Comparison of threshold determination methods for the deceleration rate to avoid a crash (DRAC)-based crash estimation
Introduction
The problems associated with the use of crash data in road safety analysis are well-documented (Sayed and Zein, 1999). There are quality and availability problems associated with crash data. As well, the use of crash data in safety analysis is a reactive approach. Crashes need to be observed over a prolonged time period to identify safety problems and to correct them. Therefore, the development of surrogate safety measures such as traffic conflicts for use in safety analysis has attracted considerable research interest. Compared to traffic crashes, traffic conflicts occur more frequently, can be easily observed, and provide better insight into the collision failure mechanism (Sayed and Zein, 1999).
Over the past decades, a number of traffic conflict indicators have been established and used in traffic safety analysis. Examples of most widely used conflict indicators include temporal and spatial proximity indicators and deceleration-based indicators. Several researchers have explicitly recognized the deceleration rate to avoid a crash (DRAC) as a useful surrogate safety measure (Archer, 2005). The DRAC is the rate at which a following vehicle must decelerate to avoid a collision with the leading vehicle (Zheng and Sayed, 2019a). It is defined as the differential speed between a following vehicle and its leading vehicle divided by their closing time (Cooper and Ferguson, 1976; Gettman and Head, 2003). Compared to the other surrogate safety measures, such as time to collision (TTC) and the post encroachment time (PET), the DRAC has some advantages, since it could reflect the evasive action (e.g., required deceleration) of the following vehicle to come to a timely stop or reach the corresponding speed of the leading vehicle and thereby avoid a collision (Cunto and Saccomanno, 2008).
The greater the DRAC value the higher the crash risk, and a crash would occur if the DRAC exceeds the maximum available deceleration rate (MADR). Therefore, the crash risk based on DRAC can be calculated given a MADR. Since the MADR is unique for individual vehicles and depends on pavement conditions, vehicle weight, tire and braking system, and other factors, it is usually assumed to follow a truncated normal distribution N(8.45, 1.42), with an upper limit of 12.68 m/s2 and a lower limit of 4.23 m/s2 (Cunto and Saccomanno, 2008). Several studies have investigated the crash risk based on DRAC by mainly using the extreme value distribution (EVT) and the truncated normal distribution. The former treats the conflict extremes characterized by DRAC as a random variable, while the latter treats the MADR as a random variable. Additionally, a threshold needs to be developed to identify traffic conflict extremes for use in the establishment of the EVT model based on DRAC.
Previous studies used different specific values for MADR to conduct DRAC-based traffic conflict safety evaluation. A value of 8 m/s2 for MADR was considered as a conservative value that all vehicles could reach in one previous study (Ceunynck, 2017). Wang et al. (2019) employed the MADR values of 8 m/s2 and 12 m/s2 for crash estimation from the bivariate extreme value models, respectively. In the study by (Zheng et al., 2019b), the mean value (i.e., 8.45 m/s2) of the truncated normal distribution N(8.45, 1.42)I(4.23, 12.68) of MADR was selected as the threshold for crash occurrence, which was applied to develop the bivariate extreme value models for crash estimation. It is noted that a specific value of MADR may lead to DRAC-based crash underestimation or overestimation at some sites. Therefore, Zheng and Sayed (2019a) used the values of DRAC sampled from the fitted extreme value distribution and the values of MADR sampled from a given truncated normal distribution N(8.45, 1.42)I(4.23, 12.68) to calculate the DRAC-based crash risk. As well, Fu et al. (2020) determined the values of MADR for DRAC-based crash estimation by sampling from the truncated normal distribution N(8.45, 1.42)I(4.23, 12.68).
In addition to the extreme value distribution, the truncated normal distribution is also used to estimate the crash risk based on DRAC. In other words, the MADR is regarded as a random variable and is assumed to follow a truncated normal distribution. The crash risk can be calculated as the probability of the MADR being less than the DRAC. Meng and Weng (2011) evaluated the rear-end crash risk measured by DRAC at work zones. They assumed that the MADR of cars and trucks followed truncated normal distributions N(8.45, 1.42)I(1.23, 12.68) and N(6.82, 1.42)I(0.60, 10.05), respectively. This approach was also applied to investigate the work zone rear-end crash risk based on DRAC for different vehicle-following patterns (Weng et al., 2014), assess the performance of two microscopic safety indices (i.e., DRAC and TTC) on safety evaluation for expressways (Qu et al., 2014), and explore the rear-end crash risk based on DRAC and TTC in work zone merging areas (Weng et al., 2015). With the updated data of the braking performance of various automobiles, a new truncated normal distribution of MADR N(9.7, 1.32)I(4.2, 12.7) was obtained (Wang and Stamatiadis, 2013, 2014). The two studies employed the distribution of MADR to calculate the probability of MADR being less than the required braking rate to avoid a crash conditional on the reaction time being less than the TTC.
The review of previous studies using DRAC in road safety analysis shows that limited research has been undertaken on selecting which threshold determination approach for DRAC-based traffic conflict analysis (i.e., setting specific values or distributions of MADR) would yield the best results in terms of accuracy and precision.
Therefore, the objective of this study is to compare the threshold determination approaches for DRAC-based crash estimation in terms of accuracy and precision. For the comparison, a Bayesian hierarchical extreme value model that combines traffic conflict extremes from different sites, incorporates the impact of various covariates, and considers the unobserved site-specific heterogeneity is developed.
The structure of this paper is organized as follows: Section 2 presents the data used in this study. Section 3 provides details of the proposed Bayesian hierarchical extreme value model, including the generalized extreme value model, the Bayesian hierarchical structure, the model estimation method, the model choice, and the crash estimation approach. The model estimation results, model comparisons, and variables interpretation are discussed in Section 4. The last section, Section 5, presents the conclusions of this study and the recommended future work.
Section snippets
Data description
The data used for this study include video data and crash data. The video data were collected from four signalized intersections in the city of Surrey, British Columbia, Canada. They are the intersections of 72 Ave & 128 St, 72 Ave & 132 St, 64 Ave & King George Blvd, and Fraser Hwy & 168 St (see Fig. 1).
Video data were collected by cameras that focus on the approaches where most of rear-end conflicts occur. Computer vision techniques were applied to identify signal cycles, extract vehicle
Generalized extreme value model
In this study, the generalized extreme value (GEV) distribution is fitted based on the block maxima (BM) extremes. The observations are aggregated into fixed blocks, and the maxima of each block is treated as extreme (Coles, 2001; Zheng et al., 2018). Suppose that is a sequence of independent random variables having a common distribution function and represents the block maxima. If there exist sequences of constants and such that
Model estimation results
With the maximum values of DRAC from different signal cycles of four signalized intersections, the multisite Bayesian hierarchical extreme value models were developed. The estimations of the proposed models were conducted using WinBUGS (Lunn et al., 2000). Two separate chains for each model parameter with diverse initial values were set to run 50,000 iterations and the first 20,000 iterations were discarded as burn-in samples. The posterior estimates were acquired from the remaining 30,000
Conclusions
This study compared the threshold determination methods (i.e., setting specific values or distributions of MADR) for DRAC-based crash estimation using Bayesian hierarchical extreme value modeling. The crash data and traffic conflicts defined by DRAC were collected from four signalized intersections in the city of Surrey, British Columbia, Canada. Four multisite Bayesian hierarchical models were developed, including a stationary model and three non-stationary models. For comparison purposes, the
CRediT authorship contribution statement
Chuanyun Fu: Conceptualization, Methodology, Investigation, Validation, Writing - original draft. Tarek Sayed: Conceptualization, Methodology, Data curation, Supervision, Writing - review & editing.
Declaration of Competing Interest
None.
Acknowledgments
This study was jointly supported by the China Scholarship Council (CSC) and the National Natural Science Foundation of China (Grant No. 71801182).
References (36)
- et al.
Calibration and validation of simulated vehicle safety performance at signalized intersections
Accid. Anal. Prev.
(2008) - et al.
Measuring direct and indirect treatment effects using safety performance intervention functions
Saf. Sci.
(2012) - et al.
Measuring safety treatment effects using full Bayes non-linear safety performance intervention functions
Accid. Anal. Prev.
(2012) - et al.
Traffic conflict models to evaluate the safety of signalized intersections at the cycle level
Transp. Res. Part C
(2018) - et al.
Full Bayesian conflict-based models for real time safety evaluation of signalized intersections
Accid. Anal. Prev.
(2019) - et al.
Multivariate Bayesian hierarchical modeling of the non-stationary traffic conflict extremes for crash estimation
Anal. Methods Accid. Res.
(2020) - et al.
Evaluation of rear-end crash risk at work zone using work zone traffic data
Accid. Anal. Prev.
(2011) - et al.
The extreme value theory approach to safety estimation
Accid. Anal. Prev.
(2006) - et al.
Evaluation of a simulation-based surrogate safety metric
Accid. Anal. Prev.
(2014) - et al.
A crash prediction method based on bivariate extreme value theory and video-based vehicle trajectory data
Accid. Anal. Prev.
(2019)
Analysis of work zone rear-end crash risk for different vehicle-following patterns
Accid. Anal. Prev.
In-depth analysis of drivers’ merging behavior and rear-end crash risks in work zone merging areas
Accid. Anal. Prev.
A generalized exponential link function to map a conflict indicator into severity index within safety continuum framework
Accid. Anal. Prev.
Bayesian hierarchical modeling of traffic conflict extremes for crash estimation: a non-stationary peak over threshold approach
Anal. Methods Accid. Res.
A bivariate Bayesian hierarchical extreme value model for traffic conflict-based crash estimation
Anal. Methods Accid. Res.
Freeway safety estimation using extreme value theory approaches: a comparative study
Accid. Anal. Prev.
Before-after safety analysis using extreme value theory: a case of left-turn bay extension
Accid. Anal. Prev.
Bayesian hierarchical modeling of the non-stationary traffic conflict extremes for crash estimation
Anal. Methods Accid. Res.
Cited by (46)
A spatio-temporal deep learning approach to simulating conflict risk propagation on freeways with trajectory data
2024, Accident Analysis and PreventionDynamic Bayesian hierarchical peak over threshold modeling for real-time crash-risk estimation from conflict extremes
2023, Analytic Methods in Accident ResearchAn integrated approach of machine learning and Bayesian spatial Poisson model for large-scale real-time traffic conflict prediction
2023, Accident Analysis and PreventionIdentification of adequate sample size for conflict-based crash risk evaluation: An investigation using Bayesian hierarchical extreme value theory models
2023, Analytic Methods in Accident Research