Validating the bivariate extreme value modeling approach for road safety estimation with different traffic conflict indicators

Accident Analysis and Prevention 123 (2019) 314–323

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Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap

Validating the bivariate extreme value modeling approach for road safety estimation with different traffic conflict indicators

T

⁎

Lai Zhenga,b, Tarek Sayedb, , Mohamed Essab a b

School of Transportation Science and Engineering, Harbin Institute of Technology, China Department of Civil Engineering, The University of British Columbia, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Bivariate extreme value model Traffic conflict Road safety estimation Indicator combination Validation

A range of conflict indicators have been developed for traffic conflict observation. The various conflict indicators have been shown in earlier studies to be of different and sometimes independent nature. Therefore, there is a need to combine different indicators to gain better understanding of the underlying severity of traffic events and for more reliable safety analysis. This study proposes a bivariate extreme value model to integrate different traffic conflict indicators for road safety estimation, and the model is validated with actual crash data. Based on video data collected from four signalized intersections in two Canadian cities, computer vision techniques were utilized to identify rear-end traffic conflicts using several indicators. The conflict indicators included: time to collision (TTC), modified time to collision (MTTC), post encroachment time (PET), and deceleration to avoid crash (DRAC). Then bivariate extreme value models were developed for combinations of each two indicators, and the numbers of crashes were estimated from the models and compared to the observed crashes. The results show that most of the estimated crashes are in the range of 95% Poisson confidence interval of observed crashes, which indicates that the bivariate extreme value model is a promising tool for road safety estimation. Moreover, the accuracy of estimated crashes are different for different indicator combinations. The results show that the estimates of TTC&PET are the most accurate, followed by TTC&MTTC, TTC&DRAC, PET&MTTC, PET&DRAC and MTTC&DRAC. A further correlation analysis suggests that a combination of two independent conflict indicators leads to better crash estimation performance.

1. Introduction Traffic conflicts have been widely used as alternative or complementary methods for crash-based road safety analysis (Mahmud et al., 2017; Johnsson et al., 2018). Compared to traffic crashes, traffic conflicts occur more frequently and are much less costly. In addition, using traffic conflicts for road safety analysis gains an advantage in timeliness and thus avoids the ethical dilemma of observing “crashes” to prevent “crashes” (Sayed and Zein, 1999; Zheng et al., 2014a). A traffic conflict is usually defined as “an observable situation in which two or more road users approach each other in space and time to such an extent that there is a risk of collision if their movements remain unchanged” (Amundsen and Hydén, 1977). To facilitate the measurement of temporal and spatial proximity, a number of traffic conflict indicators have been proposed over the past decades. Examples of the most commonly used indicators include time to collision (TTC), modified time to collision (MTTC), post encroachment time (PET), and deceleration to avoid crash (DRAC). In many traffic conflict techniques,

⁎

one single indicator is considered sufficient to evaluate all traffic interactions in terms of severity (InDev, 2016). However, one single indicator inherently represents only partial severity aspect of traffic events (Ismail et al., 2011). The early Malmö-calibration study, which compared various traffic conflict techniques with indicator of TTC, PET, or other subjective measures, showed large differences in both the number and the type of traffic conflicts (Asmussen, 1984). In view of the limitation of using one single conflict indicator, studies have proposed to combine different indicators to gain better understanding on the underlying severity of an event. There are in general two directions for the indicator combination. One direction is combining different indicators to create a new severity index, from which the severity of a traffic event can be quantitatively evaluated. For example, Ismail et al. (2011) developed map functions that combined TTC, PET, deceleration to safety time and gap time into a severity index; Wang and Stamatiadis (2014) developed an aggregate crash propensity index by combining the required braking rate and TTC. The other direction is to rate the severity of a traffic event jointly based on

Corresponding author. E-mail address: [email protected] (T. Sayed).

https://doi.org/10.1016/j.aap.2018.12.007 Received 13 July 2018; Received in revised form 2 October 2018; Accepted 6 December 2018 0001-4575/ © 2018 Elsevier Ltd. All rights reserved.

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that the bivariate model generated more accurate and less uncertain crash estimates than univariate models. This study aims at further validating the bivariate model with rear-end conflicts at signalized intersections and more common conflict indicators. Moreover, the performance of different conflict indicator combinations will be investigated.

severity levels of different conflict indicators, and such a rating can be realized either by subjective judgement or by objective cluster methods. Kaparias et al. (2010) and Salamati et al. (2011) proposed and implemented an approach that subjectively rates the severity level of a traffic conflict by combing the severity level of each of four different indicators. Xin et al. (2016) used K-means fuzzy clustering method to quantitatively classify the severity of traffic events based on several traffic conflict indicators. Tageldin and Sayed (2018) developed Ordered-response models to relate a behavior-based conflict indicator and TTC to conflict severity taking into account the unobserved heterogeneity in conflicts. Although various approaches have been proposed to combine different conflict indicators, they are limited to evaluate the relative severity of traffic events, and there is not an approach that can integrate different traffic conflict indicators in a unified framework for road safety estimation, such as estimating the crash frequency. Such an approach would be valuable for examining the validity of traffic conflict techniques as well as gaining better understanding on the relationship between traffic conflicts and crashes (Zheng et al., 2014a).

3. Traffic conflict indicators and data Among the various traffic conflict indicators, the most commonly used ones belong to TTC “family”, PET “family” and deceleration “family” (InDev, 2016). Four indicators, including TTC, MTTC, PET, and DRAC, from the three families are extracted from the data of rear-end traffic conflicts. 3.1. Description of traffic conflict indicators 3.1.1. TTC 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). It can be calculated from the following equation:

2. Previous work This section presents the review on previous work that used extreme value theory (EVT) for road safety analysis. EVT is a unique statistical discipline that develops techniques and models for enabling extrapolation from observed levels to unobserved levels (Coles, 2001), which is in line with the objective of predicting less frequent crashes from more frequent traffic conflicts. The application of EVT for road safety analysis has received considerable research interest in recent years. Songchitruksa and Tarko (2006) used EVT approach to estimate the frequency of right-angle collisions at signalized intersections based on traffic conflicts defined by PET, and a promising relationship between the estimated crashes and observed crashes was found. Tarko (2012) also applied the EVT approach to estimate the probability of run-off-the-road crash based on simulated road departure events measured by departure proximity. Zheng et al. (2014b) compared two EVT models when applying them to estimate crashes related to lane changing maneuver on freeways, and the conflicts were measured by indicator of PET. The same data together with EVT approach were also used to develop parametric safety continuum models and investigate the heterogeneity of conflict thresholds (Zheng et al., 2014c, 2016; Zheng and Ismail, 2017). Farah and Azevdo, (2017) applied the EVT to estimate crashes related to passing maneuvers on two-lane rural highways, and TTC was employed as a conflict indicator. Wang et al. (2018) used the EVT to estimate crash risks related to simulated traffic conflicts at intersections, and PET was used as the conflict indicator. Zheng et al. (2018a) used the EVT approach in traffic conflict-based before-after safety analysis of left-turn bay extension at three signalized intersections. In addition to studies that focused on one single conflict indicator, there are also studies that compared the estimation performance of different indicators. Barnes et al. (2011) used EVT to estimate road departure frequencies in a Michigan field operation test. In their study, indicators of lateral deviation, lane departure warning, and time to road edge crossing were modelled with EVT individually, and time to road edge crossing was found to be the indicator that gave the most reasonable crash estimates. Åsljung et al. (2017) compared the crash estimates from indicators of TTC and brake threat number, and they found that the estimates from brake threat number was more reasonable. Previous studies have shown that the EVT is a promising approach for road safety analysis, and these studies used univariate EVT models and were based on one single conflict indicator. In a recent study, Zheng et al. (2018b) proposed a bivariate EVT modeling methods to estimate crashes on freeway entrance merging areas using conflict indicators of PET and length proportion of merging. The results showed

TTCt =

xL, t − xF , t − DL ; ∀ (vF , t − vL, t ) > 0 vF , t − vL, t

(1)

Where, xL,t and xF,t are positions of leading vehicle and following vehicle at time t, respectively; vL,t and vF,t are their speeds at time t; DL is the length of the leading vehicle. 3.1.2. MTTC TTC is measureable only when the conflicting vehicles are in a collision course, i.e. the speed of following vehicle is higher than the speed of leading vehicle. To include other situations where conflicting vehicles are not in a collision course but may experience potential conflicts due to acceleration or deceleration discrepancies, Ozbay et al. (2008) proposed the MTTC indicator, which relaxed the constant speed assumption during the collision course and considered the accelerations of vehicles. The MTTC is calculated as:

MTTCt =

Δvt ±

Δvt2 + 2Δat (xL, t − xF , t − DL) Δat

(2)

Where, Δvt = vF , t − vL, t is the relative speed of conflicting vehicles at time t, and Δat = aF , t − aL, t is the relative acceleration of conflicting vehicles at time t. There are two outcomes to MTTC. If both of them are positive, the minimum of them is considered to be the MTTC value; if one outcome is positive while the other is negative, the positive outcome is considered to be the MTTC value. 3.1.3. PET 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). For the rear-end conflicts, it is a conflict line instead of area being used to measure the PET (Gettman and Head, 2003). The calculation of PET needs no speedrelated parameters or extrapolation of future positions, and it can be expressed as:

PETt = tF , t − tL, t

(3)

Where, tF,t is the time when the following vehicle arrives at an encroachment line and tL,t is the time when the leading vehicle leaves the encroachment line. 3.1.4. DRAC DRAC is defined as the rate at which a following vehicle must decelerate to avoid the collision with the leading vehicle, and it can be 315

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Fig. 1. Study locations and camera scenes. (a) Intersections in Surrey (b) Intersection in Edmonton

camera ranges from 110 to 130 m. Table 1 provides more details on the selected intersections that include: the intersection location; the selected approaches; the number of lanes per approach; and the date of data collection. Computer vision techniques were used to extract vehicle trajectory and vehicle length from the video footages, and the detailed procedures are described in Essa and Sayed (2018a). A Matlab code was then developed to automatically calculate the conflict indicators of TTC, MTTC, PET, and DRAC from the extracted vehicle trajectories. It is noted that, the four conflict indicators are all calculated continuously (i.e., every frame) over time for an event, and the minimum value is adopted for TTC, MTTC, and PET while for DRAC the maximum value is adopted. For each pair of consecutive vehicles on the through lane (exclusive right-tune and left-turn lanes were not included), four conflict indicator values could be obtained. To exclude normal interactions between vehicles from the analysis, only the events with TTC < 4 s, MTTC < 4 s, PET < 4 s, or DRAC > 0 m/s2 were considered for the following analysis. It is noted that, since the EVT model focuses on samples with extreme indicator values (small ones for TTC, MTTC, and PET; large ones for DRAC), the predetermined values for event selection

calculated as (Cooper and Ferguson, 1976; Gettman and Head, 2003):

DRACF , t =

(vF , t − vL, t )2 2(xL, t − xF , t − DL)

(4)

3.2. Description of data The data used in this study were collected from four signalized intersections from the study of Essa and Sayed (2018a, b). One intersection is the intersection of Stony Plain Rd & 170 St in the City of Edmonton, Alberta, Canada, while the other three intersections are intersections of 72 Ave & 128 St, 64 Ave & King George Blvd, and Fraser Hwy & 168 St in the City of Surrey, British Columbia, Canada, and as shown in Fig. 1. For all four intersections, video cameras were installed to record video-data. The video camera was installed on an existing post located either downstream the stop line or upstream the functional area of the signalized intersection. The camera scenes were generally focused on the intersection approaches where most of rear-end conflicts occur. The distance along the intersection approach that was covered by the video Table 1 Description of the study locations*. Site #

City (Province)

Intersected roads

Video-data was recorded in

Selected intersection approaches

Number of lanes per approach

1

Edmonton (AB)

Stony Plain Rd & 170 St

170 St (Northbound)

1 (Right) 1 (Left) 4 (Through)

2

Surrey (BC)

72 Ave & 128 St

72 Ave (Eastbound)

1 (Left) 1 (Through + Right) 1 (Through)

3

Surrey (BC)

64 Ave & King George Blvd

King George Blvd (Southbound)

1 (Right) 1 (Left) 2 (Through)

4

Surrey (BC)

Fraser Highway & 168 St

May 27th,2015 (2:00 – 3:00 pm) June 2nd, 2015 (2:00 – 3:00 pm) March 28th, 2012 (10:00 – 11:00 am) March 29th, 2012 (2:00 – 3:00 pm) June 10th, 2015 (1:00 – 2:00 pm) June 11th, 2015 (1:00 – 2:00 pm) June 11th, 2015 (9:00 – 10:00 am)

Fraser Highway (Southbound)

1 1 1 1

Note:*Edited from the original table provided by (Essa & Sayed, 2018). 316

(Bike lane) (Left) (Through + Right) (Through)

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Table 2 Brief statistics of the conflict and crash data.

No. of observations TTC value(s) Mean Minimum Maximum MTTC value Mean (s) Minimum Maximum PET value(s) Mean Minimum Maximum DRAC value Mean 2 (m/s ) Minimum Maximum Crash count 2013 2014 2015 Mean

1

Stony Plain Rd & 170 St

72 Ave & 128 St

64 Ave & King George Blvd

Fraser Hwy & 168 St

456 1.31 0.10 3.82 1.75 0.39 3.73 1.83 0.03 3.92 0.98 0.00 6.45 – – – –

612 1.08 0.04 3.80 1.46 0.13 3.65 1.60 0.19 4.00 1.36 0.00 7.10 8 3 3 4.7

385 0.93 0.08 3.90 1.75 0.12 3.66 1.75 0.44 4.00 1.07 0.00 6.57 9 7 2 6.0

261 0.93 0.11 3.71 1.37 0.03 3.49 1.60 0.03 4.00 1.73 0.00 7.32 4 3 3 3.3

−1

− ⎛ ⎧ ξ (X − u x ) ⎤ ξ x ⎫ ⎞ X = −⎜log 1 − ζ x ⎡1 + x ⎟ ⎢ ⎥ σx ⎟ ⎜ ⎨ ⎣ ⎦ ⎬ ⎭⎠ ⎝ ⎩ ∼

−1/ ξ

∼

and Y −1

ξ y (Y − u y ) ⎤ y ⎫ ⎞ ⎛ ⎧ = −⎜log 1 − ζ y ⎡ 1+ ⎢ ⎥ ⎜ ⎨ ⎬ ⎟⎟ σy ⎣ ⎦ ⎭⎠ ⎝ ⎩ ∼ ∼ induce a variable ( X , Y ) whose distribution function has margins that are approximately standard Fréchet distributions for X > ux and Y > uy. As detailed in Coles (2001), the joint distribution F(x, y) can be expressed as F (x , y ) ≈ G (x , y ) = exp{ −V (∼ x,∼ y )}, x > u x , y > u y (6) where ∼

∼

V (x , y ) = 2

∫0

1

w 1 − w⎞ max ⎜⎛ ∼ , ⎟ dH (w ) ∼ y ⎠ ⎝x

(7)

and H is a distribution function on [0, 1] satisfying the mean constraint

∫0

Note: “-” indicates the crash data is unavailable.

1

wdH (w ) = 1/2

(8)

The family of distributions defined by Eq. (6) is termed the class of bivariate extreme value distributions, of which V(·) represents the dependence function of two Fréchet margins, and it can also be written as

have little influence on the model estimation results as long as they are away enough from the thresholds that define extremes. A brief statistics of the conflict data is shown in Table 2. The crash data of three years (2013–2015) were collected for intersections in the City of Surrey. The crash records, which were collected from Insurance Corporation of British Columbia, included information such as location, date, time, crash type, crash severity, etc. A brief description of each crash was also included in the crash records. The description briefly described the crash occurrence and contained more details on the direction of travel of incident vehicles, the lane incident vehicle occupied, and the total number of involved vehicles. A total of 397 rear-end crashes were recorded at the three intersections during the three-year period. With these crashes, only those occurred at the target approaches and specific lanes (i.e., through lanes only) were selected. Moreover, in order to relate the crash data better to the rearend conflicts observed in the daytime period, the crash data were further reduced to the daytime only. At last, a total of 42 crashes remained, as shown in Table 2. The crash data for the intersection in the City of Edmonton was not available.

∼ ∼ 1 1 V (x , y ) = 2( ∼ + ∼ ) x y

∫0

1

∼

∼

wy (1 − w ) x max ⎜⎛ ∼ ∼ , ∼ ∼ ⎟⎞ dH (w ) x +y ⎠ ⎝x + y

Let

A (t ) =

∫0

1

max{w (1 − t ), (1 − w ) t } dH (w )

(9)

and then F(x, y) can be written as

1 1 x ⎫ F (x , y ) ≈ G (x , y ) = exp ⎧−( + ) A ( ) ⎨ x y x + y ⎬ ⎩ ⎭

(10)

where, A(t) is the Pickands dependence function that is convex and satisfies (1-t) ∨ t ≤ A(t) ≤ 1, t ∈ [0, 1]. A number of parametric families of distributions for H have been developed and the most popular one is the logistic distribution function. The bivariate threshold excess model with logistic distribution function is

G (x , y ) = exp[−(x −1/ α + y−1/ α )α ]

(11)

where α∈(0, 1) is the parameter that measures dependence. As α→1, G(x, y)→exp{-(x−1+y−1)}, corresponding to independent variables; as α →0, G(x, y)→exp{-max(x−1, y−1)}, corresponding to perfectly dependent variables. In the logistic case, the Pickands dependence function A( t) = ( t1/α+(1- t)1/α)α. Inference for the bivariate threshold excess model is complicated because a bivariate pair may exceed a specified threshold in just one of its components. A solution is to consider observations in a component (e.g., x) that is smaller than the threshold (e.g., ux) to be censored from below at the threshold (Ledford and Tawn, 1996). To be specific, depending on whether x or y exceeds the threshold ux or uy, the plane is divided into four regions, and they are

4. Bivariate extreme value model and the safety implication 4.1. Bivariate extreme value model Bivariate extreme value theory models the joint distribution of two variables at their extreme levels. Suppose (x1, y1), (x2, y2), …, (xn, yn) are independent realizations of a random variable (X, Y) with joint distribution function F(x, y). A bivariate threshold excess model approximates the joint distribution F(x, y) on regions of the form x > ux, y > uy, for large enough ux and uy. For suitable thresholds ux and uy, the marginal distributions of F each have an approximation in the form of a univariate generalized Pareto (GP) distribution, with respective parameter sets (σx, ξx) and (σy, ξy). The functional form of GP approximation is as follows:

R 0,0 = (−∞, u x ) × (−∞, u y ), R1,0 = [u x ,∞) × (−∞, u y )

R 0,1 = (−∞, u x ) × [u y ,∞), R1,1 = [u x ,∞) × [u y ,∞), 1

⎧1 − 1 + ξ (x − u) − ξ ξ ≠ 0 ⎪ σ G (x ; μ, σ , ξ ) = ⎨ x−u 1 − exp − σ ξ=0 ⎪ ⎩

(

)

(

)

So that, for instance, a point (x, y)∈R1,0 if x-component exceeds the threshold ux but the y-component is below uy. For points in R1,1, Eq. (6) ∼ applies and the density of F gives the appropriate likelihood compo∼ nent. On the other regions, F is not applicable and so it is necessary to censor the likelihood component. Supposing that (x, y)∈R1,0, then since x > ux, but y < uy, there is information in the data concerning the marginal x-component, but not the y-component. Hence, the likelihood

(5)

Defining ζ x =Pr{X > ux} and ζ y =Pr{Y > uy}, the following transformations:

317

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L. Zheng et al.

The risk of crash is usually obtained from a relatively short observation period. Assuming that this short observation period t is representative of a long period T, a year for instance, then the estimated annual number of crashes NT can be calculated by

contribution for such a point is

Pr{X = x , Y ≤ y} =

∂F ∂x

(x , uy )

as this is the only information in the datum concerning F. Applying similar considerations in the other regions, a censored likelihood function can be obtained as

NT =

n

L (θ) =

(12)

i=1

To estimate the bivariate threshold excess model, a pair of optimal thresholds has to be selected at first. In this study, simple fixed quantile rules suggested in previous studies of DuMouchel (1983) and Sun (2013) are adopted. The rules suggest to select an upper quantile, like 5% or 10%, to ensure the thresholds are large enough. Considering the sample sizes of the collected data, upper 12% is used in this study, which can ensure that the intersection with minimum sample size have large enough samples (i.e., usually more than 30 samples) for the model estimation. The selected thresholds corresponding to the 12% quantile are also further validated using diagnostics of mean residual life plot and threshold stability plots for univariate GP models (Coles, 2001; Zheng et al., 2014b). For the two types of plots, the selected thresholds all locate in the range where the mean residual life plot is almost linear and the modified scale and shape estimates seem to be constant. With the selected thresholds, bivariate threshold excess models are estimated using the censored maximum likelihood estimation method. The model estimation results are shown in Table 3. The goodness of fit of models are visually examined through QQ plots for fitted marginal distributions and Pickands dependence function plot for the dependence (i.e., logistic distribution function). An example of the diagnostic plots from PET&DRAC of intersection 64 Ave & King George Blvd is shown in Fig. 2. The QQ plots show that the fitted marginal distributions fit well with the observations; the dependence function plot conforms the goodness of fit of the dependence model, because when the t is close to 1, the strength of the asymptotic dependence is not very high (Dutfoy et al., 2014). With the estimated models, the estimated annual number of crashes corresponding to each combination of conflict indicators is calculated. The estimated annual number of crashes are calculated to facilitate the comparison of different indicator combinations. In addition, to quantify the uncertainty the 95% confidence intervals of estimated crashes are also determined, following a simulation procedure as described in Zheng et al. (2014b). The estimated crashes and their 95% confidence intervals (95% C.I.) are shown in the last column of Table 3.

Where θ ={ux, σx, ξx, uy, σy, ξy, α} denotes the parameters of F and 2

⎧∂F if (x , y ) ∈ R1,1 ⎪ ∂x ∂y (x , y) ⎪ ∂F ⎪ if (x , y ) ∈ R1,0 ψ (θ; (x i , yi )) = ∂x (x , uy) ⎨ ⎪ ∂F if (x , y ) ∈ R 0,1 ⎪ ∂x (ux , y) ⎪ F (u x , u y ) if (x , y ) ∈ R 0,0 ⎩ with each term being derived from the joint tail approximation in Eq. (6) and its parametric specifications. Estimates and standard errors for the parameters θ can be obtained by maximizing the log-likelihood. 4.2. Safety implication Extreme value theory enables the extrapolation from the distribution of observed extremal events (e.g., serious traffic conflicts) to unobserved levels (e.g., crashes). Within the bivariate framework, an event E, which is characterized by the joint behaviour of two random variables (X, Y) ∼ F(x, y), is defined as extremal if either X or Y exceeds given thresholds, given by Ex∨, y = {X > x } ∨ {Y > y} with ∨ representing OR operator. In this study, a traffic conflict is characterized by four indicators, including three proximity indicators such as TTC, MTTC, and PET and a deceleration-related indicator DRAC. By their definitions, it is clear that a traffic conflict would end up as a crash if the proximity indicator values equal to or are less than 0; for the DRAC, a crash would occur if it exceeds the maximum available deceleration rate (MADR) (Cunto and Saccomanno, 2008). Therefore, the risk of crash R can be defined as the probability of observing an event with either negated proximity indicator greater than 0 or DRAC greater than MADR, which in consequence generates six combinations:

RTTC,MTTC = FTTC,MTTC(0,0)

Pr{negated

TTC > 0

∨

DRAC > MADR} = 16. Discussion

TTC > 0

∨

negated

MTTC > 0} = 1-

6.1. Overall model performance

RTTC,PET = Pr{negated TTC > 0 ∨ negated PET > 0} = 1-FTTC,PET(0,0) RPET,DRAC = Pr{negated FPET,DRAC(0,MADR) RPET,MTTC = FPET,MTTC(0,0)

Pr{negated

(13)

5. Results

∏ ψ (θ; (xi , yi ))

RTTC,DRAC = Pr{negated FTTC,DRAC(0,MADR)

T ⋅R t

PET > 0 PET > 0

RDRAC,MTTC = Pr{DRAC > MADR FDRAC,MTTC(MADR,0)

∨

The overall performance of the bivariate extreme value model is examined by comparing the model estimated crashes to the observed crashes. The observed crashes of an intersection are assumed to follow the Poisson distribution. As described in Songchitruksa and Tarko (2006), the 95% C.I. for the true mean annual crash occurrences λ could 1 1 be obtained as λ: 2n χ22y ,0.975 ≤ λ ≤ 2n χ2(2 y + 1),0.025 , where n = 3 is the 0 0 number of years for crash data collection, and y0 is the total number of crashes in the 3-year period for an intersection. The confidence intervals of λ for intersections of 72 Ave & 128 St, 64 Ave & King George Blvd, and Fraser Hwy & 168 St are calculated to be [2.6, 7.8], [3.6, 9.5], and [1.6, 6.1], respectively. Fig. 3 shows the comparison between estimated crashes and observed crashes with their respective confidence intervals. For each intersection, there are estimated crashes that are very close to the observed ones, and a majority of the estimated crashes are in the range of Poisson confidence intervals of observed crashes. It indicates that the bivariate extreme value model is a valid approach to estimate crash

DRAC > MADR} = 1-

∨

negated

MTTC > 0} = 1-

∨

negated

MTTC > 0} = 1-

{

where, F(·) is the estimated bivariate threshold excess distribution. MADR is different for individual vehicles and depends on pavement conditions, vehicle weight, tire and braking system and other factors. It is suggested to follow a truncated normal distribution N(8.45, 1.402), bounded by minimum and maximum values of 4.23 m/s2 and 12.68 m/ s2(Cunto and Saccomanno, 2008), and the mean value 8.45 m/s2 is selected as the threshold for crash occurrence. 318

}

TTC&PET TTC&DRAC TTC&MTTC PET&DRAC PET&MTTC MTTC&DRAC TTC&PET TTC&DRAC TTC&MTTC PET&DRAC PET&MTTC MTTC&DRAC TTC&PET TTC&DRAC TTC&MTTC PET&DRAC PET&MTTC MTTC&DRAC TTC&PET TTC&DRAC TTC&MTTC PET&DRAC PET&MTTC MTTC&DRAC

72 Ave & 128 St

319

612 612 612 612 612 612 385 385 385 385 385 385 261 261 261 261 261 261 456 456 456 456 456 456

Number of observations

Note: C.I. is the abbreviation for confidence interval.

Stony Plain Rd & 170 St

Fraser Hwy & 168 St

64 Ave & King George Blvd

Indicator combination

Intersection

Table 3 Model estimation results.

70 70 70 70 70 70 46 46 46 46 46 46 31 31 31 31 31 31 54 54 54 54 54 54

−0.914 2.497 −0.875 2.497 −0.875 2.497 −0.876 2.127 −0.996 2.127 −0.996 2.127 −0.867 2.937 −0.944 2.937 −0.944 2.937 −1.033 1.941 −1.173 1.941 −1.173 1.941

−0.213 −0.213 −0.213 −0.914 −0.914 −0.875 −0.211 −0.211 −0.211 −0.876 −0.876 −0.996 −0.212 −0.212 −0.212 −0.867 −0.867 −0.944 −0.228 −0.228 −0.228 −1.033 −1.033 −1.173 70 70 70 70 70 70 46 46 46 46 46 46 30 31 31 31 31 31 54 54 54 54 54 54

20 11 22 18 33 37 12 15 14 15 19 30 3 9 12 7 6 20 13 17 16 14 26 24

Joint

x

y

x

y

Exceedance

Thresholds

0.014(0.003) 0.014(0.003) 0.014(0.003) 0.209(0.030) 0.207(0.030) 0.126(0.028) 0.014(0.004) 0.014(0.004) 0.014(0.004) 0.221(0.042) 0.222(0.042) 0.339(0.061) 0.015(0.005) 0.015(0.005) 0.015(0.005) 0.157(0.039) 0.158(0.039) 0.556(0.210) 0.022(0.005) 0.022(0.005) 0.022(0.005) 0.286(0.045) 0.288(0.045) 0.193(0.034)

σx 0.701(0.200) 0.650(0.194) 0.700(0.207) 0.063(0.086) 0.098(0.092) 0.638(0.209) 0.757(0.283) 0.792(0.291) 0.758(0.293) −0.179(0.042) −0.162(0.140) −0.012(0.175) 0.320(0.238) 0.444(0.282) 0.494(0.297) 0.038(0.177) 0.023(0.172) −0.349(0.439) 0.102(0.155) 0.141(0.174) 0.109(0.160) −0.197(0.085) −0.165(0.096) 0.186(0.119)

ξx

Estimates and stand errors

0.205(0.030) 1.557(0.287) 0.131(0.029) 1.526(0.292) 0.132(0.029) 1.290(0.268) 0.221(0.041) 1.438(0.302) 0.357(0.078) 1.490(0.132) 0.339(0.074) 1.317(0.269) 0.192(0.059) 2.345(0.531) 0.758(0.205) 2.317(0.524) 0.750(0.186) 2.428(0.606) 0.271(0.041) 0.652(0.149) 0.196(0.036) 0.645(0.146) 0.187(0.033) 0.646(0.142)

σy 0.072(0.087) −0.200(0.143) 0.506(0.199) −0.159(0.157) 0.525(0.201) 0.054(0.194) −0.196(0.119) −0.171(0.160) −0.262(0.176) −0.189(0.159) −0.185(0.185) −0.035(0.167) 0.019(0.249) −0.411(0.170) −0.798(0.274) −0.411(0.166) −0.789(0.226) −0.408(0.243) −0.187(0.078) 0.226(0.198) 0.176(0.126) 0.183(0.185) 0.225(0.133) 0.239(0.182)

ξy

0.846(0.041) 0.954(0.031) 0.801(0.043) 0.879(0.040) 0.691(0.047) 0.638(0.046) 0.869(0.049) 0.840(0.054) 0.869(0.052) 0.832(0.053) 0.763(0.057) 0.499(0.055) 0.999(0.000) 0.884(0.063) 0.831(0.069) 0.938(0.059) 0.987(0.058) 0.586(0.070) 0.879(0.045) 0.859(0.049) 0.853(0.048) 0.898(0.046) 0.676(0.053) 0.712(0.053)

α 405.3 671.4 400.8 959.0 648.2 869.7 250.1 420.9 287.7 590.1 441.2 530.7 153.1 283.0 188.9 412.0 322.9 411.6 303.8 436.9 302.6 682.8 501.2 640.9

Deviance

5.0(1.4,11.2) 2.8(0,9.3) 8.0(2.5,17.4) 2.4(0,9.3) 7.6(2.4,16.5) 8.2(017.7) 8.1(024.7) 9.0(026.4) 9.3(036.0) 0.3(012.5) 3.5(027.9) 11.5(034.4) 4.0(029.8) 2.5(030.1) 3.4(076.0) 1.5(028.4) 1.7(068.6) 17.3(082.1) 0.4(0,7.2) 1.5(011.6) 3.9(014.2) 1.1(010.8) 4.8(016.3) 4.5(015.4)

Estimated crashes with 95% C.I.

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Fig. 2. Example of goodness of fit diagnostic plots: (a) QQ plot for marginal PET; (b) QQ plot for marginal DRAC; (c) Pickands dependence function plot.

shown in Table 4. It can be found that the combination of TTC&PET has the smallest MAE, followed by TTC&DRAC, TTC&MTTC, PET&MTTC, and PET&DRAC, while the MAE of MTTC&DRAC is the largest. Overall, combinations of two different indicators would generate reasonable crash estimates, except for the combination of MTTC&DRAC. A possible reason for the less accurate estimates is that MTTC and DRAC have more in common since they both incorporate the deceleration. The scatter plot of the two indicator combination of all observations, as shown in Fig. 4(f), shows that there is a clear trend of negative correlation between MTTC and DRAC. When extrapolated to the extreme level, the dependence between MTTC and DRAC is also relatively strong, with the smallest mean α of 0.609 (Noted that a smaller value represents a stronger level of dependence). Generally, bivariate models fitted to the data that are dependent at the observed level likely to overestimate dependence at the extreme level, and hence to overestimate the probability of the occurrence of extreme events (Coles, 2001). In contrast, although conflict indicators of TTC and PET are of the proximity type, they are measured at different time of interaction process and represent pre-interaction (assumed) and post-interaction (actual) safety margins respectively. The data in this study confirms their difference. It can be found that they are independent at both the observed level (as shown in Fig. 4(a)) and the extreme level (with the largest mean α of 0.898), and their crash estimates turn out to be the most accurate. Moreover, other conflict indicators also exhibit weak level of dependence as shown in Fig. 4(b) to (e) and in Table 4, and their combinations have small MAE values. For the six combinations, the Person correlation coefficient between MAE and the mean α is

frequency based on observed conflicts. Nevertheless, there are also intersections with estimated crashes that considerably vary from the observed ones. A possible reason is the limited conflict observation period since one to two hours of observations cannot account for the variation of traffic and other factors. The short observation periods also lead to wide confidence intervals of estimated crashes. It can be found that the model estimated crashes are all with wider confidence intervals than the Poisson confidence intervals of observed crashes. The large time span differences between conflict observation period (i.e., one to two hours) and crash observation period (e.g., a year) would result in the uncertainty in crash estimates being amplified significantly. For instance, a small error of 0.01 in crash risk estimation would lead to an error of (365 × 12 ÷ 2) × 0.01 = 21.9 or (365 × 12) × 0.01 = 43.8 in the estimated annual number of crashes. 6.2. Comparison of different indicator combinations The estimated number of crashes are different for different combinations of conflict indicators. It is expectable since the four indicators are different in nature, and single indicator represents partially severity aspect of traffic events. This section compares the prediction performance of the six combinations of conflict indicators and investigates which one is better and the underlying reasons. Although the number of intersections is limited in this study, it can be found from Fig. 3 that the crash estimation results of different combinations have certain similarities across individual intersections. To quantitatively compare the combinations, the mean absolute error (MAE) between estimated crashes and observed crashes is calculated, as 320

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Fig. 3. Comparison between estimated crashes and observed crashes for: (a) 72 Ave & 128 St; (b) 64 Ave & King George Blvd; (c) Fraser Hwy & 168 St; (d) Stony Plain Rd & 170 St with no available crash data.

close to the lower boundary of 95% C.I of observed crashes, and that of PET&DRAC for intersection of 64 Ave & King George Blvd is much smaller and out of 95% C.I. of observed crashes. A possible reason for the underestimation is the large value (i.e., the mean value of a truncated normal distribution N(8.45, 1.402)) selected for MADR, which is used for distinguish traffic conflicts and crashes based on the DRAC. Different from other three indicators that have clear and fixed boundary between traffic conflicts and crashes for all observations, the boundary for DRAC is determined by the MADR or the maximum braking capability, which are influenced by various factors such as road and vehicle conditions. It is difficult to select a deterministic MADR value for all observations, and a compromise of using the mean value to simplify the analysis probably cause the underestimation of crashes.

Table 4 MAE and dependence of each indicator combination. Combination

MAE

Dependence* Extreme level (mean α)

TTC&PET TTC&DRAC TTC&MTTC PET&DRAC PET&MTTC MTTC&DRAC Pearson correlation coefficient

1.03 0.898 1.90 0.884 2.23 0.838 3.27 0.887 2.33 0.779 7.67 0.609 −0.900

Observed level (Pearson correlation coefficient) 0.263 0.372 0.396 0.159 0.466 0.429 −0.610

Note: * the dependence is between negated TTC, negated PET, negated MTTC, and DRAC.

7. Conclusions Bivariate extreme value models are developed to estimate crashes from rear-end conflicts defined by four traffic conflict indicators, TTC, MTTC, PET, and DRAC. Based on the video data collected from four signalized intersections located in two cities in Canada, the four indicators of rear-end conflicts were extracted through computer vision techniques. Bivariate extreme value models were then developed for six different combinations of conflict indicators, namely TTC&MTTC, TTC& PET, TTC&DRAC, PET&MTTC, PET&DRAC, and MTTC&DRAC. Comparisons between model estimated crashes and observed crashes were conducted to evaluate the overall model performance as well as the performance of different indicator combinations.

found to be as high as -0.900. It implies that the combination of conflict indicators that are more independent at the extreme level would generate more accurate crash estimates. A more convenient way to select an appropriate combination would be to investigate the correlation between conflict indicators at the observed level, since the dependence at the extreme level is related to the dependence at the observed level. In addition to the combination of MTTC&DRAC, it can also be found from Fig. 3 that other combinations involved DRAC tend to have smaller crash estimates. The estimated crashes of TTC&DRAC and PET& DRAC for intersections of 72 Ave & 128 St and Fraser Hwy & 168 St are

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Fig. 4. Scatter plots for different combination of conflict indicators: (a) TTC&PET; (b) TTC&DRAC; (c) TTC&MTTC; (d) MTTC&PET; (e) DRAC&PET; (f) MTTC&DRAC.

the largest estimation error. A further analysis shows that the combination of two indicators that are independent generally has better performance, and this can be taken as a practical basis for conflict indicator selection when applying the bivariate extreme value model. It is noted that there are several limitations of this study. Firstly, the traffic conflict data are limited from the perspectives of both time duration and number of intersections. It can be found that the estimated crashes are with relatively wide confidence intervals, and this limitation would probably be overcome with longer observation time for each intersection. In addition, data from more intersections are needed to further evaluate the performance of the bivariate extreme value model. Secondly, the safety implication obtained from the extreme value distribution needs a clear boundary between traffic conflicts and crashes, and such a boundary for some indicators as DRAC is not deterministic,

The results show that reasonable crash estimates are obtained by using the bivariate extreme value modeling approach, since most of the estimated crashes are in the range of 95% Poisson confidence interval of observed crashes. This result demonstrates that the bivariate extreme value modeling approach is a promising tool to establish relationship between traffic conflicts and crashes. Moreover, given that a traffic conflict can be characterized by different conflict indicators, the bivariate extreme value model provides a mathematical framework to combine two indicators that reflect different perspectives of conflict severity. The results of this study also show that the crash estimates from different indicator combinations are varied. The combination of TTC&PET has the most accurate crash estimates, combinations of TTC& MTTC, TTC&DRAC, PET&MTTC and PET&DRAC have larger but acceptable estimation errors, and the combination of MTTC&DRAC has 322

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and a subjective selection might lead to underestimated or overestimated number of crashes. Finally, the four conflict indicators represent partially and possibly very different severity aspects of traffic events, and developing a multivariate extreme value approach that can incorporate them all in a unified model is a direction deserving future research.

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