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Bayesian Travel Time Reliability
Feng Guo, Associate ProfessorDepartment of Statistics, Virginia TechVirginia Tech Transportation InstituteDengfeng 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
– : distribution under free-flow condition – : distribution under congested condition– is the proportion of the free-flow
component.
Model Parameters Vary by Time of Day
Mean Variance
Probability in congested state
90th Percentile
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
– : distribution under free-flow condition – : distribution under congested condition– is the proportion of the free-flow
component.
Link probability of travel time state with covariates
Link mean travel time of congested 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
– : distribution under free-flow condition – : distribution under congested condition– is the proportion of the free-flow
component.
where is a predefined scale parameter: How large the minimum value of comparing to
Comparing Model 1 and 2
• =1: the minimum value of congested state is the same as free flow
• =1.5: congested state is at least 50% higher than free flow
-1
Simulation Study
• To evaluate the performance of models• Based on two metrics
– Average of posterior mean– Coverage probability
1.Set n=Number of simulations we plan to run.2.For (i in 1:n){
Generate dataDo{
Markov Chain Monte Carlo}While convergenceRecord if the 95% credible intervals cover the true values
}
Simulation Study: Data Generation
• : Observed traffic volume at time interval i on day j
• : Average Traffic volume at time interval i (e.g. 8:00-9:00)
Model 1 VS Model 2: Posterior Means
Model 1 VS Model 2: Coverage Probability
Setting 3: when both and are small
Robustness
• What if…– the true value of 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?
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Simulated TravelTime
0 5 10 15 20 25 30 35
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Observed Travel Time
Hidden Markov Model
• Hidden Markov model is able to incorporate the dependency structure of the data.
• Markov chain is a sequence satisfies:
• In hidden Markov Chain, the state is not visible (i.e. latent) but the output is determined by
Hidden Markov Model
• Latent state:
• Distribution of travel time:
• and satisfy Markov property:
• If { are independent, this is exactly the traditional mixture Gaussian model we have discussed.
Model Specification
• Transition Probability: • E.g. 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:
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
0 200 400 600 800 1000
-22000
-21000
-20000
-19000
-18000
Dots: Hidden Markov Lines: Traditional Mixture
Data Set ID
Log-lik
elih
ood
• Questions?• …• Thanks!