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Investigating pedestrian waiting time at semi-controlled crossing locations: Application of multi-state models for recurrent events analysis
Accident Analysis and Prevention 137 (2020) 105437
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Investigating pedestrian waiting time at semi-controlled crossing locations: Application of multi-state models for recurrent events analysis
T
Yunchang Zhang, Yu Qiao, Jon D. Fricker* Lyles School of Civil Engineering, Purdue University, 550 Stadium Mall Dr., W. Lafayette IN, 47907, United States
A R T I C LE I N FO
A B S T R A C T
Keywords: Crosswalks Pedestrian waiting behavior Semi-Markov process
“Semi-controlled” crosswalks are unsignalized, but clearly marked with “yield to pedestrian within crosswalk” signs. Ideally, pedestrians can cross the street immediately after they arrive at the curb. However, real world observations show that pedestrians and vehicles are often involved in non-verbal “negotiations” to decide who should proceed first. This kind of “negotiation” often causes delays for both parties and may lead to unsafe situations. The study in this paper was based on video recordings of the waiting behaviors of 2059 pedestrians interacting with 1003 motorists at selected semi-controlled crosswalks. One such location experienced a conversion from one-way operation to two-way operation, which provided a rare opportunity for a before-and-after study at that location. Multi-state Markov models were introduced as a novel approach to correlate the dynamic process between recurrent events. Time-varying covariates related to pedestrian characteristics, traffic condition, and vehicle dynamics (distance and speed) turned out to be significant. The analytical method developed in this study provides a tool to dynamically model pedestrian waiting decisions with uncertainties. Model results reveal that, after the conversion from one-way to two-way operation, the probability of a pedestrian accepting a lag decreases from 69.7% to just below 60% on the same street. In addition, pedestrians are more hesitant to cross a two-way street than a one-way street. Countermeasures that increase motorist yielding rate or reduce pedestrian confusion will enhance safety such crossing locations.
1. Introduction Marked crosswalks with “State Law Yield to Pedestrian Within Crosswalk’’ signs (R1-6 sign in Fig. 1a) can be called “semi-controlled” crosswalks. Despite its presence at many locations, the meaning of “within crosswalk” on the R1-6 sign is not interpreted uniformly. It is generally understood that pedestrians have priority at the R1-6 sign, but how should motorists act when pedestrians are standing on the curb, in the gutter, etc.? The Indiana Code, the MUTCD, and the Indiana Driver’s Manual do not provide clarification as to the sign’s practical meaning. Despite the implied priority to pedestrians, it was apparent from observations that pedestrians did not always exercise that prerogative. Drivers were seldom dangerously aggressive, but were often hesitant about whether to proceed in the presence of pedestrians. Such confusion is apparent when watching pedestrians and motorists interact at “semi-controlled” crosswalks, and this confusion can lead to unsafe situations. Delay ensues as non-verbal “negotiations” take place between pedestrian and driver. Safety and delay are two primary performance measures for semicontrolled crosswalks. Safety countermeasures have been extensively
⁎
explored for the analysis of crosswalks, but there is much less literature that considers pedestrian wait time. Pedestrian wait time is an important metric to estimate efficiency or level of service (Highway Capacity Manual, 2010). It is also an indicator that can be used to evaluate potential risks between pedestrians and motorists at a crossing location (Havard and Willis, 2012). A higher pedestrian wait time is often associated with a lower driver yield rate (Gitelman et al., 2017). If average pedestrian delay is high, pedestrians tend to be impatient and take more risk (Nemeth et al., 2014; Almodfer et al., 2016). On the other hand, pedestrian wait time is an indication that a “negotiation” is taking place as pedestrians and motorists interact (Sucha et al., 2017). Inefficient negotiation between pedestrian and motorist can occur when at least one party has incorrect expectations concerning the behavior of the others (Rumar, 1990). Such miscommunication can lead to dangerous situations. The study described in this paper pays specific attention to the actions of pedestrians and motorists during interactions. Recent studies have revealed that changes to the road crossing environment can have a significant impact on observable road-crossing behavior. Existing literature has conducted before-and-after studies, such as marked versus unmarked crosswalks (Zegeer et al., 2001;
Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y. Qiao), [email protected] (J.D. Fricker).
https://doi.org/10.1016/j.aap.2020.105437 Received 10 June 2019; Received in revised form 8 January 2020; Accepted 8 January 2020 0001-4575/ © 2020 Published by Elsevier Ltd.
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Fig. 1. Semi-Controlled Crosswalk. (a) Sign at Semi-Controlled Crosswalk. (Manual on Uniform Traffic Control Devices for Streets and Highways, 2009) (b) One-way North University Street at Second Street, 2017. (Jon D. Fricker) (c) Two-way North University Street at Second Street, 2018. (Google Maps).
by 37%. However, similar to survival models, time-based gap acceptance models have difficulty in handling discrete choice data over time. Some researchers have realized the limitation of gap acceptance theory and developed simulation frameworks to facilitate the computation of pedestrian wait time. Chae (2008) incorporated pedestrian gap acceptance decision rules into simulation software and studied pedestrian waiting time at unsignalized roundabouts. Lu et al. (2016) proposed a cellular automaton (CA) framework to estimate pedestrian waiting time by incorporating pedestrian crossing behavior and vehicle yielding behavior. The modified CA model was a comprehensive simulation framework that included the prediction of pedestrian crossing behavior and pedestrian delay at the curb. Huang et al. (2016) proposed a social force model to study the interaction between cyclist and driver at unsignalized intersections. Collision avoidance and cyclist gap acceptance models were incorporated into the proposed social force model to facilitate cyclist delay estimation based on the shortest path. Other researchers developed stochastic process models to dynamically model pedestrian delay at semi-controlled crosswalks. Adams (1936) proposed an analytical method for pedestrian delay estimation at unsignalized crosswalks. He introduced vehicle arrival times as a random sequence and the gap (vehicle headway or interarrival time) between two successive vehicles was usually considered as a random variable from a specific distribution. As pedestrians find a gap sufficiently large to cross, the pedestrian delay was estimated by the Poisson process. Weiss and Maradudin (1962) assumed that successive gaps in the main traffic were uncorrelated random variables with known probability density functions, while the pedestrian crossing probability distribution was assumed to be known. Renewal theory was first used to model pedestrian delay as a distribution function by means of the convolution integral and the Laplace transform. They extended their models by means of the semi-Markov process to handle the issue of correlated successive gaps in the main traffic. HCM (2010) implemented Adams’ model by assuming a constant motorist yielding behavior when pedestrian-vehicle interaction occurs. The pedestrian delay was divided into two parts – as gap delay that was calculated by Adams’ method and as pedestrian reaction delay incurred by pedestrian-vehicle interactions. Zheng and Elefteriadou (2017) further extended the Weiss and Maradudin model (1962) and the HCM 2010 model by renewal theory. The effect of driver yielding behavior was incorporated as a function in the pedestrian delay model. However, simulation-based frameworks and stochastic process models do not address the time-varying heterogeneity caused by discrete-outcome data gathered over time. When interacting with vehicles, pedestrian waiting behavior is a dynamic process consisting of single or multiple gap acceptance
Havard and Willis, 2012; Gitelman et al., 2017), signalized versus unsignalized crosswalks (Asaithambi et al., 2016), and single-lane versus multiple-lane crosswalks (Zhang et al., 2019). In this study, the change from one-way traffic to two-way traffic provided a rare opportunity to study the negotiations of a similar population having undergone such a significant change. With this background, this study addresses the following research questions: 1 Which time-varying factors will influence pedestrian waiting behaviors at such locations? 2 How to use time-to-event data to dynamically capture pedestrian waiting behaviors at semi-controlled crossing locations? 3 How will pedestrian waiting behavior change if a one-way street is changed to two-way operation? 4 What can we learn that will guide decisions about changing control strategies at semi-controlled crosswalks? 2. Literature Review A majority of pedestrian wait time studies have been conducted using survival theory and time-based gap acceptance models. Hamed (2011) proposed a survival model to explore pedestrian wait time at the curbside of uncontrolled crosswalks. Although the survival model has limitations in dealing with recurrent events analysis and time-varying covariates, Hamed (2011) inserted a Poisson model that estimated the number of pedestrian attempts into the survival framework. The estimation results revealed that the pedestrian waiting time when crossing from one side of the street to the central median was significantly larger than the pedestrian waiting time when crossing from the median to the other side of street. Sun et al. (2003) investigated pedestrian gap acceptance behavior at several uncontrolled crosswalks. Empirical results confirmed that pedestrians who were unwilling to accept shorter gaps had to wait for a larger and safer gap, which increased their total wait time. Schroeder and Rouphail (2010a) defined an event-based modeling approach to describe a pedestrian–motorist interaction as an event. Either the pedestrian gap acceptance probability or vehicle yielding probability could be predicted by specific explanatory variables concerning the event. Consequently, pedestrian average delay was predicted as a function of the probability of encountering a crossing opportunity in the form of a vehicle yielding or a crossable gap, using a linear regression model (Schroeder and Rouphail, 2010b) at semi-controlled crosswalks. The results showed that an increase of driver yielding rate from 10% to 30% would significantly decrease the pedestrian delay for sighted pedestrians by 65% and for blind pedestrians 2
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Table 1 Explanatory Variables (for All Events). Variable Explanatory Variable Pedestrian Waiting time Independent Variables Pedestrian characteristics Sex Young Indicator Distraction Indicator Traffic Condition Near Side Indicator Bus/Truck Indicator Veh-to-Ped Distance Approach Speed Adjacent Vehicle Indicator Vehicle Close Follower Indicator
Description
Duration (in seconds) between the time a pedestrian reaches the curb area and the time the pedestrian starts crossing (One-way: mean = 2.67 s, sd = 2.61 s; Two-way: mean = 3.02 s, sd = 3.08 s)
1 if the pedestrian is Male (51.1%), 0 if Female (48.9%). 1 if the pedestrian appears to be younger than 30 years old (77%); 0 otherwise (23%). 1 if the pedestrian is using a cellphone (9%) or talking to others (8.4%) when waiting at the curb, 0 otherwise (82.6%). 1 if the critical vehicle is in the near lane (53%), 0 if the critical vehicle is in the far lane (47%). 1 if the critical vehicle is a bus or large truck (16.7%), 0 otherwise (83.3%). Distance (ft) between the pedestrian and the interacted vehicle when an interaction begins (mean = 61.8 ft, sd = 51.7 ft). If an interaction occurs when a pedestrian arrives at the curb, the video is paused to calculate the distance to vehicle using Google Maps. Speed (ft/s) of the interacted vehicle when an interaction begins (mean = 12.4 ft/s, sd = 11.2 ft/s). Using distances along the street found with Google Maps, the distance covered by a vehicle every 34 milliseconds (one video frame) is converted into a speed. 1 if one or more vehicles are present in the adjacent lane within the area of influence when the motorist begins to interact with a pedestrian (32.3%), 0 otherwise (67.7%). 1 if there is at least one vehicle closely following the subject vehicle (31.9%), 0 otherwise (68.1%). The object vehicle has a close follower (CloseFollow = 1) if the headway is less than 4 seconds, as defined in Schroeder and Rouphail (2011).
Other Pedestrians Group Size Group ID Nr Ped Waiting
Number of people in the pedestrian group (45.7% of pedestrians came alone, 54.3% came in a group; mean = 2.31, sd = 2.03) Identifies the group to which the subject pedestrian belongs (748 in one-way; 636 in two-way). Number of pedestrians waiting at the curb as a pedestrian arrives at the curb (no pedestrians waiting in 52.8% of cases, at least one pedestrian was waiting at the curb in 47.2% of cases; mean = 0.84, sd = 1.21) Nr Ped Crossing Number of other pedestrians crossing the street as a pedestrian arrives at the curb (48.5% of cases, no pedestrians crossing; 51.5% of cases, at least one other pedestrian crossing in crosswalk; mean = 1.27, sd = 1.99) Note: mean = average value; sd = standard deviation
bicycle lanes and a speed limit of 25 mph. The two sets of video recordings were made at four different time periods (7:40-8:20; 12:4013:25; 13:20-14:00; 16:20-17:00), when low-to-moderate traffic volumes and pedestrian flows could be observed. On one-way N. University Street, we observed 1126 pedestrians interacting with 498 motorists over 155 min. With two-way street operation, there were 927 pedestrians interacting with 505 motorists in 150 min. The pedestrian and vehicle rates did not change much between the one-way and twoway cases. The pedestrian rates across N. University Street are probably higher than at the typical “semi-controlled” crosswalk location, but the higher rate gives us more interactions to analyze, without pedestrians dominating the situation.
decisions. In pedestrian delay studies, pedestrians are observed over time and covariates are quantified on different occasions (MeiraMachado et al., 2009). We hope to use such time-varying covariates from longitudinal data to empirically test the proposed method and understand its implications and sensitivity. An extension of the stochastic process approach is likely to solve the problem by using a multistate semi-Markov model. The multi-state semi-Markov model is a continuous time stochastic process that allows individuals to move among a finite number of states (Zhang, 2019). In transportation studies, multi-state Markov models have not been widely used. Researchers (Malyshkina and Mannering, 2009, 2010; Xiong et al., 2014) have developed Markov switching models to account for the severity of highway injuries. The proposed models provide critical insights into the temporal impacts of weather conditions on the likelihood of accidents and their resulting injury severities. Multi-state Markovian models have also been used in bridge deterioration modeling (Baik et al., 2006; Thomas and Sobanjo, 2016), mode choice (Xiong et al., 2015) and tourism management (Xia et al., 2011).
3.2. Definitions 3.2.1. Interaction Each pedestrian was documented from the time he/she arrived at the curb area until he/she crossed the street. A pedestrian-motorist interaction occurred when a pedestrian and a motorist were in the area of influence of each other (Fricker and Zhang, 2019):
3. Methodology
• A pedestrian enters the curb area and intends to cross. • The subject driver is aware of the pedestrian’s intention to cross and
3.1. Data collection The data for this study were collected at a “semi-controlled” crossing location (North University Street at the T intersection with Second Street) on the Purdue University campus. The two crosswalks are clearly marked and "State Law Yield to Pedestrian Within Crosswalk” signs are present (Fig. 1(a)). The studied intersection is primarily used by students walking to/from class and by faculty and staff walking from/to nearby parking garages. Two sets of video recordings were made to collect data on both pedestrians’ and drivers’ behaviors (2059 pedestrians and 1003 vehicles). The first set of videos was made in Spring 2017, when North University Street was a one-way northbound street. See Fig. 1(b). Under one-way operation, the street had two 10-ft wide lanes (plus a 4-foot bicycle lane) with a speed limit of 25 mph. By the time the second set of videos was made in Spring 2018, the street had been converted to two-way operation. See Fig. 1(c). After the conversion, the street is 34 ft wide, with two 5-foot
responds to the pedestrian.
There are two types of driver outcomes for each interaction:
• The driver yields to the pedestrian. • The subject driver fails to yield to the pedestrian. The important variables that were extracted from the videos when an interaction occurred are summarized in Table 1. 3.2.2. Lag and gap The lag begins when a pedestrian enters the curb area and interacts with a motorist who (a) is aware of the pedestrian’s intention to cross and then (b) responds to the pedestrian. If the pedestrian rejects the lag, the lag ends when the motorist passes the crosswalk. If the pedestrian 3
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Fig. 2. Multi-State Framework for Pedestrian Waiting Behavior.
chain in terms of conditional probability, is:
accepts the lag, the lag ends when the pedestrian enters the crosswalk. A gap for pedestrians begins after a rejected lag or gap. If the pedestrian rejects a gap, the gap ends when the vehicle passes the crosswalk. If the pedestrian accepts the gap, the gap ends when the pedestrian enters the crosswalk.
Fhj (d ) = P (Sn + 1 ≤ d|Jn + 1 = j, Jn = h) =
αhj = lim
• State 1: A pedestrian arrives at the curb area and interacts with the first approaching vehicle. • State 2: The pedestrian rejects the lag and keeps waiting for subsequent (recurrent) gaps. • State 3: The pedestrian accepts the lag (or gap) and enters the
Δd → 0
=
3.2.4. Transition The potential transitions for this model are defined as:
• Transition 1-3: pedestrian accepts the lag (indicating that first interacted vehicle yields to subject pedestrian). • Transition 1-2: pedestrian rejects the lag (indicating the first interacted vehicle fails to yield). • Transition 2-2: pedestrian rejects all subsequent gaps. (This transition is not considered in this paper.) • Transition 2-3: pedestrian accepts a subsequent gap (indicating a
fhj (d ) 1 − Fhj (d )
(4)
k x k − 1 −( x ) ⎛ ⎞ e λ λ ⎝λ⎠
k
(5)
According to Eq.s (4) and (5), the hazard ratio for the Weibull distribution is:
4. Modeling Framework
αhj =
4.1. Multi-state semi-markov model
fhj (d ) 1 − Fhj (d )
=
k x k−1 ⎛ ⎞ λ ⎝λ⎠
(6)
Fig. 3 shows the probability density function for every transition. Transition 1-2-3 (reject the lag, then accept the next gap) can be exx pressed as the convolution product f1 − 2 − 3 = ∫0 f1 − 2 (u) f2 − 3 (x − u) du , where x = the total waiting time during the transition 1-2-3.
Consider a Markov renewal process (Jn, Tn), where T0 < T1 < … < Tn < ∞ are the successive times of entry to States J0, J1,…, Jn. If Sn = Tn − Tn − 1 is the sojourn time (gap time or lag time), the Markov renewal kernel Qhj (d ) is a cumulative distribution function of time:
4.1.2. Parameterization Let Zhj be a vector of explanatory variables related to the transition from State h to j and βh,j be the vector of estimated regression parameters. By the Cox proportional hazard model,
Qhj (d ) = P (Jn + 1 = j, Sn + 1 ≤ d J0 , J1, …, Jn = h, S1, S2, …, Sn ) (1)
J0, J1,…, Jn is an embedded homogeneous Markov chain taking values in a finite state space with transition probability:
Table 2 Weibull Distribution Duration Parameters. Duration Parameters in Weibull Distribution
(2)
Transition
We define the distribution function of the sojourn time in State h by: s
∑ j=1 Qhj (t ), ∀ s ∈
fhj (d ) Pr(d < Sn + 1 ≤ d + Δd|Jn + 1 = j, Jn = h, Sn + 1 > d ) = Δd Shj (d )
f (x|λ, k ) =
subsequent vehicle yields to pedestrian).
Hh (t ) =
(3)
4.1.1. Distribution of durations For the semi-Markov process, we need to first assume that sojourn time (gap time or lag time) belongs to a specific parametric distribution. The SemiMarkov package in R software (Listwon-Krol and Saint-Pierre, 2015) offers three distributions – Exponential, Weibull and Exponential Weibull. Based on maximum likelihood estimation, the Weibull distribution was chosen to model the sojourn time from State h to State j. Table 2 shows the estimated coefficients and standard errors for Weibull distributions. Table 3 shows the results of the Wald test.
crosswalk.
phj = P (Jn + 1 = j|Jn = h) = limt →∞Qhj (t ), n ∈ N
phj
Fhj (d ) is a cumulative probability distribution of a sojourn time in State h if the next state will be j. Based on Eq. (3), we can write the probability density function as fhj (d ) . The hazard function αhj of Fhj (d ) will be:
3.2.3. States We identify 3 observable states of pedestrian’s waiting behavior (See Fig. 2).
= P (Jn + 1 = j, Sn + 1 ≤ d Jn = h)
Qhj (d )
(3)
1→2 1→3 2→3
The probability distribution function of sojourn time (gap time or lag time), through the transition probabilities of the embedded Markov 4
One-Way λ Estim. 2.864 2.014 1.881
SE 0.12 0.04 0.11
k Estim. 1.364 1.843 1.329
SE 0.06 0.04 0.07
Two-Way λ Estim. SE 3.021 0.09 2.229 0.06 1.67 0.11
k Estim. 1.785 1.794 1.075
SE 0.07 0.05 0.05
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Table 3 Wald Test of Weibull Distribution Duration Parameters. One-Way
Two-Way
Transition 1→2 1→3 2→3
αhj (d|Zh, j ) =
Wald Test 43.53 357.06 19.84
P-value < 0.0001 < 0.0001 < 0.0001
Transition 1→2 1→3 2→3
Wald Test 128.59 259.26 1.89
αhj (d )exp(βhT, j Zh, j )
P-value < 0.0001 < 0.0001 0.1692
In Eq. (7), we used the transition-specific variables Zh,j defined in Table 1. Using the Semi-Markov package in R, the significant factors were retained. 4.1.3. Hazard of semi-markov process The hazard rate of a semi-Markov process is defined as the probability of transition towards State j between the time d and d+Δd, given that the process is in State h for a duration d.
5. Results 5.1. Estimation results Table 4 presents the estimation results for the multi-state semiMarkov model (called simply the Markov model in the following sections). In the Markov model, the proportional hazard (Eq. 7) is used. Therefore, a negative coefficient indicates that the increase of an explanatory variable increases the hazard rate and consequently decreases pedestrian waiting time of a specific transition, and vice versa.
γhj (d ) = lim
5.2. Findings
(8)
5.2.1. Comparison of one-way to two-way street operation It was found that (1) in one-way street operation, pedestrians who accepted the lag (Transition 1–3) tend to experience shorter delay in the curb area than for two-way operation; (2) in the one-way case, pedestrians who reject the lag (Transition 1-2–3) tend to have longer waiting time than in the two-way case before entering the crosswalk. See Fig. 4.
Note that Eq. (6) and Eq. (8) demonstrate two different hazards. The hazard in Eq. (6) is defined as the hazard, given a transition from State h to State j. The hazard defined by Eq. (8) is the hazard of the semiMarkov process, which represents the immediate probability of going to State j given State h in a small-time interval [d, d+Δd] (De Dominicis and Manca, 1984). Therefore, for the state space I = {1, 2, 3}, we can use Eq. (9) to calculate the “staying” probability for the case h=j:
p11 (d ) = e− ∫0
d
p22 (d ) = e− ∫0 p33 (d ) = 1
d
(10)
P1–3 and P1-2-3 in Eq. (10) are the two most common behavior of pedestrians waiting in the curb area. Out of 1132 transitions observed in the one-way case, 966 were for Transitions 1–3 and 1-2-3. The remaining behavior type is 1-2-…-2-3, which is not explicitly considered here, but has been tested to be statistically similar to 1-2-3. Therefore, we can use the transition probability of the semi-Markov process P1–3 + P1-2-3 to explain the variables in Table 1.
(7)
Pr(Jn + 1 = j, d < Sn + 1 ≤ d + Δd|Jn = h, Sn + 1 > d ) Δd → 0 Δd ( ) p f d q hj hj hj ⎧ ⎪ 1 − H (d ) = 1 − H (d ) h h ⎪ = phj (1 − Fhj (d )) αhj ⎨ = ifphj > 0 andHh (d ) < 1 ⎪ 1 − Hh (d ) ⎪ ⎩ 0otherwise
d
∫0 p11 (τ ) γ1−2 (τ ) p22 (d − τ ) dτ d p1 − 3 (d ) = ∫ p11 (τ ) γ1 − 3 (τ ) p33 (d − τ ) dτ 0 d p2 − 3 (d ) = ∫ p22 (τ ) γ2 − 3 (τ ) p33 (d − τ ) dτ 0 d p1 − 2 − 3 (d ) = ∫ p11 (τ ) γ1 − 2 (τ ) p2 − 3 (d − τ ) dτ 0 p1 − 2 (d ) =
5.2.2. Vehicle dynamics 5.2.2.1. Pedestrian-to-vehicle distance. According to our data, the squared terms of the vehicle-to-pedestrian distance are highly significant in both the one-way and two-way cases, indicating that there exists a non-monotonic relationship between pedestrian delay and the pedestrian-to-vehicle distance: the delay increases at first, then decreases as the distance increases. Such a relationship is presented in Fig. 5(a) for the one-way and two-way cases. The longest pedestrian
γ12 (τ ) + γ13 (τ ) dτ γ23 (τ ) dτ
(9)
Consequently, we can calculate the probabilities of each transition in a Markov chain as:
Fig. 3. Density Functions Between Different Transitions. 5
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Table 4 Multi-State Markov Model Results. Coefficients
Sex Young Distraction Vehicle Speed Distance Bus Indicator Near or Far Side Close Follower Indicator Adjacent Vehicle Group Size Nr. Ped. Crossing Nr. Ped. Waiting Log-Likelihood
One-Way
Two-Way
1→3
1→2
2→3
1→3
1→2
2→3
– 0.495*** – −0.031*** 0.251*** – – – −0.388*** – 0.086*** −0.143*** −1208.360
– 0.346** – 0.059*** −0.769*** −0.727*** – – – 0.159** – –
0.473** – – – – – – – 0.462* –
– 0.480*** – −0.012* – 0.386* – – – 0.088** 0.072*** −0.077* −1137.986
– – – 0.034*** −0.467*** – – – – 0.117. −0.096. –
0.309* – −0.348^ – – – – – – 0.216* – –
–
Notes; Significance codes: ‘***’: 0.001; ‘**’: 0.01; ‘*’: 0.05; ‘^’: 0.1.
of the first vehicle he/she interacts with is higher. However, this is true only for pedestrians who accepted the lag (Transition 1–3 in Fig. 6). For pedestrians who reject the lag (Transition 1–2) and keep waiting for the subsequent gaps, an opposite relationship is observed: the total pedestrian delay is shorter if the first approaching vehicle comes at a higher speed. This indicates that when the vehicle is too close or too fast upon the pedestrian’s arrival, “Let the car go first” saves time for both pedestrians and drivers.
delay was found at a distance of 38.41 ft in the one-way case and a distance of 50.13 ft in the two-way case using regression analysis. This means that a longer distance generally results in shorter pedestrian waiting time (because pedestrians are less likely to wait). However, if the vehicle is already too close to yield (less than 38.41 ft or 50.13 ft), this short pedestrian-to-vehicle distance also reduces pedestrian delay (because the pedestrian is likely to let the vehicle go first). Such a nonmonotonic relationship makes sense, but has not been quantified in previous studies. The semi-Markov model results (Fig. 5(b)) indicate that, for pedestrians who accepted the lag of the first vehicle (Transition 1–3), larger pedestrian-to-vehicle distance leads to a shorter pedestrian delay. However, for pedestrians who rejected the first gap (Transition 1–2), their waiting time increases as the distance increases. First, if a vehicle is too close to yield to the subject pedestrian, the normal pedestrian choice is to “let the vehicle go first”, and it will cause little delay to the pedestrian. See Transition 1-2-3. If a vehicle is too far away, the normal pedestrian choice is to cross without any hesitation, because the pedestrian will feel safe. (See Transition 1–3 in Fig. 5(b)).
5.3.1. Pedestrian characteristics 5.3.1.1. Young Indicator. Fig. 7 reveals that the waiting behavior of younger pedestrians was only observed as significant when the pedestrian accepted/rejected a lag (Transition 1–3 or 1–2). When making decisions on the subsequent gaps (2–3), pedestrians from different age groups didn’t behave significantly differently. 5.3.1.2. Sex. In Fig. 8, male pedestrians were found to generally wait less than do female pedestrians, but only after they reject the lag and wait for subsequent gaps (Transition 2–3). This indicates that male and female pedestrians don’t behave differently when they interact with the first approaching vehicle. However, when they are waiting and looking for a sufficiently large gap in the subsequent vehicles, male pedestrians are found to behave more “aggressively”.
5.3. Vehicle speed It is not surprising that a pedestrian tends to wait longer if the speed
Fig. 4. Empirical Transition Probability for Different Street Operations. 6
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Fig. 5. Effects of Pedestrian-to-Vehicle Distance on Pedestrian Waiting Time.
Fig. 6. Effects of Speed on Pedestrian Waiting Time (One-way).
7
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Fig. 7. Effects of Pedestrian’s Age on Pedestrian Waiting Time (One-way).
5.4. Future research directions
5.3.1.3. Group size. The Markov model indicates that, when crossing a two-way street, pedestrians entering the curb area with a larger group tend to wait less than those who come alone or come with a smaller group. However, such “herd behavior” was not as prevalent in the oneway case. See Fig. 9.
The pedestrian-motorist interaction can be formulated as a game (Chen et al., 2016; Bjørnskau, 2017) that consists of decision-making by both pedestrians and motorists. We are considering a comprehensive game framework to characterize pedestrian-motorist interactions in terms of delay and risk. The probability of pedestrian delay for each strategy can be transformed into expected pedestrian delay costs (Pd). Expected pedestrian risk costs (Pr) can be obtained by estimating the probability of accepting the lag or gap at a specific time by the semiMarkov process model with specific covariates. Both delay costs and risk costs can be translated into “payoffs” to pedestrians when interacting with motorists. See the payoff matrix in Table 5. Without any countermeasure, the information exchanges between pedestrians and motorists are not perfect and both parties display “bounded rationality”. With the implementation of controls or technologies, we can simulate the “equilibrium” and regime change between risk costs and delay costs. This approach has promise in forecasting the safety or efficiency effects at semi-controlled crosswalks.
5.3.2. Other variables 5.3.2.1. Bus/truck Indicator. The model reveals that pedestrians tend to wait longer if the first approaching vehicle is a bus or truck. However, in the two-way case (Markov model), an opposite relationship was found for pedestrians who accepted the lag of the first approaching vehicle: they tend to wait less for Transition 1–3 (0.386* in Table 4) if the first vehicle is a bus or truck. 5.3.2.2. Distraction. The effect of Distraction (Cellphone or Talking) on pedestrian waiting time is found to be significant only in the two-way case in the Markov model. This is because, when crossing a two-way street (which is wider and has more complicated traffic movements), a pedestrian has to exercise more care to ensure a safe crossing. It was not unusual to see pedestrians let distractions prevent them from paying enough attention to find an acceptable lag/gap on a two-way street.
Fig. 8. Effect of Sex on Pedestrian Waiting Time (Two-way). 8
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Fig. 9. Effects of Group Size on Pedestrian Waiting Time (Two-way).
Fig. 5(a)) can enhance safety at semi-controlled crosswalks. Such a control strategy will facilitate the pedestrian-motorist negotiation in the most confusing situation.
Table 5 Payoff Matrix in Evolutionary Framework. Pedestrian
Cross Yield
Motorist Cross
Yield
[-(Pr + Pd), -(Mr + Md)] [-Pd, -Md]
[-Pd, -Md] [-Pd, -Md]
6.3. One-way vs. Two-way street operations The crosswalks studied in this paper are located on a street that experienced a conversion from one-way to two-way operation. Several important findings are listed:
6. Conclusions 6.1. Recurrent events analysis – semi-Markov approach
• Model results reveal that, after the conversion to two-way operation,
The multi-state semi-Markov model produces estimates of pedestrian delay that match observed field data. Based on the estimation results, the proposed multi-state semi-Markov model has several advantages:
•
• The Markov model allows us to estimate the separate effects of each •
factor (time-varying unobserved heterogeneity) on (i) rejecting the lag (Transition 1–2), (ii) accepting the lag (Transition 1–3), and (iii) accepting the repeated gap (Transition 2–3). The model is applicable to situations where pedestrian delay is governed by pedestrian gap acceptance or driver yielding behavior. Moreover, the proposed analytical method can be further generalized to behavioral studies of the interaction between any two road users at unsignalized intersections.
the probability of a pedestrian choosing Transition 1–3 decreases from 69.7% to just below 60%. This means that the probability of a pedestrian rejecting a lag and then accepting the subsequent gap (Transition 1-2–3) increases from 30.3% to about 40%. The model reveals that pedestrians who accept a lag (Transition 1–3) experience shorter delay in the one-way case than those in the two-way case; however, those who reject a lag (Transition 1-2–3) tend to wait longer at the one-way street than those at the two-way street.
The main implication is that pedestrians are more hesitant when interacting with motorists in the two-way case than in the one-way case. For two-way street operation, placing traffic calming measures ahead of the crossing (near 50 ft in Fig. 5(a)) is likely to increase driver yielding behavior and reduce confusion between pedestrians and motorists. Another implication is that pedestrians tend to behave more aggressively (wait less) when crossing as part of a group. This behavior is more significant in the two-way case than in the one-way case.
6.2. Implications of some critical variables
6.4. Study limitations
This paper investigates the impacts of various factors that influence pedestrian waiting behavior. A principal finding is that, if the vehicle is neither too far from the crosswalk nor too close, the “negotiations” between pedestrian and motorist will be more complicated and result in some amount of pedestrian delay. The distance and speed variables indicate that, although pedestrians have the priority over motorists, a large proportion of pedestrians consider the tradeoff between risk costs and delay costs before making a decision. See Figs. 5 and 6. Consequently, the mere existence of a sign granting right-of-way to pedestrians may not guarantee their safety. The main implication for practice is that reducing the speed of approaching vehicles a certain intermediate distance from a crossing location (such as 40 ft for one-way streets and 50 ft for two-way streets in
The two sets of video recordings were made at four different time periods (7:40 - 8:20; 12:40 - 13:25; 13:20 - 14:00; 16:20 - 17:00) on different weekdays. This is a location with periods of high pedestrian volume, especially for students attending classes. Although it is likely that some pedestrians were observed more than once in our videos, the number would be so small as to make the “shared variance” problem negligible. Most of the explanatory variables in Table 1 are easy to define and measure. Two exceptions may be Vehicle Speed and Distraction. Different individuals watching the same video may estimate speeds with different values – even if using the same method – but the differences 9
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should be minor and probably consistent. The judgment of the researchers was that Distraction as an explanatory variable was similar – whether the distraction was caused by cell phone use or by talking with others – but this assumption should be tested.
motorist behavior at semi-controlled crosswalks: the effect of a change from one-way to two-way street operation”. J. Trans. Res. Board 2019https://doi.org/10.1177/ 0361198119850142. Article first published online: June 14Issue published: November 1, 2019. Gitelman, V., Carmel, R., Pesahov, F., Hakkert, S., 2017. An examination of the influence of crosswalk marking removal on pedestrian safety as reflected in road user behaviours. Transp. Res. Part F Traffic Psychol. Behav. 46, 342–355. Hamed, M.M., 2011. Analysis of pedestrians’ behavior at pedestrian crossings. Saf. Sci. 38 (1), 63–82 2011. Highway Capacity Manual, 2010. Transportation Research Board. National Research, Washington, DC 2010. Havard, C., Willis, A., 2012. Effects of installing a marked crosswalk on road crossing behaviour and perceptions of the environment. Transp. Res. Part F Traffic Psychol. Behav. 15 (3), 249–260. Huang, L., Wu, J., You, F., Lv, Z., Song, H., 2016. Cyclist social force model at unsignalized intersections with heterogeneous traffic. IEEE Trans. Industr. Inform. 13 (2), 782–792. Lu, L., Ren, G., Wang, W., Chan, C.Y., Wang, J., 2016. A cellular automaton simulation model for pedestrian and vehicle interaction behaviors at unsignalized mid-block crosswalks. Accid. Anal. Prev. 95, 425–437. Malyshkina, N.V., Mannering, F.L., 2009. Markov switching multinomial logit model: an application to accident-injury severities. Accid. Anal. Prev. 41 (4), 829–838. Malyshkina, N.V., Mannering, F.L., 2010. Zero-state Markov switching count-data models: an empirical assessment. Accid. Anal. Prev. 42 (1), 122–130. Manual on Uniform Traffic Control Devices for Streets and Highways, 2009. Edition. FHWA, U.S. Department of Transportation, Washington D.C., pp. 30–44 December 2009. Meira-Machado, L., de Uña-Álvarez, J., Cadarso-Suarez, C., Andersen, P.K., 2009. Multistate models for the analysis of time-to-event data. Stat. Methods Med. Res. 18 (2), 195–222. Nemeth, B., Tillman, R., Melquist, J., Hudson, A., 2014. Uncontrolled Pedestrian Crossing Evaluation Incorporating Highway Capacity Manual Unsignalized Pedestrian Crossing Analysis Methodology. Rumar, K., 1990. The basic driver error: late detection. Ergonomics 33 (10-11), 1281–1290. Schroeder, B.J., Rouphail, N.M., 2010a. Event-based modeling of driver yielding behavior at unsignalized crosswalks. J. Transp. Eng. 137 (7), 455–465. Schroeder, B.J., Rouphail, N.M., 2010b. Mixed-priority pedestrian delay models at singlelane roundabouts. Transp. Res. Rec. 2182 (1), 129–138. Sucha, M., Dostal, D., Risser, R., 2017. Pedestrian-driver communication and decision strategies at marked crossings. Accid. Anal. Prev. 102, 41–50. Sun, D., Ukkusuri, S.V.S.K., Benekohal, R.F., Waller, S.T., 2003. January). Modeling of motorist-pedestrian interaction at uncontrolled mid-block crosswalks. In: In 82nd Annual Meeting of the Transportation Research Board. Washington, DC. Thomas, O., Sobanjo, J., 2016. Semi-Markov models for the deterioration of bridge elements. J. Infrastruct. Syst. 22 (3), 04016010. Weiss, G.H., Maradudin, A.A., 1962. Some problems in traffic delay. Oper. Res. 10 (1), 74–104. Xia, J.C., Zeephongsekul, P., Packer, D., 2011. Spatial and temporal modelling of tourist movements using Semi-Markov processes. Tour. Manag. 32 (4), 844–851. Xiong, C., Chen, X., He, X., Guo, W., Zhang, L., 2015. The analysis of dynamic travel mode choice: a heterogeneous hidden Markov approach. Transportation 42 (6), 985–1002. Xiong, Y., Tobias, J.L., Mannering, F.L., 2014. The analysis of vehicle crash injury-severity data: a Markov switching approach with road-segment heterogeneity. Transp. Res. Part B Methodol. 67, 109–128. Zegeer, C.V., Richard Stewart, J., Huang, H., Lagerwey, P., 2001. Safety effects of marked versus unmarked crosswalks at uncontrolled locations: analysis of pedestrian crashes in 30 cities. Transp. Res. Rec. 1773 (1), 56–68. Zhang, C., Chen, F., Wei, Y., 2019. Evaluation of pedestrian crossing behavior and safety at uncontrolled mid-block crosswalks with different numbers of lanes in China. Accid. Anal. Prev. 123, 263–273. Zhang, Y., 2019. Pedestrian-Vehicle Interactions at Semi-Controlled Crosswalks: Explanatory Metrics and Models. Masters thesis, Purdue University Graduate School. Zheng, Y., Elefteriadou, L., 2017. A model of pedestrian delay at unsignalized intersections in urban networks. Transp. Res. Part B Methodol. 100, 138–155.
Author statement Author statement for Manuscript Ref: AAP_2019_763,”Investigating Pedestrian Wait Time at Semi-Controlled Crossing Locations: Application of Multi-State Models for Recurrent Events Analysis”. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported as part of the Center for Connected and Automated Transportation (CCAT) Region V University Transportation Center funded by the U.S. Department of Transportation, Award #69A3551747105. Cost share was provided by the Joint Transportation Research Project SPR-4155. A copy of the technical report for SPR-4155 can be obtained from the electronic repository http://docs.lib.purdue.edu/jtrp/. Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.aap.2020.105437. References Almodfer, R., Xiong, S., Fang, Z., Kong, X., Zheng, S., 2016. Quantitative analysis of lanebased pedestrian-vehicle conflict at a non-signalized marked crosswalk. Transp. Res. Part F Traffic Psychol. Behav. 42, 468–478. Asaithambi, G., Kuttan, M.O., Chandra, S., 2016. Pedestrian road crossing behavior under mixed traffic conditions: a comparative study of an intersection before and after implementing control measures. Transp. Dev. Econ. 2 (2), 14. Adams, W.F., 1936. Road traffic considered as a random series. J. Ice 4 (1), 121–130. https://doi.org/10.1680/ijoti.1936.14802. November 1936 Published online June 05, 2015. Baik, H.S., Jeong, H.S., Abraham, D.M., 2006. Estimating transition probabilities in Markov chain-based deterioration models for management of wastewater systems. J. Water Resour. Plan. Manag. 132 (1), 15–24. Bjørnskau, T., 2017. The Zebra Crossing Game–Using game theory to explain a discrepancy between road user behaviour and traffic rules. Saf. Sci. 92, 298–301. Chae, K.S., 2008. Simulation of Pedestrian-vehicle Interactions at Roundabouts. https:// repository.lib.ncsu.edu/bitstream/handle/1840.16/5877/etd.pdf?sequence=1. Chen, P., Wu, C., Zhu, S., 2016. Interaction between vehicles and pedestrians at uncontrolled mid-block crosswalks. Saf. Sci. 82, 68–76. De Dominicis, R., Manca, R., 1984. An algorithmic approach to non-homogeneous semiMarkov processes. Commun. Statistics-Simulation and Computation 13 (6), 823–838. Fricker, J.D., Zhang, Y., 2019. Transportation research record“Modeling pedestrian and
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Investigating pedestrian waiting time at semi-controlled crossing locations: Application of multi-state models for recurrent events analysis
T
Yunchang Zhang, Yu Qiao, Jon D. Fricker* Lyles School of Civil Engineering, Purdue University, 550 Stadium Mall Dr., W. Lafayette IN, 47907, United States
A R T I C LE I N FO
A B S T R A C T
Keywords: Crosswalks Pedestrian waiting behavior Semi-Markov process
“Semi-controlled” crosswalks are unsignalized, but clearly marked with “yield to pedestrian within crosswalk” signs. Ideally, pedestrians can cross the street immediately after they arrive at the curb. However, real world observations show that pedestrians and vehicles are often involved in non-verbal “negotiations” to decide who should proceed first. This kind of “negotiation” often causes delays for both parties and may lead to unsafe situations. The study in this paper was based on video recordings of the waiting behaviors of 2059 pedestrians interacting with 1003 motorists at selected semi-controlled crosswalks. One such location experienced a conversion from one-way operation to two-way operation, which provided a rare opportunity for a before-and-after study at that location. Multi-state Markov models were introduced as a novel approach to correlate the dynamic process between recurrent events. Time-varying covariates related to pedestrian characteristics, traffic condition, and vehicle dynamics (distance and speed) turned out to be significant. The analytical method developed in this study provides a tool to dynamically model pedestrian waiting decisions with uncertainties. Model results reveal that, after the conversion from one-way to two-way operation, the probability of a pedestrian accepting a lag decreases from 69.7% to just below 60% on the same street. In addition, pedestrians are more hesitant to cross a two-way street than a one-way street. Countermeasures that increase motorist yielding rate or reduce pedestrian confusion will enhance safety such crossing locations.
1. Introduction Marked crosswalks with “State Law Yield to Pedestrian Within Crosswalk’’ signs (R1-6 sign in Fig. 1a) can be called “semi-controlled” crosswalks. Despite its presence at many locations, the meaning of “within crosswalk” on the R1-6 sign is not interpreted uniformly. It is generally understood that pedestrians have priority at the R1-6 sign, but how should motorists act when pedestrians are standing on the curb, in the gutter, etc.? The Indiana Code, the MUTCD, and the Indiana Driver’s Manual do not provide clarification as to the sign’s practical meaning. Despite the implied priority to pedestrians, it was apparent from observations that pedestrians did not always exercise that prerogative. Drivers were seldom dangerously aggressive, but were often hesitant about whether to proceed in the presence of pedestrians. Such confusion is apparent when watching pedestrians and motorists interact at “semi-controlled” crosswalks, and this confusion can lead to unsafe situations. Delay ensues as non-verbal “negotiations” take place between pedestrian and driver. Safety and delay are two primary performance measures for semicontrolled crosswalks. Safety countermeasures have been extensively
⁎
explored for the analysis of crosswalks, but there is much less literature that considers pedestrian wait time. Pedestrian wait time is an important metric to estimate efficiency or level of service (Highway Capacity Manual, 2010). It is also an indicator that can be used to evaluate potential risks between pedestrians and motorists at a crossing location (Havard and Willis, 2012). A higher pedestrian wait time is often associated with a lower driver yield rate (Gitelman et al., 2017). If average pedestrian delay is high, pedestrians tend to be impatient and take more risk (Nemeth et al., 2014; Almodfer et al., 2016). On the other hand, pedestrian wait time is an indication that a “negotiation” is taking place as pedestrians and motorists interact (Sucha et al., 2017). Inefficient negotiation between pedestrian and motorist can occur when at least one party has incorrect expectations concerning the behavior of the others (Rumar, 1990). Such miscommunication can lead to dangerous situations. The study described in this paper pays specific attention to the actions of pedestrians and motorists during interactions. Recent studies have revealed that changes to the road crossing environment can have a significant impact on observable road-crossing behavior. Existing literature has conducted before-and-after studies, such as marked versus unmarked crosswalks (Zegeer et al., 2001;
Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y. Qiao), [email protected] (J.D. Fricker).
https://doi.org/10.1016/j.aap.2020.105437 Received 10 June 2019; Received in revised form 8 January 2020; Accepted 8 January 2020 0001-4575/ © 2020 Published by Elsevier Ltd.
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Fig. 1. Semi-Controlled Crosswalk. (a) Sign at Semi-Controlled Crosswalk. (Manual on Uniform Traffic Control Devices for Streets and Highways, 2009) (b) One-way North University Street at Second Street, 2017. (Jon D. Fricker) (c) Two-way North University Street at Second Street, 2018. (Google Maps).
by 37%. However, similar to survival models, time-based gap acceptance models have difficulty in handling discrete choice data over time. Some researchers have realized the limitation of gap acceptance theory and developed simulation frameworks to facilitate the computation of pedestrian wait time. Chae (2008) incorporated pedestrian gap acceptance decision rules into simulation software and studied pedestrian waiting time at unsignalized roundabouts. Lu et al. (2016) proposed a cellular automaton (CA) framework to estimate pedestrian waiting time by incorporating pedestrian crossing behavior and vehicle yielding behavior. The modified CA model was a comprehensive simulation framework that included the prediction of pedestrian crossing behavior and pedestrian delay at the curb. Huang et al. (2016) proposed a social force model to study the interaction between cyclist and driver at unsignalized intersections. Collision avoidance and cyclist gap acceptance models were incorporated into the proposed social force model to facilitate cyclist delay estimation based on the shortest path. Other researchers developed stochastic process models to dynamically model pedestrian delay at semi-controlled crosswalks. Adams (1936) proposed an analytical method for pedestrian delay estimation at unsignalized crosswalks. He introduced vehicle arrival times as a random sequence and the gap (vehicle headway or interarrival time) between two successive vehicles was usually considered as a random variable from a specific distribution. As pedestrians find a gap sufficiently large to cross, the pedestrian delay was estimated by the Poisson process. Weiss and Maradudin (1962) assumed that successive gaps in the main traffic were uncorrelated random variables with known probability density functions, while the pedestrian crossing probability distribution was assumed to be known. Renewal theory was first used to model pedestrian delay as a distribution function by means of the convolution integral and the Laplace transform. They extended their models by means of the semi-Markov process to handle the issue of correlated successive gaps in the main traffic. HCM (2010) implemented Adams’ model by assuming a constant motorist yielding behavior when pedestrian-vehicle interaction occurs. The pedestrian delay was divided into two parts – as gap delay that was calculated by Adams’ method and as pedestrian reaction delay incurred by pedestrian-vehicle interactions. Zheng and Elefteriadou (2017) further extended the Weiss and Maradudin model (1962) and the HCM 2010 model by renewal theory. The effect of driver yielding behavior was incorporated as a function in the pedestrian delay model. However, simulation-based frameworks and stochastic process models do not address the time-varying heterogeneity caused by discrete-outcome data gathered over time. When interacting with vehicles, pedestrian waiting behavior is a dynamic process consisting of single or multiple gap acceptance
Havard and Willis, 2012; Gitelman et al., 2017), signalized versus unsignalized crosswalks (Asaithambi et al., 2016), and single-lane versus multiple-lane crosswalks (Zhang et al., 2019). In this study, the change from one-way traffic to two-way traffic provided a rare opportunity to study the negotiations of a similar population having undergone such a significant change. With this background, this study addresses the following research questions: 1 Which time-varying factors will influence pedestrian waiting behaviors at such locations? 2 How to use time-to-event data to dynamically capture pedestrian waiting behaviors at semi-controlled crossing locations? 3 How will pedestrian waiting behavior change if a one-way street is changed to two-way operation? 4 What can we learn that will guide decisions about changing control strategies at semi-controlled crosswalks? 2. Literature Review A majority of pedestrian wait time studies have been conducted using survival theory and time-based gap acceptance models. Hamed (2011) proposed a survival model to explore pedestrian wait time at the curbside of uncontrolled crosswalks. Although the survival model has limitations in dealing with recurrent events analysis and time-varying covariates, Hamed (2011) inserted a Poisson model that estimated the number of pedestrian attempts into the survival framework. The estimation results revealed that the pedestrian waiting time when crossing from one side of the street to the central median was significantly larger than the pedestrian waiting time when crossing from the median to the other side of street. Sun et al. (2003) investigated pedestrian gap acceptance behavior at several uncontrolled crosswalks. Empirical results confirmed that pedestrians who were unwilling to accept shorter gaps had to wait for a larger and safer gap, which increased their total wait time. Schroeder and Rouphail (2010a) defined an event-based modeling approach to describe a pedestrian–motorist interaction as an event. Either the pedestrian gap acceptance probability or vehicle yielding probability could be predicted by specific explanatory variables concerning the event. Consequently, pedestrian average delay was predicted as a function of the probability of encountering a crossing opportunity in the form of a vehicle yielding or a crossable gap, using a linear regression model (Schroeder and Rouphail, 2010b) at semi-controlled crosswalks. The results showed that an increase of driver yielding rate from 10% to 30% would significantly decrease the pedestrian delay for sighted pedestrians by 65% and for blind pedestrians 2
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Table 1 Explanatory Variables (for All Events). Variable Explanatory Variable Pedestrian Waiting time Independent Variables Pedestrian characteristics Sex Young Indicator Distraction Indicator Traffic Condition Near Side Indicator Bus/Truck Indicator Veh-to-Ped Distance Approach Speed Adjacent Vehicle Indicator Vehicle Close Follower Indicator
Description
Duration (in seconds) between the time a pedestrian reaches the curb area and the time the pedestrian starts crossing (One-way: mean = 2.67 s, sd = 2.61 s; Two-way: mean = 3.02 s, sd = 3.08 s)
1 if the pedestrian is Male (51.1%), 0 if Female (48.9%). 1 if the pedestrian appears to be younger than 30 years old (77%); 0 otherwise (23%). 1 if the pedestrian is using a cellphone (9%) or talking to others (8.4%) when waiting at the curb, 0 otherwise (82.6%). 1 if the critical vehicle is in the near lane (53%), 0 if the critical vehicle is in the far lane (47%). 1 if the critical vehicle is a bus or large truck (16.7%), 0 otherwise (83.3%). Distance (ft) between the pedestrian and the interacted vehicle when an interaction begins (mean = 61.8 ft, sd = 51.7 ft). If an interaction occurs when a pedestrian arrives at the curb, the video is paused to calculate the distance to vehicle using Google Maps. Speed (ft/s) of the interacted vehicle when an interaction begins (mean = 12.4 ft/s, sd = 11.2 ft/s). Using distances along the street found with Google Maps, the distance covered by a vehicle every 34 milliseconds (one video frame) is converted into a speed. 1 if one or more vehicles are present in the adjacent lane within the area of influence when the motorist begins to interact with a pedestrian (32.3%), 0 otherwise (67.7%). 1 if there is at least one vehicle closely following the subject vehicle (31.9%), 0 otherwise (68.1%). The object vehicle has a close follower (CloseFollow = 1) if the headway is less than 4 seconds, as defined in Schroeder and Rouphail (2011).
Other Pedestrians Group Size Group ID Nr Ped Waiting
Number of people in the pedestrian group (45.7% of pedestrians came alone, 54.3% came in a group; mean = 2.31, sd = 2.03) Identifies the group to which the subject pedestrian belongs (748 in one-way; 636 in two-way). Number of pedestrians waiting at the curb as a pedestrian arrives at the curb (no pedestrians waiting in 52.8% of cases, at least one pedestrian was waiting at the curb in 47.2% of cases; mean = 0.84, sd = 1.21) Nr Ped Crossing Number of other pedestrians crossing the street as a pedestrian arrives at the curb (48.5% of cases, no pedestrians crossing; 51.5% of cases, at least one other pedestrian crossing in crosswalk; mean = 1.27, sd = 1.99) Note: mean = average value; sd = standard deviation
bicycle lanes and a speed limit of 25 mph. The two sets of video recordings were made at four different time periods (7:40-8:20; 12:4013:25; 13:20-14:00; 16:20-17:00), when low-to-moderate traffic volumes and pedestrian flows could be observed. On one-way N. University Street, we observed 1126 pedestrians interacting with 498 motorists over 155 min. With two-way street operation, there were 927 pedestrians interacting with 505 motorists in 150 min. The pedestrian and vehicle rates did not change much between the one-way and twoway cases. The pedestrian rates across N. University Street are probably higher than at the typical “semi-controlled” crosswalk location, but the higher rate gives us more interactions to analyze, without pedestrians dominating the situation.
decisions. In pedestrian delay studies, pedestrians are observed over time and covariates are quantified on different occasions (MeiraMachado et al., 2009). We hope to use such time-varying covariates from longitudinal data to empirically test the proposed method and understand its implications and sensitivity. An extension of the stochastic process approach is likely to solve the problem by using a multistate semi-Markov model. The multi-state semi-Markov model is a continuous time stochastic process that allows individuals to move among a finite number of states (Zhang, 2019). In transportation studies, multi-state Markov models have not been widely used. Researchers (Malyshkina and Mannering, 2009, 2010; Xiong et al., 2014) have developed Markov switching models to account for the severity of highway injuries. The proposed models provide critical insights into the temporal impacts of weather conditions on the likelihood of accidents and their resulting injury severities. Multi-state Markovian models have also been used in bridge deterioration modeling (Baik et al., 2006; Thomas and Sobanjo, 2016), mode choice (Xiong et al., 2015) and tourism management (Xia et al., 2011).
3.2. Definitions 3.2.1. Interaction Each pedestrian was documented from the time he/she arrived at the curb area until he/she crossed the street. A pedestrian-motorist interaction occurred when a pedestrian and a motorist were in the area of influence of each other (Fricker and Zhang, 2019):
3. Methodology
• A pedestrian enters the curb area and intends to cross. • The subject driver is aware of the pedestrian’s intention to cross and
3.1. Data collection The data for this study were collected at a “semi-controlled” crossing location (North University Street at the T intersection with Second Street) on the Purdue University campus. The two crosswalks are clearly marked and "State Law Yield to Pedestrian Within Crosswalk” signs are present (Fig. 1(a)). The studied intersection is primarily used by students walking to/from class and by faculty and staff walking from/to nearby parking garages. Two sets of video recordings were made to collect data on both pedestrians’ and drivers’ behaviors (2059 pedestrians and 1003 vehicles). The first set of videos was made in Spring 2017, when North University Street was a one-way northbound street. See Fig. 1(b). Under one-way operation, the street had two 10-ft wide lanes (plus a 4-foot bicycle lane) with a speed limit of 25 mph. By the time the second set of videos was made in Spring 2018, the street had been converted to two-way operation. See Fig. 1(c). After the conversion, the street is 34 ft wide, with two 5-foot
responds to the pedestrian.
There are two types of driver outcomes for each interaction:
• The driver yields to the pedestrian. • The subject driver fails to yield to the pedestrian. The important variables that were extracted from the videos when an interaction occurred are summarized in Table 1. 3.2.2. Lag and gap The lag begins when a pedestrian enters the curb area and interacts with a motorist who (a) is aware of the pedestrian’s intention to cross and then (b) responds to the pedestrian. If the pedestrian rejects the lag, the lag ends when the motorist passes the crosswalk. If the pedestrian 3
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Fig. 2. Multi-State Framework for Pedestrian Waiting Behavior.
chain in terms of conditional probability, is:
accepts the lag, the lag ends when the pedestrian enters the crosswalk. A gap for pedestrians begins after a rejected lag or gap. If the pedestrian rejects a gap, the gap ends when the vehicle passes the crosswalk. If the pedestrian accepts the gap, the gap ends when the pedestrian enters the crosswalk.
Fhj (d ) = P (Sn + 1 ≤ d|Jn + 1 = j, Jn = h) =
αhj = lim
• State 1: A pedestrian arrives at the curb area and interacts with the first approaching vehicle. • State 2: The pedestrian rejects the lag and keeps waiting for subsequent (recurrent) gaps. • State 3: The pedestrian accepts the lag (or gap) and enters the
Δd → 0
=
3.2.4. Transition The potential transitions for this model are defined as:
• Transition 1-3: pedestrian accepts the lag (indicating that first interacted vehicle yields to subject pedestrian). • Transition 1-2: pedestrian rejects the lag (indicating the first interacted vehicle fails to yield). • Transition 2-2: pedestrian rejects all subsequent gaps. (This transition is not considered in this paper.) • Transition 2-3: pedestrian accepts a subsequent gap (indicating a
fhj (d ) 1 − Fhj (d )
(4)
k x k − 1 −( x ) ⎛ ⎞ e λ λ ⎝λ⎠
k
(5)
According to Eq.s (4) and (5), the hazard ratio for the Weibull distribution is:
4. Modeling Framework
αhj =
4.1. Multi-state semi-markov model
fhj (d ) 1 − Fhj (d )
=
k x k−1 ⎛ ⎞ λ ⎝λ⎠
(6)
Fig. 3 shows the probability density function for every transition. Transition 1-2-3 (reject the lag, then accept the next gap) can be exx pressed as the convolution product f1 − 2 − 3 = ∫0 f1 − 2 (u) f2 − 3 (x − u) du , where x = the total waiting time during the transition 1-2-3.
Consider a Markov renewal process (Jn, Tn), where T0 < T1 < … < Tn < ∞ are the successive times of entry to States J0, J1,…, Jn. If Sn = Tn − Tn − 1 is the sojourn time (gap time or lag time), the Markov renewal kernel Qhj (d ) is a cumulative distribution function of time:
4.1.2. Parameterization Let Zhj be a vector of explanatory variables related to the transition from State h to j and βh,j be the vector of estimated regression parameters. By the Cox proportional hazard model,
Qhj (d ) = P (Jn + 1 = j, Sn + 1 ≤ d J0 , J1, …, Jn = h, S1, S2, …, Sn ) (1)
J0, J1,…, Jn is an embedded homogeneous Markov chain taking values in a finite state space with transition probability:
Table 2 Weibull Distribution Duration Parameters. Duration Parameters in Weibull Distribution
(2)
Transition
We define the distribution function of the sojourn time in State h by: s
∑ j=1 Qhj (t ), ∀ s ∈
fhj (d ) Pr(d < Sn + 1 ≤ d + Δd|Jn + 1 = j, Jn = h, Sn + 1 > d ) = Δd Shj (d )
f (x|λ, k ) =
subsequent vehicle yields to pedestrian).
Hh (t ) =
(3)
4.1.1. Distribution of durations For the semi-Markov process, we need to first assume that sojourn time (gap time or lag time) belongs to a specific parametric distribution. The SemiMarkov package in R software (Listwon-Krol and Saint-Pierre, 2015) offers three distributions – Exponential, Weibull and Exponential Weibull. Based on maximum likelihood estimation, the Weibull distribution was chosen to model the sojourn time from State h to State j. Table 2 shows the estimated coefficients and standard errors for Weibull distributions. Table 3 shows the results of the Wald test.
crosswalk.
phj = P (Jn + 1 = j|Jn = h) = limt →∞Qhj (t ), n ∈ N
phj
Fhj (d ) is a cumulative probability distribution of a sojourn time in State h if the next state will be j. Based on Eq. (3), we can write the probability density function as fhj (d ) . The hazard function αhj of Fhj (d ) will be:
3.2.3. States We identify 3 observable states of pedestrian’s waiting behavior (See Fig. 2).
= P (Jn + 1 = j, Sn + 1 ≤ d Jn = h)
Qhj (d )
(3)
1→2 1→3 2→3
The probability distribution function of sojourn time (gap time or lag time), through the transition probabilities of the embedded Markov 4
One-Way λ Estim. 2.864 2.014 1.881
SE 0.12 0.04 0.11
k Estim. 1.364 1.843 1.329
SE 0.06 0.04 0.07
Two-Way λ Estim. SE 3.021 0.09 2.229 0.06 1.67 0.11
k Estim. 1.785 1.794 1.075
SE 0.07 0.05 0.05
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Table 3 Wald Test of Weibull Distribution Duration Parameters. One-Way
Two-Way
Transition 1→2 1→3 2→3
αhj (d|Zh, j ) =
Wald Test 43.53 357.06 19.84
P-value < 0.0001 < 0.0001 < 0.0001
Transition 1→2 1→3 2→3
Wald Test 128.59 259.26 1.89
αhj (d )exp(βhT, j Zh, j )
P-value < 0.0001 < 0.0001 0.1692
In Eq. (7), we used the transition-specific variables Zh,j defined in Table 1. Using the Semi-Markov package in R, the significant factors were retained. 4.1.3. Hazard of semi-markov process The hazard rate of a semi-Markov process is defined as the probability of transition towards State j between the time d and d+Δd, given that the process is in State h for a duration d.
5. Results 5.1. Estimation results Table 4 presents the estimation results for the multi-state semiMarkov model (called simply the Markov model in the following sections). In the Markov model, the proportional hazard (Eq. 7) is used. Therefore, a negative coefficient indicates that the increase of an explanatory variable increases the hazard rate and consequently decreases pedestrian waiting time of a specific transition, and vice versa.
γhj (d ) = lim
5.2. Findings
(8)
5.2.1. Comparison of one-way to two-way street operation It was found that (1) in one-way street operation, pedestrians who accepted the lag (Transition 1–3) tend to experience shorter delay in the curb area than for two-way operation; (2) in the one-way case, pedestrians who reject the lag (Transition 1-2–3) tend to have longer waiting time than in the two-way case before entering the crosswalk. See Fig. 4.
Note that Eq. (6) and Eq. (8) demonstrate two different hazards. The hazard in Eq. (6) is defined as the hazard, given a transition from State h to State j. The hazard defined by Eq. (8) is the hazard of the semiMarkov process, which represents the immediate probability of going to State j given State h in a small-time interval [d, d+Δd] (De Dominicis and Manca, 1984). Therefore, for the state space I = {1, 2, 3}, we can use Eq. (9) to calculate the “staying” probability for the case h=j:
p11 (d ) = e− ∫0
d
p22 (d ) = e− ∫0 p33 (d ) = 1
d
(10)
P1–3 and P1-2-3 in Eq. (10) are the two most common behavior of pedestrians waiting in the curb area. Out of 1132 transitions observed in the one-way case, 966 were for Transitions 1–3 and 1-2-3. The remaining behavior type is 1-2-…-2-3, which is not explicitly considered here, but has been tested to be statistically similar to 1-2-3. Therefore, we can use the transition probability of the semi-Markov process P1–3 + P1-2-3 to explain the variables in Table 1.
(7)
Pr(Jn + 1 = j, d < Sn + 1 ≤ d + Δd|Jn = h, Sn + 1 > d ) Δd → 0 Δd ( ) p f d q hj hj hj ⎧ ⎪ 1 − H (d ) = 1 − H (d ) h h ⎪ = phj (1 − Fhj (d )) αhj ⎨ = ifphj > 0 andHh (d ) < 1 ⎪ 1 − Hh (d ) ⎪ ⎩ 0otherwise
d
∫0 p11 (τ ) γ1−2 (τ ) p22 (d − τ ) dτ d p1 − 3 (d ) = ∫ p11 (τ ) γ1 − 3 (τ ) p33 (d − τ ) dτ 0 d p2 − 3 (d ) = ∫ p22 (τ ) γ2 − 3 (τ ) p33 (d − τ ) dτ 0 d p1 − 2 − 3 (d ) = ∫ p11 (τ ) γ1 − 2 (τ ) p2 − 3 (d − τ ) dτ 0 p1 − 2 (d ) =
5.2.2. Vehicle dynamics 5.2.2.1. Pedestrian-to-vehicle distance. According to our data, the squared terms of the vehicle-to-pedestrian distance are highly significant in both the one-way and two-way cases, indicating that there exists a non-monotonic relationship between pedestrian delay and the pedestrian-to-vehicle distance: the delay increases at first, then decreases as the distance increases. Such a relationship is presented in Fig. 5(a) for the one-way and two-way cases. The longest pedestrian
γ12 (τ ) + γ13 (τ ) dτ γ23 (τ ) dτ
(9)
Consequently, we can calculate the probabilities of each transition in a Markov chain as:
Fig. 3. Density Functions Between Different Transitions. 5
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Table 4 Multi-State Markov Model Results. Coefficients
Sex Young Distraction Vehicle Speed Distance Bus Indicator Near or Far Side Close Follower Indicator Adjacent Vehicle Group Size Nr. Ped. Crossing Nr. Ped. Waiting Log-Likelihood
One-Way
Two-Way
1→3
1→2
2→3
1→3
1→2
2→3
– 0.495*** – −0.031*** 0.251*** – – – −0.388*** – 0.086*** −0.143*** −1208.360
– 0.346** – 0.059*** −0.769*** −0.727*** – – – 0.159** – –
0.473** – – – – – – – 0.462* –
– 0.480*** – −0.012* – 0.386* – – – 0.088** 0.072*** −0.077* −1137.986
– – – 0.034*** −0.467*** – – – – 0.117. −0.096. –
0.309* – −0.348^ – – – – – – 0.216* – –
–
Notes; Significance codes: ‘***’: 0.001; ‘**’: 0.01; ‘*’: 0.05; ‘^’: 0.1.
of the first vehicle he/she interacts with is higher. However, this is true only for pedestrians who accepted the lag (Transition 1–3 in Fig. 6). For pedestrians who reject the lag (Transition 1–2) and keep waiting for the subsequent gaps, an opposite relationship is observed: the total pedestrian delay is shorter if the first approaching vehicle comes at a higher speed. This indicates that when the vehicle is too close or too fast upon the pedestrian’s arrival, “Let the car go first” saves time for both pedestrians and drivers.
delay was found at a distance of 38.41 ft in the one-way case and a distance of 50.13 ft in the two-way case using regression analysis. This means that a longer distance generally results in shorter pedestrian waiting time (because pedestrians are less likely to wait). However, if the vehicle is already too close to yield (less than 38.41 ft or 50.13 ft), this short pedestrian-to-vehicle distance also reduces pedestrian delay (because the pedestrian is likely to let the vehicle go first). Such a nonmonotonic relationship makes sense, but has not been quantified in previous studies. The semi-Markov model results (Fig. 5(b)) indicate that, for pedestrians who accepted the lag of the first vehicle (Transition 1–3), larger pedestrian-to-vehicle distance leads to a shorter pedestrian delay. However, for pedestrians who rejected the first gap (Transition 1–2), their waiting time increases as the distance increases. First, if a vehicle is too close to yield to the subject pedestrian, the normal pedestrian choice is to “let the vehicle go first”, and it will cause little delay to the pedestrian. See Transition 1-2-3. If a vehicle is too far away, the normal pedestrian choice is to cross without any hesitation, because the pedestrian will feel safe. (See Transition 1–3 in Fig. 5(b)).
5.3.1. Pedestrian characteristics 5.3.1.1. Young Indicator. Fig. 7 reveals that the waiting behavior of younger pedestrians was only observed as significant when the pedestrian accepted/rejected a lag (Transition 1–3 or 1–2). When making decisions on the subsequent gaps (2–3), pedestrians from different age groups didn’t behave significantly differently. 5.3.1.2. Sex. In Fig. 8, male pedestrians were found to generally wait less than do female pedestrians, but only after they reject the lag and wait for subsequent gaps (Transition 2–3). This indicates that male and female pedestrians don’t behave differently when they interact with the first approaching vehicle. However, when they are waiting and looking for a sufficiently large gap in the subsequent vehicles, male pedestrians are found to behave more “aggressively”.
5.3. Vehicle speed It is not surprising that a pedestrian tends to wait longer if the speed
Fig. 4. Empirical Transition Probability for Different Street Operations. 6
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Fig. 5. Effects of Pedestrian-to-Vehicle Distance on Pedestrian Waiting Time.
Fig. 6. Effects of Speed on Pedestrian Waiting Time (One-way).
7
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Fig. 7. Effects of Pedestrian’s Age on Pedestrian Waiting Time (One-way).
5.4. Future research directions
5.3.1.3. Group size. The Markov model indicates that, when crossing a two-way street, pedestrians entering the curb area with a larger group tend to wait less than those who come alone or come with a smaller group. However, such “herd behavior” was not as prevalent in the oneway case. See Fig. 9.
The pedestrian-motorist interaction can be formulated as a game (Chen et al., 2016; Bjørnskau, 2017) that consists of decision-making by both pedestrians and motorists. We are considering a comprehensive game framework to characterize pedestrian-motorist interactions in terms of delay and risk. The probability of pedestrian delay for each strategy can be transformed into expected pedestrian delay costs (Pd). Expected pedestrian risk costs (Pr) can be obtained by estimating the probability of accepting the lag or gap at a specific time by the semiMarkov process model with specific covariates. Both delay costs and risk costs can be translated into “payoffs” to pedestrians when interacting with motorists. See the payoff matrix in Table 5. Without any countermeasure, the information exchanges between pedestrians and motorists are not perfect and both parties display “bounded rationality”. With the implementation of controls or technologies, we can simulate the “equilibrium” and regime change between risk costs and delay costs. This approach has promise in forecasting the safety or efficiency effects at semi-controlled crosswalks.
5.3.2. Other variables 5.3.2.1. Bus/truck Indicator. The model reveals that pedestrians tend to wait longer if the first approaching vehicle is a bus or truck. However, in the two-way case (Markov model), an opposite relationship was found for pedestrians who accepted the lag of the first approaching vehicle: they tend to wait less for Transition 1–3 (0.386* in Table 4) if the first vehicle is a bus or truck. 5.3.2.2. Distraction. The effect of Distraction (Cellphone or Talking) on pedestrian waiting time is found to be significant only in the two-way case in the Markov model. This is because, when crossing a two-way street (which is wider and has more complicated traffic movements), a pedestrian has to exercise more care to ensure a safe crossing. It was not unusual to see pedestrians let distractions prevent them from paying enough attention to find an acceptable lag/gap on a two-way street.
Fig. 8. Effect of Sex on Pedestrian Waiting Time (Two-way). 8
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Fig. 9. Effects of Group Size on Pedestrian Waiting Time (Two-way).
Fig. 5(a)) can enhance safety at semi-controlled crosswalks. Such a control strategy will facilitate the pedestrian-motorist negotiation in the most confusing situation.
Table 5 Payoff Matrix in Evolutionary Framework. Pedestrian
Cross Yield
Motorist Cross
Yield
[-(Pr + Pd), -(Mr + Md)] [-Pd, -Md]
[-Pd, -Md] [-Pd, -Md]
6.3. One-way vs. Two-way street operations The crosswalks studied in this paper are located on a street that experienced a conversion from one-way to two-way operation. Several important findings are listed:
6. Conclusions 6.1. Recurrent events analysis – semi-Markov approach
• Model results reveal that, after the conversion to two-way operation,
The multi-state semi-Markov model produces estimates of pedestrian delay that match observed field data. Based on the estimation results, the proposed multi-state semi-Markov model has several advantages:
•
• The Markov model allows us to estimate the separate effects of each •
factor (time-varying unobserved heterogeneity) on (i) rejecting the lag (Transition 1–2), (ii) accepting the lag (Transition 1–3), and (iii) accepting the repeated gap (Transition 2–3). The model is applicable to situations where pedestrian delay is governed by pedestrian gap acceptance or driver yielding behavior. Moreover, the proposed analytical method can be further generalized to behavioral studies of the interaction between any two road users at unsignalized intersections.
the probability of a pedestrian choosing Transition 1–3 decreases from 69.7% to just below 60%. This means that the probability of a pedestrian rejecting a lag and then accepting the subsequent gap (Transition 1-2–3) increases from 30.3% to about 40%. The model reveals that pedestrians who accept a lag (Transition 1–3) experience shorter delay in the one-way case than those in the two-way case; however, those who reject a lag (Transition 1-2–3) tend to wait longer at the one-way street than those at the two-way street.
The main implication is that pedestrians are more hesitant when interacting with motorists in the two-way case than in the one-way case. For two-way street operation, placing traffic calming measures ahead of the crossing (near 50 ft in Fig. 5(a)) is likely to increase driver yielding behavior and reduce confusion between pedestrians and motorists. Another implication is that pedestrians tend to behave more aggressively (wait less) when crossing as part of a group. This behavior is more significant in the two-way case than in the one-way case.
6.2. Implications of some critical variables
6.4. Study limitations
This paper investigates the impacts of various factors that influence pedestrian waiting behavior. A principal finding is that, if the vehicle is neither too far from the crosswalk nor too close, the “negotiations” between pedestrian and motorist will be more complicated and result in some amount of pedestrian delay. The distance and speed variables indicate that, although pedestrians have the priority over motorists, a large proportion of pedestrians consider the tradeoff between risk costs and delay costs before making a decision. See Figs. 5 and 6. Consequently, the mere existence of a sign granting right-of-way to pedestrians may not guarantee their safety. The main implication for practice is that reducing the speed of approaching vehicles a certain intermediate distance from a crossing location (such as 40 ft for one-way streets and 50 ft for two-way streets in
The two sets of video recordings were made at four different time periods (7:40 - 8:20; 12:40 - 13:25; 13:20 - 14:00; 16:20 - 17:00) on different weekdays. This is a location with periods of high pedestrian volume, especially for students attending classes. Although it is likely that some pedestrians were observed more than once in our videos, the number would be so small as to make the “shared variance” problem negligible. Most of the explanatory variables in Table 1 are easy to define and measure. Two exceptions may be Vehicle Speed and Distraction. Different individuals watching the same video may estimate speeds with different values – even if using the same method – but the differences 9
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should be minor and probably consistent. The judgment of the researchers was that Distraction as an explanatory variable was similar – whether the distraction was caused by cell phone use or by talking with others – but this assumption should be tested.
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Author statement Author statement for Manuscript Ref: AAP_2019_763,”Investigating Pedestrian Wait Time at Semi-Controlled Crossing Locations: Application of Multi-State Models for Recurrent Events Analysis”. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported as part of the Center for Connected and Automated Transportation (CCAT) Region V University Transportation Center funded by the U.S. Department of Transportation, Award #69A3551747105. Cost share was provided by the Joint Transportation Research Project SPR-4155. A copy of the technical report for SPR-4155 can be obtained from the electronic repository http://docs.lib.purdue.edu/jtrp/. Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.aap.2020.105437. References Almodfer, R., Xiong, S., Fang, Z., Kong, X., Zheng, S., 2016. Quantitative analysis of lanebased pedestrian-vehicle conflict at a non-signalized marked crosswalk. Transp. Res. Part F Traffic Psychol. Behav. 42, 468–478. Asaithambi, G., Kuttan, M.O., Chandra, S., 2016. Pedestrian road crossing behavior under mixed traffic conditions: a comparative study of an intersection before and after implementing control measures. Transp. Dev. Econ. 2 (2), 14. Adams, W.F., 1936. Road traffic considered as a random series. J. Ice 4 (1), 121–130. https://doi.org/10.1680/ijoti.1936.14802. November 1936 Published online June 05, 2015. Baik, H.S., Jeong, H.S., Abraham, D.M., 2006. Estimating transition probabilities in Markov chain-based deterioration models for management of wastewater systems. J. Water Resour. Plan. Manag. 132 (1), 15–24. Bjørnskau, T., 2017. The Zebra Crossing Game–Using game theory to explain a discrepancy between road user behaviour and traffic rules. Saf. Sci. 92, 298–301. Chae, K.S., 2008. Simulation of Pedestrian-vehicle Interactions at Roundabouts. https:// repository.lib.ncsu.edu/bitstream/handle/1840.16/5877/etd.pdf?sequence=1. Chen, P., Wu, C., Zhu, S., 2016. Interaction between vehicles and pedestrians at uncontrolled mid-block crosswalks. Saf. Sci. 82, 68–76. De Dominicis, R., Manca, R., 1984. An algorithmic approach to non-homogeneous semiMarkov processes. Commun. Statistics-Simulation and Computation 13 (6), 823–838. Fricker, J.D., Zhang, Y., 2019. Transportation research record“Modeling pedestrian and
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