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

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 …

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

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

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 1 + 1−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.

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

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?

Model Specification: Model 1 𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 1 + 1−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

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

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

Model Specification: Model 2 𝑓 𝑦 ~𝜆 𝑓 1 𝑦 𝜇 1 , 𝜎 1 + 1−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

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

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 }

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 )

Model 1 VS Model 2: Posterior Means

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

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

Robustness

Robustness

Robustness: Coverage Probabilities

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

Study Corridor

Modeling Results

Real Data Analysis

Probability of Congested State as A Function of Traffic Volume

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

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

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

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.

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 11 = 𝛽 0,1 + 𝛽 1,1 ∗𝑥 log 𝑃 22 𝑃 21 = 𝛽 0,2 + 𝛽 1,2 ∗𝑥

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 578.8 and 972.6 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.

Simulation Study

Questions? … Thanks!