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category: literaturenote citekey: furandomparametersbayesianhierarchical2021 title: Random parameters Bayesian hierarchical modeling of traffic conflict extremes for crash estimation authors: "Fu, Chuanyun; Sayed, Tarek" year: 2021 date: 2021-00-00 2021 doi: 10.1016/j.aap.2021.106159 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2021.106159" zotero_key: 8J5KUNAH zotero_storage: 8NQ9VEZG collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Fu, Chuanyun"

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Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Random parameters Bayesian hierarchical modeling of traffic conflict extremes for crash estimation

Chuanyun Fu a,b, *, Tarek Sayed b

  • a School of Transportation and Logistics, Southwest Jiaotong University, China
  • b Department of Civil Engineering, The University of British Columbia, Canada

ARTICLE INFO

Keywords: Traffic conflict Crash estimation Random parameters Univariate extreme value Bayesian hierarchical structure Random intercepts

ABSTRACT

The use of Extreme Value Theory (EVT) models for traffic conflict-based crash estimation is becoming increasingly popular with considerable recent advances achieved. The latest advances include developing EVT models that combine several conflict indicators and the use of data from several sites to increase the sample size of conflict extremes. Nevertheless, one important issue while developing EVT models is accounting for the unobserved heterogeneity across different conflict observation sites and road user behaviours which can lead to biased and inefficient parameter estimates and erroneous inferences. This study proposes a random parameters (RP) Bayesian hierarchical extreme value modeling approach to account for the unobserved heterogeneity. The proposed approach is applied to estimate rear-end crashes from traffic conflicts collected from four signalized intersections in the city of Surrey, British Columbia. Traffic conflicts were characterized by four indicators: time to collision (TTC), modified TTC (MTTC), post-encroachment time (PET), and deceleration rate to avoid a crash (DRAC). MTTC was used to fit the generalized extreme value distribution, while the other three conflict indicators were treated as covariates. Six covariates including TTC, PET, DRAC, traffic volume, shock wave area, and platoon ratio were considered to account for non-stationarity in conflict extremes. Several RP, random intercepts (RI), and fixed parameters (FP) Bayesian hierarchical univariate extreme value models were developed. The results indicate that the RP model outperforms both the RI model and the FP model in terms of crash estimation accuracy and precision. Such superiority may be due to the ability of the RP model to better account for the unobserved heterogeneity.

1. Introduction

The use of the traffic conflict technique as a proactive surrogate tool for evaluating road safety has recently gained considerable popularity (Sayed and Zein, 1999; Fu et al., 2020; Fu and Sayed, 2021a,b). The focus of recent research in traffic conflict analysis is on conflict-based crash estimation to overcome the availability, quality and scarcity issues associated with crash data (Zheng et al., 2021). The main assumption in conflict-based crash estimation is that, usually, the same failure mechanism in the driving process leads to the occurrence of both conflicts and crashes (Hyd´en, 1987; Tarko, 2018b). One modeling approach that enables the extrapolation from observed levels (conflicts) to unobserved levels (crashes) is extreme value theory (EVT) (Coles, 2001).

Traffic conflicts can be measured by several indicators, such as temporal proximal indicators, spatial proximal indicators, decelerationbased indicators, among others (Zheng et al., 2014b; Mahmud et al., 2017). Several studies have developed EVT models using the conflict extremes identified by different conflict indicators (Songchitruksa and Tarko, 2006; Tarko, 2012; Åsljung et al., 2017; Farah and Azevedo, 2017; Tarko, 2018a). It is noted that most of the EVT models in previous studies were developed for individual sites using a single conflict indicator. Recently, significant advances have been achieved in this area including developing EVT models to integrate several conflict indicators (Fu et al., 2020; Zheng and Sayed, 2020; Fu and Sayed, 2021a) and developing Bayesian hierarchical extreme value models incorporating conflict data from several sites to account for the scarcity and non-stationarity of conflict extremes (Zheng and Sayed, 2019b; Zheng et al., 2019b; Fu et al., 2020; Zheng and Sayed, 2020; Fu and Sayed, 2021a,b). As pointed out by Tarko (2012), a statistical modeling approach for combing the data from several sites/subjects is important to address the unobserved heterogeneity rather than grouping the data

* Corresponding author at: School of Transportation and Logistics, Southwest Jiaotong University, China. E-mail addresses: fcy1128@mail.ubc.ca (C. Fu), tsayed@civil.ubc.ca (T. Sayed).

in terms of sites/subjects to ensure homogeneity. Site-specific factors that affect conflict occurrence could be explicitly accounted for by being incorporated as covariates in non-stationary EVT models. These non-stationary models were showed to provide better accuracy and precision conflict-based crash estimation compared to the stationary models (Farah and Azevedo, 2017; Zheng et al., 2019b; Cavadas et al., 2020). However, issues related to unobserved heterogeneity can still be problematic and need to be accounted for. Decades of experience developing safety analysis models have clearly shown the importance of accounting for unobserved heterogeneity to yield efficient parameter estimates and valid inferences (Mannering and Bhat, 2014; Mannering et al., 2016; Liu et al., 2021). Some recent studies have attempted to account for unobserved heterogeneity using random effect terms representing between-site variances, which are consistent for extremes in the same site but vary between different sites (Zheng and Sayed, 2019b; Zheng et al., 2019b; Fu et al., 2020; Zheng and Sayed, 2020; Fu and Sayed, 2021a,b). This approach, while representing an improvement over models that do not account for unobserved heterogeneity, still has limitations. The main shortcoming is the assumption that the unobserved site-specific heterogeneity is completely unrelated to the covariates (Zheng et al., 2019b). This is a rather strong assumption indicating that the estimated parameters for all the covariates are fixed across different sites. Actually, the influence of certain covariates may vary across different sites due to unobserved factors. Additionally, as discussed in Mannering et al. (2016), the random parameters (RP) models are more flexible than the random intercepts (RI) models, since the former can be estimated with cross-sectional data as well as panel data, while the latter typically requires panel data.

To address the limitations discussed above, this paper introduces RP to a Bayesian hierarchical univariate extreme value approach for traffic conflict-based crash estimation. The proposed approach uses random coefficients and RI to better account for the unobserved heterogeneity across different conflict observation sites. Moreover, the proposed approach provides a simpler way to integrate several conflict indicators by including some conflict indicators as covariates.

2. Previous work on EVT models

The application of the EVT approach for road safety analysis was first proposed by Campbell et al. (1996), who reported that the EVT approach is a robust tool to estimate the probability of a crash based on the frequency of small crash margin. Songchitruksa (2004) and Songchitruksa and Tarko (2006) proposed detailed EVT modeling approaches, which facilitated the development of EVT models in road safety analysis. After more than two decades of rapid development, some significant advances have been achieved which primarily include the following three aspects.

2.1. Development of non-stationary EVT models

EVT models can be classified into two main groups: stationary and non-stationary. Stationary EVT models have an underlying assumption that the observed traffic conflict extremes are from the same distribution. Due to unavailable data of influence factors or the simplification of model development, traffic conflict extremes have often been assumed to be stationary (Tarko, 2012; Åsljung et al., 2017; Tarko, 2018a; Wang et al., 2018; Zheng et al., 2018a; Orsini et al., 2019; Wang et al., 2019; Zheng and Sayed, 2019c;d; Zheng et al., 2019a; Borsos et al., 2020; Orsini et al., 2020).

However, the assumption of stationarity is difficult to justify given the many factors that can affect conflict occurrence. Without accounting for non-stationarity, traffic conflict-based crash estimation will be biased. Hence, a growing body of studies has developed non-stationary EVT models for crash estimation. Songchitruksa and Tarko (2006) considered the impact of through and left-turn volumes to develop generalized extreme value (GEV) models based on post-encroachment time (PET) for predicting right-angle crashes. Zheng et al. (2014a) developed the PET GEV models and the PET generalized Pareto distribution (GPD) models that accounted for the influence of traffic volume, large vehicle proportion, merging vehicle type, and other factors to estimate lane changing maneuver-related crashes. Farah and Azevedo (2017) accounted for the influence of passing gap characteristics and the speeds of the front vehicle, passing vehicle, and opposite vehicle to develop the GEV model and the GPD model based on time to collision (TTC) for predicting head-on crashes. Zheng et al. (2018b) considered the influence of traffic volume, number of conflicts with TTC less than 1.5 s, and number of conflicts with TTC less than 3.0 s in the development of TTC GEV models for conflict-based before-after safety analysis. Zheng and Sayed (2019a) employed the full Bayes approach to develop a TTC GEV model that incorporated the influence of treatment indicator, time indicator, interact indicator, and number of conflicts with TTC less than 1.5 s to conduct the before-after safety evaluation. Zheng et al. (2019b) used the Bayesian hierarchical modeling approach to develop the GEV models based on the modified TTC (MTTC) for estimating rear-end crashes at signalized intersections. The developed Bayesian hierarchical models incorporated the influence of traffic volume, shock wave area, average shock wave speed, and platoon ratio of each cycle. These covariates were also considered in their other EVT modeling studies (Zheng and Sayed, 2019b; Fu et al., 2020; Zheng and Sayed, 2020; Fu and Sayed, 2021a,b). Cavadas et al. (2020) accounted for the effect of driver specific characteristics and road infrastructure conditions to develop the GEV models based on TTC and the time headway between the front of the passed vehicle and the rear of the passing vehicle (denoted as THW) for estimating the head-on and rear-end collisions in passing maneuvers. The findings of the above-mentioned studies confirmed the superiority of non-stationary EVT models compared to the stationary EVT models.

2.2. Development of multivariate EVT models

Most existing EVT models are univariate which are developed based on a single traffic conflict indicator (Campbell et al., 1996; Songchitruksa, 2004; Songchitruksa and Tarko, 2006; Tarko, 2012; Zheng et al., 2014a; Åsljung et al., 2017; Farah and Azevedo, 2017; Tarko, 2018a; Wang et al., 2018; Zheng et al., 2018b; Orsini et al., 2019; Zheng and Sayed, 2019a,b,c,d; Zheng et al., 2019b; Borsos et al., 2020; Orsini et al., 2020). It is noted that a single traffic conflict indicator only represents a partial severity aspect of a traffic interaction (Ismail et al., 2011). To represent a more complete severity aspect of a traffic interaction, a growing number of bivariate EVT models based on two conflict indicators have been developed by researchers. Zheng et al. (2018a) constructed the bivariate threshold excess models based on PET and length proportion of merging to estimate crashes on freeway entrance merging areas. Zheng and Sayed (2019c) developed the bivariate GEV models based on PET & TTC to predict left-turn crashes at signalized intersections. Zheng et al. (2019a) used four conflict indicators TTC, MTTC, PET, and deceleration rate to avoid a crash (DRAC) to develop several bivariate GEV models for estimating rear-end crashes at signalized intersections. Wang et al. (2019) built the bivariate threshold excess models based on time-to-accident (TA) & PET, TA & maximum deceleration rate, and TA & minimum TTC to estimate angle crashes and rear-end crashes, respectively. Cavadas et al. (2020) established the bivariate GEV models based on TTC & THW to predict head-on and rear-end collisions in passing maneuvers. Zheng and Sayed (2020) applied the Bayesian hierarchical modeling approach to develop a bivariate GEV model based on MTTC & PET to estimate rear-end crashes at signalized intersections. These studies indicated that considerable improvement could be achieved from univariate EVT models to bivariate EVT models. However, as pointed out by Zheng and Sayed (2019c), two traffic conflict indicators may be still insufficient to cover all severity aspects of a traffic interaction. Therefore, in order to integrate more than two conflict indicators, a dependence parameter (Fu et al., 2020) and Gaussian copula (Fu and Sayed, 2021a) were utilized to develop multivariate Bayesian hierarchical extreme value models to predict rear-end crashes at signalized intersections. These two studies argued that the high dimensional multivariate (e.g., trivariate) Bayesian hierarchical EVT model outperforms both the low dimensional multivariate (e.g., bivariate) model and the univariate model in terms of crash estimation accuracy.

2.3. Development of multi-site EVT models

Multi-site EVT models that combine traffic conflicts from several sites and account for the unobserved site-specific heterogeneity were developed in Zheng et al. (2019b); Zheng and Sayed (2019b); Fu et al. (2020); Zheng and Sayed (2020), and Fu and Sayed (2021a). These studies indicate a remarkable improvement in model estimation due to handling the unobserved heterogeneity. However, it is noted that the unobserved heterogeneity in the developed multi-site EVT models is captured by using random effect terms representing between-site variances. This may be insufficient, because the unobserved heterogeneity will likely exist in the effect of some covariates across different sites.

3. Methodology

3.1. GEV distribution

One fundamental method to estimate EVT models is the block maxima approach, in which observations are aggregated into fixed blocks over time and the maxima within each block is regarded as an extreme (Coles, 2001). Suppose that *X1, X2,* …, Xn is a sequence of independent random variables having a common distribution and Mn = max{*X1, X2,* …, Xn} denotes a block maxima of n values. If the normalized variable M∗ n = (Mn − μ)/σ converges, then it will converge to a GEV distribution (Zheng et al., 2019b), which is given by

$$G(z) = Pr\left{M_n^* \le z\right} = exp\left{-\left[1 + \xi\left(\frac{z - \mu}{\sigma}\right)\right]^{-1/\xi}\right}$$ (1)

defined on {z : 1 + ξ(z − μ)/σ > 0}, where μ is the location parameter, − ∞ < μ < ∞; σ is the scale parameter, σ > 0; ξ is the shape parameter, − ∞ < ξ < ∞. It is noted that the GEV distributions with ξ > 0 and ξ < 0 respectively correspond to the Fr´echet and Weibull distributions, while the GEV distribution with ξ = 0 corresponds to the Gumbel distribution (Coles, 2001).

3.2. RP Bayesian hierarchical structure

Following the approach of (Zheng and Sayed, 2019b; Zheng et al., 2019b; Fu et al., 2020; Zheng and Sayed, 2020; Fu and Sayed, 2021a,b), this study develops a RP Bayesian hierarchical structure for the univariate extreme value model, including data layer, process layer, and prior layer. The data layer uses the GEV distribution to model traffic conflict extremes. The process layer models the latent process that employs like functions to relate the location parameter, scale parameter, and shape parameter of GEV distribution to several covariates. The prior layer determines the prior distributions of parameters in the latent process.

According to the Bayes theorem, the posterior inference for the parameters θ of the univariate extreme value model with a three-layer hierarchical structure given the data Z is

$$p(\boldsymbol{\theta} \mid \boldsymbol{Z}) \propto \underbrace{p_1(\boldsymbol{Z} \mid \boldsymbol{\theta}1)}{Data} \underbrace{p_2(\boldsymbol{\theta}_1 \mid \boldsymbol{\theta}2)}{Process} \underbrace{p_3(\boldsymbol{\theta}2)}{Prior} \tag{2}$$

where ps is the density function corresponding to the level s (s = 1, 2, 3) of Bayesian hierarchical structure and depends on parameters θε(ε = 1, 2).

3.2.1. Data layer

A GEV distribution expressed by Eq. (1) is employed to model the data in the first layer of the three-layer hierarchy. Due to the positive property of scale parameter, the GEV (μ, σ, ξ) is reparameterized as GEV (μ, ψ, ξ) where ψ = logσ. Let zij denotes the i th block maxima at site j, with j = 1, 2*,…, k* and i = 1, 2, …, nj. Given zij representing the block maxima, it can be described by the GEV distribution whose parameters depend on the site. Hence, letting *μj, ψj, ξj*denote the parameters at site j generates

$$G(z_{ij} < z | \mu_j, \psi_j, \xi_j) = exp \left{ -\left[1 + \xi_j \frac{(z - \mu_j)}{\exp(\psi_j)}\right]^{-1/\xi_j} \right}$$ (3)

By differentiating the function given by Eq. (3), the "Data" piece of Eq. (2) can be expressed as:

$$p{1}(\mathbf{Z}|\boldsymbol{\theta}{1}) = \prod{j=1}^{k} \prod{i=1}^{n{j}} \left(\frac{1}{\exp(\psi{j})}\right) exp\left(-\left(1 + \xi{j} \frac{(z{ij} - \mu{j})}{\exp(\psi{j})}\right)^{-1/\xi_{j}}\right)$$

$$\left(1 + \xi{j} \frac{(z{ij} - \mu{j})}{\exp(\psi{j})}\right)^{-1-1/\xi_{j}}$$

$$(4)$$

where θ1 = [μ, ψ, ξ].

3.2.2. Process layer

A latent Gaussian process underlying extremes is characterized by applying the identity link functions to relate the parameters in the data layer to certain covariates. What is noteworthy is that since it is difficult to accurately estimate the shape parameter ξ of GEV distribution, it is unrealistic to construct a smooth function of covariates for the shape parameter ξ. Generally, the shape parameter ξ of GEV distribution is hypothesized to be unknown but not impacted by covariates during the process (Coles, 2001; Cooley et al., 2006). Therefore, the parameters of GEV distribution in the data layer can be expressed as:

$$\begin{cases} \mu{j} = \beta{\mu 0} + \beta{\mu} Y \ \psi{j} = \beta{\theta 0} + \beta{\psi} Y \ \xi{j} = \beta{\xi 0} \end{cases}$$ (5)

where *βμ*0, *βψ*0, and *βξ*0 are intercepts; βμ and βψ are vectors of estimated parameters; Y is the vector of covariates.

In practice, there usually exists unobserved heterogeneity over observations. In the field of traffic safety analysis, the existing models typically account for unobserved heterogeneity among analyst-specified groups (such as road segments, intersections, and corridors) instead of individual observations (Mannering et al., 2016). For this current study, the unobserved heterogeneity among observation sites (i.e., signalized intersections) is considered. There are two ways to capture the site-specific unobserved heterogeneity. One is to introduce RI with an assumption that the unobserved site-specific heterogeneity is completely unrelated to the explanatory variables (i.e., covariates) (Zheng et al., 2019b). By introducing RI, the parameters of GEV distribution in the data layer can be written as:

$$\begin{cases} \mu{j} = \beta{\mu 0j} + \beta{\mu}Y \ \psi{j} = \beta{\psi 0j} + \beta{\psi}Y \ \xi{j} = \beta{\psi 0j} \end{cases}$$

$$(6)$$

where *βμ*0j, *βψ*0j, *βξ*0*j*are site-specific RI, which are allowed to vary between different sites to capture the heterogeneity caused by unobserved factors that are consistent for extremes at the same site. Additionally, it is noted that βμ and βψ are vectors of fixed parameters (one parameter estimate for all observational sites).

In fact, the effect of a certain covariate on extremes may also vary across sites due to the unobserved heterogeneity. It is insufficient to introduce only RI to capture the unobserved heterogeneity. Hence, another is to introduce random coefficients as well as RI, and thus the

model for the process layer can also be written as:

$$\begin{cases} \mu{j} = \beta{j,0,j} + \beta{pj}Y \ \psi{j} = \beta{q,0,j} + \beta{qj}Y \ \xi{j} = \beta{20,j} \end{cases}$$

$$(7)$$

where βμj and βψj are vectors of site-specific random coefficients that vary across different sites but are common for each site.

With the specified GEV parameters, the "process" piece of Eq. (2) can be given by

$$p{2}(\boldsymbol{\theta}{1}|\boldsymbol{\theta}{2}) = \frac{1}{\sqrt{2\pi\delta{\mu}^{2}}} exp\left[-\frac{(\mu-\mu{j})^{2}}{2\delta{\mu}^{2}}\right] \times \frac{1}{\sqrt{2\pi\delta{\psi}^{2}}} exp\left[-\frac{(\psi-\psi{j})^{2}}{2\delta_{\psi}^{2}}\right]$$

$$\times \frac{1}{\sqrt{2\pi\delta{\xi}^{2}}} exp\left[-\frac{(\xi-\xi{j})^{2}}{2\delta_{\xi}^{2}}\right]$$ (8)

where θ2 = [*βμ, βψ, βμ0, βψ0, βξ*0].

3.2.3. Prior layer

In this layer, a series of priors are assigned to the parameters in the process layer. Moreover, it is hypothesized that each parameter in the process layer is independent of the others. Due to no available prior information on how the parameters of GEV distribution related to covariates, uninformative priors for parameters (i.e., *βμ, βψ, βμ0, βψ*0) related to the location and scale parameters are used. All the regression parameters βμ and βψ are assumed to follow normal distributions with mean ζ and variance *ϱ*2. The mean ζ of a parameter is assumed to follow a normal distribution with zero mean and large variance, i.e., N(0, 106 ), while the distribution of the variance *ϱ*2 of a parameter is set as inverse-Gamma (0.001, 0.001). If the variance of the distribution for a parameter is not significantly different from zero, which suggests that it is statistically appropriate to set this parameter as a fixed parameter (Mannering et al., 2016). With respect to the fixed parameters, they are assumed to be normally distributed with zero mean and large variance, i. e., N(0, 106 ). As well, the parameters *βμ*0 and *βψ*0 are assumed to follow normal distributions with zero mean and large variance, i.e., N(0, 106 ). In previous relevant studies such as (Songchitruksa and Tarko, 2006) and (Zheng et al., 2014a), the estimated shape parameter of GEV distribution is usually within the range of (-1.0, 1.0). Hence, the prior distributions of shape parameters are set as uniform distributions, i.e., Unif(-1.0, 1.0).

With the specified priors, the "prior" piece of Eq. (2) can be written as:

$$p_3(\theta2) = p{\beta{\mu}}(\boldsymbol{\beta{\mu}}) \times p{\beta{\psi}}(\boldsymbol{\beta{\psi}}) \times p{\beta{\mu 0}}(\boldsymbol{\beta{\mu 0}}) \times p{\beta{\psi 0}}(\boldsymbol{\beta{\psi 0}}) \times p{\beta{z 0}}(\boldsymbol{\beta{\xi 0}})$$ (9)

3.2.4. Model inference

For the Bayesian hierarchical models, the likelihood functions are developed on the basis of the probability distributions defined in the data layer and process layer as well as the prior distributions specified in the prior layer. The posterior distributions of model parameters are acquired by combining Eqs. (4),(8), and (9). The posterior distributions of model parameters can be inferred by virtue of Markov Chain Monte Carlo (MCMC) simulation using Gibbs Sampling.

3.3. Model choice

When developing the RP Bayesian hierarchical univariate model with linking the location parameter and scale parameter of GEV distribution to several covariates, there will be a lot of models to choose from. In the Bayesian framework, the Deviance Information Criterion (DIC) is extensively used as the model selection criterion. The basic principle of DIC is parsimony, obtaining the simplest model that explains as much of the variation in the data as possible (Spiegelhalter et al., 2002). The DIC is expressed as:

$$DIC = \overline{D} + p_D \tag{10}$$

where D denotes the posterior mean deviance that measures the model fitting; pD represents the effective number of parameters in the model. Typically, a model with a smaller DIC value is better. Moreover, a difference greater than 10 in the DIC value can definitely rule out the model with the higher DIC value, the difference of DIC value ranging from 5 to 10 is considered substantial, and a difference less than 5 in the DIC value make the models competitive (El-Basyouny and Sayed, 2012a,b).

3.4. Crash estimation approach

Considering a traffic conflict measured by MTTC as an example, a smaller MTTC indicates a more severe traffic conflict (Zheng and Sayed, 2019b). Once the MTTC is less than or equal to zero, a crash occurs. Therefore, based on the fitted GEV distribution of negated MTTC, the crash risk can be calculated as:

$$R_i = \Pr(Z_i \ge 0) = 1 - \Pr(Z_i < 0) = 1 - \exp\left{-\left[1 - \xi_i \mu_i / \exp(\psi_i)\right]^{-1/\xi_i}\right}$$ (11)

where Ri is the crash risk for block i; Zi represents the maximum negated MTTC for block i.

To compare the estimated crashes with the observed crashes, it is hypothesized that the traffic conflict observation period *ηj*with nj blocks at site j is representative for a long period T, a year for example, and then the predicted annual number of crashes Nj for such period T can be calculated as:

$$N_j = \frac{T}{\etaj} \sum{i=1}^{nj} R{ij} \tag{12}$$

where Rij denotes crash risk for block i at site j.

4. Traffic conflict and crash data

4.1. Traffic conflict indicators

Several conflict indicators are used in this study. TTC is defined as "the time required for two vehicles to collide if they continue at their present speeds and on the same path" (Hayward, 1972). The TTC for the following vehicle (F) at time t with respect to a leading vehicle (L) can be calculated as:

$$TTC{t} = \frac{X{L,t} - X{F,t} - D{L}}{V{F,t} - V{L,t}}; \ \forall (V{F,t} - V{L,t}) > 0$$ (13)

where *XL,t*and *XF,t*are positions of the leading and following vehicles at time t, respectively; *VF,t*and *VL,t*are speeds of the following and leading vehicles at time t, respectively; DL represents the leading vehicle length. It is noted that the TTC is valid only if the speed difference (VF,t − VL,t) between the following vehicle and the leading vehicle is positive.

As pointed out by Ozbay et al. (2008), the assumption of the positive constant speed difference for TTC ignores many potential conflicts due to acceleration or deceleration discrepancies. Therefore, Ozbay et al. (2008) proposed a modified TTC, named MTTC, that considered relative position, relative speed, and relative acceleration of the conflicting vehicles. The MTTC can be calculated as:

$$MTTC{t} = \frac{\Delta V{t} \pm \sqrt{\Delta V{t}^{2} + 2\Delta a{t}(\Delta X{t} - D{L})}}{\Delta a_{t}}$$ (14)

where ΔXt is the relative position of conflicting vehicles at time t; ΔVt denotes the relative speed of conflicting vehicles at time t; Δat represents the relative acceleration of conflicting vehicles at time t. There are two outcomes of MTTC calculation. If both of them are positive, the smaller

one is selected as the MTTC value; if one outcome is positive while another is negative, the positive outcome is selected as the MTTC value.

PET is defined as "the time difference between the moment an 'offending' vehicle passes out of the area of potential collision and the moment of arrival at the potential collision point by the 'conflicted' vehicle possessing the right-of-way" (Cooper, 1984; Zheng et al., 2019a). For the car-following scenario, PET is typically measured based on a conflict line rather than a conflict area (Gettman and Head, 2003). Accordingly, given that the conflict line coincides with the rear of a leading vehicle at time t, the corresponding PET is calculated as:

$$PET{t} = M{F,t} - M_{L,t} \tag{15}$$

where *MF,t*is the moment when the following vehicle arrives at the conflict line and *ML,t*denotes the moment when the leading vehicle leaves the conflict line.

The DRAC is defined as the rate at which the following vehicle must decelerate to avoid the collision with the leading vehicle (Cooper and Ferguson, 1976; Gettman and Head, 2003). It is calculated as:

$$DRAC_{FJ} = \frac{(V{FJ} - V{LJ})^2}{2(X{LJ} - X{F_J} - D_L)}$$ (16)

4.2. Data description

Traffic conflict data were extracted from video data collected from the selected approaches of four signalized intersections in the city of Surrey, British Columbia, Canada. They are the intersection approaches where the majority of rear-end conflicts occur as shown in Fig.1. A total of 24 h of video data were collected: 4 h for the intersection of 72 Ave & 128 St, 3 h for the intersection of 72 Ave & 132 St, 6 h for the intersection of 64 Ave & King George Blvd, and 11 h for the intersection of Fraser Hwy & 168 St.

A detailed description of the extraction and identification of traffic conflict indicators, traffic signal cycles, vehicle trajectories, and spacetime diagram of vehicles can be found in (Essa and Sayed, 2018, 2019). According to the video-analysis procedure, different traffic parameters and conflict indicators for each traffic signal cycle were extracted.

The traffic parameters included traffic volume, shock wave area, and platoon ratio. Traffic volume is the number of vehicles for each cycle on the through lane (exclusive right-turn and left-turn lanes excluded) of the studied intersection approach (Essa and Sayed, 2018; Zheng and Sayed, 2020). As illustrated in Fig. 2, the shock wave area is the area that consists of backward forming shock wave, backward recovery shock wave, and frontal stationary shock wave. It also refers to the area of traffic flow state II in terms of classical four traffic flow states theory (May, 1990). The platoon ratio is defined as the proportion of vehicles arriving during the green signal divided by the ratio of the effective green time to the signal cycle length of the subject movement (Highway Capacity Manual, 2010; Essa and Sayed, 2018).

Based on the trajectories of each pair of consecutive vehicles, four conflict indicators TTC, MTTC, PET, and DRAC were computed continuously (i.e., frame by frame with one second equals to 30 frames). Among the calculated values, the minimum values were adopted for TTC, MTTC, and PET, whereas the maximum value was adopted for DRAC. The development of EVT models requires the identification of extreme conflicts within each block. These extremes were identified from the events with TTC≤ 4 s, MTTC ≤ 4 s, PET≤ 4 s, or DRAC> 0 m/s2 It is worth noting that, as argued by Zheng et al. (2019a), because the EVT models focus on the extreme values of conflict indicators (block minimum values for TTC, MTTC, and PET; block maximum value for DRAC), the predefined values for event selection have little effect on the results of model estimation only if they are far away from the thresholds of extreme determination. Eventually, a total of 7,998 traffic conflicts with their four indicators were obtained from 654 signal cycles of four signalized intersections: 1,447 of which for the intersection of 72 Ave & 128 St, 795 of which for the intersection of 72 Ave & 132 St, 2,065 of which for the intersection of 64 Ave & King George Blvd, and 3,691 of which for the intersection of Fraser Hwy & 168 St.

In addition to traffic conflict data, the crash records of three years (2013–2015) were obtained for the four signalized intersections in the city of Surrey. The crash data, which were obtained from the Insurance Corporation of British Columbia, included detailed crash information (such as location, date, time, crash type, crash severity, among others) and also contained a short description for each crash. Such description further presented more details on the process of crash occurrence, the

Fig. 1. Study sites and video scenes.

Fig. 2. Traffic flow states and shock wave phenomena at a signalized intersection.

specific lane incident vehicle occupied, the traveling directions of incident vehicles at the moment of crash occurrence, the types of incident vehicles, and the total number of incident vehicles. To better relate the crash data to the rear-end conflicts observed in the daytime period, only rear-end crashes that happened during the same period in the through lanes of subject intersection approaches were chosen. Moreover, since the conflicts were observed under good weather conditions, rear-end crashes that took place during adverse weather conditions (e.g., raining or snowing) were not included. In the end, a total of 45 rear-end crashes remained. The summary statistics of the conflict and crash data is shown in Table 1.

5. Results and discussion

5.1. Model estimation results

Since previous studies showed that the MTTC outperforms the other

three indicators TTC, PET, and DRAC in terms of crash estimation accuracy (Zheng and Sayed, 2019d), it was used to fit the GEV distribution, while the other three conflict indicators were treated as covariates. For all the covariates of conflict indicators and traffic parameters, the variance inflation factor of each covariate is less than 2, indicating no multicollinearity among covariates. Therefore, all the six covariates can be included in the model.

Several RP Bayesian hierarchical univariate models were developed. For comparison, a series of RI and fixed parameters (FP) Bayesian hierarchical univariate models were also developed, respectively. The developed models were estimated using the WinBUGS tool. For each model parameter, two separate simulation chains with different 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 iterations. The trace plots of parameter chains were visually inspected to check the model convergence (El-Basyouny and Sayed, 2009; Fu et al., 2020). Additionally, the

Table 1 Summary of data statistics.

Site 72 Ave & 128 St 72 Ave & 132 St 64 Ave & King George Blvd Fraser Hwy & 168 St
Observation date March 28, 2012 (10− 12 am; 2− 4 pm) April 3, 2012 (9− 12 am) June 10, 2015 (1− 7 pm) June 10, 2015 (9 am− 8 pm)
Number of cycles 127 71 179 277
Selected approach Eastbound Westbound Southbound Eastbound
Traffic conflict indicators
Mean 0.75 0.58 0.49 0.63
TTC (s) Min 0.04 0.06 0.10 0.07
Max 3.77 3.15 3.76 3.43
Mean 1.38 1.62 1.74 1.51
MTTC (s) Min 0.12 0.15 0.12 0.11
Max 3.82 3.75 3.66 3.74
Mean 1.54 1.76 1.75 1.61
PET (s) Min 0.06 0.30 0.44 0.07
Max 4.00 4.00 4.00 4.00
Mean 1.59 0.99 1.08 1.69
DRAC (m/s2
)		 Min 0.00 0.00 0.00
Max 7.10 6.38 6.57 0.00
7.24
15.08
3
23
1.25
0.00
5.66
1.12
0.32
1.75
4
3
3
3.3
Traffic parameters
Mean 15.33 14.47 11.51
V (vehicles/lane/cycle) Min 2 6 3
Max 28 25 16
Mean 1.76 0.97 0.64
A (km∙s) Min 0.00 0.00 0.00
Max 4.45 2.90 1.80
Mean 1.12 1.13 1.59
P Min 0.00 0.14 0.55
Max 1.88 1.81 2.27
Crash records
2013 8 0 9
2014 3 1 7
Observed crashes 2015 3 0 2
Average 4.7 0.3 6.0

Brooks-Gelman-Rubin (BGR) statistic of two chains for each parameter was also calculated to check the model convergence (Gelman and Rubin, 1992; Fu and Sayed, 2021a). Moreover, it is common to examine the changing behavior of BGR statistic over iteration time rather than calculating a single value (Lunn et al., 2012). Two chains of each model parameter were found to mix well and the BGR statistic values of all model parameters were also found to stabilize around 1 (less than 1.1), both of which indicate model convergence.

By following the procedure described in Zheng et al. (2019b), the RP model with the smallest DIC value was selected as the best fitted model. As well, the best fitted RI model and FP model were obtained separately. The estimation results of the best fitted FP, RI, and RP models are shown in Table 2. The DIC values are 256.9, 222.1, and 195.3 for the FP, RI, and RP models, respectively. Among these three models, the RP model has the smallest DIC value and the differences between it and the other two models are greater than 10. This implies that the RP model substantially outperforms other models in terms of goodness of fit. The superior fit of RP model also indicates that the introduced random parameters can better capture the unobserved heterogeneity across different traffic conflict observation sites.

As presented in Table 2, the results show that the shock wave area, platoon ratio, TTC, PET, and DRAC are significantly associated with the location parameter of GEV distribution of MTTC, while the shock wave area, platoon ratio, and TTC are significantly related to the scale parameter of GEV distribution of MTTC. It is noted that all the three covariates shock wave area, platoon ratio, and TTC have opposite relationships with the location parameter and scale parameter of GEV distribution of MTTC. In addition, the estimation results of RP model indicate that for both location parameter and scale parameter, the variances of parameter distributions (i.e., the chosen normal distribution) for TTC are significantly different from zero, which suggests that a random coefficient for TTC is statistically appropriate.

5.2. Comparison between FP, RI, and RP models

With the estimated model parameters, the best fitted FP, RI, and RP Bayesian hierarchical univariate models were employed to calculate the annual number of crashes for each intersection based on Eqs. (12), where T = 365 × 12 = 4380, and η = 4, 3, 6, and 11 for intersections of 72 Ave & 128 St, 72 Ave & 132 St, 64 Ave & King George Blvd, and Fraser Hwy & 168 St, respectively. Similar to previous studies (Fu et al., 2020; Zheng and Sayed, 2020), the uncertainties of predicted crashes were estimated directly within WinBUGS during the process of model posterior inference via using different quantiles. This method can include all uncertainties from the data and model parameters. The crash estimation results from the FP, RI, and RP models are shown in Table 3. In the table, the 95 % confidence intervals (95 % C.I.) of observed crashes are also presented, given that the observed crashes at each intersection are just one observation of the underlying count process (Zheng and Sayed, 2020). The observed crashes were hypothesized to follow the Poisson distribution, and then the 95 % confidence intervals were computed as { 2(*ϑ0+1),0.*025}

were computed as $$\left{\varphi: \frac{1}{2m}\chi^2_{2\theta0,0.975} \leq \varphi \leq \frac{1}{2m}\chi^2{2(\theta_0+1),0.025}\right}$$ , where $m=3$ is the number of years for crash data collection, and $\theta_0$ is the total number of crashes in the 3-year period for an intersection (Songchitruksa and Tarko, 2006).

The comparison of estimated crashes between FP, RI, and RP models is shown in Fig. 3. With respect to the FP model, although the estimated crashes of two intersections are within the 95 % confidence intervals of observed crashes, the estimated crashes of the other two intersections are outside the 95 % confidence intervals of observed crashes. As to both the RI model and the RP model, the estimated crashes of four intersections are all inside the 95 % confidence intervals of observed crashes. It indicates that the FP model is worse than both the RI model and the RP model in terms of crash estimation accuracy. This is likely due to not accounting for the unobserved heterogeneity across different

Table 2 Model estimation results.

| | | FP model | | | | RI model | | | | RP model | | | | | |-----------------|--------------|-----------|--------|----------|----------|-----------|--------|----------------|----------|-----------|--------|----------|----------|--| | Model parameter | | | | 95 % HDI | | Mean | | 95 % HDI | | | S.D. | 95 % HDI | | | | | | Mean | S.D. | Lower | Upper | | S.D. | Lower
Upper | | Mean | | Lower | Upper | | | | βμ0 | − 1.049* | 0.0697 | − 1.192 | − 0.9151 | – | – | – | – | – | – | – | – | | | | βμ0[1] | – | – | – | – | − 0.9509* | 0.0710 | − 1.093 | − 0.8156 | − 1.082* | 0.0768 | − 1.23 | − 0.9336 | | | | βμ0[2] | – | – | – | – | − 1.159* | 0.0771 | − 1.315 | − 1.013 | − 1.144* | 0.0895 | − 1.317 | − 0.9678 | | | | βμ0[3] | – | – | – | – | − 0.9954* | 0.0741 | − 1.147 | − 0.8541 | − 0.9764* | 0.0847 | − 1.138 | − 0.8069 | | | | βμ0[4] | – | – | – | – | − 1.081* | 0.0723 | − 1.228 | − 0.944 | − 1.108* | 0.0759 | − 1.255 | − 0.9592 | | | | βμ(A) | 0.046* | 0.0128 | 0.0204 | 0.0707 | 0.0375* | 0.0122 | 0.0137 | 0.0614 | 0.0365* | 0.0123 | 0.0129 | 0.0604 | | | | βμ(P) | − 0.1453* | 0.0362 | − 0.2166 | − 0.0752 | − 0.1699* | 0.0406 | − 0.2505 | − 0.0910 | − 0.172* | 0.0420 | − 0.2547 | − 0.0903 | | | μ | βμ(TTC) | 0.6024* | 0.0831 | 0.4442 | 0.771 | 0.5717* | 0.0756 | 0.4275 | 0.7232 | – | – | – | – | | | | βμ(TTC) [1] | – | – | – | – | – | – | – | – | − 0.0161 | 0.1039 | − 0.1837 | 0.2323 | | | | βμ(TTC) [2] | – | – | – | – | – | – | – | – | 0.7789* | 0.1671 | 0.5609 | 1.227 | | | | βμ(TTC) [3] | – | – | – | – | – | – | – | – | 0.8428* | 0.2295 | 0.4496 | 1.343 | | | | βμ(TTC) [4] | – | – | – | – | – | – | – | – | 0.5953* | 0.0883 | 0.4146 | 0.7626 | | | | ζ(βμ(TTC)) | – | – | – | – | – | – | – | – | 0.5516* | 0.3671 | 0.0554 | 1.271 | | | | ϱ2(βμ(TTC)) | – | – | – | – | – | – | – | – | 0.5102* | 2.156 | 0.0392 | 2.536 | | | | βμ(PET) | 0.1977* | 0.0353 | 0.1277 | 0.2679 | 0.1827* | 0.0333 | 0.117 | 0.2452 | 0.1742* | 0.0331 | 0.108 | 0.2382 | | | | βμ(DRAC) | 0.1044* | 0.0086 | 0.0877 | 0.1211 | 0.1054* | 0.0092 | 0.087 | 0.1235 | 0.1071* | 0.0087 | 0.0897 | 0.1244 | | | | βψ0 | − 1.675* | 0.1208 | − 1.915 | − 1.437 | – | – | – | – | – | – | – | – | | | | βψ0[1] | – | – | – | – | − 1.704* | 0.1496 | − 1.983 | − 1.396 | − 1.682* | 0.1454 | − 1.954 | − 1.378 | | | | βψ0[2] | – | – | – | – | − 1.506* | 0.1533 | − 1.793 | − 1.199 | − 1.369* | 0.1709 | − 1.697 | − 1.025 | | | | βψ0[3] | – | – | – | – | − 1.514* | 0.141 | − 1.776 | − 1.232 | − 1.507* | 0.1307 | − 1.758 | − 1.24 | | | | βψ0[4] | – | – | – | – | − 1.754* | 0.1314 | − 1.998 | − 1.492 | − 1.741* | 0.1291 | − 1.989 | − 1.481 | | | | βψ (A) | − 0.0827* | 0.0311 | − 0.1456 | − 0.0239 | − 0.0749* | 0.0340 | − 0.1406 | − 0.0078 | − 0.0796* | 0.0325 | − 0.1434 | − 0.0146 | | | | βψ (P) | 0.3434* | 0.0822 | 0.1834 | 0.5001 | 0.2758* | 0.0868 | 0.0951 | 0.442 | 0.2567* | 0.0832 | 0.1006 | 0.4249 | | | ψ = log(σ) | βψ (TTC) | − 0.4133* | 0.0852 | − 0.5938 | − 0.2552 | − 0.4083* | 0.0869 | − 0.5877 | − 0.252 | – | – | – | – | | | | βψ (TTC) [1] | – | – | – | – | – | – | – | – | 0.0147 | 0.2547 | − 0.4814 | 0.4547 | | | | βψ (TTC) [2] | – | – | – | – | – | – | – | – | 0.2343 | 0.4039 | − 0.4945 | 0.923 | | | | βψ (TTC) [3] | – | – | – | – | – | – | – | – | − 0.4153* | 0.1334 | − 0.6998 | − 0.172 | | | | βψ (TTC) [4] | – | – | – | – | – | – | – | – | − 0.3808* | 0.1373 | − 0.6779 | − 0.1362 | | | | ζ(βψ (TTC)) | – | – | – | – | – | – | – | – | − 0.1366 | 0.3888 | − 0.7334 | 0.618 | | | | ϱ2(βψ (TTC)) | – | – | – | – | – | – | – | – | 0.5041* | 5.812 | 0.0015 | 2.535 | | | ξ | βξ0 | − 0.2566* | 0.0193 | − 0.2943 | − 0.2185 | – | – | – | – | – | – | – | – | | | | βξ0[1] | – | – | – | – | − 0.2612* | 0.0573 | − 0.3697 | − 0.1435 | − 0.2452* | 0.0518 | − 0.3427 | − 0.1391 | | | | βξ0[2] | – | – | – | – | − 0.316* | 0.0685 | − 0.4473 | − 0.1755 | − 0.3702* | 0.0745 | − 0.5167 | − 0.2212 | | | | βξ0[3] | – | – | – | – | − 0.356* | 0.0372 | − 0.4293 | − 0.2831 | − 0.3521* | 0.0373 | − 0.4258 | − 0.2791 | | | | βξ0[4] | – | – | – | – | − 0.1799* | 0.0305 | − 0.2359 | − 0.1165 | − 0.1768* | 0.0312 | − 0.2332 | − 0.1118 | | | D | | 245.5 | | | | 202.3 | | | | 171.1 | | | | | | pD | | 11.4 | | | | 19.8 | | | | 24.2 | | | | | | DIC | | 256.9 | | | | 222.1 | | | | 195.3 | | | | |

Note: "–" means the covariate is not included; "*" means the covariate is significant at 0.05 level; HDI stands for the highest density interval.

Table 3 Crash estimation results of FP, RI, and RP models.

Model Intersection Observed crashes per year Mean and %-quantiles of estimated crashes per year
Mean 95 % C.I. Mean 2.5 5.0 10.0 25.0 50.0 75.0 90.0 95.0 97.5
FP model 72 Ave &128 St 4.7 [2.6, 7.8] 6.4 0.0 0.0 0.0 0.4 5.6 210.9 315.8 402.0 495.1
72 Ave & 132 St 0.3 [0.0, 1.9] 1.9 0.0 0.0 0.0 0.0 0.6 129.8 195.2 250.4 312.1
64 Ave & KG Blvd 6.0 [3.6, 9.5] 3.1 0.0 0.0 0.0 0.0 2.4 96.9 166.3 223.3 284.5
Fraser Hwy & 168 St 3.3 [1.6, 6.1] 2.2 0.0 0.0 0.0 0.0 0.8 84.2 138.8 183.8 232.3
RI model 72 Ave &128 St 4.7 [2.6, 7.8] 6.1 0.0 0.0 0.0 0.0 2.6 48.0 154.2 270.2 414.2
72 Ave & 132 St 0.3 [0.0, 1.9] 0.5 0.0 0.0 0.0 0.0 0.0 0.5 14.0 51.1 125.3
64 Ave & KG Blvd 6.0 [3.6, 9.5] 7.8 0.0 0.0 0.0 0.8 4.0 197.8 397.3 556.9 726.2
Fraser Hwy & 168 St 3.3 [1.6, 6.1] 5.6 0.0 0.0 0.0 0.3 3.8 180.9 249.6 299.4 348.5
RP model 72 Ave &128 St 4.7 [2.6, 7.8] 5.4 0.0 0.0 0.0 0.0 0.0 3.9 62.5 142.7 253.8
72 Ave & 132 St 0.3 [0.0, 1.9] 0.1 0.0 0.0 0.0 0.0 0.0 0.0 2.8 11.5 35.2
64 Ave & KG Blvd 6.0 [3.6, 9.5] 6.8 0.0 0.0 0.0 0.1 3.7 48.7 187.5 321.9 471.1
Fraser Hwy & 168 St 3.3 [1.6, 6.1] 5.7 0.0 0.0 0.0 0.4 4.9 147.8 204.2 246.7 289.6

sites in FP model (Mannering and Bhat, 2014; Mannering et al., 2016). Compared to the RI model, the estimated crashes of three intersections from the RP model are closer to the observed crashes. Moreover, the 95 % confidence intervals of estimated crashes of all the four intersections from the RP model are narrower. Accordingly, in terms of accuracy and

precision, the RP model outperforms the RI model. Such superiority in crash estimation may be due to the RP model with the use of random coefficients and RI can better capture the unobserved site-specific heterogeneity.

Fig. 3. Comparison of estimated crashes between RP, RI, and FP models.

6. Summary and conclusions

This study proposes a RP Bayesian hierarchical extreme value modeling approach which consists of a RP Bayesian hierarchical structure and a univariate extreme value model. The main contribution of the study is accounting for unobserved heterogeneity in EVT models for traffic conflict-based crash estimation using the RP approach. Moreover, this study attempts to integrate several conflict indicators in a simpler way that takes some conflict indicators as covariates. The proposed approach is applied to estimate rear-end crashes from traffic conflicts collected from four signalized intersections in the city of Surrey, British Columbia. Traffic conflicts were characterized by four indicators TTC, MTTC, PET, and DRAC. MTTC was used to fit the GEV distribution, while the other three conflict indicators were treated as covariates. With different covariates (including TTC, PET, DRAC, traffic volume, shock wave area, and platoon ratio) added to the location parameter and scale parameter of GEV distribution, several FP, RI, and RP Bayesian hierarchical univariate models were developed, respectively. The best fitted FP, RI, and RP models with the smallest DIC values were chosen to estimate crashes separately. The estimated crash frequencies and their 95 % confidence intervals from the best fitted FP, RI, and RP models were compared to the observed crashes, respectively.

The results show that: 1) both the RP model and the RI model outperform the FP model in terms of crash estimation accuracy; 2) the RP model outperforms the RI model in terms of crash estimation accuracy and precision. The superiority of the RP model in crash estimation may be due to the use of random coefficients and RI better account for unobserved heterogeneity.

Future research directions are recommended to address several limitations of this study. First, there are still large uncertainties in estimated crashes from the developed RP Bayesian hierarchical univariate model. More traffic conflict data collected with a longer observation period may be needed to address this issue (Songchitruksa and Tarko, 2006). Second, developing RP Bayesian hierarchical multivariate models is worthy of exploration, since multivariate models were shown to outperform univariate models in several recent studies (Fu et al., 2020; Fu and Sayed, 2021a). Third, more crash data from a larger number of sites and other crash types is required to validate the developed RP Bayesian hierarchical univariate model, since only rear-end crashes collected from four signalized intersections were used in this study. Finally, it may be interesting to develop a RP Bayesian hierarchical threshold excess model since the peak over threshold approach were shown to have efficient use of data and could improve the EVT model performance (Zheng et al., 2014a; Tarko, 2018a).

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

The authors report no declarations of interest.

Acknowledgments

This study was jointly supported by the National Natural Science Foundation of China (Grant Nos. 71801182; 61703352) and the China Scholarship Council (Grant No. 201907005017).

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