Analytical models developed using field traffic datacan 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 theMarkov Chain theory and random coefficients logistic regression modelin 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

- Traffic data like many other social science data are
**nested**in nature- Vehicles in different lanes
- Traffic in different
*days of the week*, etc

- As a results
**disparities**arise in data characteristics- May be Monday has different pattern than Wednesday
- May be inside lanes (near median) has
**few**vehicles than outside lane (near shoulder)

- Models without accounting these effect lead to
**inconsistency conclusion**

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*

**Markov Chain theory**was used in the analysisOnly current traffic regime has influence on the next regime

Mathematically:

\[P_{ij}=(P_{t+1}=j|P_{t}=i)\]

\[P_{ij} = \left[\begin{array} {rr} P_{ff} & P_{fc} \\ P_{cf} & P_{cc} \\ \end{array}\right] \]

- Dynamic Characteristics
- Markovian Property

\[\sum_{j=1}^{2} P_{ij}=1\]

\[P_{ij} = \left[\begin{array} {rr} 1-P_{fc} & P_{fc} \\ 1-P_{cc} & P_{cc} \\ \end{array}\right] \]

**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

- Four series of random coefficient models were developed:
- Varying Intercept Model (RIO)
- Varying Intercept and Constant Slope Model (RICS)
- Constant Intercept and Varying Slope Model (CIRS)
- Varying Intercept Model and Varying Slope Model (RC)

- Varying Intercept Model and Varying Slope Model

\[ Y_{ij} = Bernoulli(\pi_{ij})\] \[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \beta X + \epsilon_{k}\]

- Prior

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

- Prior

\[ \sigma_{1} \sim unif(0, 100)\] \[ \beta \sim N(\mu=0, \sigma = 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]

- Varying Intercept Model

\[ Y_{ij} = Bernoulli(\pi_{ij})\]

\[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \epsilon_{k}\]

- Prior

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

- Prior \[ \sigma_{1} \sim unif(0, 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]

- Widely Information criterion (WAIC)
- Penalize model with large number of parameters
- Fully Bayesian approach - uncertainty incorporated

\[ WAIC = -2*lppd + 2*pWAIC\]

where,

```
lppd is the log point-wise posterior density
pWAIC is the effective number of parameters
```

- Estimated Speed threshold

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 |

- Speed occupancy relationship