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The tolerable waiting time: A generalized Pareto distribution model with empirical investigation
Computers & Industrial Engineering 137 (2019) 106019
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The tolerable waiting time: A generalized Pareto distribution model with empirical investigation
T
Hui Xionga, Lu Mab, Mengxi Ningc, Xu Zhaod, , Jinxian Wenge ⁎
a
School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China MOT Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Beijing Jiaotong University, Beijing 100044, China c Department of Administration, Tianfu College of SWUFE, Mianyang 621000, China d College of Applied Sciences, Beijing University of Technology, Beijing 100124, China e College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China b
ARTICLE INFO
ABSTRACT
Keywords: Waiting time Generalized Pareto distribution Signalized intersection Parameter estimation
In China, it is more common for pedestrians than vehicles to disobey traffic signals, resulting in a high risk of pedestrian-vehicle accidents. Pedestrian waiting time are the most critical indicator of the tendency to violate traffic signals. A statistical analysis based on 4027 field-collected samples showed that the length of time that pedestrians are prepared to wait depends on the type of pedestrian traffic signal. Compared to a countdown-type signal, pedestrians were more likely to violate conventional-type signals. Furthermore, pedestrians were willing to wait longer during peak hours than during off-peak hours. There were no significant differences between the waiting times of male and female travelers. To predict pedestrian waiting time, we propose a generalized Pareto distribution (GPD) model and calibrated it based on our field data. Monte Carlo simulations showed that the maximum likelihood estimation (MLE), Bayesian MLE (BMLE), and weighted nonlinear least squares (WNLS) models are the best methods for estimating the scale and shape parameters of the GPD model. Several empirical results were output from the models. For example, at countdown-type signals, the 85th quantile of the tolerable waiting time in off-peak and peak hours was 51.5 and 54.4 s, respectively; the respective values for males and females were 55.4 and 55.0 s. At conventional signals, the tolerable waiting time was approximately 42.5 s. These findings are useful for the planning, design, and operation of pedestrian facilities.
1. Introduction Pedestrians are vulnerable road users at relatively high risk of being killed in traffic accidents. Road traffic caused 1.35 million deaths globally in 2016, with pedestrian deaths accounting for 23% of all fatalities (WHO, 2018). In 2013, 4735 pedestrians were killed and approximately 66,000 were injured in traffic accidents in the United States (National Center for Statistics and Analysis, 2015). In China, 16,281 pedestrians were killed in traffic accidents, representing 25% of all road fatalities in 2010 (CRTASR, 2011). Furthermore, the percentage of fatal accidents involving pedestrians is much higher in urban areas than in rural areas. In London, pedestrians accounted for 46% of road traffic fatalities. In New York, pedestrian fatalities made up 52% of traffic deaths between 2005 and 2009, and as many as 74% of pedestrian deaths or severe injuries occurred in accidents at intersections. Collisions caused by violation of pedestrian traffic signals are more likely to be fatal than those that occur when pedestrians are crossing in
accordance with the signal (Viola, Roe, & Shin, 2010). In Canada, pedestrian traffic rule and signal violations accounted for 13% of all pedestrian fatalities. Pedestrians expose themselves to severe risk when crossing the street during the “stop” or “red-man” phases of traffic signals. However, disobeying traffic signals is prevalent among pedestrians in China (Guo, Wang, Guo, Jiang, & Bubb, 2012; Yang, Deng, Wang, Li, & Wang, 2006). Brosseau, Zangenehpour, Saunier, and Miranda-Moreno (2013) found 10 factors that contribute to the violation of pedestrian traffic signals. These factors can be classified into human characteristics and external factors. The human characteristics include age, sex, personality, education, profession, and history of traffic accident involvement. Female pedestrians are less likely to cross the street against the red-man signal and tend to wait for longer than do males. The elderly exhibit a similar tendency, in contrast to young and middle-aged pedestrians (Guo et al., 2012; Hamed, 2001; Keegan & O’Mahony, 2003; Rosenbloom, Nemrodov, & Barkan, 2004; Tiwari, Bangdiwala, Saraswat, & Gaurav,
Corresponding author. E-mail addresses: [email protected] (H. Xiong), [email protected] (L. Ma), [email protected] (M. Ning), [email protected] (X. Zhao), [email protected] (J. Weng). ⁎
https://doi.org/10.1016/j.cie.2019.106019 Received 3 May 2019; Received in revised form 26 July 2019; Accepted 13 August 2019 Available online 14 August 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.
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2007; Yagil, 1998, 2000; Yang, Abdel-Atyb, Huan, Peng, & Gao, 2015). Furthermore, the more people there are in a group waiting at the curbside, the less likely the group members are to violate pedestrian traffic signals (Guo, Gao, Yang, & Jiang, 2011; Hamed, 2001; Rosenbloom, 2009; Yagil, 2000). External factors include traffic rules and regulations, signal type, trip purpose, intersection layout, vehicle headway, weather conditions, and lighting conditions. Tapiro, OronGilad, and Parmet (2018) found that pedestrians tended to choose smaller crossing gaps and took longer time to decide whether to cross when they encountered environmental distractions. Most recently, Zhao, Malenje, Tang, and Han (2019) modeled pedestrians' gap acceptance considering gap size, crossing distance, number of waiting pedestrians, waiting time, vehicle traffic volume and position of pedestrian. To evaluate the influence of traffic signals on pedestrian crossing behaviors, Keegan and O’Mahony (2003) carried out a questionnaire and video survey in Dublin, Ireland. They observed a significant reduction (from 35% to 24%) in the likelihood of violating pedestrian traffic signals after installing a countdown signal. Many recent studies reported similar results (e.g. Xiong, Xiong, Deng, & Wang, 2014; Lipovac, Vujanic, Maric, & Nesic, 2013). The vehicle time headway directly affects the likelihood of safely crossing the street during the red-man phase (Li, 2013). Cohen, Dearnaley, and Hansel (1955) found that 98.57% of pedestrians crossed the road when the gap was 10.5 s or longer, while no-one crossed when the gap was less than 1.5 s. Wilson and Grayson (1980) found that 3.4% of males and 2.1% of females accepted gaps of less than 2 s when crossing without a “protected pedestrian phase”. Pedestrians waiting at refuge islands or road medians were likely to accept shorter gaps than those waiting at other types of crossings (Das, Manski, & Manuszak, 2005). The characteristics of a crosswalk, including its length and the type of median, significantly affect pedestrians’ crossing behavior; the longer the crosswalk, the fewer pedestrians are likely to cross against the signal (Supernak, Verma, & Supernak, 2013). Empirical studies have also observed differences in behavior depending on whether pedestrians were crossing narrow or wide refuge islands (Rosenbloom & Pereg, 2012). In addition, pedestrians on their way to work were more likely to maintain shorter waiting times than those making trips for other purposes (Hamed, 2001). Weather and lighting conditions are the main environmental factors affecting pedestrian crossing behavior (Andrew, 1991; Yagil, 2000). On rainy days, pedestrians may be more conservative when crossing (Sun, Zhuang, Wu, Zhao, & Zhang, 2015). Haleem, Alluri, and Gan (2015) reported that clear weather conditions were associated with a 3.06% reduction in the probability of severe injuries, in comparison to other weather conditions. Poor light is associated with an increase in the probability of severe injuries. Darkness is perceived to be a factor that increases the tendency for pedestrians to wait longer for the “go” or “green-man” signal (Yagil, 2000). The waiting times when crossing several streets in succession are also related. Hamed (2001) reported that pedestrians who spend more time waiting for the first crosswalk tend to attempt the second crosswalk earlier. Rosenbloom and Pereg (2012) analyzed the waiting time before entering a third crosswalk. In the case of streets with a wide refuge island, a positive correlation was detected between the waiting time of the first and second crossings. Furthermore, a positive correlation was also observed between the waiting time of the second and third crossings. Yang et al. (2006) established a pedestrian crossing model to characterize two types of pedestrians: law abiding and opportunistic. Time-gap distributions were used to assess the criteria that pedestrians used to decide whether to walk, and when such action conflicted with traffic flow. Li (2013) noted that the distribution of pedestrian waiting time during the red-man phase was U-shaped. Furthermore, Li (2014) developed two hierarchical generalized linear models to measure the impact of different factors on pedestrian waiting time during the red-man phase. Cao, Zhuang, and Ma (2019) studied the impact of tempo and pitch on audible pedestrian signals for waiting time estimation.
Waiting time is a key factor in pedestrians’ tendency to violate traffic signals. Van Houten, Ellis, and Kim (2007) investigated the relation between pedestrian waiting time and violation at two signalized midblock crosswalks. They reported that the violation rate increased as the waiting time increased. In fact, long waiting time caused pedestrians to lose patience, resulting in illegal crossings. Tiwari et al. (2007) observed that the probability of violation varied depending on the waiting time. Sun, Liu, Li, and Yang (2010) concluded that the 85th percentile of the tolerable waiting time is approximately 90 s. More specifically, Brosseau et al. (2013) reported that a 10% increase in waiting time increased the probability of violations by 7.9%. Zhuang, Wu, and Ma (2018) studied pedestrians' crossing behavior when arriving at pedestrian clearance phase. Field data showed that as many as 85.2% of walkers chose to cross as opposed to 14.8% of total abided by the signal. However, there have been few empirical studies on pedestrian waiting time in recent years. Very little information has been published on the methodological aspects of modeling pedestrian waiting time at signalized intersections (Li, 2013). Toward this end, we adopted a generalized Pareto distribution (GPD) approach (Pickands, 1975) to model pedestrian waiting time, which are extreme values from the perspective of probability theory. Extreme-value theory is a popular approach for modeling extreme values, by extracting observations that exceed a certain threshold magnitude. It is generally referred to as the peaks-over-threshold method, where a distribution is fitted describing how far a variable exceeds a high threshold. For random variables satisfying certain regularity conditions, the distribution of the exceedance of a threshold tends to follow the GPD (Beirlant, Goegebeur, Segers, & Teugels, 2004; Coles, 2001; Kang & Song, 2017; McNeil & Saladin, 1997). GPD is commonly applied to model observations that exceed a given threshold in a variety of fields, such as insurance, hydrology, economics, finance, energy, and transportation. For example, Brazauskas and Kleefeld (2009) fitted Danish insurance data to the GPD and studied a range of risk factors and ratemaking exercises based on the fitted model. Niroomandi, Maa, Ye, Lou, and Xue (2018) carried out an extreme value analysis based on generalized extreme values and GPD functions to evaluate wave heights with different return periods. Liang, Shao, Li, Shao, and Lee (2019) adopted a 40-year hindcast dataset of tropical cyclone waves in the South China Sea and reported a stable threshold range. In economics and finance, the GPD has been employed to estimate excess returns (Fang & Egan, 2018), stock market returns (Gencay & Selcuk, 2004), share returns (Tolikas & Gettinby, 2009), and market risks (Riedel & Wagner, 2009). In wind energy, the GPD was introduced to analyze extreme wind speeds (D'Amico, Petroni, & Prattico, 2015; Holmes & Moriarty, 1999). In transportation engineering, Prietoa, Gómez-Dénizb, and Sarabia (2014) modeled Spanish road crash and fatality data with discrete GPD and Lomax distributions. Zheng, Ismail, and Meng (2014) applied a shifted gamma-GPD model of the safety continuum and estimated crashes due to lane change maneuvers on freeways. Most recently, Zhou, Schmidt, Toutlemonde, and Jacob (2016) predicted an extreme value distribution for bridge traffic load effects with the GPD, and stated that the GPD is an efficient method for conducting extreme value analyses of data with mixture probability distribution functions. We had two main objectives in this study. First, we aimed to understand the behavioral impact of different types of signals on pedestrian waiting time, contingent on the violation of traffic signals. The second objective was to estimate waiting times under different scenarios. The findings of this study could be used to address the conditions that increase the chance of crossing streets without abiding by traffic regulations. This information will be useful for the planning, design, and operation of pedestrian facilities. To realize the above two objectives, we collected field data on pedestrian waiting time at two signalized intersections: one with a pedestrian countdown signal and the other with a pedestrian conventional signal. The rest of the paper is structured as follows. In Section 2, we detail the survey method for field data collection and present the observations 2
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Fig. 1. Intersection A at Jianbao St and Dongdan N St, Beijing.
from our field survey. In Section 3, we develop a GPD model of pedestrian waiting time at both countdown and conventional traffic signals. Finally, the paper concludes with empirical and theoretical findings pertaining to pedestrian waiting time, as presented in Section 4.
using a video camera mounted on an adjacent building at the target site. Pedestrians crossing during the red-man phase were classified as violators who did or did not wait. This classification is similar to that of Yang et al. (2015), who denoted the former as immediate crossing behavior and the latter as waiting behavior. The duration of the redman phase has no influence on the waiting behaviors of the former. As the objective of this study was to elucidate the behavior of pedestrians who are willing to wait, we omitted violators who crossed the street without pausing (zero waiting time) from our analysis. More precisely, this study used only data from pedestrians who arrived at the curbside during the red-man phase and started to cross after a certain time, i.e., during the same red-man phase. The pedestrian waiting time is defined in Eq. (1). Here, t1 is the time that the pedestrian arrives at the curbside and t2 is the time that the pedestrian starts to cross. This definition is similar to that of Tiwari et al. (2007), Brosseau et al. (2013), and Li (2013).
2. Methods 2.1. Site selection Two intersections with pedestrian traffic signals in Beijing, China, were selected for field data collection. One used a pedestrian countdown signal, and the other a conventional signal. Both intersections met the following criteria. First, there were sufficient pedestrians within a given period to meet the requirements of statistical analyses. Second, the two intersections had a similar layout. Third, there were no traffic police or wardens at the site, which would influence pedestrians’ waiting behavior. Furthermore, pedestrians were not affected by poor weather conditions or darkness when they crossed the streets. The intersection with a pedestrian countdown signal had a four-leg four-way controlled signal located at Jinbao Street and Dongdan North Street (hereinafter referred to as intersection A), as shown in Fig. 1. Northbound and southbound pedestrians were observed on its west side. The intersection with a conventional pedestrian signal was also a four-leg four-way controlled signal, located at Zengguang Road and Shouti South Street (hereinafter referred to as intersection B), as shown in Fig. 2. We observed eastbound and westbound pedestrians on its north side. Crosswalks at intersections A and B are marked by parallel white lines perpendicular to the walking direction. At intersection A, the signal cycle for pedestrians is 140 s with a red-man phase of 77 s and a green-man phase of 63 s. At intersection B, the signal cycle for pedestrians is also 140 s, but with a red-man phase of 95 s and a greenman phase of 45 s.
t = t2
t1
(1)
Fig. 3 shows an example of the procedure used to calculate the pedestrian waiting time. In Fig. 3(a), the pedestrian in the red circle was approaching the crosswalk and the signal was in the green-man phase. At 0′35″, the pedestrian in the red circle arrived at the curbside and the signal had already changed to the red-man phase (Fig. 3(b)). In Fig. 3(c) and 3(d), the pedestrian was waiting for the green-man signal. At 1′37″, the pedestrian started to cross during the red-man phase (Fig. 3(e)). In Fig. 3(f), the crosswalk was heavily occupied. According to Equation (1), the waiting time was calculated as 1′37″ − 80′35″ = 1′02″. 2.3. Sample data Data were collected at intersection A during both off-peak hours (15:00–17:00) and peak hours (17:00–19:00) on a workday, i.e., August 20, 2015. To study sex differences, we further classified these pedestrians as male or female during peak hours at intersection A. At intersection B, we collected data only for off-peak hours (15:00–17:00) on a workday, i.e., August 19, 2015. The numbers of pedestrians observed at
2.2. Survey method The entire process of pedestrians crossing the street was recorded
Fig. 2. Intersection B at Zengguang Rd and Shouti S St., Beijing. 3
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Fig. 3. The procedure for measuring the time waited by each pedestrian.
2.4. Descriptive analyses based on field data
Table 1 Numbers of observations at intersections A and B. Type of pedestrian
Total Signal abider Signal violator Signal violator without wait Signal violator with wait
Intersection A
2.4.1. Influence of signal type on pedestrian waiting time We used the data for intersections A and B during off-peak hours to study the influence of signal type, including 1376 pedestrians at intersection A and 543 pedestrians at intersection B. At intersection A, 1158 (84.2%) pedestrians abided by the traffic signal, whereas 218 (15.8%) violated the signal. Among the violators, 58 (26.6%) crossed the street without waiting, and 160 (73.4%) waited. In the case of the 160 pedestrians who crossed after waiting, the mean waiting time was 29.6 s, with a standard deviation of 19.2 s. At intersection B, 317 (58.4%) pedestrians abided by the traffic signal, and 226 (41.6%) violated it. Among the signal violators, 63 (27.9%) crossed the street without waiting and 163 (72.1%) crossed after waiting. For the 163 pedestrians who crossed after waiting, the mean waiting time was 23.4 s, with a standard deviation of 16.7 s. For the countdown signal, 15.8% of pedestrians did not comply with the traffic signal. In the case of the conventional signal, the percentage was 41.6%, which is consistent with previous findings (Keegan & O’Mahony, 2003; Lipovac et al., 2013; Xiong et al., 2014). An
Intersection B
Off-peak hours
Peak hours
Off-peak hours
Sum
Males
Females
1376 1158 218 58
2108 1617 491 96
1043 798 245 74
1065 819 246 22
543 317 226 63
160
395
171
224
163
the two intersections are reported in Table 1. As an example, Fig. 4 shows the fluctuations in waiting time sequence based on 395 observation records from intersection A during peak hours.
Fig. 4. The waiting time sequence of pedestrians during peak hours at intersection A. 4
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independent-sample t-test was carried out to compare pedestrian waiting time between the two types of signals. The results indicate a significant difference in waiting times between the two conditions ( p = 0.002 ), indicating that the signal type influences the waiting time. Pedestrians tend to wait longer before violating the traffic signal at intersections with a countdown signal.
2012). The results of the t-test indicate that there were no significant differences between male and female waiting times ( p = 0.128). T-tests were performed for pairwise comparisons between different signal types, time of day, and sex groups. Levene’s tests were also applied to compare the equality of variance among groups. The results of these statistical tests are reported in Table 3.
2.4.2. Influence of the time of day on pedestrian waiting time Travel during peak hours is more likely to be for work purposes, whereas travel during off-peak hours is typically for non-work purposes. Empirical results show that pedestrians traveling for work purposes show shorter waiting times than those with non-work purposes (Hamed, 2001). We observed 2108 pedestrians during peak hours and 1376 during off-peak hours at intersection A. In the samples collected during peak hours, 1617 (76.7%) pedestrians complied with traffic signals, whereas 491 (23.3%) were violators. Among the violators, 96 (19.6%) crossings occurred without waiting, and 395 (80.4%) pedestrians waited. In the case of samples collected during off-peak hours, 1158 (84.2%) pedestrians complied with the traffic signal, whereas 218 (15.8%) pedestrians were violators. Among the violators, 58 (26.6%) crossings occurred without waiting, and 160 (73.4%) pedestrians waited. The percentage of signal abiders was lower during peak hours than during off-peak hours. This finding is consistent with those of previous studies (Guo et al., 2011; Hamed, 2001). Visual observations of the video record revealed that pedestrians traveling during peak hours had a higher tendency to stand on crosswalks at the curbside and wait for an opportunity to cross. Statistical analysis indicated a significant difference in waiting time between the two times of day ( p = 0.009 ), which implies that waiting time may be strongly influenced by trip purpose. One possible reason for this is that, during peak hours, vehicular traffic demand is high, leading to smaller gaps in which pedestrians can cross (Hamed, 2001). Pedestrians are more likely to find an acceptable gap for crossing under low-volume conditions than under high-volume conditions (Yang et al., 2015).
3. Model development and application 3.1. Generalized Pareto distribution model The distributional characteristics of pedestrian waiting time are important to transportation planners and traffic operators implementing traffic policies that reduce the risks associated with pedestrians crossing the street. Toward this end, a GPD model was adopted to model pedestrian waiting time. Let X be a random variable with an unknown distribution function F (x ) = Pr (Xi x ) ; then, the distribution function of X over a threshold u is derived according to Eq. (2).
Fu (x ) = Pr (X
u
F (x + u) F (u) 1 F (u)
x X > u) =
(2)
Pickands (1975) showed that the above distribution can be approximated by a GPD. Its cumulative distribution function (CDF) and R ), location ( µ R ) probability density function (PDF) with shape ( and scale ( > 0 ) parameters are defined in Eqs. (3) and (4), respec0 and µ < x < µ + tively. Here, x µ for for > 0 .
F (x ; , µ , ) =
1 1 1
f (x ; , µ , ) = 1
2.4.3. Impact of gender on pedestrian waiting time To assess the influence of sex on pedestrian waiting time, 2108 samples were classified into male and female pedestrians based on data collected during peak hours at intersection A. There were 1043 males and 1065 females. Among the males, 798 (76.5%) crossed during the green signal, and 245 (23.5%) crossed during the red signal. Among these violators, 74 (30.2%) crossings occurred without waiting, and 171 (69.8%) occurred with waiting. Among females, 819 (76.9%) crossed during the green signal, and 246 (23.1%) during the red signal. Among the female violators, 22 (8.9%) crossings occurred without waiting, and 224 (91.1%) crossings involved waiting, indicating that female violators were more conservative than male violators in terms of when they crossed the street during the red signal. In Table 2, we summarize descriptive statistics of the waiting time for violators who waited before crossing. The mean waiting times for male and female pedestrians were 32.5 and 35.3 s, with standard deviations of 18.4 and 17.5 s, respectively. These findings are consistent with those of previous studies (Hamed, 2001; Rosenbloom & Pereg,
(1 + exp ( (1 + exp (
x
µ
x
µ
) )
) )
x
µ
x
µ
1
1 +1
0 =0
(3)
0 =0
(4)
3.2. Simulation studies The three parameters , µ , and must be estimated for the GPD model. Estimators for the GPD parameters abound in the literature. The method of moments (MOM) is the simplest approach, calculating estimators by equating the sample mean and variance with the population mean and variance (Hosking & Wallis, 1987; Mackay, Challenor, & Bahaj, 2011). There are several commonly used approaches for obtaining estimators of the GPD parameters. Among these methods, the probability weighted moment (PWM) method, method of least squares (MLS), maximum likelihood estimation (MLE), likelihood moment estimation (LME), and L-moments (LMOM) are representative and have been widely applied to fit GPD models. Dupuis (1996) reported that the PWM method may produce inconsistent estimates of the upper bound of the GPD. Simulation studies found that the MLS, in general, has a lower root mean square error (RMSE) than other methods. The MLS also performs best in terms of estimation bias in cases where the shape
Table 2 Descriptive statistics of waiting times for pedestrian signal violators. Scenario
Site
Signal type
Time of day
Coverage
Countdown Countdown Countdown Countdown Conventional
Off-peak Peak Peak Peak Off-peak
Total Total Male Female Total
A A A A B
Sample size
160 395 171 224 163
5
Descriptive statistics of waiting time Min
Max
Mean
SD
2 1 1 1 2
76 75 70 75 78
29.6 34.1 32.5 35.3 23.4
19.2 17.9 18.4 17.5 16.7
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Table 3 Statistical test results for the equality of variance and mean waiting times. Scenario
Countdown vs. conventional signals Off-peak vs. peak hours Male vs. female pedestrians
Levene’s test for the equality of variance
T-test for the equality of means
F
p -value
t
df
p -value
3.682 0.194 0.340
0.056 0.660 0.560
3.112 −2.632 −1.524
321 553 393
0.002 0.009 0.128
parameter is greater than zero, while the PWM method performs best when the shape parameter is less than zero (Castillo & Serra, 2015; Moharram, Gosain, & Kapoor, 1993). The MLE is the most efficient method for estimating the parameters of the GPD. However, in the case of small samples with less than 100 observations, the fit is not always good. Moreover, numerical algorithms can fail to converge to local maxima (Hosking & Wallis, 1987). When the shape parameter of the GPD is not large than −1, the MLE does not exist. In contrast, no matter what the value of the shape parameter, the LME always exists. Furthermore, the LME is computationally easy and has high asymptotic efficiency (Zhang, 2007). The LMOM was formally introduced by Hosking (1990), as linear combinations of the order statistics of a population. Hosking (1990) stated that LMOM are more robust to outliers than conventional moments and enable more secure inferences from small sample sizes. Sankarasubramanian and Srinivasan (1999) reported that with a reasonable small sample size, L-skewness is found to be nearly unbiased. The Bayesian MLE (BMLE) is based on MLE, but with a data-based prior that ensures that the estimates always exist. Studies of bias and efficiency have shown that BMLE performs very well compared to some other estimators. Numerical examples of fitting peaks over thresholds demonstrated that the BMLE is very effective (Zhang & Stephens, 2009). Adapted from the nonlinear least squares estimator proposed by Song and Song (2012), Park and Kim (2016) developed the weighted nonlinear least squares (WNLS) method, which minimizes the sum of the squared deviations between the empirical distribution function and the theoretical GPD for data exceeding the tail threshold. Results of simulated samples show that the WNLS outperformed the alternative estimators, particularly when the shape parameter of the GPD and the tail threshold increased. We carried out extensive Monte Carlo simulations to investigate the performances of various methods for estimating and . We considered four methods: the MOM, MLE, LME, and LMOM. The results of the four principal methods were compared to two recently proposed predominant estimators: the BMLE and WNLS. All evaluations were based on the criteria of empirical bias and MSE, measured from 10,000 simulated trials. The simulations were carried out according to the steps below. The first step was to generate independently and identically distributed (IID) observations that follow the GPD with parameters( , ) . Six different pairs of parameters ( , ) were used in this study: (0.2, 1), (1.5, 1), (5, 1), (0.2, 10), (1.5, 10), and (5, 10). In the second step, we applied the six aforementioned methods to estimate the parameters( , ) . In the third step, the previous steps were repeated 1000 times to compute the bias and MSE of each method. Table 4 reports the performance of parameters when the shape parameter takes on certain values. To evaluate the performance of different estimation methods comprehensively, another simulation was carried out to estimate the bias and MSE with respect to the shape parameter. Here, we considered ranging from 0 to 5. The evaluation was based on the bias and MSE estimated from 1000 trials. In this simulation study, the scale parameter and sample size n were fixed to 1 and 100, respectively. The results can be seen in Figs. 5 and 6. In Table 4, we summarize the performances of the six estimators for specific values of the shape parameters. To provide a comprehensive evaluation of the effectiveness of each estimator, the variation in the
scale parameter was examined as the shape parameter was varied within a certain range. Fig. 5 illustrates the MSE and bias in with sample size n = 100. The MSE represents the difference between the fitting value and the observed data. It is clear that the estimators of the MOM and LMOM perform poorly in all cases. The performance of the WNLS, LME, and BMLE is better in some cases and the MLE behaves as the best estimator in most of those cases. Fig. 6 shows the MSE and bias in with a sample size of n = 100. The MOM and LMOM perform poorly in most cases. Estimators of the LME and BMLE are much better in some cases, especially in the case of in a certain region. In most of these cases, the estimators of the MLE and WNLS perform best. Broadly, regarding the MSE and bias under each parameter group, the MOM, LMOM, and LME are relatively poor estimators in all cases (see Table 4). The performance of the WNLS and BMLE is much better in some cases and the MLE is the best estimator in most cases. 3.3. Model estimation and validation The steps for estimating the parameters and are explained as follows. In the first step, we selected a temporary threshold u based on a goodness of fit test. In each iteration, the smallest value was removed until the updated samples followed the GPD. The updated samples were ranked in descending order and the smallest was set as the threshold u . In the second step, parameters were estimated using the MLE, WNLS and BMLE methods. In the third step, the value at risk (VaR ) was calculated using Eq. (5), which is equivalent to the waiting time.
VaRp = ^tp = u +
n (1 Nu
p)
1
(5)
Here, Nu is the number of samples exceeding the threshold u and n is the total number of samples. and are estimators of and , respectively. The p -quantile means that the probability that the waiting time does not exceed t^p is equal to p . For each sample set, we selected the 97th, 98th and 99th sample quantiles for threshold u , and the estimated quantiles were taken asVaR 0.97 , VaR 0.98 , and VaR 0.99 . Tables 5 and 6 show the estimates of parameters and for off-peak and peak hours, respectively, in the case of countdown signals. Based on the estimates in Tables 5 and 6, the fitting curves are shown in Fig. 7. Fig. 7(a), (b) and (c) show comparisons between the waiting times between the field data and the fitted model based on the MLE, WNLS, and BMLE during off-peak hours, respectively. Fig. 7(d), (e) and (f) show comparisons of the waiting times between the field data and fitted model based on the MLE, WNLS, and BMLE during peak hours, respectively. It is clear that the performances of the MLE and BMLE were approximately equivalent. When the number of observations was less than 20 or more than 40, the MLE and BMLE outperformed the WNLS. For other numbers of observations, the WNLS behaved best. In the case of peak hours, Fig. 7(d), (e) and (f) show that there was no evident difference in terms of the estimator performance among the MLE, WNLS, and BMLE. Furthermore, the estimators performed better when there were more observations. We applied the proposed estimators to predict pedestrian waiting time. In this study, we considered the time of day (peak and off-peak 6
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Table 4 The MSE and bias of parameters with sample size n = 50 and n = 100. ( , )
Methods
n = 50
n = 100
MSE
(0.2, 1)
(1.5, 1)
(5, 1)
(0.2, 10)
(1.5, 10)
(5, 10)
MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE
0.027 0.040 0.032 0.054 0.035 0.032 1.101 0.132 0.467 0.211 0.138 0.120 20.4 0.729 16.2 0.858 0.927 0.691 0.027 0.040 0.032 0.054 0.035 0.032 1.102 0.131 0.470 0.181 0.138 0.119 20.4 0.723 16.2 0.823 0.929 0.683
Bias
0.055 0.065 0.070 0.059 0.057 0.049 3.4e+13 0.130 20.6 0.152 0.149 0.133 1.8e+47 0.443 6.1e+31 0.609 1.285 1.886 5.509 6.402 6.978 5.832 5.591 4.853 2.0e+11 12.5 1739.6 13.1 14.4 12.7 1.3e+50 44.4 3.5e+26 47.0 134.2 204.2
MSE
−0.088 −0.053 −0.034 −0.081 −0.039 0.010 −1.048 −0.037 −0.670 −0.163 −0.054 −0.064 −4.515 −0.066 −4.019 −0.291 −0.138 −0.492 −0.088 −0.052 −0.035 −0.071 −0.038 0.011 −1.049 −0.040 −0.672 −0.145 −0.057 −0.066 −4.515 −0.059 −4.019 −0.284 −0.124 −0.489
0.102 0.063 0.041 0.026 0.046 −0.003 5.9e+4 0.077 2.181 0.082 0.101 0.103 4.4e+21 0.184 8.0e+13 0.181 0.411 0.851 0.980 0.577 0.400 0.160 0.412 −0.078 8730.8 0.724 21.6 0.615 0.959 0.980 1.6e+23 1.812 3.1e+11 1.674 3.977 8.573
0.014 0.017 0.016 0.028 0.016 0.015 1.054 0.065 0.396 0.090 0.069 0.062 20.3 0.351 16.1 0.397 0.455 0.314 0.014 0.017 0.016 0.028 0.016 0.015 1.053 0.065 0.396 0.083 0.069 0.062 20.3 0.361 16.1 0.407 0.458 0.317
Bias
0.027 0.028 0.034 0.030 0.026 0.024 6.2e+13 0.058 19.8 0.064 0.065 0.059 1.8e+52 0.165 1.5e+27 0.172 0.383 0.526 2.663 2.806 3.315 3.072 2.654 2.428 2.7e+11 5.781 3.2e+04 5.933 6.501 5.863 3.7e+56 15.8 6.7e+30 16.6 37.1 52.3
−0.054 −0.026 −0.018 −0.056 −0.019 0.007 −1.026 −0.020 −0.622 −0.082 −0.030 −0.035 −4.507 −0.041 −4.009 −0.169 −0.073 −0.324 −0.055 −0.026 −0.019 −0.049 −0.019 0.007 −1.026 −0.020 −0.622 −0.076 −0.030 −0.035 −4.507 −0.033 −4.009 −0.164 −0.061 −0.318
0.062 0.029 0.021 0.021 0.022 −0.005 8.2e+4 0.039 2.074 0.039 0.052 0.054 1.5e+24 0.085 5.4e+11 0.084 0.181 0.455 0.634 0.294 0.215 0.173 0.224 −0.042 1.2e+04 0.358 21.8 0.313 0.484 0.501 1.9e+26 0.806 2.6e+13 0.796 1.735 4.554
MSE, mean square error; MOM, method of moments; LMOM, L-moments; WNLS, weighted nonlinear least squares; LME, likelihood moment estimation; BMLE, Bayesian MLE.
hours), sex (male or female), and pedestrian signal type (countdown or conventional signals) as the main factors affecting waiting times. In Table 7, we summarize the predicted waiting times with various quantiles for both the countdown and conventional signals. For a given quantile, the waiting times predicted by the three estimators were quite close, illustrating the consistency between the chosen estimators and
the reliability of the GPD model. For example, in the case of the 85th quantile at the countdown signalized intersection, the waiting time during off-peak hours was 50.1 to 52.4 s (51.5 s on average), whereas during peak hours, it was approximately 54.1–54.7 s (54.4 s on average). In the case of male pedestrians, the waiting time was approximately 55.2–55.6 s (55.4 s on average), while female pedestrians
Fig. 5. The bias and mean square error (MSE) of the six estimators of . 7
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Fig. 6. The bias and MSE of the six estimators of .
waited for similar times to males, varying from 54.5 to 55.4 s (55.0 s on average). In the case of conventional signal intersections, the waiting time was approximately 41.9–42.9 s (42.5 s on average), which is significantly lower than at the countdown signal intersection.
confirms the consistency between the estimators and reliability of the GPD model. Understanding pedestrians’ behaviors toward their waiting tolerance prior to street crossing could be useful due to its policy and practical implications, e.g., planning and investment of crossing facilities and traffic policies with time-differentiated priorities for pedestrians. In contrast to vehicular traffic, disobeying traffic signals is more prevalent among pedestrians in many countries, causing a high risk of pedestrian-vehicle crashes. The data from this field survey show how long pedestrians are prepared to wait prior to violating traffic lights to cross the street. With the aid of the developed GPD model, we could evaluate the quantitative relationship between the change in waiting time and the change in the proportion of pedestrian signal violators. This will be important for planners, investors, and policymakers so that they can evaluate the likely impact of pedestrian-friendly facilities. It could also provide guidance for configuring the timing of traffic lights, and will probably promote investment in permissive crossing facilities for pedestrians at certain intersections. On the other hand, the results of this study raise the possibility of implementing traffic policies with time-differentiated priorities for pedestrians, as their waiting times vary between peak and off-peak hours. Furthermore, the calibrated distribution of the pedestrian waiting time is extremely useful in microscopic simulations of traffic flow, specifically for scenarios with pedestrian crossing behaviors. It would also be useful to investigate the relationship between a pedestrian’s waiting time and chance of being in a traffic crash based on additional data on traffic accidents, and to further evaluate the safety of intersections for pedestrians. Factors influencing pedestrians’ behaviors are complex. This study did not consider the waiting behaviors of a cohort of pedestrians in the current study. Pedestrians were actually hesitating between waiting at the curbside and violating the signal, and some of them were readily influenced by surrounding pedestrians. A certain proportion of pedestrians would decide not to wait and follow other violators. This is an interesting phenomenon and may be an important topic for future research. Methodologically, it is useful to compare the performance of
4. Discussion and conclusions In this study, we focused on the empirical and theoretical aspects of pedestrian waiting time prior to the violation of traffic signals. Our statistical findings show that the signal type significantly affects pedestrian waiting time. According to the field data, at the intersection with a countdown signal, 15.8% of pedestrians violated the traffic signal. At the intersection with a conventional signal, the violation rate was as high as 41.6%. Moreover, pedestrians tended to wait longer before they violated traffic signals at intersections with countdown signals. The percentage of signal abiders during peak hours was 76.7%, which is less than during off-peak hours (84.2%). Notably, pedestrians tended to wait longer before they violated traffic signals during peak hours than during off-peak hours. One important reason for this is that vehicle traffic is relatively heavy during peak hours, so it takes longer for pedestrians to find acceptable gaps. Furthermore, we detected no significant differences between males and females in terms of the waiting times during peak hours at countdown signal intersections. Modeling waiting time is a typical application of extreme value theory, as we focus on the tail of the distribution. Due to the ability to model observations that exceed thresholds, we used the GPD to estimate the extreme quantiles of the waiting time. Of the many GPD parameter estimation methods that have been used in the literature, we selected four traditional methods, i.e., the MOM, MLE, LME, and LMOM, in addition to two recently proposed estimators, i.e., the BMLE and WNLS. The results of Monte Carlo simulations show that the MLE, WNLS, and BMLE were most accurate in terms of the bias and MSE of the shape and scale parameters. Based on the field data, we obtained the pedestrian waiting time at various quantiles in five scenarios. The three estimators output very similar waiting times in most cases, which Table 5 Parameter estimates (off-peak hours with a countdown signal), u = 10 . Methods
MLE WNLS BMLE
VaR 0.97
−0.518 −0.381 −0.468
37.220 31.813 35.457
VaR 0.99
VaR 0.98
Value
RMSE
Bias
Value
RMSE
Bias
Value
RMSE
Bias
69.036 69.994 69.801
2.068 4.572 1.991
−1.183 −0.146 0.254
71.462 73.353 72.556
2.116 5.613 2.150
−1.248 −0.021 0.633
74.592 78.019 76.210
2.246 7.601 2.600
−1.328 0.251 1.247
MLE, maximum likelihood estimation; WNLS, weighted nonlinear least squares; BMLE, Bayesian MLE. 8
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Table 6 Parameter estimates (peak hours with a countdown signal), u = 35. Methods
MLE WNLS BMLE
VaR 0.97
−0.580 −0.570 −0.555
23.647 22.807 23.085
VaR 0.99
VaR 0.98
Value
RMSE
Bias
Value
RMSE
Bias
Value
RMSE
Bias
67.475 66.640 67.548
1.413 3.083 1.036
−1.120 0.249 0.457
69.213 68.366 69.377
1.389 3.858 1.064
−1.102 0.388 0.518
71.383 70.532 71.692
1.369 5.284 1.154
−1.063 0.645 0.612
RMSE, root mean squared error; MLE, maximum likelihood estimation; WNLS, weighted nonlinear least squares; BMLE, Bayesian MLE.
Fig. 7. Empirically distributed curves and fitted models of the MLE, WNLS, and BMLE during off-peak and peak hours.
9
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Table 7 Different quantiles of pedestrian waiting time. p
Methods
Countdown signal Off-peak hours
Peak hours
Male
Female
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Conventional signal
60%
MLE WNLS BMLE
32.9 30.5 32.2
38.5 38.4 38.4
37.4 37.1 36.7
39.5 39.3 39.4
24.6 24.3 24.2
65%
MLE WNLS BMLE
36.1 33.6 35.4
41.3 41.1 41.2
40.8 40.5 40.1
42.2 41.9 42.0
27.5 27.2 27.1
70%
MLE WNLS BMLE
39.6 37.0 38.9
44.2 43.9 44.1
44.3 44.0 43.6
45.1 44.7 44.9
30.7 30.2 30.3
75%
MLE WNLS BMLE
43.4 40.8 42.7
47.4 47.0 47.2
47.9 47.6 47.3
48.3 47.6 47.9
34.2 33.6 33.9
80%
MLE WNLS BMLE
47.6 45.1 47.0
50.8 50.3 50.6
51.7 51.3 51.1
51.6 50.9 51.3
38.2 37.5 37.9
85%
MLE WNLS BMLE
52.4 50.1 51.9
54.7 54.1 54.5
55.6 55.3 55.2
55.4 54.5 55.1
42.9 41.9 42.7
90%
MLE WNLS BMLE
58.0 56.3 57.7
59.1 58.4 58.9
59.7 59.4 59.5
59.8 58.7 59.5
48.7 47.4 48.7
95%
MLE WNLS BMLE
65.2 65.0 65.5
64.6 63.8 64.6
64.3 64.0 64.4
65.2 64.0 65.2
56.8 54.9 57.3
100%
MLE WNLS BMLE
81.8 93.4 85.7
75.8 75.0 76.6
70.2 70.0 71.1
76.2 75.0 77.7
84.1 78.0 89.1
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The tolerable waiting time: A generalized Pareto distribution model with empirical investigation
T
Hui Xionga, Lu Mab, Mengxi Ningc, Xu Zhaod, , Jinxian Wenge ⁎
a
School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China MOT Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Beijing Jiaotong University, Beijing 100044, China c Department of Administration, Tianfu College of SWUFE, Mianyang 621000, China d College of Applied Sciences, Beijing University of Technology, Beijing 100124, China e College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China b
ARTICLE INFO
ABSTRACT
Keywords: Waiting time Generalized Pareto distribution Signalized intersection Parameter estimation
In China, it is more common for pedestrians than vehicles to disobey traffic signals, resulting in a high risk of pedestrian-vehicle accidents. Pedestrian waiting time are the most critical indicator of the tendency to violate traffic signals. A statistical analysis based on 4027 field-collected samples showed that the length of time that pedestrians are prepared to wait depends on the type of pedestrian traffic signal. Compared to a countdown-type signal, pedestrians were more likely to violate conventional-type signals. Furthermore, pedestrians were willing to wait longer during peak hours than during off-peak hours. There were no significant differences between the waiting times of male and female travelers. To predict pedestrian waiting time, we propose a generalized Pareto distribution (GPD) model and calibrated it based on our field data. Monte Carlo simulations showed that the maximum likelihood estimation (MLE), Bayesian MLE (BMLE), and weighted nonlinear least squares (WNLS) models are the best methods for estimating the scale and shape parameters of the GPD model. Several empirical results were output from the models. For example, at countdown-type signals, the 85th quantile of the tolerable waiting time in off-peak and peak hours was 51.5 and 54.4 s, respectively; the respective values for males and females were 55.4 and 55.0 s. At conventional signals, the tolerable waiting time was approximately 42.5 s. These findings are useful for the planning, design, and operation of pedestrian facilities.
1. Introduction Pedestrians are vulnerable road users at relatively high risk of being killed in traffic accidents. Road traffic caused 1.35 million deaths globally in 2016, with pedestrian deaths accounting for 23% of all fatalities (WHO, 2018). In 2013, 4735 pedestrians were killed and approximately 66,000 were injured in traffic accidents in the United States (National Center for Statistics and Analysis, 2015). In China, 16,281 pedestrians were killed in traffic accidents, representing 25% of all road fatalities in 2010 (CRTASR, 2011). Furthermore, the percentage of fatal accidents involving pedestrians is much higher in urban areas than in rural areas. In London, pedestrians accounted for 46% of road traffic fatalities. In New York, pedestrian fatalities made up 52% of traffic deaths between 2005 and 2009, and as many as 74% of pedestrian deaths or severe injuries occurred in accidents at intersections. Collisions caused by violation of pedestrian traffic signals are more likely to be fatal than those that occur when pedestrians are crossing in
accordance with the signal (Viola, Roe, & Shin, 2010). In Canada, pedestrian traffic rule and signal violations accounted for 13% of all pedestrian fatalities. Pedestrians expose themselves to severe risk when crossing the street during the “stop” or “red-man” phases of traffic signals. However, disobeying traffic signals is prevalent among pedestrians in China (Guo, Wang, Guo, Jiang, & Bubb, 2012; Yang, Deng, Wang, Li, & Wang, 2006). Brosseau, Zangenehpour, Saunier, and Miranda-Moreno (2013) found 10 factors that contribute to the violation of pedestrian traffic signals. These factors can be classified into human characteristics and external factors. The human characteristics include age, sex, personality, education, profession, and history of traffic accident involvement. Female pedestrians are less likely to cross the street against the red-man signal and tend to wait for longer than do males. The elderly exhibit a similar tendency, in contrast to young and middle-aged pedestrians (Guo et al., 2012; Hamed, 2001; Keegan & O’Mahony, 2003; Rosenbloom, Nemrodov, & Barkan, 2004; Tiwari, Bangdiwala, Saraswat, & Gaurav,
Corresponding author. E-mail addresses: [email protected] (H. Xiong), [email protected] (L. Ma), [email protected] (M. Ning), [email protected] (X. Zhao), [email protected] (J. Weng). ⁎
https://doi.org/10.1016/j.cie.2019.106019 Received 3 May 2019; Received in revised form 26 July 2019; Accepted 13 August 2019 Available online 14 August 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.
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2007; Yagil, 1998, 2000; Yang, Abdel-Atyb, Huan, Peng, & Gao, 2015). Furthermore, the more people there are in a group waiting at the curbside, the less likely the group members are to violate pedestrian traffic signals (Guo, Gao, Yang, & Jiang, 2011; Hamed, 2001; Rosenbloom, 2009; Yagil, 2000). External factors include traffic rules and regulations, signal type, trip purpose, intersection layout, vehicle headway, weather conditions, and lighting conditions. Tapiro, OronGilad, and Parmet (2018) found that pedestrians tended to choose smaller crossing gaps and took longer time to decide whether to cross when they encountered environmental distractions. Most recently, Zhao, Malenje, Tang, and Han (2019) modeled pedestrians' gap acceptance considering gap size, crossing distance, number of waiting pedestrians, waiting time, vehicle traffic volume and position of pedestrian. To evaluate the influence of traffic signals on pedestrian crossing behaviors, Keegan and O’Mahony (2003) carried out a questionnaire and video survey in Dublin, Ireland. They observed a significant reduction (from 35% to 24%) in the likelihood of violating pedestrian traffic signals after installing a countdown signal. Many recent studies reported similar results (e.g. Xiong, Xiong, Deng, & Wang, 2014; Lipovac, Vujanic, Maric, & Nesic, 2013). The vehicle time headway directly affects the likelihood of safely crossing the street during the red-man phase (Li, 2013). Cohen, Dearnaley, and Hansel (1955) found that 98.57% of pedestrians crossed the road when the gap was 10.5 s or longer, while no-one crossed when the gap was less than 1.5 s. Wilson and Grayson (1980) found that 3.4% of males and 2.1% of females accepted gaps of less than 2 s when crossing without a “protected pedestrian phase”. Pedestrians waiting at refuge islands or road medians were likely to accept shorter gaps than those waiting at other types of crossings (Das, Manski, & Manuszak, 2005). The characteristics of a crosswalk, including its length and the type of median, significantly affect pedestrians’ crossing behavior; the longer the crosswalk, the fewer pedestrians are likely to cross against the signal (Supernak, Verma, & Supernak, 2013). Empirical studies have also observed differences in behavior depending on whether pedestrians were crossing narrow or wide refuge islands (Rosenbloom & Pereg, 2012). In addition, pedestrians on their way to work were more likely to maintain shorter waiting times than those making trips for other purposes (Hamed, 2001). Weather and lighting conditions are the main environmental factors affecting pedestrian crossing behavior (Andrew, 1991; Yagil, 2000). On rainy days, pedestrians may be more conservative when crossing (Sun, Zhuang, Wu, Zhao, & Zhang, 2015). Haleem, Alluri, and Gan (2015) reported that clear weather conditions were associated with a 3.06% reduction in the probability of severe injuries, in comparison to other weather conditions. Poor light is associated with an increase in the probability of severe injuries. Darkness is perceived to be a factor that increases the tendency for pedestrians to wait longer for the “go” or “green-man” signal (Yagil, 2000). The waiting times when crossing several streets in succession are also related. Hamed (2001) reported that pedestrians who spend more time waiting for the first crosswalk tend to attempt the second crosswalk earlier. Rosenbloom and Pereg (2012) analyzed the waiting time before entering a third crosswalk. In the case of streets with a wide refuge island, a positive correlation was detected between the waiting time of the first and second crossings. Furthermore, a positive correlation was also observed between the waiting time of the second and third crossings. Yang et al. (2006) established a pedestrian crossing model to characterize two types of pedestrians: law abiding and opportunistic. Time-gap distributions were used to assess the criteria that pedestrians used to decide whether to walk, and when such action conflicted with traffic flow. Li (2013) noted that the distribution of pedestrian waiting time during the red-man phase was U-shaped. Furthermore, Li (2014) developed two hierarchical generalized linear models to measure the impact of different factors on pedestrian waiting time during the red-man phase. Cao, Zhuang, and Ma (2019) studied the impact of tempo and pitch on audible pedestrian signals for waiting time estimation.
Waiting time is a key factor in pedestrians’ tendency to violate traffic signals. Van Houten, Ellis, and Kim (2007) investigated the relation between pedestrian waiting time and violation at two signalized midblock crosswalks. They reported that the violation rate increased as the waiting time increased. In fact, long waiting time caused pedestrians to lose patience, resulting in illegal crossings. Tiwari et al. (2007) observed that the probability of violation varied depending on the waiting time. Sun, Liu, Li, and Yang (2010) concluded that the 85th percentile of the tolerable waiting time is approximately 90 s. More specifically, Brosseau et al. (2013) reported that a 10% increase in waiting time increased the probability of violations by 7.9%. Zhuang, Wu, and Ma (2018) studied pedestrians' crossing behavior when arriving at pedestrian clearance phase. Field data showed that as many as 85.2% of walkers chose to cross as opposed to 14.8% of total abided by the signal. However, there have been few empirical studies on pedestrian waiting time in recent years. Very little information has been published on the methodological aspects of modeling pedestrian waiting time at signalized intersections (Li, 2013). Toward this end, we adopted a generalized Pareto distribution (GPD) approach (Pickands, 1975) to model pedestrian waiting time, which are extreme values from the perspective of probability theory. Extreme-value theory is a popular approach for modeling extreme values, by extracting observations that exceed a certain threshold magnitude. It is generally referred to as the peaks-over-threshold method, where a distribution is fitted describing how far a variable exceeds a high threshold. For random variables satisfying certain regularity conditions, the distribution of the exceedance of a threshold tends to follow the GPD (Beirlant, Goegebeur, Segers, & Teugels, 2004; Coles, 2001; Kang & Song, 2017; McNeil & Saladin, 1997). GPD is commonly applied to model observations that exceed a given threshold in a variety of fields, such as insurance, hydrology, economics, finance, energy, and transportation. For example, Brazauskas and Kleefeld (2009) fitted Danish insurance data to the GPD and studied a range of risk factors and ratemaking exercises based on the fitted model. Niroomandi, Maa, Ye, Lou, and Xue (2018) carried out an extreme value analysis based on generalized extreme values and GPD functions to evaluate wave heights with different return periods. Liang, Shao, Li, Shao, and Lee (2019) adopted a 40-year hindcast dataset of tropical cyclone waves in the South China Sea and reported a stable threshold range. In economics and finance, the GPD has been employed to estimate excess returns (Fang & Egan, 2018), stock market returns (Gencay & Selcuk, 2004), share returns (Tolikas & Gettinby, 2009), and market risks (Riedel & Wagner, 2009). In wind energy, the GPD was introduced to analyze extreme wind speeds (D'Amico, Petroni, & Prattico, 2015; Holmes & Moriarty, 1999). In transportation engineering, Prietoa, Gómez-Dénizb, and Sarabia (2014) modeled Spanish road crash and fatality data with discrete GPD and Lomax distributions. Zheng, Ismail, and Meng (2014) applied a shifted gamma-GPD model of the safety continuum and estimated crashes due to lane change maneuvers on freeways. Most recently, Zhou, Schmidt, Toutlemonde, and Jacob (2016) predicted an extreme value distribution for bridge traffic load effects with the GPD, and stated that the GPD is an efficient method for conducting extreme value analyses of data with mixture probability distribution functions. We had two main objectives in this study. First, we aimed to understand the behavioral impact of different types of signals on pedestrian waiting time, contingent on the violation of traffic signals. The second objective was to estimate waiting times under different scenarios. The findings of this study could be used to address the conditions that increase the chance of crossing streets without abiding by traffic regulations. This information will be useful for the planning, design, and operation of pedestrian facilities. To realize the above two objectives, we collected field data on pedestrian waiting time at two signalized intersections: one with a pedestrian countdown signal and the other with a pedestrian conventional signal. The rest of the paper is structured as follows. In Section 2, we detail the survey method for field data collection and present the observations 2
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Fig. 1. Intersection A at Jianbao St and Dongdan N St, Beijing.
from our field survey. In Section 3, we develop a GPD model of pedestrian waiting time at both countdown and conventional traffic signals. Finally, the paper concludes with empirical and theoretical findings pertaining to pedestrian waiting time, as presented in Section 4.
using a video camera mounted on an adjacent building at the target site. Pedestrians crossing during the red-man phase were classified as violators who did or did not wait. This classification is similar to that of Yang et al. (2015), who denoted the former as immediate crossing behavior and the latter as waiting behavior. The duration of the redman phase has no influence on the waiting behaviors of the former. As the objective of this study was to elucidate the behavior of pedestrians who are willing to wait, we omitted violators who crossed the street without pausing (zero waiting time) from our analysis. More precisely, this study used only data from pedestrians who arrived at the curbside during the red-man phase and started to cross after a certain time, i.e., during the same red-man phase. The pedestrian waiting time is defined in Eq. (1). Here, t1 is the time that the pedestrian arrives at the curbside and t2 is the time that the pedestrian starts to cross. This definition is similar to that of Tiwari et al. (2007), Brosseau et al. (2013), and Li (2013).
2. Methods 2.1. Site selection Two intersections with pedestrian traffic signals in Beijing, China, were selected for field data collection. One used a pedestrian countdown signal, and the other a conventional signal. Both intersections met the following criteria. First, there were sufficient pedestrians within a given period to meet the requirements of statistical analyses. Second, the two intersections had a similar layout. Third, there were no traffic police or wardens at the site, which would influence pedestrians’ waiting behavior. Furthermore, pedestrians were not affected by poor weather conditions or darkness when they crossed the streets. The intersection with a pedestrian countdown signal had a four-leg four-way controlled signal located at Jinbao Street and Dongdan North Street (hereinafter referred to as intersection A), as shown in Fig. 1. Northbound and southbound pedestrians were observed on its west side. The intersection with a conventional pedestrian signal was also a four-leg four-way controlled signal, located at Zengguang Road and Shouti South Street (hereinafter referred to as intersection B), as shown in Fig. 2. We observed eastbound and westbound pedestrians on its north side. Crosswalks at intersections A and B are marked by parallel white lines perpendicular to the walking direction. At intersection A, the signal cycle for pedestrians is 140 s with a red-man phase of 77 s and a green-man phase of 63 s. At intersection B, the signal cycle for pedestrians is also 140 s, but with a red-man phase of 95 s and a greenman phase of 45 s.
t = t2
t1
(1)
Fig. 3 shows an example of the procedure used to calculate the pedestrian waiting time. In Fig. 3(a), the pedestrian in the red circle was approaching the crosswalk and the signal was in the green-man phase. At 0′35″, the pedestrian in the red circle arrived at the curbside and the signal had already changed to the red-man phase (Fig. 3(b)). In Fig. 3(c) and 3(d), the pedestrian was waiting for the green-man signal. At 1′37″, the pedestrian started to cross during the red-man phase (Fig. 3(e)). In Fig. 3(f), the crosswalk was heavily occupied. According to Equation (1), the waiting time was calculated as 1′37″ − 80′35″ = 1′02″. 2.3. Sample data Data were collected at intersection A during both off-peak hours (15:00–17:00) and peak hours (17:00–19:00) on a workday, i.e., August 20, 2015. To study sex differences, we further classified these pedestrians as male or female during peak hours at intersection A. At intersection B, we collected data only for off-peak hours (15:00–17:00) on a workday, i.e., August 19, 2015. The numbers of pedestrians observed at
2.2. Survey method The entire process of pedestrians crossing the street was recorded
Fig. 2. Intersection B at Zengguang Rd and Shouti S St., Beijing. 3
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Fig. 3. The procedure for measuring the time waited by each pedestrian.
2.4. Descriptive analyses based on field data
Table 1 Numbers of observations at intersections A and B. Type of pedestrian
Total Signal abider Signal violator Signal violator without wait Signal violator with wait
Intersection A
2.4.1. Influence of signal type on pedestrian waiting time We used the data for intersections A and B during off-peak hours to study the influence of signal type, including 1376 pedestrians at intersection A and 543 pedestrians at intersection B. At intersection A, 1158 (84.2%) pedestrians abided by the traffic signal, whereas 218 (15.8%) violated the signal. Among the violators, 58 (26.6%) crossed the street without waiting, and 160 (73.4%) waited. In the case of the 160 pedestrians who crossed after waiting, the mean waiting time was 29.6 s, with a standard deviation of 19.2 s. At intersection B, 317 (58.4%) pedestrians abided by the traffic signal, and 226 (41.6%) violated it. Among the signal violators, 63 (27.9%) crossed the street without waiting and 163 (72.1%) crossed after waiting. For the 163 pedestrians who crossed after waiting, the mean waiting time was 23.4 s, with a standard deviation of 16.7 s. For the countdown signal, 15.8% of pedestrians did not comply with the traffic signal. In the case of the conventional signal, the percentage was 41.6%, which is consistent with previous findings (Keegan & O’Mahony, 2003; Lipovac et al., 2013; Xiong et al., 2014). An
Intersection B
Off-peak hours
Peak hours
Off-peak hours
Sum
Males
Females
1376 1158 218 58
2108 1617 491 96
1043 798 245 74
1065 819 246 22
543 317 226 63
160
395
171
224
163
the two intersections are reported in Table 1. As an example, Fig. 4 shows the fluctuations in waiting time sequence based on 395 observation records from intersection A during peak hours.
Fig. 4. The waiting time sequence of pedestrians during peak hours at intersection A. 4
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independent-sample t-test was carried out to compare pedestrian waiting time between the two types of signals. The results indicate a significant difference in waiting times between the two conditions ( p = 0.002 ), indicating that the signal type influences the waiting time. Pedestrians tend to wait longer before violating the traffic signal at intersections with a countdown signal.
2012). The results of the t-test indicate that there were no significant differences between male and female waiting times ( p = 0.128). T-tests were performed for pairwise comparisons between different signal types, time of day, and sex groups. Levene’s tests were also applied to compare the equality of variance among groups. The results of these statistical tests are reported in Table 3.
2.4.2. Influence of the time of day on pedestrian waiting time Travel during peak hours is more likely to be for work purposes, whereas travel during off-peak hours is typically for non-work purposes. Empirical results show that pedestrians traveling for work purposes show shorter waiting times than those with non-work purposes (Hamed, 2001). We observed 2108 pedestrians during peak hours and 1376 during off-peak hours at intersection A. In the samples collected during peak hours, 1617 (76.7%) pedestrians complied with traffic signals, whereas 491 (23.3%) were violators. Among the violators, 96 (19.6%) crossings occurred without waiting, and 395 (80.4%) pedestrians waited. In the case of samples collected during off-peak hours, 1158 (84.2%) pedestrians complied with the traffic signal, whereas 218 (15.8%) pedestrians were violators. Among the violators, 58 (26.6%) crossings occurred without waiting, and 160 (73.4%) pedestrians waited. The percentage of signal abiders was lower during peak hours than during off-peak hours. This finding is consistent with those of previous studies (Guo et al., 2011; Hamed, 2001). Visual observations of the video record revealed that pedestrians traveling during peak hours had a higher tendency to stand on crosswalks at the curbside and wait for an opportunity to cross. Statistical analysis indicated a significant difference in waiting time between the two times of day ( p = 0.009 ), which implies that waiting time may be strongly influenced by trip purpose. One possible reason for this is that, during peak hours, vehicular traffic demand is high, leading to smaller gaps in which pedestrians can cross (Hamed, 2001). Pedestrians are more likely to find an acceptable gap for crossing under low-volume conditions than under high-volume conditions (Yang et al., 2015).
3. Model development and application 3.1. Generalized Pareto distribution model The distributional characteristics of pedestrian waiting time are important to transportation planners and traffic operators implementing traffic policies that reduce the risks associated with pedestrians crossing the street. Toward this end, a GPD model was adopted to model pedestrian waiting time. Let X be a random variable with an unknown distribution function F (x ) = Pr (Xi x ) ; then, the distribution function of X over a threshold u is derived according to Eq. (2).
Fu (x ) = Pr (X
u
F (x + u) F (u) 1 F (u)
x X > u) =
(2)
Pickands (1975) showed that the above distribution can be approximated by a GPD. Its cumulative distribution function (CDF) and R ), location ( µ R ) probability density function (PDF) with shape ( and scale ( > 0 ) parameters are defined in Eqs. (3) and (4), respec0 and µ < x < µ + tively. Here, x µ for for > 0 .
F (x ; , µ , ) =
1 1 1
f (x ; , µ , ) = 1
2.4.3. Impact of gender on pedestrian waiting time To assess the influence of sex on pedestrian waiting time, 2108 samples were classified into male and female pedestrians based on data collected during peak hours at intersection A. There were 1043 males and 1065 females. Among the males, 798 (76.5%) crossed during the green signal, and 245 (23.5%) crossed during the red signal. Among these violators, 74 (30.2%) crossings occurred without waiting, and 171 (69.8%) occurred with waiting. Among females, 819 (76.9%) crossed during the green signal, and 246 (23.1%) during the red signal. Among the female violators, 22 (8.9%) crossings occurred without waiting, and 224 (91.1%) crossings involved waiting, indicating that female violators were more conservative than male violators in terms of when they crossed the street during the red signal. In Table 2, we summarize descriptive statistics of the waiting time for violators who waited before crossing. The mean waiting times for male and female pedestrians were 32.5 and 35.3 s, with standard deviations of 18.4 and 17.5 s, respectively. These findings are consistent with those of previous studies (Hamed, 2001; Rosenbloom & Pereg,
(1 + exp ( (1 + exp (
x
µ
x
µ
) )
) )
x
µ
x
µ
1
1 +1
0 =0
(3)
0 =0
(4)
3.2. Simulation studies The three parameters , µ , and must be estimated for the GPD model. Estimators for the GPD parameters abound in the literature. The method of moments (MOM) is the simplest approach, calculating estimators by equating the sample mean and variance with the population mean and variance (Hosking & Wallis, 1987; Mackay, Challenor, & Bahaj, 2011). There are several commonly used approaches for obtaining estimators of the GPD parameters. Among these methods, the probability weighted moment (PWM) method, method of least squares (MLS), maximum likelihood estimation (MLE), likelihood moment estimation (LME), and L-moments (LMOM) are representative and have been widely applied to fit GPD models. Dupuis (1996) reported that the PWM method may produce inconsistent estimates of the upper bound of the GPD. Simulation studies found that the MLS, in general, has a lower root mean square error (RMSE) than other methods. The MLS also performs best in terms of estimation bias in cases where the shape
Table 2 Descriptive statistics of waiting times for pedestrian signal violators. Scenario
Site
Signal type
Time of day
Coverage
Countdown Countdown Countdown Countdown Conventional
Off-peak Peak Peak Peak Off-peak
Total Total Male Female Total
A A A A B
Sample size
160 395 171 224 163
5
Descriptive statistics of waiting time Min
Max
Mean
SD
2 1 1 1 2
76 75 70 75 78
29.6 34.1 32.5 35.3 23.4
19.2 17.9 18.4 17.5 16.7
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Table 3 Statistical test results for the equality of variance and mean waiting times. Scenario
Countdown vs. conventional signals Off-peak vs. peak hours Male vs. female pedestrians
Levene’s test for the equality of variance
T-test for the equality of means
F
p -value
t
df
p -value
3.682 0.194 0.340
0.056 0.660 0.560
3.112 −2.632 −1.524
321 553 393
0.002 0.009 0.128
parameter is greater than zero, while the PWM method performs best when the shape parameter is less than zero (Castillo & Serra, 2015; Moharram, Gosain, & Kapoor, 1993). The MLE is the most efficient method for estimating the parameters of the GPD. However, in the case of small samples with less than 100 observations, the fit is not always good. Moreover, numerical algorithms can fail to converge to local maxima (Hosking & Wallis, 1987). When the shape parameter of the GPD is not large than −1, the MLE does not exist. In contrast, no matter what the value of the shape parameter, the LME always exists. Furthermore, the LME is computationally easy and has high asymptotic efficiency (Zhang, 2007). The LMOM was formally introduced by Hosking (1990), as linear combinations of the order statistics of a population. Hosking (1990) stated that LMOM are more robust to outliers than conventional moments and enable more secure inferences from small sample sizes. Sankarasubramanian and Srinivasan (1999) reported that with a reasonable small sample size, L-skewness is found to be nearly unbiased. The Bayesian MLE (BMLE) is based on MLE, but with a data-based prior that ensures that the estimates always exist. Studies of bias and efficiency have shown that BMLE performs very well compared to some other estimators. Numerical examples of fitting peaks over thresholds demonstrated that the BMLE is very effective (Zhang & Stephens, 2009). Adapted from the nonlinear least squares estimator proposed by Song and Song (2012), Park and Kim (2016) developed the weighted nonlinear least squares (WNLS) method, which minimizes the sum of the squared deviations between the empirical distribution function and the theoretical GPD for data exceeding the tail threshold. Results of simulated samples show that the WNLS outperformed the alternative estimators, particularly when the shape parameter of the GPD and the tail threshold increased. We carried out extensive Monte Carlo simulations to investigate the performances of various methods for estimating and . We considered four methods: the MOM, MLE, LME, and LMOM. The results of the four principal methods were compared to two recently proposed predominant estimators: the BMLE and WNLS. All evaluations were based on the criteria of empirical bias and MSE, measured from 10,000 simulated trials. The simulations were carried out according to the steps below. The first step was to generate independently and identically distributed (IID) observations that follow the GPD with parameters( , ) . Six different pairs of parameters ( , ) were used in this study: (0.2, 1), (1.5, 1), (5, 1), (0.2, 10), (1.5, 10), and (5, 10). In the second step, we applied the six aforementioned methods to estimate the parameters( , ) . In the third step, the previous steps were repeated 1000 times to compute the bias and MSE of each method. Table 4 reports the performance of parameters when the shape parameter takes on certain values. To evaluate the performance of different estimation methods comprehensively, another simulation was carried out to estimate the bias and MSE with respect to the shape parameter. Here, we considered ranging from 0 to 5. The evaluation was based on the bias and MSE estimated from 1000 trials. In this simulation study, the scale parameter and sample size n were fixed to 1 and 100, respectively. The results can be seen in Figs. 5 and 6. In Table 4, we summarize the performances of the six estimators for specific values of the shape parameters. To provide a comprehensive evaluation of the effectiveness of each estimator, the variation in the
scale parameter was examined as the shape parameter was varied within a certain range. Fig. 5 illustrates the MSE and bias in with sample size n = 100. The MSE represents the difference between the fitting value and the observed data. It is clear that the estimators of the MOM and LMOM perform poorly in all cases. The performance of the WNLS, LME, and BMLE is better in some cases and the MLE behaves as the best estimator in most of those cases. Fig. 6 shows the MSE and bias in with a sample size of n = 100. The MOM and LMOM perform poorly in most cases. Estimators of the LME and BMLE are much better in some cases, especially in the case of in a certain region. In most of these cases, the estimators of the MLE and WNLS perform best. Broadly, regarding the MSE and bias under each parameter group, the MOM, LMOM, and LME are relatively poor estimators in all cases (see Table 4). The performance of the WNLS and BMLE is much better in some cases and the MLE is the best estimator in most cases. 3.3. Model estimation and validation The steps for estimating the parameters and are explained as follows. In the first step, we selected a temporary threshold u based on a goodness of fit test. In each iteration, the smallest value was removed until the updated samples followed the GPD. The updated samples were ranked in descending order and the smallest was set as the threshold u . In the second step, parameters were estimated using the MLE, WNLS and BMLE methods. In the third step, the value at risk (VaR ) was calculated using Eq. (5), which is equivalent to the waiting time.
VaRp = ^tp = u +
n (1 Nu
p)
1
(5)
Here, Nu is the number of samples exceeding the threshold u and n is the total number of samples. and are estimators of and , respectively. The p -quantile means that the probability that the waiting time does not exceed t^p is equal to p . For each sample set, we selected the 97th, 98th and 99th sample quantiles for threshold u , and the estimated quantiles were taken asVaR 0.97 , VaR 0.98 , and VaR 0.99 . Tables 5 and 6 show the estimates of parameters and for off-peak and peak hours, respectively, in the case of countdown signals. Based on the estimates in Tables 5 and 6, the fitting curves are shown in Fig. 7. Fig. 7(a), (b) and (c) show comparisons between the waiting times between the field data and the fitted model based on the MLE, WNLS, and BMLE during off-peak hours, respectively. Fig. 7(d), (e) and (f) show comparisons of the waiting times between the field data and fitted model based on the MLE, WNLS, and BMLE during peak hours, respectively. It is clear that the performances of the MLE and BMLE were approximately equivalent. When the number of observations was less than 20 or more than 40, the MLE and BMLE outperformed the WNLS. For other numbers of observations, the WNLS behaved best. In the case of peak hours, Fig. 7(d), (e) and (f) show that there was no evident difference in terms of the estimator performance among the MLE, WNLS, and BMLE. Furthermore, the estimators performed better when there were more observations. We applied the proposed estimators to predict pedestrian waiting time. In this study, we considered the time of day (peak and off-peak 6
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Table 4 The MSE and bias of parameters with sample size n = 50 and n = 100. ( , )
Methods
n = 50
n = 100
MSE
(0.2, 1)
(1.5, 1)
(5, 1)
(0.2, 10)
(1.5, 10)
(5, 10)
MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE MOM MLE LMOM WNLS LME BMLE
0.027 0.040 0.032 0.054 0.035 0.032 1.101 0.132 0.467 0.211 0.138 0.120 20.4 0.729 16.2 0.858 0.927 0.691 0.027 0.040 0.032 0.054 0.035 0.032 1.102 0.131 0.470 0.181 0.138 0.119 20.4 0.723 16.2 0.823 0.929 0.683
Bias
0.055 0.065 0.070 0.059 0.057 0.049 3.4e+13 0.130 20.6 0.152 0.149 0.133 1.8e+47 0.443 6.1e+31 0.609 1.285 1.886 5.509 6.402 6.978 5.832 5.591 4.853 2.0e+11 12.5 1739.6 13.1 14.4 12.7 1.3e+50 44.4 3.5e+26 47.0 134.2 204.2
MSE
−0.088 −0.053 −0.034 −0.081 −0.039 0.010 −1.048 −0.037 −0.670 −0.163 −0.054 −0.064 −4.515 −0.066 −4.019 −0.291 −0.138 −0.492 −0.088 −0.052 −0.035 −0.071 −0.038 0.011 −1.049 −0.040 −0.672 −0.145 −0.057 −0.066 −4.515 −0.059 −4.019 −0.284 −0.124 −0.489
0.102 0.063 0.041 0.026 0.046 −0.003 5.9e+4 0.077 2.181 0.082 0.101 0.103 4.4e+21 0.184 8.0e+13 0.181 0.411 0.851 0.980 0.577 0.400 0.160 0.412 −0.078 8730.8 0.724 21.6 0.615 0.959 0.980 1.6e+23 1.812 3.1e+11 1.674 3.977 8.573
0.014 0.017 0.016 0.028 0.016 0.015 1.054 0.065 0.396 0.090 0.069 0.062 20.3 0.351 16.1 0.397 0.455 0.314 0.014 0.017 0.016 0.028 0.016 0.015 1.053 0.065 0.396 0.083 0.069 0.062 20.3 0.361 16.1 0.407 0.458 0.317
Bias
0.027 0.028 0.034 0.030 0.026 0.024 6.2e+13 0.058 19.8 0.064 0.065 0.059 1.8e+52 0.165 1.5e+27 0.172 0.383 0.526 2.663 2.806 3.315 3.072 2.654 2.428 2.7e+11 5.781 3.2e+04 5.933 6.501 5.863 3.7e+56 15.8 6.7e+30 16.6 37.1 52.3
−0.054 −0.026 −0.018 −0.056 −0.019 0.007 −1.026 −0.020 −0.622 −0.082 −0.030 −0.035 −4.507 −0.041 −4.009 −0.169 −0.073 −0.324 −0.055 −0.026 −0.019 −0.049 −0.019 0.007 −1.026 −0.020 −0.622 −0.076 −0.030 −0.035 −4.507 −0.033 −4.009 −0.164 −0.061 −0.318
0.062 0.029 0.021 0.021 0.022 −0.005 8.2e+4 0.039 2.074 0.039 0.052 0.054 1.5e+24 0.085 5.4e+11 0.084 0.181 0.455 0.634 0.294 0.215 0.173 0.224 −0.042 1.2e+04 0.358 21.8 0.313 0.484 0.501 1.9e+26 0.806 2.6e+13 0.796 1.735 4.554
MSE, mean square error; MOM, method of moments; LMOM, L-moments; WNLS, weighted nonlinear least squares; LME, likelihood moment estimation; BMLE, Bayesian MLE.
hours), sex (male or female), and pedestrian signal type (countdown or conventional signals) as the main factors affecting waiting times. In Table 7, we summarize the predicted waiting times with various quantiles for both the countdown and conventional signals. For a given quantile, the waiting times predicted by the three estimators were quite close, illustrating the consistency between the chosen estimators and
the reliability of the GPD model. For example, in the case of the 85th quantile at the countdown signalized intersection, the waiting time during off-peak hours was 50.1 to 52.4 s (51.5 s on average), whereas during peak hours, it was approximately 54.1–54.7 s (54.4 s on average). In the case of male pedestrians, the waiting time was approximately 55.2–55.6 s (55.4 s on average), while female pedestrians
Fig. 5. The bias and mean square error (MSE) of the six estimators of . 7
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Fig. 6. The bias and MSE of the six estimators of .
waited for similar times to males, varying from 54.5 to 55.4 s (55.0 s on average). In the case of conventional signal intersections, the waiting time was approximately 41.9–42.9 s (42.5 s on average), which is significantly lower than at the countdown signal intersection.
confirms the consistency between the estimators and reliability of the GPD model. Understanding pedestrians’ behaviors toward their waiting tolerance prior to street crossing could be useful due to its policy and practical implications, e.g., planning and investment of crossing facilities and traffic policies with time-differentiated priorities for pedestrians. In contrast to vehicular traffic, disobeying traffic signals is more prevalent among pedestrians in many countries, causing a high risk of pedestrian-vehicle crashes. The data from this field survey show how long pedestrians are prepared to wait prior to violating traffic lights to cross the street. With the aid of the developed GPD model, we could evaluate the quantitative relationship between the change in waiting time and the change in the proportion of pedestrian signal violators. This will be important for planners, investors, and policymakers so that they can evaluate the likely impact of pedestrian-friendly facilities. It could also provide guidance for configuring the timing of traffic lights, and will probably promote investment in permissive crossing facilities for pedestrians at certain intersections. On the other hand, the results of this study raise the possibility of implementing traffic policies with time-differentiated priorities for pedestrians, as their waiting times vary between peak and off-peak hours. Furthermore, the calibrated distribution of the pedestrian waiting time is extremely useful in microscopic simulations of traffic flow, specifically for scenarios with pedestrian crossing behaviors. It would also be useful to investigate the relationship between a pedestrian’s waiting time and chance of being in a traffic crash based on additional data on traffic accidents, and to further evaluate the safety of intersections for pedestrians. Factors influencing pedestrians’ behaviors are complex. This study did not consider the waiting behaviors of a cohort of pedestrians in the current study. Pedestrians were actually hesitating between waiting at the curbside and violating the signal, and some of them were readily influenced by surrounding pedestrians. A certain proportion of pedestrians would decide not to wait and follow other violators. This is an interesting phenomenon and may be an important topic for future research. Methodologically, it is useful to compare the performance of
4. Discussion and conclusions In this study, we focused on the empirical and theoretical aspects of pedestrian waiting time prior to the violation of traffic signals. Our statistical findings show that the signal type significantly affects pedestrian waiting time. According to the field data, at the intersection with a countdown signal, 15.8% of pedestrians violated the traffic signal. At the intersection with a conventional signal, the violation rate was as high as 41.6%. Moreover, pedestrians tended to wait longer before they violated traffic signals at intersections with countdown signals. The percentage of signal abiders during peak hours was 76.7%, which is less than during off-peak hours (84.2%). Notably, pedestrians tended to wait longer before they violated traffic signals during peak hours than during off-peak hours. One important reason for this is that vehicle traffic is relatively heavy during peak hours, so it takes longer for pedestrians to find acceptable gaps. Furthermore, we detected no significant differences between males and females in terms of the waiting times during peak hours at countdown signal intersections. Modeling waiting time is a typical application of extreme value theory, as we focus on the tail of the distribution. Due to the ability to model observations that exceed thresholds, we used the GPD to estimate the extreme quantiles of the waiting time. Of the many GPD parameter estimation methods that have been used in the literature, we selected four traditional methods, i.e., the MOM, MLE, LME, and LMOM, in addition to two recently proposed estimators, i.e., the BMLE and WNLS. The results of Monte Carlo simulations show that the MLE, WNLS, and BMLE were most accurate in terms of the bias and MSE of the shape and scale parameters. Based on the field data, we obtained the pedestrian waiting time at various quantiles in five scenarios. The three estimators output very similar waiting times in most cases, which Table 5 Parameter estimates (off-peak hours with a countdown signal), u = 10 . Methods
MLE WNLS BMLE
VaR 0.97
−0.518 −0.381 −0.468
37.220 31.813 35.457
VaR 0.99
VaR 0.98
Value
RMSE
Bias
Value
RMSE
Bias
Value
RMSE
Bias
69.036 69.994 69.801
2.068 4.572 1.991
−1.183 −0.146 0.254
71.462 73.353 72.556
2.116 5.613 2.150
−1.248 −0.021 0.633
74.592 78.019 76.210
2.246 7.601 2.600
−1.328 0.251 1.247
MLE, maximum likelihood estimation; WNLS, weighted nonlinear least squares; BMLE, Bayesian MLE. 8
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Table 6 Parameter estimates (peak hours with a countdown signal), u = 35. Methods
MLE WNLS BMLE
VaR 0.97
−0.580 −0.570 −0.555
23.647 22.807 23.085
VaR 0.99
VaR 0.98
Value
RMSE
Bias
Value
RMSE
Bias
Value
RMSE
Bias
67.475 66.640 67.548
1.413 3.083 1.036
−1.120 0.249 0.457
69.213 68.366 69.377
1.389 3.858 1.064
−1.102 0.388 0.518
71.383 70.532 71.692
1.369 5.284 1.154
−1.063 0.645 0.612
RMSE, root mean squared error; MLE, maximum likelihood estimation; WNLS, weighted nonlinear least squares; BMLE, Bayesian MLE.
Fig. 7. Empirically distributed curves and fitted models of the MLE, WNLS, and BMLE during off-peak and peak hours.
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Table 7 Different quantiles of pedestrian waiting time. p
Methods
Countdown signal Off-peak hours
Peak hours
Male
Female
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Conventional signal
60%
MLE WNLS BMLE
32.9 30.5 32.2
38.5 38.4 38.4
37.4 37.1 36.7
39.5 39.3 39.4
24.6 24.3 24.2
65%
MLE WNLS BMLE
36.1 33.6 35.4
41.3 41.1 41.2
40.8 40.5 40.1
42.2 41.9 42.0
27.5 27.2 27.1
70%
MLE WNLS BMLE
39.6 37.0 38.9
44.2 43.9 44.1
44.3 44.0 43.6
45.1 44.7 44.9
30.7 30.2 30.3
75%
MLE WNLS BMLE
43.4 40.8 42.7
47.4 47.0 47.2
47.9 47.6 47.3
48.3 47.6 47.9
34.2 33.6 33.9
80%
MLE WNLS BMLE
47.6 45.1 47.0
50.8 50.3 50.6
51.7 51.3 51.1
51.6 50.9 51.3
38.2 37.5 37.9
85%
MLE WNLS BMLE
52.4 50.1 51.9
54.7 54.1 54.5
55.6 55.3 55.2
55.4 54.5 55.1
42.9 41.9 42.7
90%
MLE WNLS BMLE
58.0 56.3 57.7
59.1 58.4 58.9
59.7 59.4 59.5
59.8 58.7 59.5
48.7 47.4 48.7
95%
MLE WNLS BMLE
65.2 65.0 65.5
64.6 63.8 64.6
64.3 64.0 64.4
65.2 64.0 65.2
56.8 54.9 57.3
100%
MLE WNLS BMLE
81.8 93.4 85.7
75.8 75.0 76.6
70.2 70.0 71.1
76.2 75.0 77.7
84.1 78.0 89.1
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