- Analytical models developed using field traffic data can be utilized with acceptable confidence to reliably represent, predict, and evaluate the operational characteristics of the highway systems.
- Establishing a model to estimate the stochastic evolution of traffic conditions is important for prediction and developing effective solution to address roadway congestion.
- Using a freeway segment of I-295 located in Jacksonville, Florida as a case study, an advanced statistical method developed in the Bayesian modeling framework was developed to account for the day of the week and lane lateral location disparities effect in influencing the dynamic evolution characteristics of traffic regimes.
- Specifically, this study applied the Markov Chain theory and random coefficients logistic regression model in the analysis.
More information: https://link.springer.com/article/10.1007/s40534-019-00199-2
Random coefficient models best fit data which are nested/clustered
The correlation within the group can be analyzed
Unlike independence assumption in basic regression models
Also disparities between the groups can be quantified
AKA:
Hierarchical models
Multilevel models
Mixed-effect models
Random-effect models
Variance-component models
Partial pulled models
In the absence of a Bayesian hierarchical model, there are two approaches for this problem:
Independently compute influence of factors on the response variable for each group (no pooling)
Compute an overall average, under the assumption that every group has the same underlying average (complete pooling)
Complete pooling and No pooling
Data observations are independent
Underlying characteristics is combined to a grand mean
One homogeneous model is fitted
Random coefficient provide a tradeoff between no pooling and complete pooling
Resulting model also known as Partial pooled model
Partial pooled model/Random coefficient
Data observations are nested to their group
Underlying characteristics are nested in groups
Heterogeneous models are fitted
Modeling the dynamic characteristics of traffic conditions
Investigate disperity associated with lane lateral loaction
Also, the effect of day of the week
Dynamic Modeling of traffic conditions
Only current traffic regime has influence on the next regime
Mathematically:
\[P_{ij}=(P_{t+1}=j|P_{t}=i)\]
\[P_{ij} = \left[\begin{array} {rr} P_{ff} & P_{fc} \\ P_{cf} & P_{cc} \\ \end{array}\right] \]
\[\sum_{j=1}^{2} P_{ij}=1\]
\[P_{ij} = \left[\begin{array} {rr} 1-P_{fc} & P_{fc} \\ 1-P_{cc} & P_{cc} \\ \end{array}\right] \]
Time-varying effect
Regression to model Markov chain Transition probability
Model is more flexible
Heterogeneous issue by time is addressed
Lane and day of the week varying-effect
Random coefficient models
Address heterogeneous characteristics associated by lane lateral location
Address heterogeneous characteristics associated by day of the week
\[ Y_{ij} = Bernoulli(\pi_{ij})\] \[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \beta X + \epsilon_{k}\]
\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]
\[ \sigma_{1} \sim unif(0, 100)\] \[ \beta \sim N(\mu=0, \sigma = 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]
\[ Y_{ij} = Bernoulli(\pi_{ij})\]
\[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \epsilon_{k}\]
\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]
\[ WAIC = -2*lppd + 2*pWAIC\]
where,
lppd is the log point-wise posterior density
pWAIC is the effective number of parameters
Random coefficient | Lane lateral location |
---|---|
Median lane | 46.72 |
Inner-left lane | 51.42 |
Inner-right lane | 50.93 |
Shoulder lane | 57.26 |
Overall | 52.23 |
No significant difference in model fit between CIRS and RC for the staying in the congested regime data
RC was the best in fitting Breakdown data
Comparison of the estimated probability at different flow rate
Breakdown probability
Comparison of the estimated probability at different flow rate
Symmetrical properties of logistic model
Based on the symmetrical properties of logistic model, the recovery probabilities and congested regime probabilities are the same but in opposite signs (reference).
Recovery process
Comparison of the estimated probability at different flow rate
- The results show that there are considerable evidence that there are heterogeneity characteristics of evolution characteristics associated with different lane lateral location.
- However, the day of the week influence on the disparities of the estimated transition phenomenon was found only to influence the breakdown transition process.
- The lane lateral location disparity effect was quantified and found that shoulder lane is prone to both breakdown or SCR compared to lanes near the median.
- The results of the random coefficient logistic model in estimating the Markov Chain transition probability show that flow rate is significant at 95% credible interval (CI) in predicting the likelihood of transition from one traffic regime to the next.
- For breakdown process, an “S” shaped pattern was revealed for the relationship between the flow rate and the breakdown probability while an increasing “concave downward” shaped pattern was found in the process of staying in the congested regime (SCR).