡 =1 𝑓 2 , 𝑖𝑓 𝑤 𝑡 =2
𝑤 𝑡 and 𝑤 𝑡−1 satisfy Markov property:
𝑃 𝑤 𝑡+1 =𝑤 𝑤 1 , 𝑤 2 ,… 𝑤 𝑡 =𝑃 𝑤 𝑡+1 =𝑤 𝑤 𝑡
If { 𝑤 𝑡 } are independent, this is exactly the traditional mixture Gaussian model we have discussed.

30. Model Specification Transition Probability: 𝑃 𝑖𝑗 =𝑃( 𝑤 𝑡+1 =𝑗| 𝑤 𝑡 =𝑖)
E.g. 𝑃 12 is the probability that the travel time is jumping from free-flow state to congested state.
We use logit link function to model the transition probabilities with traffic volume:
log 𝑃 12 𝑃 = 𝛽 0,1 + 𝛽 1,1 ∗𝑥
log 𝑃 22 𝑃 = 𝛽 0,2 + 𝛽 1,2 ∗𝑥

31. Preliminary Results
When the traffic volume is higher, the congested state will be more likely to stay and free-flow state will be more likely to make a jump.
The mean travel time of the two states are and seconds.
If we calculate the stationary distribution, the proportion of congested state is around 11.3%.
AIC indicates that hidden Markov model is superior to traditional mixture Gaussian model.

32. Simulation Study

33. Questions? …
Thanks!