# Accident Rate Prediction Model for Urban Expressway Underwater Tunnel

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## Abstract

**:**

## 1. Introduction

- (1)
- From the dimension of tunnel geometry and control measures, we analyzed the characteristics of tunnel traffic accidents, and quantified the coupling relationship between multiple factors and the incidence of tunnel traffic accidents.
- (2)
- Based on the negative binomial regression model, we established the prediction model of the Jiaozhou Bay undersea tunnel accident rate, which can provide theoretical support for further traffic safety research such as tunnel safety risk identification.

## 2. Materials and Methods

#### 2.1. Screening of Accident Influence Factors

#### 2.1.1. Accident Characteristics Analysis

- (1)
- Accident time distribution

- (2)
- Accident types

- (3)
- Type of vehicle involved

- (4)
- Accident spatial distribution

#### 2.1.2. Analysis of the Correlation between Influential Variables and Accident Rate

- (1)
- Influence of slope on the occurrence of accidents

- (2)
- Influence of the curve radius on accident occurrence

- (3)
- Influence of slope length on accident occurrence

- (4)
- Influence of the percentage of road length from the bottom of the slope on the occurrence of accidents

- (5)
- Influence of geometric two-factor combination conditions on the occurrence of accidents

- (6)
- Impact of control measures on the occurrence of accidents

#### 2.2. Accident Rate Modeling

#### 2.2.1. Model Form

#### 2.2.2. Regression Model

- (1)
- Possion regression model

- (2)
- Negative binomial regression model

#### 2.3. Model Feasibility Analysis

#### 2.3.1. Goodness-of-Fit Test

#### 2.3.2. Parameter Testing

^{2}distribution.

#### 2.3.3. Model Evaluation Indexes

- (1)
- Absolute Error (AE) is a term used to describe the difference between the theoretical value and the actual value; its formula is as follows.

- (2)
- Mean Absolute Error (MAE) is the average absolute error between the theoretical value and the actual value. In general, the smaller the MAE value, the greater the accuracy of the regression model, which can more accurately reflect the magnitude of the theoretical value’s error, and its calculation formula is as follows.

- (3)
- Percent Error (PE) is the percentage of the ratio of absolute error to the actual value. When the value of PE is smaller, it indicates that the regression model has a higher level of accuracy.

## 3. Results and Discussion

#### 3.1. Data Processing

#### 3.1.1. Road Segmentation

#### 3.1.2. Data Processing

#### 3.2. Example Calculation

#### 3.2.1. Model Form

#### 3.2.2. Accident Rate Model for the Left Line Based on Geometric Conditions

- (1)
- Model goodness of fit

- (2)
- Estimation of model parameters

- (3)
- The left-lane accident occurrence rate model based on geometric conditions

#### 3.2.3. Accident Rate Model of the Right Lane Based on Geometric Conditions

#### 3.3. Model Feasibility Analysis

#### 3.3.1. Left-Line Accident Rate Model

#### 3.3.2. Right-Line Accident Rate Model

## 4. Conclusions

- (1)
- Factors influencing tunnel accidents: The study analyzes the influence of road characteristics, traffic characteristics, traffic control methods, and environmental characteristics on tunnel accidents. It identifies traffic volume, road geometry, and traffic characteristics as the primary factors affecting accident occurrence.
- (2)
- Effectiveness of control measures: Quantitative analysis shows that comprehensive control measures, such as the implementation of variable lane markings, significantly reduce the rate of tunnel traffic accidents. The reduction rate ranges between 30% and 40%. This highlights the effectiveness of control measures in mitigating accident risks.
- (3)
- Model construction for extra-long tunnels: The study focuses on constructing a prediction model for accident occurrence in extra-long tunnels. Using the Jiaozhou Bay underwater tunnel in Qingdao as an example, the characteristics of tunnel accident data are combined with a negative binomial regression model. The model considers factors such as the slope of the tunnel’s left line, the percentage of road length from the bottom of the slope, the slope of the right line, and the inverse of the curve radius, which significantly affect the tunnel’s traffic accident rate.
- (4)
- Model validation: After the second delineation of the Qingdao Jiaozhou Bay Subsea Tunnel, the study selects relevant data to validate the model’s viability. By comparing and analyzing the actual and theoretical values of the tunnel accident rate, the study confirms the consistency between the model predictions and the observed accident rates. The percent error falls within an acceptable range, with the model performing well for both the left and right lanes of the tunnel.

- (1)
- This study focuses on the impact of geometric conditions and control measures on traffic accidents in extra-long tunnels. Future research can explore additional factors, such as traffic status and facilities, to analyze the patterns of traffic accidents in urban expressway tunnels.
- (2)
- The accident rate prediction model in this study considers comprehensive control measures, but further research is needed to assess the specific influence of individual measures like traffic markings and intelligent traffic facilities on traffic safety.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Turner-Fairbank Highway Research Center. Interactive Highway Safety Design Midel: Getting Started Guide; Federal Highway Adiministration: Washington, DC, USA, 2003. [Google Scholar]
- Aidoo, E.N.; Amoh-Gyimah, R.; Ackaah, W. The Effect of Road and Environmental Characteristics on Pedestrian Hit-and-Run Accidents in Ghana. Accid Anal Prev.
**2013**, 53, 23–27. [Google Scholar] [CrossRef] - Pan, F.Q.; Xing, Y.; Yang, J.S.; Zhang, L.X.; Yang, X.X.; Xu, Q.N. Factors Affecting Traffic Accidents in Sub-sea Tunnels and Prevention and Control Strategies. Transp. Technol. Econ.
**2020**, 22, 1–5, 56. [Google Scholar] - Tanishita, M.; van Wee, B. Impact of Vehicle Speeds and Changes in Mean Speeds on Per Vehicle-kilometer Traffic Accident Rates in Japan. IATSS Res.
**2016**, 41, 107–112. [Google Scholar] [CrossRef] [Green Version] - Wen, H.; Zhang, X.; Zeng, Q.; Sze, N.N. Bayesian Spatial-temporal Model for the Main and Interaction Effects of Roadway and Weather Characteristics on Freeway Crash Incidence. Accid. Anal. Prev.
**2019**, 132, 105249. [Google Scholar] [CrossRef] - Mohammadi, M.A.; Samaranayake, V.A.; Bham, G.H. Crash Frequency Modeling Using Negative Binomial Models: An application of Generalized Estimating Equation to Longitudinal Data. Anal. Methods Accid. Res.
**2014**, 2, 52–69. [Google Scholar] [CrossRef] - Leng, Q.; Huang, H. Bayesian Spatial Joint Modeling of Trattic Crashes on an Urban Road Network. Accid. Anal. Prev.
**2014**, 67, 105–112. [Google Scholar] - Tang, Z.G.; Fu, Y.G.; Dong, W.W.; Liu, Y.K. Analysis of Speed Characteristics of Highway Tunnel Sections. J. Chongqing Jiaotong Univ. (Nat. Sci. Ed.)
**2020**, 39, 25–32. [Google Scholar] - Wang, X.Y.; Ma, Z.Y.; Dong, X.Y. Analysis of Tunnel Traffic Accidents and Management Countermeasures in China. Traffic Eng.
**2017**, 17, 33–37. [Google Scholar] - Du, B.Y.; Sun, P.; Liu, K.F. Highway Tunnel Alignment Design Based on Operational Safety. Highway
**2018**, 63, 278–282. [Google Scholar] - Zhang, Z.Z.; Zhu, S.H.; Zhu, K.N.; Liu, H.X.; Zhu, T. Simulation of Traffic Flow and Safety in the Entrance Section of Highway Tunnels. J. Saf. Environ.
**2015**, 15, 146–150. [Google Scholar] - Hadi, M.A.; Aruldhas, J.; Chow, L.F.; Wattleworth, J.A. Estimating Safety Effects of Cross-section Design for Various Highway Types Using Negative Binomial Regression. Transp. Res. Rec.
**1995**, 1500, 169–177. [Google Scholar] - Caliendo, C.; Guida, M.; Parisi, A. A Crash-prediction Model for Multilane Roads. Accid. Anal. Prev.
**2007**, 39, 657–670. [Google Scholar] [CrossRef] - Zeng, Q.; Sun, J.; Wen, H. Bayesian Hierarchical Modeling Monthly Crash Counts on Freeway Segments with Temporal Correlation. J. Adv. Transp.
**2017**, 2017, 5391054. [Google Scholar] [CrossRef] [Green Version] - Hosseinlou, M.H.; Mahdavi, A.; Nooghabi, M.J. Validation of the Influencing Factors Associated with Traffic Violations and Crashes on Freeways of Developing Countries: A Case Study of Iran. Accid. Anal. Prev.
**2018**, 121, 358–366. [Google Scholar] [CrossRef] - Ture Kibar, F.; Celik, F.; Wegman, F. Analyzing Truck Accident Data on the Interurban Road Ankara-Aksaray-Eregli in Turkey: Comparing the Performances of Negative Binomial Regression and the Artificial Neural Networks Models. J. Transp. Saf. Secur.
**2019**, 11, 129–149. [Google Scholar] [CrossRef] - Lord, D.; Mannering, F. The Statistical Analysis of Crash-frequency Data: A Review and Assessment of Methodological Alternatives. Transp. Res. Part A Policy Pract.
**2010**, 44, 291–305. [Google Scholar] [CrossRef] [Green Version] - Kim, D.; Lee, Y. Modelling Crash Frequencies at Signalized Intersections with a Truncated Count Data model. Int. J. Urban Sci.
**2013**, 17, 85–94. [Google Scholar] [CrossRef] - Ma, Z.; Zhang, H.; Chien, S.I.J.; Wang, J.; Dong, C. Predicting Expressway Crash Frequency Using a Random Effect Negative Binomial Model: A Case Study in China. Accid. Anal. Prev.
**2017**, 98, 214–222. [Google Scholar] [CrossRef] - Zou, Y.; Wu, L.; Lord, D. Modeling over-dispersed crash data with a long tail: Examining the accuracy of the dispersion parameter in Negative Binomial models. Anal. Methods Accid. Res.
**2015**, 5–6, 1–16. [Google Scholar] [CrossRef] - Shirazi, M.; Lord, D.; Dhavala, S.S.; Geedipally, S.R. A Semiparametric Negative Binomial Generalized Linear Model for Modeling Over-dispersed Count Data with a Heavy Tail: Characteristics and Applications to Crash Data. Accid. Anal. Prev.
**2016**, 91, 10–18. [Google Scholar] [CrossRef] - Hou, Q.; Meng, X.; Leng, J.; Yu, L. Application of a Random Effects Negative Binomial Model to Examine Crash Frequency for Freeways in China. Phys. A Stat. Mech. Appl.
**2018**, 509, 937–944. [Google Scholar] [CrossRef] - Li, L.; Wang, W.; Liu, P.; Bai, L.; Du, M. Analysis of Crash Risks by Collision Type at Freeway Diverge Area Using Multivariate Modeling Technique. J. Transp. Eng.
**2015**, 144, 04015002. [Google Scholar] [CrossRef]

**Figure 8.**Influence of the percentage of road length from the bottom of the slope on the occurrence of accidents.

**Figure 13.**Tunnel traffic accidents under the combination of slope and road length from the bottom of the slope.

**Figure 14.**Effect of the combination of slope and road length from the bottom of the slope on the traffic accidents in the tunnel.

**Figure 18.**Schematic diagram of the second delineation position of Qingdao Jiaozhou Bay Subsea Tunnel.

**Figure 19.**Diagram of the comparison analysis between the theoretical and actual values of the accident rate model of the left line.

**Figure 20.**Diagram of the comparison analysis between the theoretical and actual values of the accident rate model of the right line.

**Figure 21.**Diagram of the comparison analysis between theoretical and actual values of the traffic accident rate in the left lane.

**Figure 22.**Absolute error between theoretical and actual values of the traffic accident rate in the left lane.

**Figure 24.**Diagram of comparison analysis between theoretical and actual values of the traffic accident rate in the right lane.

**Figure 25.**Absolute error between theoretical and actual values of traffic accident rate in the right lane.

Number | Location | First Scribing Scheme |
---|---|---|

1 | ZK4 + 980~ZK5 + 480 | Solid marking line to dummy marking line, length 500 m |

2 | ZK3 + 721~ZK4 + 321 | The real marking line is adjusted to the right real left virtual marking line, length 600 m |

3 | ZK3 + 272~ZK3 + 472 | The right solid left dummy marker is extended 200 m towards Huangdao |

4 | ZK1 + 376~ZK1 + 676 | The right solid left dummy marker is extended 300 m towards Huangdao |

5 | YK3 + 580~YK3 + 980 | The real marking line is changed to the virtual marking line, the length is 400 m |

6 | YK5 + 785~YK6 + 485 | The actual marker line is adjusted to the right solid left virtual marker line, length 700 m |

7 | YK7 + 800~YK8 + 600 | The solid marker line is adjusted to the right solid left virtual marker line, length 800 m |

**Table 2.**Accident rate calculation results for the delineated section of Qingdao Jiaozhou Bay Subsea Tunnel.

Number | Location | Road Length/m | Before Delineation | After Delineation |
---|---|---|---|---|

1 | ZK4 + 980~ZK5 + 480 | 500 | 1.410934 | 0.945626 |

2 | ZK3 + 721~ZK4 + 321 | 600 | 0.881834 | 0.525348 |

3 | ZK3 + 272~ZK3 + 472 | 200 | 1.763668 | 1.576044 |

4 | ZK1 + 376~ZK1 + 676 | 300 | 1.763668 | 1.050696 |

5 | YK3 + 580~YK3 + 980 | 400 | 7.495591 | 5.516154 |

6 | YK5 + 785~YK6 + 485 | 700 | 3.527337 | 2.026342 |

7 | YK7 + 800~YK8 + 600 | 800 | 2.645503 | 1.77305 |

**Table 3.**The Jiaozhou Bay tunnel delineation section accident rate’s independent sample test results.

Accident Rate | Levene’s Test for Variance Equations | t-Test for the Mean Equation | |||||||
---|---|---|---|---|---|---|---|---|---|

F | Sig | t | Degree of Freedom | Sig (Double Tail) | Mean Value Difference | Standard Error | Difference 95% Confidence Interval | ||

Lower Limit | Upper Limit | ||||||||

Suppose the variances are equal | 5.305 | 0.049 | 2.165 | 10 | 0.056 | 2.052 | 0.95 | −0.06 | 4.16 |

Suppose the variances are not equal | 2.165 | 5.312 | 0.080 | 2.052 | 0.95 | −0.34 | 4.44 |

Number | Location | Second Delineation Program |
---|---|---|

1 | ZK1 + 376~ZK1 + 676 | Right solid left dashed marker line to Huangdao direction extended 300 m |

2 | ZK3 + 705~ZK4 + 355 | Dashed line, length 650 m |

3 | ZK4 + 955~ZK6 + 155 | Dashed line, length 1200 m |

4 | YK5 + 785~YK7 + 045 | Dashed line, length 1260 m |

5 | YK7 + 800~YK8 + 600 | Dashed line, length 800 m |

Section Number | Accident Rate | Section Number | Accident Rate |
---|---|---|---|

Segment 1 | 2.40396 | Segment 19 | 5.620458 |

Segment 2 | 2.898638 | Segment 20 | 6.02192 |

Segment 3 | 2.274568 | Segment 21 | 11.04019 |

Segment 4 | 2.600138 | Segment 22 | 12.04384 |

Segment 5 | 15.25553 | Segment 23 | 9.534706 |

Segment 6 | 7.795365 | Segment 24 | 10.83946 |

Segment 7 | 15.76736 | Segment 25 | 12.04384 |

Segment 8 | 2.606892 | Segment 26 | 5.046942 |

Segment 9 | 1.022397 | Segment 27 | 1.338204 |

Segment 10 | 0.514694 | Segment 28 | 1.50548 |

Segment 12 | 0.63322 | Segment 30 | 1.338204 |

Segment 13 | 1.209221 | Segment 31 | 1.003653 |

Segment 14 | 0.266929 | Segment 32 | 1.50548 |

Segment 15 | 0.495631 | Segment 33 | 1.338204 |

Segment 16 | 0.581828 | Segment 34 | 0.501827 |

Segment 17 | 4.014613 | Segment 35 | 0.501827 |

Segment 18 | 3.345511 |

Section Number | Accident Rate | Section Number | Accident Rate |
---|---|---|---|

Segment 1 | 0.585221 | Segment 23 | 11.04019 |

Segment 2 | 1.029388 | Segment 24 | 11.37474 |

Segment 3 | 1.760795 | Segment 25 | 11.54201 |

Segment 4 | 0.981568 | Segment 26 | 6.02192 |

Segment 5 | 0.627283 | Segment 27 | 5.352818 |

Segment 6 | 0.766148 | Segment 28 | 8.029226 |

Segment 7 | 1.550044 | Segment 29 | 7.360124 |

Segment 8 | 2.124134 | Segment 30 | 7.5274 |

Segment 9 | 7.806192 | Segment 31 | 1.50548 |

Segment 10 | 11.34873 | Segment 32 | 1.338204 |

Segment 11 | 4.9933 | Segment 33 | 1.50548 |

Segment 12 | 6.766202 | Segment 34 | 1.605845 |

Segment 13 | 2.606892 | Segment 35 | 1.672755 |

Segment 14 | 2.90213 | Segment 36 | 2.007307 |

Segment 15 | 1.43379 | Segment 37 | 1.338204 |

Segment 16 | 2.007307 | Segment 38 | 1.50548 |

Segment 17 | 1.077749 | Segment 39 | 1.605845 |

Segment 18 | 1.544082 | Segment 40 | 1.338204 |

Segment 19 | 1.574358 | Segment 41 | 0.802923 |

Segment 20 | 1.254567 | Segment 42 | 1.338204 |

Segment 21 | 0.528239 | Segment 43 | 1.003653 |

Segment 22 | 3.01096 | Segment 44 | 1.204384 |

Number of Cases | Minimum Value | Maximum Value | Average Value | Standard Deviation | Variance | |
---|---|---|---|---|---|---|

Number of left-line accidents | 35 | 3 | 386 | 84.03 | 110.446 | 12,198.382 |

Number of effective cases | 35 | - | - | - | - | - |

**Table 8.**Goodness-of-fit indicators for the left-lane traffic accident rate model based on geometric conditions.

Indicators | Degree of Freedom | Possion Returns | Negative Binomial Regression |
---|---|---|---|

Deviance | 30 | 7.8683 | 1.3391 |

Pearson ${\chi}^{2}$ | 30 | 239.1663 | 41.5097 |

AIC | - | 433.6176 | 302.9171 |

BIC | - | 441.3943 | 321.2492 |

**Table 9.**Estimated parameters of the left-lane traffic accident rate model based on geometric conditions.

Parameters | Degree of Freedom | Estimated | Standard Error | Wald 95% Confidence Limit | Wald Cardinality | Pr > Cardinality | |
---|---|---|---|---|---|---|---|

Intercept | 1 | 1.8781 | 0.2405 | 1.4068 | 2.3494 | 61.01 | <0.0001 |

$i$ | 1 | −0.3347 | 0.0279 | −0.3893 | −0.2801 | 144.26 | <0.0001 |

$\frac{1}{R}$ | 1 | 136.5531 | 346.3367 | −542.254 | 815.3605 | 0.16 | 0.6934 |

$w$ | 1 | −2.7089 | 0.3358 | −3.3671 | −2.0507 | 65.06 | <0.0001 |

$D$ | 1 | 0.6474 | 0.3937 | −0.1242 | 1.419 | 2.7 | 0.1001 |

Number of Cases | Minimum Value | Maximum Value | Average Value | Standard Deviation | Variance | |
---|---|---|---|---|---|---|

Number of right-line incidents | 42 | 2 | 285 | 65.02 | 71.982 | 5181.341 |

Number of active cases | 42 | - | - | - | - | - |

**Table 11.**Goodness-of-fit indicators for the right-lane traffic accident rate model based on geometric conditions.

Indicators | Degree of Freedom | Possion Returns | Negative Binomial Regression |
---|---|---|---|

Deviance | 37 | 4.3391 | 1.1727 |

Pearson ${\chi}^{2}$ | 37 | 161.6143 | 47.0172 |

AIC | - | 397.3384 | 338.7095 |

BIC | - | 406.0267 | 349.1355 |

**Table 12.**Estimated parameters of the right-lane traffic accident rate model based on geometric conditions.

Parameters | Degree of Freedom | Estimated | Standard Error | Wald 95% Confidence Limit | Wald Cardinality | Pr > Cardinality | |
---|---|---|---|---|---|---|---|

Intercept | 1 | 1.8057 | 0.179 | 1.4548 | 2.1566 | 101.73 | <0.0001 |

$i$ | 1 | −0.3114 | 0.0286 | −0.3674 | −0.2554 | 118.78 | <0.0001 |

$\frac{1}{R}$ | 1 | −2404.06 | 215.9038 | −2827.23 | −1980.9 | 123.99 | <0.0001 |

$w$ | 1 | 0.1336 | 0.2943 | −0.4432 | 0.7104 | 0.21 | 0.6498 |

$D$ | 1 | 0.1584 | 0.2908 | −0.4115 | 0.7282 | 0.3 | 0.586 |

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## Share and Cite

**MDPI and ACS Style**

Xing, R.; Li, Z.; Cai, X.; Yang, Z.; Zhang, N.; Yang, T.
Accident Rate Prediction Model for Urban Expressway Underwater Tunnel. *Sustainability* **2023**, *15*, 10730.
https://doi.org/10.3390/su151310730

**AMA Style**

Xing R, Li Z, Cai X, Yang Z, Zhang N, Yang T.
Accident Rate Prediction Model for Urban Expressway Underwater Tunnel. *Sustainability*. 2023; 15(13):10730.
https://doi.org/10.3390/su151310730

**Chicago/Turabian Style**

Xing, Ruru, Zimu Li, Xiaoyu Cai, Zepeng Yang, Ningning Zhang, and Tao Yang.
2023. "Accident Rate Prediction Model for Urban Expressway Underwater Tunnel" *Sustainability* 15, no. 13: 10730.
https://doi.org/10.3390/su151310730