# An application of Bayesian multilevel model to evaluate variations in stochastic and dynamic transition of traffic conditions

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## Abstract

This study seeks to investigate the variations associated with lane lateral locations and days of the week in the stochastic and dynamic transition of traffic regimes (DTTR). In the proposed analysis, hierarchical regression models fitted using Bayesian frameworks were used to calibrate the transition probabilities that describe the DTTR. Datasets of two sites on a freeway facility located in Jacksonville, Florida, were selected for the analysis. The traffic speed thresholds to define traffic regimes were estimated using the Gaussian mixture model (GMM). The GMM revealed that two and three regimes were adequate mixture components for estimating the traffic speed distributions for Site 1 and 2 datasets, respectively. The results of hierarchical regression models show that there is considerable evidence that there are heterogeneity characteristics in the DTTR associated with lateral lane locations. In particular, the hierarchical regressions reveal that the breakdown process is more affected by the variations compared to other evaluated transition processes with the estimated intra-class correlation (ICC) of about 73%. The transition from congestion on-set/dissolution (COD) to the congested regime is estimated with the highest ICC of 49.4% in the three-regime model, and the lowest ICC of 1% was observed on the transition from the congested to COD regime. On the other hand, different days of the week are not found to contribute to the variations (the highest ICC was 1.44%) on the DTTR. These findings can be used in developing effective congestion countermeasures, particularly in the application of intelligent transportation systems, such as dynamic lane-management strategies.

## Keywords

Dynamic transition of traffic regimes Hierarchical model Bayesian frameworks Lane lateral locations Days of the week Disparity effect## 1 Introduction

Establishing models that estimate the stochastic and dynamic transition of traffic regimes (DTTR) is important for predicting future traffic conditions and developing timely effective countermeasures to address congestion. For example, when two major traffic regimes—free-flow and congested regimes—are analyzed, the DTTR involves four transition phenomena. These include evolving from the free-flow to congested regime (breakdown), staying in the congested regime, congested to the free-flow regime (recovery), and staying in the free-flow regime in the next observation period. Since time is a major factor in their occurrences, the four transition processes can be referred to as the traffic regimes’ dynamic transition.

The DTTR is complex in nature, which is influenced by several factors, such as driver behavior, demand, vehicle mix, and weather conditions. Furthermore, the DTTR can vary greatly by day of the week and lateral lane locations on the same highway. Understanding the impact of these factors is useful for implementing advanced traffic management strategies such as variable speed limit, variable message signs, congestion pricing, and ramp-metering to improve the efficiency of traffic operation [1, 2].

Among the DTTR phenomena, the breakdown process is well-studied in the literature and its theory has recently been introduced in the roadway capacity estimation [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. One major limitation of many previous investigations on the breakdown phenomenon is the fact that they ignore the operational differences due to lateral lane locations on the freeways. In the analysis, the multi-lane facility’s traffic data are usually aggregated and implicitly treated as one unit [1, 13]. The resulting model is also called a complete-pooled model [14, 15], which indicates that the operational characteristics are averaged across lanes. In practice, however, the operational characteristics of freeway segments may vary significantly across lanes [1, 13, 16], which is sometimes influenced by the operational policies. For instance, in urban areas, some states in the USA restrict heavy vehicles to use lanes near the shoulder. Also, some states discourage drivers using lanes near the median unless passing slow moving vehicles. Moreover, the operational characteristics of the lanes near the shoulder can be significantly influenced by weaving (merging to the freeway and diverging to exit a freeway) than lanes near the median [13, 17]. These introduce variations in the operating characteristics of a highway [18, 19]. Developing a model that does not take into account these characteristics and constrains the effect of influencing factors on the breakdown process to be the same across all lanes may lead to incorrect conclusions.

Recognizing the operational variations across different lanes and thus the breakdown process on the freeway, some empirical studies evaluated individual lanes separately. One study compared the complete-pooled and the lane-based approach to estimate the breakdown phenomenon on the diverging sections [1]. The study shows that using the lane-based approach significantly improves the accuracy of the extracted breakdown flow rate, while the aggregated approach underestimates the breakdown flow rate. Another study evaluated individual lane breakdown behavior on the merging freeway sections [16]. It also concludes that there is a significant difference in breakdown phenomenon among lanes.

Separating data and developing a model for each group are also referred to as the no-pooled model [14]. One outstanding drawback of using this model is that the operational characteristics of lanes are assumed to be independent, which as well implicitly suggests that data are coming from completely different sources or different portions of data. Such a model assumes that the operational characteristics of one lane do not affect other lanes. However, it may not be the case in traffic operations. The breakdown usually starts with one lane, generally on a lane near shoulder, and then other lanes follow [20]. Consequently, dependence on operational characteristics as well as some similarities across different lanes exist. Instead of conducting a separate analysis for each lane, some studies have utilized the hierarchical model (random effect) to estimate the breakdown phenomenon [13]. This type of model is also referred to as a partial-pooled model. This model provides a trade-off between the complete-pooled and no-pooled model properties by accounting for both the between-group and within-group variations [15, 21]. The hierarchical model also recognizes the group similarities and integrates such information in the parameter estimates [14, 21]. Using the hierarchical Weibull model, the study in [13] indicates that there is a significant variation in operational characteristics across different lanes on the freeway. Further, the study suggests that aggregating data could potentially ignore the possibility of one lane being congested, while the rest of the lanes are not congested on the same freeway segment (partial breakdown or semi-congested state).

In summary, despite the growing literature in evaluating the probabilistic characteristics of the breakdown process, quantifying the disparity effects on the other transition phenomena that describe the DTTR is not studied in the literature. As a result, this study attempts to fill the research gap by developing hierarchical regression models to calibrate the transition probabilities that describe the DTTR and quantify the associated variations due to different lateral lane locations and days of the week. The parameters’ posterior distributions of the proposed models are all fitted via the Bayesian framework to account for model and parameter uncertainties. Moreover, the transition phenomena that define the DTTR are identified on the basis of the number of traffic regimes, which are estimated using the Gaussian mixture model (GMM). This study uses one-year traffic data collected from a freeway facility located in Jacksonville, Florida. To the best of the authors’ knowledge, the approach herein has not been presented in the existing literature.

## 2 Study sites and data description

Figure 1b shows the 24-h time series of speed variable at Sites 1 and 2 for all data (one-year data) used in modeling, respectively. Evaluating these figures reveals that both sites experience congestion only in the morning peak period. As seen in the figures, the peak period is from 6 a.m. to 9 a.m. Further assessing the traffic speed variable in Fig. 1, one can say that Site 1 has a relatively lower speed than Site 2. The higher data density in the time series scatter plot for Site 1 is between 59 and 68 mph, while for Site 2 is between 61 and 81 mph in the free-flow state.

Descriptive statistics of flow parameters during the peak period

Variable | Metric | Site 1 | Site 2 | |||||
---|---|---|---|---|---|---|---|---|

Lane near median | Middle lane | Lane near shoulder | Lane near median | Inner-left lane | Inner-right lane | Lane near shoulder | ||

Speed (mph) | Mean | 53.3 | 54.0 | 61.7 | 62.6 | 60.8 | 60.6 | 64.8 |

Median | 60.5 | 60.3 | 66.9 | 71.8 | 69.3 | 67.3 | 69.5 | |

SD | 13.7 | 12.3 | 12.1 | 22.4 | 19.9 | 17.9 | 18.9 | |

Minimum | 18.5 | 20.5 | 19.1 | 5.5 | 5.2 | 3.0 | 11.1 | |

Maximum | 71.1 | 71.4 | 84.6 | 91.1 | 87.5 | 86.9 | 98.4 | |

Flow (veh/h/lane) | Mean | 1606.2 | 1637.9 | 1156.1 | 1359.7 | 1296.8 | 1110.6 | 791.2 |

Median | 1644 | 1648 | 1144 | 1364.0 | 1304.0 | 1132.0 | 856.0 | |

SD | 339.2 | 219.1 | 308.1 | 372.3 | 250.3 | 306.1 | 482.4 | |

Minimum | 492 | 672 | 304 | 40.0 | 180.0 | 12.0 | 12.0 | |

Maximum | 2528 | 2216 | 1856 | 2420.0 | 2152.0 | 1852.0 | 1828.0 | |

Number of observations | 2297 | 2300 | 2272 | 8071 | 8079 | 8082 | 7867 |

## 3 Speed thresholds for clustering traffic states

*n*is the total number of the Gaussian distributions in the mixture model, and \(w_{i}\) is the mixing probability of component \(i\),

Two GMM models were developed in the PyMC3 package, Python programming language, to detect the speed thresholds for clustering traffic conditions for Site 1 and Site 2 dataset. The GMM model parameters were estimated using the Markov chain Monte Carlo (MCMC) simulation through the No-U-Turns (NUTS) step. As indicated in Eq. 1, the non-informative prior distributions were used in the model. The mixing probabilities were assumed to follow the Dirichlet distribution similar to [27, 28] studies. For the mean parameters, the prior distribution was assigned to follow the normal distribution with zero mean and standard deviation of 100, \(N\left( {0, 100^{2} } \right).\) Also, the standard deviation parameters in the model were assumed to follow the half-Cauchy distribution, \({\text{HalfCauchy}}\left( {0, 10} \right).\) In the analysis, a total of 10,000 iterations were sampled in each model, whereby the initial 5000 iterations were discarded as warm-up samples, while the last 5000 iterations were used for inference. The convergences were monitored using the Gelman–Rubin statistic and trace plots.

For Site 2 dataset, three components were found to best estimate the data distributions for each lane corresponding to free-flow, congestion on-set/dissolution (COD) or transitional flow condition, and congested regimes. As seen in Fig. 2b, the expected posterior distributions approximate well the field data distributions. As opposed to Site 1, the modeling results suggest that the lane near the median has the highest speed threshold (56.2 mph) followed by the inner-left lane (55.7 mph) and then the inner-right lane (55 mph), and the lane near shoulder had the lowest speed (51 mph) for the COD and congested regimes. A similar pattern was seen on the thresholds that separate COD and free-flow regime. The estimated trend for Site 2 dataset mirrors what was revealed in one of the previous studies [13].

## 4 Modeling the dynamic transition of the traffic regimes

To analyze the dynamic transition of the estimated traffic regimes by the GMM, two Markov chain (MC) models were developed. The first model was the two-regime MC regression for Site 1 and the second model was the three-regime MC regression for Site 2 dataset. The discussions of the two MC regressions are presented in the following subsections.

### 4.1 Two-regime MC model

*i*to

*j*, \({\text{Prob}}(\,)\) is the probability function, \(S_{t}\) is the current observed traffic regime, \(S_{t + 1}\) is the next traffic regime, and \(S_{j}^{'}\) is the future estimated traffic regime.

*L*lanes and

*m*vehicles observed in each lane in each day (

*m*= 1,..,

*M*, and

*M*is the total number of vehicles on the freeway). The transition process of the traffic regime \(R_{ij}\) can be predicted as follows:

*k*= 1,…,5.

### 4.2 Three-regime MC model

*i*to

*j*, \(\beta_{0lv}\) is the random intercept for the transition process \(v\), \(\beta_{1v}\) represents the flow rate parameter for the transition process \(v\), and \(\varepsilon_{kv}\) is the random-effect term for the transition process \(v\).

### 4.3 Parameter estimation for the two- and three-regime MC regressions

## 5 Results

Parameters posterior distributions summaries for Site 1 models

Binary logistic hierarchical regression | ||||||||
---|---|---|---|---|---|---|---|---|

Coefficients | Breakdown process \(\left( {P_{\text{fc}} } \right)\) | Stay in the congested regime \(\left( {P_{\text{cc}} } \right)\) | ||||||

Posterior mean | Posterior SD | 95% credible intervals | Posterior mean | Posterior SD | 95% credible intervals | |||

Intercept | − 65.10 | 3.29 | − 71.10 | − 58.30 | − 11.50 | 2.90 | − 17.0 | − 5.80 |

Log of traffic flow | 8.68 | 0.36 | 8.00 | 9.35 | 1.80 | 0.40 | 1.06 | 2.50 |

Dispersion \(\sigma_{1}\) | 3.02 | 1.74 | 0.75 | 6.79 | 0.87 | 1.00 | 0.07 | 2.80 |

Dispersion \(\sigma_{2}\) | 0.25 | 0.19 | 0.01 | 0.59 | 0.17 | 0.20 | 0.00 | 0.50 |

Parameters posterior distributions summaries for Site 2 models

Binary logistic hierarchical regression | ||||||||
---|---|---|---|---|---|---|---|---|

Free-flow to COD \(\left( {\pi_{\text{fo}} } \right)\) | Congested regime to COD \(\left( {\pi_{\text{co}} } \right)\) | |||||||

Coefficients | Posterior mean | Posterior SD | 95% credible intervals | Posterior mean | Posterior SD | 95% credible intervals | ||

Intercept | − 8.73 | 0.71 | − 10.03 | − 7.36 | − 3.67 | 0.47 | − 4.58 | − 2.81 |

Log of traffic flow | 1.02 | 0.08 | 0.86 | 1.18 | 0.28 | 0.06 | 0.17 | 0.4 |

Dispersion \(\sigma_{1}\) | 0.62 | 0.55 | 0.13 | 1.54 | 0.18 | 0.28 | 0.03 | 0.5 |

Dispersion \(\sigma_{2}\) | 0.21 | 0.13 | 0.05 | 0.46 | 0.22 | 0.12 | 0.05 | 0.49 |

Multinomial logistic hierarchical regression | ||||||||
---|---|---|---|---|---|---|---|---|

COD to free-flow \(\left( {\pi_{\text{of}} } \right)\) | COD to congested regime \(\left( {\pi_{\text{oc}} } \right)\) | |||||||

Coefficients | Posterior mean | Posterior SD | 95% credible intervals | Posterior mean | Posterior SD | 95% credible intervals | ||

Intercept | 15.72 | 1.11 | 13.56 | 17.84 | − 7.01 | 1.24 | − 9.41 | − 4.52 |

Log of traffic flow | − 2.39 | 0.13 | − 2.64 | − 2.12 | 0.82 | 0.09 | 0.65 | 0.99 |

Dispersion \(\sigma_{1}\) | 0.9 | 0.72 | 0.19 | 2.23 | 1.8 | 1.13 | 0.51 | 4.05 |

Dispersion \(\sigma_{2}\) | 0.07 | 0.09 | 0 | 0.2 | 0.17 | 0.12 | 0.01 | 0.39 |

### 5.1 Results of regression models for site 1

Two regression models were fitted to calibrate the transition probabilities of the breakdown and the stay in the congested regime processes. As presented in Table 2, the logarithm of the flow rate coefficient has a positive sign, which potentially indicates that when the flow rate increases the probability of traffic to breakdown also increases. The estimate of this coefficient suggests that a 1% increase in the log-transformed flow rate increases the likelihood of breakdown by 8.68%. The CI of this estimate does not contain zero as one of the credible values, and thus it is statistically significant at 95% CIs.

It is noteworthy to know that the stay in the free-flow and the recovery transition processes (congestion to free-flow) are not presented because these were considered as the base category in the model. To clarify this, the stay in the free-flow and breakdown probabilities in the transition matrix presented in Eq. 3 sum up to 1. Since the logit link function was used in the hierarchical regression to fit the transition matrix, the breakdown estimates and the stay in the free-flow regime are the same but in opposite sign (negative vs. positive). Similarly, the estimate of the stay in the congested regime and the recovery transition processes are the same but with different signs.

### 5.2 Results of regression models for site 2

Also presented in Table 3, the results for the COD to congested transition were significant at the 95% CI. The estimate of the logarithm of traffic flow is 0.82, which indicates that a 1% increase in the logarithm of flow rate would cause the likelihood of COD to congested transition to increase by 0.82% relative to staying in the COD regime.

### 5.3 Disparity effects caused by different lane lateral locations and days of the week

Similar analyses were conducted for Site 2, and the estimates indicate that the lateral lane location has the largest impact on the COD to congested transition process (ICC = 49.4%) followed by the COD to free-flow transition (ICC = 19.7%), the free-flow to COD transition (ICC = 10.5%), and the congested to COD transition (ICC = 1%). For different days of the week, the congested to COD transition has the highest variation (ICC = 1.44%) followed by the free-flow to COD transition (ICC = 1.2%), COD to congested transition (ICC = 0.5%), and COD to free-flow transition (ICC = 0.1%).

In summary, there is considerable evidence that lane lateral locations contribute a significant amount of variation to the DTTR than different days of the week (considering only weekdays). This observation is consistent across the two sites. Moreover, the highest disparity estimate associated with different days of the week is 1.44%. Based on this estimate, one may conclude that different days of the week are insignificantly causing variability in the DTTR. Even though the study in [36] investigated the difference in flow capacity due to different days of the week using the analysis of variance (ANOVA) approach, the same conclusions were made that there is no variation attributed to different days of the week on estimated capacity flow.

## 6 Discussion

This study has presented an empirical approach aimed at investigating disparity effects of the lateral lane locations and days of the week on the dynamic transition of traffic regimes (DTTR). In the analysis, the Markov chain theory and hierarchical regressions were integrated to describe the transition processes and the dependence of traffic regimes and capture the hierarchical structure of observations of the traffic data. The historical traffic flow parameters—speed and flow—collected for 1 year (2015–2016) from two sites on the freeway highway, were applied.

Using the GMM, the speed threshold of each lane that defines traffic conditions was identified in the analysis. Overall, the results of the hierarchical regressions in estimating the MC transition probabilities indicated that the log-transformed flow rate is the significant variable, at 95% posterior credible intervals, in predicting the likelihood of evolving from one traffic regime to the next. The lane near shoulder was estimated to have the highest likelihood of transitioning from one regime to the next compared to other lanes at a similar flow rate. Using the intra-class correlation coefficient (ICC) analysis, it was revealed that different lane lateral locations contribute a significant percentage to the total variations in the DTTR for Site 1 dataset. More specifically, the breakdown process was found to be more influenced by the variations than the rest evaluated transition processes (ICC = 73%). For Site 2 dataset, the largest variation due to lateral lane location was observed on the transition from the COD to the congested regime (ICC = 49.4%). Different days of the week, on the contrary, were found not to cause variations in the transition probabilities describing the DTTR. The highest estimate of the ICC among the fitted hierarchal models for both Site 1 and 2 was 1.44%.

The findings from this study can be possibly used to enhance the lane-distribution strategy in the application of the intelligent transportation systems, particularly in the dynamic lane-management to improve operations efficiency. Furthermore, results are anticipated to increase the awareness of the variation associated with different lateral lane locations and days of the week in traffic operations to both researchers and practitioners. This information is also useful to transportation agencies in developing other congestion countermeasures.

One limitation that could be further improved in this study is that the data that were used in modeling the DTTR from the detectors were not filtered to remove data that had overlapping bottlenecks between the exit and entrance ramps. It would be the future research task to consider this situation in the analysis. Also, more research using data with different site characteristics is required to validate the conclusion made in the current study. In addition, it is not clear if a similar conclusion will be made if different data resolution is used in modeling, such as 2 min, 5 min. In the future work, different data resolutions can be used in the model and compared with the current study results. Another future work would be the analysis of effects of the spatial heterogeneity, vehicle mix, weather, and driving characteristics on the DTTR and the number of traffic regimes in the GMM. Although the two sites evaluated in this study have different geometric characteristics and two regimes were identified on Site 1, while three regimes optimally describe the operating speed for Site 2, it is not yet clear if sites with similar geometric characteristics will yield a similar number of traffic regimes.

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