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!