Accident Analysis and Prevention xxx (2015) xxx–xxx

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Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras Bar-Gera H. a, * , Musicant O. b , Schechtman E. a , Ze’evi T a a b

Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel Department of Industrial Engineering and Management, Ariel University, Ariel 40700, Israel

A R T I C L E I N F O

A B S T R A C T

Article history: Received 1 November 2014 Received in revised form 5 March 2015 Accepted 30 March 2015 Available online xxx

The yellow signal driver behavior, reﬂecting the dilemma zone behavior, is analyzed using naturalistic data from digital enforcement cameras. The key variable in the analysis is the entrance time after the yellow onset, and its distribution. This distribution can assist in determining two critical outcomes: the safety outcome related to red-light-running angle accidents, and the efﬁciency outcome. The connection to other approaches for evaluating the yellow signal driver behavior is also discussed. The dataset was obtained from 37 digital enforcement cameras at non-urban signalized intersections in Israel, over a period of nearly two years. The data contain more than 200 million vehicle entrances, of which 2.3% (5 million vehicles) entered the intersection during the yellow phase. In all non-urban signalized intersections in Israel the green phase ends with 3 s of ﬂashing green, followed by 3 s of yellow. In most non-urban signalized roads in Israel the posted speed limit is 90 km/h. Our analysis focuses on crossings during the yellow phase and the ﬁrst 1.5 s of the red phase. The analysis method consists of two stages. In the ﬁrst stage we tested whether the frequency of crossings is constant at the beginning of the yellow phase. We found that the pattern was stable (i.e., the frequencies were constant) at 18 intersections, nearly stable at 13 intersections and unstable at 6 intersections. In addition to the 6 intersections with unstable patterns, two other outlying intersections were excluded from subsequent analysis. Logistic regression models were ﬁtted for each of the remaining 29 intersection. We examined both standard (exponential) logistic regression and four parameters logistic regression. The results show a clear advantage for the former. The estimated parameters show that the time when the frequency of crossing reduces to half ranges from1.7 to 2.3 s after yellow onset. The duration of the reduction of the relative frequency from 0.9 to 0.1 ranged from 1.9 to 2.9 s. ã 2015 Elsevier Ltd. All rights reserved.

Keywords: Dilemma Zone Trafﬁc Signals Safety Digital Enforcement Cameras Yellow Onset Crossing Probability

1. Introduction Driver behavior during the yellow signal phase inﬂuences both the safety and the efﬁciency of signalized intersections. The main two safety outcomes are: (i) the impact on angle accidents as a result of red-light-running1 ; and (ii) the impact on rear-end accidents as a result of conﬂicting decisions by consecutive drivers. Efﬁciency refers to trafﬁc throughput (or capacity) of the intersection, which is inﬂuenced by the changes in trafﬁc ﬂow during the yellow phase. Red-light-running accidents are a major concern. In 2012, 683 people were killed and an estimated 133,000 were injured in accidents that involved red light running in the US (IIHS 2015).

* Corresponding author. E-mail address: [email protected] (H. Bar-Gera). 1 With permissive left-turns red-light-running may lead to left-turn head-on accidents as well, but in Israel all signalized left-turns are protected.

Rear-end accidents at signalized accidents require attention as well. Devlin et al. (2011, Fig. 4) report that during 2004–2009 in West Australia 20% of serious casualty accidents at signalized intersections were rear-end, compared with only 4% at signcontrolled intersections. Characterization of driver behavior during the yellow phase can be done in several related but not completely equivalent ways. One way that has been explored fairly extensively in the literature is by estimating the conditional probability that a driver will cross given the state of the vehicle at yellow onset, which is described primarily by two key factors: current speed, and distance from the stop line. Another way that has been considered less often and less thoroughly (e.g. Köll et al., 2004) is by the distribution of entrance times after the yellow onset. The main contribution of this paper is a demonstration of the possibility to model statistically the distribution of entrance time during the yellow phase as well as at the beginning of the red phase, from information collected by loop detectors at signalized

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intersections. In this particular case the loop detectors are part of a digital enforcement system. This system also includes on-site digital camera and a link to a control center that maintains a log of the intersection entrance data. In principle similar data can be obtained from loop detectors installed for operational (or other) purposes, an opportunity that will be further discussed in Section 7. A secondary contribution of this paper (see Section 6) is a conceptual discussion on the conditions under which both characterizations of driver behavior during the yellow phase (conditional crossing probability and entrance time distribution) contain equivalent information. When such conditions are satisﬁed, the advantage of the proposed methodology is mainly the ability to collect large sample sizes at lower costs. Otherwise, the two approaches are complementary, and the importance of the proposed method is even more acute. Red light cameras have been in use for nearly 50 years, traditionally using wet ﬁlm technologies, while the ﬁrst digital enforcement camera was introduced in 1997 (GATSO, 2014). Previous studies of red light cameras (e.g. Miller et al., 2006; Shin and Washington, 2007; Retting et al., 2008; Erke, 2009) focused primarily on the safety beneﬁts of camera installations. To the best of our knowledge, the potential of datasets generated by enforcement cameras, and especially by digital systems, was not thoroughly explored. Digital red-light cameras typically have direct communication to a control center, thus allowing storage of data about each passing vehicle. One such data element, stored by some digital red light camera systems, is the exact stop line crossing time after the yellow onset. Our proposed methodology for quantifying yellow behavior relies on this data element. The remainder of the paper is organized as follows. Section 2 presents background on yellow signal driver behavior studies. Section 3 describes the methodology and deﬁnes the notation used in our analyses. Section 4 presents the dataset. Section 5 presents the results. Section 6 describes a conceptual framework for evaluating the conditions under which the distribution of entrance times contains equivalent information to the conditional probability of crossing. Conclusions and opportunities for future research are discussed in Section 7. 2. Background Studies of yellow behavior commonly focus on three main variables of analysis: two state variables – speed and distance from the stop line at yellow onset, and a binary variable representing the decision of the driver to cross or to stop. It is also quite common to consider the ratio between distance and speed at yellow onset (with appropriate conversion of units). This ratio can be interpreted as extrapolated entrance time (EET), especially if drivers maintain (nearly) constant speed, as it was at yellow onset. (Previous studies used various alternative terminologies for EET, as indicated hereon.) Because EET is determined at yellow onset (it is derived from the two original state variables), it is not affected by driver’s response to the signal transition which comes later. In that sense its use as explanatory variable for driver decisions is as appropriate as the use of speed and distance. A brief overview of the literature on yellow behavior is presented next, covering three topics: Examples of simulator and other lab-type studies (Subsection 2.1); Examples of naturalistic studies (Subsection 2.2); and a discussion about the need for large sample sizes (Subsection 2.3), which is one of the key advantages of the methodology proposed here. 2.1. Simulator and experimental studies Driver behaviors at yellow onset have been studied extensively in simulator and lab-type studies. Haque et al. (2013) used a

driving simulator to study the effect of cell-phone usage on yellow behavior of 58 drivers at urban intersections with 72 km/h (45 mph) speed limit. The information collected for each participant involved 5–6 scenarios with different EETs (denoted there as TSL) and phone usage (Yes/No). Yellow duration was a continuous variable between 2.46 and 3.85 s. The simulator program was designed so that the switch from green to yellow will occur when the EET is equal to the yellow duration, thus requiring drivers to cope with a situation of dilemma. The researchers analyzed the data with classiﬁcation trees and with a repeated measures logistic regression model. The explanatory variables in the latter model were EET, age, gender and cell-phone usage. All variables had ﬁrst order signiﬁcant effect, some of their interactions were also signiﬁcant. The odds ratio for EET was 0.1037 (CI = [0.0145, 0.7403]), implying that larger EET results in smaller likelihood to cross. Drivers using the phone were much less likely to cross (odds ratio = 0.0005, CI = [1.1E-6, 0.2372]). Yet this effect was relaxed (interaction) as the EET was larger (odds ratio = 10.616, CI = [1.55, 72.88]). Bar-Gera et al. (2013) studied the effect of an in-vehicle stopping decisions advisory system. Twenty drivers participated in the driving simulator experiment. Each participant drove through 28 interurban trafﬁc lights, 14 with and 14 without the evaluated system. The key parameter in the experiment was EET (denoted there as AOT), set at seven values, from 2 to 5 s at 0.5 s increments. Designated speed limit was 90 km/h, and yellow duration was 4 s. The probability of crossing was evaluated as a function of EET by logistic regression. The time range between 0.9 crossing probability and 0.1 crossing probability was 1.03 s in the base scenario, and 0.31 s with the advisory system. This means that the proposed advisory system is effective in reducing driver behavior uncertainty, and thus can potentially improve safety. Rakha et al. (2007) studied perception-reaction times and the probability to stop following yellow onset using an empty 2.1 km test track. Sixty drivers were instructed to drive an instrumented car at 72 km/h (45 mph), passing through trafﬁc lights 24 times. 4 s yellow was initiated when the car was at 32 m, 55 m, 66 m, 88 m, or 111 m from the stop line. (At 72 km/h these correspond to EET values of 1.6–5.6 s.) Crossing probabilities were 0.91, 0.41, 0.27, 0.01 and 0.0 respectively. This study was conducted with real vehicles, yet its setup was artiﬁcial, so in many ways it is closer to a designed experiment than to a naturalistic study. We note that using lab/simulator studies in trafﬁc safety research has advantages as well as limitations. Primary advantages of driving simulator studies are: (i) the ability to obtain clear identiﬁcation of crossing decisions, precise representation of the vehicle state at yellow onset, as well as many other measures of interest; and (ii) the ability to design the distributions of various attributes. Key limitations of simulator studies are: (i) driver behavior is not necessarily exactly the same as in reality; (ii) sample size is usually modest (typical samples are of up to 100 drivers and up to 1000 yellow crossing events); and (iii) since conditions are created to explore drivers’ decisions, outcome frequencies (e.g. RLR crossings) cannot be identiﬁed. 2.2. Naturalistic studies Köll et al. (2004) studied the impact of ﬂashing green on yellow behavior by collecting naturalistic data at 10 intersections over 5000 cycles. Yellow durations were 3 s at all intersections, except for one intersection with 4 s yellow. Peak volumes ranged from 400 to 2500 veh/h. Speed limits ranged from 40 to 70 km/h. Video records covering 85 m upstream of the signal were analyzed by AutoScope to identify vehicles and determine their speeds. Additional software was used to trace the vehicles. In total, the researchers identiﬁed 1621 crossings during yellow and 133 during

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red. In Austria (with ﬂashing green) there were 271, 204, and 79 crossings during the ﬁrst, second, and third second of the yellow, and 23 red crossings. In Munich, Germany (without ﬂashing green) there were 186, 201, and 210 crossings2 during the ﬁrst, second, and third second of the yellow, and 92 red crossings. In about half of the cycles (2661) a vehicle was observed at yellow onset. Within these relevant cycles, the probabilities for yellow crossing were 0.46 in Austria and 0.97 in Munich. Red crossing rates (per 100 relevant cycles) were 1.90 in Austria and 14.94 in Germany. The range from 0.8 to 0.2 crossing probability was 2.2 s in Austria and 1.2 s in Munich. Gates et al. (2007) studied six intersections, with yellow durations ranging from 3.5 to 5.0 s, all-red durations ranging from 1.0 to 3.0 s, and speed limits ranging from 40 to 80 km/h (25 to 50 mph). Video cameras were used to determine the main attributes of 1001 relevant vehicles (either last-to-go or ﬁrst-tostop). In each site, 4 to 14 h of video were collected, during which 90–250 relevant vehicles were observed. These data were used to analyze response times, deceleration rates, and crossing probabilities. Logistic regression analysis showed that EET is the best single predictor of crossing probabilities, with 77% correct predictions, compared to 82% correct predictions for a full model that included: EET, yellow duration, action of adjacent vehicle, vehicle type, presence of side-street trafﬁc, and cycle length. Papaioannou (2007) studied a T intersection in Greece, with trafﬁc volumes ranging from 50 to 500 veh/h. During 126 h of video, 6480 cycles were monitored, 18,271 vehicles were observed, of which the arrival of 2452 occurred while the green phase changed to yellow. Their average approach speed was 51.9 km/h. The research results (Papaioannou, 2007, Fig. 4) indicate behavior uncertainty for drivers positioned 20–70 m from the intersection at yellow onset. Hurwitz et al. (2011) examined driver behavior at 10 approaches of 5 intersections with yellow durations of 3.5 or 4 s, all-red times of 2 or 3 s, speed limits ranging from 72 to 88 km/h (45 to 55 mph), and average daily trafﬁc ranging from 4000 to 9000 vehicles. Data was collected over 48 to 72 h. Pneumatic detectors 122 m (400 ft) upstream of the stop line were used to measure speeds accurately (Hurwitz, 2014). Video cameras were used to determine stop/cross decisions as well as vehicle positions at yellow onset on a 15 m (50 ft) grid. This setup was used to obtain “a sample size of approximately 1900 vehicles that experienced an incursion with the change interval while approaching one of the signalized intersections.” The number of observations per approach ranged from 40 to 200. The authors estimated for each intersection the crossing probabilities as a function of the distance to the intersection at yellow onset. These examples of naturalistic studies demonstrate their potential, as well as the practical challenges of data collection. One critical challenge is the complexity of setting up video equipment, and analyzing its output. This challenge limits sample sizes, and practical values are relatively modest, up to 1000–2000 yellow crossings. Another challenge is the precision of speed measurements. In the papers mentioned here estimates of speed accuracy from video records were not provided. One study used pneumatic detectors in addition to video in order to obtain more accurate speed estimates (Hurwitz, 2014). This approach has a limitation as well, because the pneumatic detectors are placed at a speciﬁc ﬁxed location,

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and therefore the speed of the vehicle is known when crossing the detectors, rather than at yellow onset. 2.3. The need for a large sample size Sample sizes of previous studies were sufﬁcient to obtain many meaningful results, such as measures of “width” for the yellow behavior uncertainty. Yet there are important questions that require larger sample sizes by orders of magnitude. For example: 1. What functional form is the most appropriate one for predicting

crossing probability? Should it be four-parameters logistic or standard (exponential) logistic? Is there symmetry between the transition from nearly certain crossing and the transition to nearly certain stopping? 2. How do estimated parameters change when the data is divided into categories such as weekdays vs. weekends, daytime vs. nighttime, peak vs. off-peak, by weather conditions, variations over time, etc. To the best of our knowledge, to date there have not been any studies that evaluated 100,000 yellow crossings or more, which is needed in order to address questions of the type mentioned above. In other contexts, loop detectors have been used to collect extensive trafﬁc data, with much larger sample sizes (e.g. PEMS, 2004). In this research we demonstrate how loop detector data can be used also for quantifying driver behavior at yellow signals. 3. Methodology The actual entrance time (AET) of vehicle i (time elapsed from yellow onset until entrance) is denoted by ti. Given a chosen time resolution, dt (0.01 s in our actual data), the frequencies of entrances with AET in the interval [t,t + dt) are deﬁned by: yðtÞ ¼ jfi : t <¼ ti < t þ dtgj

(1)

where |S| indicates the number of elements in set S. Let mðtÞ ¼ EðyðtÞÞ be the expected values of these random variables. Our analysis focuses on intersections where it is plausible to assume that is stable (or nearly stable) during the beginning of the yellow phase (i.e. there is a time t0 such that m(t)=m0 for 0 t t0). There are two reasons for this focus. First, if entrance frequency during the beginning of yellow is stable, it can serve as a reasonable estimate for the entrance frequency towards the end of the green, thus enabling an evaluation of the yellow phase contribution to efﬁciency. Second, as will be discussed in Section 6, such stability is one of the conditions under which entrance time distribution and conditional crossing probability contain equivalent information. The ﬁrst stage in our methodology is therefore to determine whether m(t) is reasonably stable during the beginning of the yellow phase. This goal is addressed by linear regression with a breakpoint, limited to the yellow phase, as discussed in Section 3.1. The second stage in the analysis is to estimate m(t) by logistic regressions (both exponential and 4-parameter logistic regressions), for the intersections that meet the ﬁrst criterion, as discussed in Section 3.2. 3.1. Linear regression with a breakpoint

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The authors note that “In Austria, the number of crossings decreases with the duration of the amber . . . In Munich, they are spread equally across the whole duration.” The marginal increase in Munich may be due to Poisson random variation. The authors do not provide further explanation for these different patterns.

Fig. 1 illustrates the distribution of entrance time after the yellow onset, i.e. the values of y(t), using one camera site as an example. (The time resolution in this case is 0.01 s.) To examine whether frequencies are stable during the beginning of the yellow

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Fig. 1. Distribution of entrance time after the yellow onset at one intersection.

phase, a continuous piecewise linear model with one breakpoint was ﬁtted to the data for the yellow phase, i.e. 0 t 3 s, ^ ½t t þ g ^ðtÞ ¼ a ^ ½t tBP þ ^ þb y BP þ

(2)

^ is the estimated where tBP is a selected breakpoint time; a ^ frequency at the breakpoint; b is the estimated slope of the ﬁrst

^ is the estimated slope of the second segment segment [0 s, tBP]; g ½tBP ; 3s; and ½xþ ¼ maxð0; xÞ is a notation for the positive component. A second model was estimated in which the slope before the breakpoint was constrained to zero: ^ðtÞ ¼ a ^c þ g ^ c ½t tBP þ y

(3)

To choose a breakpoint all possible breakpoint values were examined, the constrained model was estimated for each breakpoint, and the breakpoint leading to the smallest BIC (Bayesian Information Criterion) value was selected. To examine whether entrance frequencies were stable during the beginning of the yellow phase, two alternative decision tools were considered: (a) formally test whether the slope before the breakpoint is signiﬁcantly different from zero; and (b) BIC values for the constrained and unconstrained models were compared, using the chosen breakpoint as describes above. When the BIC of the constrained model was lower than that of the unconstrained model (i.e. a negative difference), the constrained model should be preferred, implying that initial frequencies can be considered as stable. Otherwise, in cases where the difference was positive, it could be useful to distinguish between relatively low BIC differences, considered as nearly stable, and higher (positive) BIC differences which we consider as unstable. The last group is excluded from subsequent analyses.

^ is the estimated point; ^c is the estimated inﬂection point and d lower asymptote. All parameters are constrained to be nonnegative. The (exponential) logistic model is deﬁned by: ^ ðtÞ ¼ m

^ K ^ ^ 1 þ eBðtMÞ

^ is the estimated upper asymptote; B ^ is the estimated where K ^ is the estimated growth rate (i.e. a sensitivity parameter) and M 3 time of steepest slope . Our preliminary results show that the exponential logistic model ﬁts the data better than the 4PL (see Section 5), and therefore the following derivations refer primarily to the exponential logistic model (5), although similar derivations can be done for the 4PL model as well. Evaluation of safety and efﬁciency outcomes is based on interpretations focusing on relative frequencies. Denote by t r the time when the estimated expected entrance frequency reduces by ^ A key attribute from the safety ^ ðt r Þ ¼ r K. a factor of r, so that m perspective is the duration of the reduction process in relative entrance frequencies, referred to subsequently as entrance frequency reduction duration, and denoted, for example, by t 0:10:9 ¼ t 0:1 t 0:9 . In the (exponential) logistic model, it can be determined explicitly as follows: ln 1 1 ^ t 0:1 ¼ 0:1^ (6) þM B

t 0:9 ¼

1 ln 0:9 1 ^ þM ^ B

3.2. Logistic regressions Once the issue of stability at the beginning of the yellow phase has been established, the next step is to model the entire pattern, with particular focus on the end of the yellow phase and the beginning of the red phase. In most intersections the pattern departs from the linear downward slope towards the end of the yellow phase, continuing smoothly to a decaying tail during the red phase. To represent this pattern, two types of logistic-regression models were evaluated. The four parameters logistic (4PL) is deﬁned by: ^ ðtÞ ¼ m

^ ^d a ^ þd ^ b 1 þ ^t c

(4)

^ is the estimated “hill’s ^ is the estimated upper asymptote; b where a slope” which refers to the steepness of the curve at the inﬂection

(5)

t 0:10:9 ¼ t 0:1 t 0:9 ¼ ¼

lnð81Þ 4:4 ﬃ ^ ^ B B

(7)

lnð9Þ ^ lnð1=9Þ ^ þM þM ^ ^ B B (8)

The efﬁciency outcome can be examined by considering the effective green (see detailed deﬁnition in HCM, 2010). Simply stated, the actual amount of trafﬁc crossings during a green phase can be represented as the product of effective green time and the saturation ﬂow rate (which is the maximum ﬂow rate that could be sustained over a very long green phase). The yellow phase contribution to effective green time can be measured by the ratio

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Both logistic regression models were estimated by the ‘nls’ function of R.

Please cite this article in press as: Bar-Gera, H., et al., Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras. Accid. Anal. Prev. (2015), http://dx.doi.org/10.1016/j.aap.2015.03.040

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of non-green crossings to the average number of crossings per one second of green. This contribution can be approximated by t 0:5 (see details in Appendix A). In the exponential logistic model ^ ¼ 0:5 K, ^ so M ^ ¼ t 0:5 . ^ ðMÞ m 4. The data The digital enforcement system allows storage of data about each passing vehicle. Data elements stored in each passage refer to the exact time in which the vehicle crossed the stop line and include the following: the vehicle location (site code, lane), the vehicle speed, the vehicle length, the date and time of stop line crossing with 1 s resolution, and the stop line crossing time after the onset of each of the trafﬁc lights (i.e. three separate values after the green, yellow and red light onsets), with 0.01 s resolution. In Israel, at all intersections the yellow duration is constant and set to 3 s. However the duration of the green and red phases may vary from time to time, from site to site, and from one cycle to the next (most non-urban trafﬁc signals are semiactuated). Furthermore, at all non-urban intersections the red phase ends with a short transition phase of simultaneous red and yellow lights indicating transition to the green phase. Likewise, the green phase ends with 3 s of ﬂashing green indicating transition to the yellow phase. In order to avoid overlapping between the crossings during the yellow phase and those which occurred during the red–yellow phase, the data which refers to the crossings during the yellow phase include only data records for which the stop line crossing time after the yellow onset was greater than zero and the stop line crossing time after the red onset was equal to zero. The dataset was obtained from 37 digital enforcement cameras at non-urban signalized intersections in Israel over a period of nearly two years (2012–2013). In most non-urban signalized roads in Israel the posted speed limit is 90 km/h. The data contain approximately 223 million vehicle entrances, of which approximately 5.2 million (2.3%) entered the intersection during the yellow phase, and 288,296 (0.13%) entered during the red phase. Our analysis focuses on crossings during the yellow phase and the ﬁrst 1.5 s of the red phase. In most cases frequencies by time after yellow onset were fairly smooth, except for a few outliers. In 15 intersections the value at 0.01 s was at least twice higher than the average value between 0.03 and 0.05 s. Therefore, the 0.01 s values at all intersections were omitted from the analysis. Eleven other values (in 6 intersections) were at least twice higher than the average of the nearest nonoutlying values on both sides. These outliers were also omitted from the analysis. Fig. 2 portrays the proportion of yellow and red crossings at different intersections. In general, as can be expected, the higher the proportion of yellow crossings, the higher the proportion of red ones. While the proportion of yellow crossings lies between 0.8% and 4.5% (with one outlying intersection with 7.5%), the range of proportions for red crossings lies between 0.1% and 0.3% with one outlying intersection with 0.5%. 5. Results The analysis was conducted in three steps. In the ﬁrst step we examined the issue of frequency stability in the beginning of the yellow phase, using the piecewise linear regression results, and determined which intersections to include in subsequent analyses (Subsection 5.1). In the second step we compared the 4PL model with the (exponential) logistic model for the intersections that passed the ﬁrst criterion (Subsection 5.2). The third step focuses on the implied behavioral patterns, demonstrating differences

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between intersections as well as between daytime and nighttime behaviors using the qualiﬁed intersections (Subsection 5.3). 5.1. Piecewise linear regression results Piecewise linear regression results were used to validate the assumption of stable crossing frequency during the initial stage of the yellow phase. As discussed in Section 3.1, this is done by comparing constrained and unconstrained models, as illustrated for three cases in Fig. 3. In the ﬁrst two cases the two alternatives produce similar predictions, while in the third case the predictions are visually different, and the instability of crossing frequency is also apparent. Statistical comparison was based primarily on the difference in BIC values between the constrained and unconstrained models. In total for 18 intersections (group A) the BIC difference was negative, indicating stable initial frequencies. At signiﬁcance level of 0.01 the null hypothesis (H0: b = 0) was not rejected for any of them. We divided the remaining intersections into two groups, 13 intersections with BIC differences between 1.9 and 8.6 in group B, and 6 intersections with BIC differences above 27 in group C. The null hypothesis of zero slope was statistically rejected in all intersections from groups B and C, except for one. The examples in Fig. 3 show one case from each BIC range group. In view of these ﬁndings, we consider the initial frequencies in group B as nearly stable, and include them in subsequent analyses, while the initial frequencies in group C are considered unstable and excluded from subsequent analysis. In addition, we examined the actual breakpoint value for each intersection. At most intersections the breakpoint was in the range of 0.56–1.58 s, except for two outliers with breakpoint times of 0.03 s and 0.35 s. We excluded these two outlying intersections from subsequent analyses as well. In the following analyses we distinguish between intersections in group A (DBIC < 0) and intersections in group B (0 < DBIC < 10). 5.2. Logistic regression results Figs. 4 and 5 illustrate the results of the logistic regressions for the same intersection examined in Fig. 3a. Fig. 4 shows the 4PL regression line (4a) and the residuals of this model (4b). At a ﬁrst glance the ﬁt seems quite satisfactory, but the model tends to overestimate during the red phase. Fig. 5 shows the (exponential) logistic regression line (5a) and the corresponding residuals (5b). This ﬁgure shows a better ﬁt, since the tail of the estimated distribution is narrower. The advantage of the (exponential) logistic regression in this case is veriﬁed by comparing the sum of squared residuals, 157E + 3 vs. 240E + 3 for the 4PL model. We performed similar comparison for all intersections, and the results in most cases demonstrated a similar pattern. In particular, in all cases but four, the sum of squared residuals was smaller for the (exponential) logistic model, as shown in Fig. 6. (In Fig. 6 different symbols are used to distinguish the ranges of DBIC between the constrained and unconstrained piecewise linear models. As the ﬁgure shows, the behavior is largely the same for both groups of intersections.) Therefore, we decided to use the (exponential) logistic model for the comparisons in the next stage of the analysis. In all ^ was up to 0.02 s and the estimated logistic regressions std. err. of M ^ was up to 0.07 s1. std. err. of B 5.3. Intersection behavior characterization Using the (exponential) logistic regression parameters, yellow signal driver behavior in each intersection can be described

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Fig. 2. Yellow and red crossing proportions at 37 intersections.

Fig. 3. Constrained and Unconstrained Piecewise Linear Models for Three Examples. Frequencies in the Beginning of the Yellow are: stable (7a), nearly stable (7b) and unstable (7c).

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Fig. 4. 4PL regression line (a) and residuals (b); SSE ﬃ 240 103.

Fig. 5. Exponential logistic regression line (a) and residuals (b); SSE ﬃ 157 103.

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Fig. 6. Residual sum of squares comparison: logistic vs. 4PL regression.

succinctly by two main characteristics: t 0:5 (contribution to effective green time) and t 0:10:9 (entrance frequency reduction duration), as shown in Fig. 7. As the ﬁgure shows, the contribution to effective green time is about 2 s in most of the intersections with a few exceptions that may deserve additional exploration. A wider variation can be observed in the duration of entrance frequency reduction, ranging from 1.9 s to 2.9 s. This variation deserves consideration when setting trafﬁc signal timing at the different intersections, as well as further research about the factors inﬂuencing these differences. The next analysis compared patterns during night times (24:00–05:59) with day time patterns. Sample sizes at night-time were above 1000 at all intersections. For night-time estimated ^ was up to 0.1 s and the std. err. of B ^ was parameters the std. err. of M 1

up to 0.3 s . The ratios of daytime to nighttime for both characteristics (t 0:5 and t 0:10:9 ) are presented in Fig. 8. In 23 out of 29 the changes are within 10%, thus exhibiting fairly consistent behavior, especially relative to the differences between intersections demonstrated in Fig. 7. However, in three cases the differences in t 0:10:9 between daytime and nighttime reach 30–40%, which may be quite important from a practical perspective, and deserves further exploration.

Fig. 7. Summary of characteristics: contribution to effective green time vs. the entrance frequency reduction duration (29 intersections).

Consider as relevant vehicle arrivals only those with di dmax, where dmax is the maximal sight distance (e.g. 300 m.) Assume that all relevant vehicles are free to choose whether to cross or not. This choice is depicted by a binary variable Ci, with values of TRUE or FALSE respectively. In the case of stopping, AET will be greater than the red duration, denoted by Tred, hence FALSE ti > T red (9) Ci ¼ TRUE ti T red Given a chosen time resolution, dt (0.01 s in our data), the frequency of arrivals with EET in the interval [t, t + dt) can be deﬁned by: ye ðtÞ ¼ jfi : t te;i < t þ dtgj

(10)

Similarly, the frequencies of arrivals which result in crossing with EET in the interval (t, t + dt) can be deﬁned by: yec ðtÞ ¼ jfi : C i ¼ TRUE and t te;i < t þ dtgj

(11)

6. Conceptual framework for comparing entrance time distribution with conditional crossing probability In Section 1 two primary alternatives for quantifying yellow signal driver behavior were mentioned: one by the conditional probability of stopping as a function of EET, and the other by the relative frequency of entrances as a function of the time after the yellow onset. These characterizations are related but not equivalent. We show here that under certain naïve assumptions the two are in fact equivalent, enabling the use of both approaches interchangeably. Otherwise, appropriate conversions may be needed for relaxing these assumptions. 6.1. The framework The deﬁnition of actual entrance time (AET), denoted by ti for vehicle i, was presented in Section 3, as well as entrance frequency, y(t), and its expected value mðtÞ. The primary variables describing a state of vehicle i at yellow onset are its speed, si, and its distance from the stop line, di (as discussed in Section 2). The ratio between them, te,i = di / si, is viewed as the extrapolated entrance time (EET).

Fig. 8. Day to night ratios: contribution to effective green time vs. the entrance frequency reduction duration (29 intersections).

Please cite this article in press as: Bar-Gera, H., et al., Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras. Accid. Anal. Prev. (2015), http://dx.doi.org/10.1016/j.aap.2015.03.040

G Model AAP 3769 No. of Pages 11

H. Bar-Gera et al. / Accident Analysis and Prevention xxx (2015) xxx–xxx

9

Given ye ðtÞ ¼ k we have yec ðtÞ Binomðk; PðtÞÞ, hence Eðyec ðtÞjye ðtÞ ¼ kÞ ¼ k PðtÞ: A general fact in probability theory is that EðXÞ ¼ EðEðXjYÞÞ. Therefore, the connection between the crossing probability function and the expected frequencies is given by:

intersection4 , and the second is the effect of gaps between extrapolated entrance time and actual stop line crossing time on differences between the frequencies of crossings. We claim that the latter effect is in a sense “residual”. We will illustrate this via a speciﬁc (hypothetical) example where gaps between AET and EET exist, but they do not cause any differences between expected frequencies of crossings, i.e. mðtÞ ¼ mec ðtÞ, over a certain range of values of t. To present the example we need the following additional notations:

(16) vt;k ¼ j i : t ti t þ dt; t þ k dt te;i t þ ðk þ 1Þ dt j

mec ðtÞ ¼ Eðyec ðtÞÞ ¼ E½Eðyec ðtÞjye ðtÞÞ ¼ Eðye ðtÞ PðtÞÞ ¼ PðtÞme ðtÞ

ct;k ¼ P t ti t þ dtjt þ k dt te;i t þ ðk þ 1Þ dt

We assume that both ye(t) and yec(t) are random variables, with expected values of me ðtÞ and mec ðtÞ respectively. The conditional crossing probability function, given extrapolated entrance time t, is: PðtÞ ¼ ProbðC ¼ TRUEjt te;i < t þ dtÞ

(12)

(13)

We want to examine the connection between mðtÞ, me ðtÞ, and mec ðtÞ. In the considered range (t < Tred) all vehicles counted in y(t) crossed the intersection. Now consider the following assumptions: (i) P(0) = 1. In other words, a vehicle that reached the stop line

right at yellow onset will cross. m(t) = mec(t). In other words, the expected frequency of stopline-crossings when the actual entrance time is equal to t is the same as the expected frequency of stop-line-crossings when the extrapolated entrance time is equal to t. See additional discussion in Subsection 6.2. (iii) me(t) = l for all t. That is, the expected frequency of arrival times is constant for all t. This is the most problematic assumption of the three, as will be discussed in Subsection 6.2. (ii)

Based on these assumptions we deﬁne the relative expected frequency of stop-line-crossing times at time t as: Pr ðtÞ ¼ mðtÞ=mð0Þ

The key assumptions leading to mðtÞ ¼ mec ðtÞ are that ct;k is symmetric in k, and that mec ðtÞ is a linear function of t. It is natural for mec ðtÞ to be a decreasing function of t. To simplify the exposition we consider a speciﬁc (arbitrary) illustrative example where ct;k ¼ 0:05 0:0025jkj, regardless of t, for 0:5 t 2:5 and jkj 20; and mec ðtÞ ¼ mec ð0Þ ð1 t=3Þ. Gaps between EET and AET in this case are as high as 20 dt, i.e. 0.2 s if dt = 0.01 s as is the case in our data, thus corresponding to 10% speed changes. We now show that in this case mec(t) = m(t) for 0.5 t 2.5.

mðtÞ ¼

20 X

mec ðt þ k dtÞ ct;k

k¼20

(14)

¼ mec ð0Þ

20 X

k¼20

6.2. Plausibility of assumptions The analysis in Subsection 6.1 is based on three quantitative assumptions: (i) P(0) = 1; (ii) mðtÞ ¼ mec ðtÞ; and (iii) me ðtÞ ¼ l for all t, and two qualitative assumptions: (iv) all relevant vehicles are free to choose to cross if they wish to do so; and (v) all vehicles that cross the stop line will enter the intersection and cross it. In most countries the transition from green to yellow is instantaneous. Under such circumstances the ﬁrst assumption is almost trivial, as drivers who entered the intersection at yellow onset cannot stop. In Israel (the case here), at all non-urban intersections the green ends with 3 s of ﬂashing green, followed by 3 s of yellow. It is therefore feasible for drivers to make the decision to stop even before yellow onset. As discussed in Section 3.1, we can examine whether entrance frequencies are stable during the beginning of the yellow phase. If they are stable, it can be interpreted as a supporting evidence for the conjecture that during the same interval crossing probabilities are stable, and possibly close to one. Assumption (ii) involves two issues. The ﬁrst is whether a vehicle that crossed the stop line will necessarily enter the

1

t þ k dt 3

ð0:05 0:0025jkjÞ

t ¼ mec ð0Þ 0:05 1 3 20 X t þ k dt t k dt þ1 1 ð0:05 0:0025jkjÞ þ 3 3 k¼1

(15)

The ﬁrst transition is by assumption (ii), the second transition is by assumption (i), and the third transition is by assumption (iii). Note that Pr(t) can be calculated from the data while P(t) cannot. The quality of Pr(t) as an approximation of the crossing probability function, P(t), is therefore dictated by the degree to which each of the three assumptions holds.

(17)

Note that: vt;k jðyec ðt þ k dtÞ ¼ nÞ Binomðn; ct;k Þ; ¼ E E vt;k jyec ðt E vt;k jðyec ðt þ k dtÞ ¼ n ¼ n ct;k ;Eðvt;k Þ þk dtÞÞÞ ¼ E yec ðt þ k dtÞ ct;k ¼ mec ðt þ k dtÞ ct;k ; and P P1 mðtÞ ¼ EðyðtÞÞ ¼ E 1 v m ðt þ k d tÞ c ¼ ec t;k k¼1 t;k k¼1

If the above three assumptions are satisﬁed, then: Pr ðtÞ ¼ mðtÞ=mð0Þ ¼ mec ðtÞ=mec ð0Þ ¼ mec ðtÞ=me ð0Þ ¼ mec ðtÞ=me ðtÞ ¼ PðtÞ

"

# X 20 t t ¼ mec ð0Þ 0:05 1 2 1 þ ð0:05 0:0025jkjÞ 3 3 k¼1 # " 20 X t 2 ð0:05 0:0025jkjÞ 0:05 þ ¼ mec ð0Þ 1 3 k¼1 t ¼ mec ð0Þ 1 ¼ mec ðtÞ 3 Differences vanish in this case due to the combination of linearity of mec ðtÞ, the symmetry of the gap distribution (ct;k ) with respect to k, and its independence with respect to the EET value. In general, a similar mechanism may reduce part of the inﬂuence of gaps on differences in frequencies of crossings, leaving only “residual” differences. The question is when frequency differences can be

4

The digital camera system in Israel takes two pictures (1 s apart) in every event that a vehicle crossed the stop line 1 s or more into the red phase. Out of these documented events, 45% lead to citations. One of the reasons for not issuing a citation is if the pictures show that the vehicle stopped after the stop line, and did not enter the intersection. This is an indication that crossing the stop line while not entering the intersection is not a negligible phenomenon.

Please cite this article in press as: Bar-Gera, H., et al., Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras. Accid. Anal. Prev. (2015), http://dx.doi.org/10.1016/j.aap.2015.03.040

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10

H. Bar-Gera et al. / Accident Analysis and Prevention xxx (2015) xxx–xxx

neglected, or alternatively whether some conversion can be used to obtain mec ðtÞ from m(t) and vice versa. These issues are beyond the scope of the present paper. The main concern is assumption (iii), of uniform extrapolated entrance times. Various factors may lead to deviations from this assumption, including: signal coordination, actuated or semiactuated trafﬁc signal control, cycle failures, etc. Quantitative analysis of each of these effects, and development of methodologies to address them, remain a subject for future studies. While we cannot rule out the possibility of substantial deviations from the uniformity assumption, we can examine its plausibility by considering the stability of the frequencies of stopline-crossings during the beginning of the yellow phase. If the uniformity assumption is valid, we should expect stability of initial frequencies. Indeed such stability is not a proof that the assumption of uniformity holds, but it is a positive indicator that can be determined practically. Speciﬁc options to examine this stability were discussed in Subsection 3.1. Only intersections for which the stability assumption was found reasonable were included in the analyses. 7. Conclusions and future research This paper shows how loop detector data can be used to naturalistically quantify driver behavior following the onset of the yellow trafﬁc signal phase. The dataset used was obtained from 37 digital enforcement cameras at non-urban signalized intersections in Israel, over a period of nearly two years. The data contain more than 200 million vehicle entrances, of which 2.3% (5 million vehicles) entered the intersection during the yellow phase. Piecewise linear regression was used to check whether the distribution of the entrance times is stable at the beginning of the yellow phase. The results show that in most intersections under study the frequencies of entrance time after the yellow onset are relatively stable during the beginning of the yellow phase. For these intersections logistic regression ﬁts the data well, and the ﬁt of exponential logistic regression is better than the ﬁt of four parameters logistic regression. The duration of frequency reduction from 90% to 10% varies considerably across intersections, and ranges from 1.9 s to 2.9 s. Yet in most intersections the night-time behavior is rather similar to the day-time behavior. The methodology presented here opens many opportunities for future research. It can be used to examine the inﬂuence on driver behavior of changes in enforcement strategies. Examples for such studies include transferring cameras between active and nonactive installations, or evaluating the inﬂuence of new installations (assuming that the loops are installed before the camera). Furthermore, it can be used to examine causes of heterogeneity, both within and across locations, which may include geometric and trafﬁc characteristics, as well as signal timing schemes. Another possible opportunity for future research is to examine the connection between crash data and driver behavior as quantiﬁed by the proposed method. It should be mentioned that the speciﬁc results presented here are not necessarily transferrable, due to the fact that at the intersections examined in this study the green phase ends with 3 s ﬂashing green. Yet the underlying behavior could be quite similar, and therefore the methodology is likely to be applicable in other countries too. Comparisons of results from different places using this methodology can potentially lead to interesting insights. Possible improvements in the data collection conﬁguration include: (1) identiﬁcation of entrance frequencies towards the end of the green, and during the ﬂashing green; and (2) record signal phase duration and cycle times as potential explanatory variables.

Another possibility for collecting distributions of vehicle entrance times after the yellow or red onset for off-line analysis may be by utilizing existing loop detectors connected to controllers at actuated and semi-actuated trafﬁc signals. This might be a challenging direction, because existing controllers may or may not have the abilities to record individual vehicle entrance times, and because the location of the loop detectors may not necessarily be ideal for the purpose of identifying intersection entrance times. In addition, if such conﬁguration is used to identify red-lightrunning, it is also necessary to verify by video or otherwise that the vehicles indeed crossed the intersection and did not stop after the detector. Precise entrance time records may be useful not only for scientiﬁc purposes, but also in order to address several other needs: 1. Site-speciﬁc determination of the necessary all-red duration; 2. Off-line signal timing performance evaluation; and 3. Accident investigations.

Perhaps the combined motivations may justify the necessary investment in controller technology. Acknowldgement This research was partially funded by the Israeli National Road Safety Authority. Appendix A. We claim that t 0.5 is a reasonable approximation for the yellow contribution to effective green time. This claim can be demonstrated in two ways: intuitive justiﬁcation and formal derivation. Both types of arguments rely on the consideration of the asymptotic logistic behavior ﬁtted to the data as representative of the ﬂow during the last seconds of the green phase, i.e. the count per one second of green is K 0 ¼ K=dt. This is based on the assumption of stability during the beginning of the yellow phase. The intuitive argument is based only on the symmetry property of the logistic functional form around the inﬂection point t 0.5, i.e. the fact that mðt 0:5 þ DÞ þ mðt 0:5 DÞ ¼ K. The overall total number of crossings after the green, S, can be divided into two components: S1 – the number of crossings from t ¼ 0 until t ¼ 2t 0:5 , and S2 – the number of crossings from t ¼ 2t 0:5 þ dt onwards. Due to the exponential nature of the logistic regression, the tail in S2 is usually relatively small, and can be neglected. Therefore: X

2t 0:5=dt

S1 ¼

mðk dtÞ ¼ K t 0:5 =dt ¼ K 0 t 0:5

(18)

k¼0

and the resulting approximation for effective green contribution is geff ¼

S S1 ¼ t 0:5 K0 K0

(19)

To make a formal argument we switch to continuous time, and let yc ð½t1 ; t2 Þ be the count of crossings during a general interval [t1, t2]. The expected value is mc ð½t1 ; t2 Þ ¼ Eðyc ð½t1 ; t2 ÞÞ, and its density with respect to t is md ðtÞ ¼ K 0 =½1 þ expðBðt MÞÞ. These deﬁnitions are consistent with the estimated discrete model because tþdt Z

mc ð½t; t þ dtÞ ¼

tþdt Z

md ðtÞ dt ¼ t

K0 dt 1 þ expðBðt MÞÞ

t

K0 K dt ¼ ¼ mðtÞ 1 þ expðBðt MÞÞ 1 þ expðBðt MÞÞ

(20)

Please cite this article in press as: Bar-Gera, H., et al., Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras. Accid. Anal. Prev. (2015), http://dx.doi.org/10.1016/j.aap.2015.03.040

G Model AAP 3769 No. of Pages 11

H. Bar-Gera et al. / Accident Analysis and Prevention xxx (2015) xxx–xxx

We want to ﬁnd S ¼ mc ð½0; T Red Þ mc ð½0; 1Þ. For this purpose, consider the general integral Zt2

mc ð½t1 ; t2 Þ ¼

Zt2

md ðtÞ dt ¼ t1

K0 dt 1 þ expðBð1 MÞÞ

(21)

t1

This integral can be computed by substituting h ¼ expðBðt MÞÞ, as follows: dh ¼ B expðBðt MÞÞ dt

(22)

dh ¼ dt Bh

(23) Zh2

mc ð½t1 ; t2 Þ ¼ h1

lnð1 þ hÞhh21

Zh2 K 0 dh K0 1 1 K0 ¼ dh ¼ ½lnðhÞ h 1þh 1 þ hB h B B h1

K0 h2 1 þ h1 þ ln ¼ ln B 1 þ h2 h1

(24)

where h1 ¼ expðBðt1 MÞÞ and h2 ¼ expðBðt2 MÞÞ. Substituting t1 ¼ 0 and t2 ! 1, i.e. h1 ¼ expðBMÞ and h2 ! 1, we get: h2 (25) ¼ lnð1Þ ¼ 0 lim In 1 þ h2 h2 !1

S ¼ mc ð½0; 1Þ ¼ ¼

K0 1 þ h1 K0 1 ln ¼ ln þ1 h1 B h1 B

K0 lnðexpðBMÞ þ 1Þ B

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By Taylor’s ﬁrst order expansion lnðexpðBMÞ þ 1Þ lnðexpðBMÞÞ þ expðBMÞ ¼ BM þ expðBMÞ 1 K0 0 ðBM þ expðBMÞÞ K0 K B expðBMÞ M ¼Mþ B

geff ¼

(27)

mc ð½0; 1Þ

11

This derivation also provides a ﬁrst order estimate for the precision of the approximation of geff by M ¼ t 0:5. For example, typical values in our results are M ¼ B ¼ 2. For these values the ﬁrst order correction term is expðBMÞ=B 0:01s. References Devlin, A., Candappa, N., Corben, B., Logan, D., 2011. Designing safer roads to accommodate driver error. Curtin-Monash Accident Research Center. Project 09-006RSC. Erke, A., 2009. Red light for red-light cameras?: a meta-analysis of the effects of red-light cameras on crashes. Accid. Anal. Prev. 31 (5), 897–905. Gates, T.J., Noyce, D.A., Laracuente, L., Nordheim, E.V., 2007. Analysis of driver behavior in dilemma zones at signalized intersections. Transp. Res. Rec. J. Transp. Res. Board 2030 (1), 29–39. GATSO, 2014, GATSO MILLIA, http://www.gatso.com/upload/ 126904133751c0452706a09.pdf, (accessed 31.10.2014). Md. Mazharul Haque, Amanda D. Ohlhauser, Simon Washington, Linda Ng Boyle, 2013. Examination of distracted driving and yellow light running: analysis of simulator data. Transp. Res. Board 13–1710. HCM, 2010. The Highway Capacity Manual. Transportation Research Board, Washington. Hurwitz, D.S., 2014. Personal Communication . Hurwitz, D.S., Knodler, M. A, M.A., Nyquist, B., 2011. Evaluation of driver behavior in type II dilemma zones at high-speed signalized intersections. ASCE J. Transp. Eng. 137 (4), 277–286. IIHS, 2015. Insurance Institute for Highway Safety. Q&A: Red Light Cameras. http:// www.iihs.org/iihs/topics/t/red-light-running/qanda#red-light-running (accessed 04.03.2015). Köll, H., Bader, M., Axhausen, K.W., 2004. Driver behavior during ﬂashing green before amber: a comparative study. Accid. Anal. Prev. 36 (2), 273–280. Miller, J.S., Khandelwal, R., Garber, J.N., 2006. Safety impacts of photo-red enforcement at suburban signalized intersections an empirical Bayes approach. Transp. Res. Rec. J. Transp. Res. Board 1853, 27–34. Papaioannou, P., 2007. Driver behavior, dilemma zone and safety effects at urban signalized intersections in Greece. Accid. Anal. Prev. 39 (1), 147–158. PEMS, 2004, Caltrans Performance Measurement System, http://pems.dot.ca.gov/ (accessed 31.10.2014). Rakha, H., El-Shawarby, I., Setti, J.R., 2007. Characterizing driver behavior on signalized intersection approaches at the onset of a yellow-phase trigger. IEEE Trans. Intell. Transp. Syst. 8 (4), 630–640. Retting, R.A., Ferguson, S.A., Farmer, C.M., 2008. Reducing red light running through longer yellow signal timing and red light camera enforcement: results of a ﬁeld investigation. Accid. Anal. Prev. 40 (1), 327–333. Shin, K., Washington, S., 2007. The impact of red light cameras on safety in Arizona. Accid. Anal. Prev. 39 (1), 1212–1221.

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Please cite this article in press as: Bar-Gera, H., et al., Quantifying the yellow signal driver behavior based on naturalistic data from digital enforcement cameras. Accid. Anal. Prev. (2015), http://dx.doi.org/10.1016/j.aap.2015.03.040

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