category: literaturenote citekey: lucellularautomatonsimulationmodel2016 title: A cellular automaton simulation model for pedestrian and vehicle interaction behaviors at unsignalized mid-block crosswalks authors: "Lu, Lili; Ren, Gang; Wang, Wei; Chan, Ching-Yao; Wang, Jian" year: 2016 date: 2016-00-00 2016 doi: 10.1016/j.aap.2016.04.014 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2016.04.014" zotero_key: K5DSUACW zotero_storage: GFZ6GQ5M collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Lu, Lili"

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Accident Analysis and Prevention

j ournal homepage: www.elsevier.com/locate/aap

A cellular automaton simulation model for pedestrian and vehicle interaction behaviors at unsignalized mid-block crosswalks

Lili Lu^a,b,^∗, Gang Renc, Wei Wangc, Ching-Yao Chand, Jian Wang^e

^a Faculty of Maritime and Transportation, Ningbo University, China
^b Jiangsu Province Collaborative Innovation Center for Modern Urban Traffic Technologies, China
^c Jiangsu Key Laboratory of Urban ITS, Southeast University, Si Pai Lou #2, Nanjing 210096, Jiangsu Province, China
^d University of California, Berkeley, Partners for Advanced Transportation Technology,1357 S.46th Street, Bldg. 452, Richmond, CA 94804-464, United States
^e School of Civil Engineering, Purdue University, West Lafayette, IN 47907, United States

a r t i c l e i n f o

Article history: Received 7 February 2015 Received in revised form 8 March 2016 Accepted 12 April 2016 Available online 18 May 2016

Keywords: Unsignalized crosswalks Yielding behavior Crossing behavior Cellular automaton model Simulation

a b s t r a c t

At unsignalized crosswalks, interactions between pedestrians and vehicles often lead to traffic safety hazards due to absence of traffic control and unclear right-of-ways. To address this safety problem, there is a need to understand the interaction behaviors of pedestrians and vehicles that are complicated by a variety of traffic and roadway attributes. The prime objective of this study is to establish a reliable simulation model to represent the vehicle yielding and pedestrian crossing behaviors at unsignalized crosswalks in a realistic way. The model is calibrated with detailed behavioral data collected and extracted from field observations. The capability of the calibrated model in predicting the pedestrian-interaction events as well as estimating the driver yielding rate and pedestrian delay are also tested and demonstrated. Meanwhile, the traffic dynamics in the vicinity of the crosswalk can be meaningfully represented with simulation results based on the model. Moreover, with the definitions of the vehicle-pedestrian conflicts, the proposed model is capable to evaluate the pedestrian safety. Thereby, the simulation model has the potential to serve as a useful tool for assessing safety performance and traffic operations at existing facilities. Furthermore, the model can enable the evaluation of policy effectiveness and the selection of engineering treatments at unsignalized crosswalks to improve safety and efficiency of pedestrian crossing.

1. Introduction

With a growing emphasis on the development of multi-modal transportation around the world, pedestrian safety at crossings has drawn considerable attention from transportation professionals and engineers. Specifically, the safety concern arises because pedestrians are more likely to expose themselves to the risk of being hit by vehicles at unsignalized crosswalks due to the lack of signal control and unclear right of way. Many countries have already established legislations and traffic regulations that vehicles should yield for pedestrians at crosswalks. Despite the existence and enforcement of these laws, a substantial proportion of vehicles still fail to yield to pedestrians and the yielding pattern varies across locations. Previous studies revealed that the average per-

E-mail address: lulili seu@163.com (L. Lu).

centage of vehicle yielding to pedestrians was 66% with a range of 8% to 100% at 11 sites in the United States (Fitzpatrick and Turner Brewer, 2007). In China, the yielding rate at mid-block unsignalized crosswalks is about 56% based on our field observations in the city of Nanjing. Failure to yield the right of way at crossings raises a lot of pedestrian safety issues (Bertulis and Dulaski, 2013). Moreover, pedestrians sometimes may have illusions that approaching vehicles should yield to them and make decisions to accept risks incautiously. To address these safety problems, there is a need to better understand the behaviors of pedestrians and vehicles and their interactions at unsignalized crosswalks.

Over the years, extensive research has been conducted on modeling vehicle yielding and pedestrian crossing behavior at unsignalized crosswalks. A majority of the prior work has been dedicated to identifying significant factors that affect these behaviors and developing suitable models to predict the likelihood of their occurrences. Harrell (1994) found that vehicles tended to yield for pedestrians with bright clothes and assertive attitude to enter the crosswalk (Harrell, 1994). Bertulis and Dulaski (2013)

^∗ Corresponding author at: Faculty of Maritime and Transportation, Ningbo University, Zhejiang Province. Fenghua road 818#, Ningbo 315211, China. Tel.: +86 18758362393.

evaluated the effects of vehicle speed on driver yielding rate to pedestrians at crosswalks (Bertulis and Dulaski, 2013). It was found that there was a strong correlation between the vehicle speed and yielding rate and vehicles with higher speeds were less likely to yield for pedestrians. Schroeder and Rouphail (2011) developed an estimation model for driver yielding behavior through logistic regression techniques (Schroeder and Rouphail, 2011). Their study suggested that dynamic profiles of vehicles and pedestrian attributes as well as other concurrent events at the crosswalk jointly affected drivers' yielding decisions. Moreover, it pointed out that pedestrian assertiveness was a very influential factor for vehicle yielding to the pedestrian. In addition, some researchers found that certain treatments such as advanced yield pavement markings and "yield here to pedestrians" signs were effective in promoting driver yielding behavior (Turner et al., 2006; Huybers and Van Houten Malenfant, 2004; Gómez et al., 2011).

Meanwhile, researchers attempted to develop gap acceptance models to represent pedestrian crossing behaviors. Using the probit and binary logit approach, Sun and Ukkusuri Benekohal (2003) established models to estimate the probability of pedestrian gap acceptance (Sun and Ukkusuri Benekohal, 2003). The model showed that gap size, number of waiting pedestrians, and pedestrian age were the vital attributes for pedestrian crossing behaviors. Another research conducted by Yannis and Papadimitriou Theofilatos (2013) concluded that pedestrian speed, vehicle speed, crossing direction, gap size and pedestrian age all influenced the decision-making of pedestrian crossing (Yannis and Papadimitriou Theofilatos, 2013). Numerous researchers also focus on developing methods for estimating the critical gap since it is a very important variable in predicting the pedestrian crossing probability. The critical gap is the time between consecutive vehicles on the road at which a pedestrian waiting for crossing has equal tendency to accept or reject the time gap. The methodologies for critical gap estimation include the HCM2010 method, graphical method, regression method, maximum likelihood method and Ramsey-Routledge method. Schroeder (2008) discussed the applicability of these methods in pedestrian critical gap estimation (Schroeder, 2008).

The potential of using simulation models for the representation and investigation oftraffic behaviors has received considerable attention from a number of researchers (Yang et al., 2006; Wan and Rouphail, 2004; Chai and Wong, 2014; Antonini et al., 2006; He et al., 2015) due to their robustness and good performance in mimicking the real traffic condition. However, since interactions between vehicle yielding and pedestrian crossing behavior at unsignalized crosswalks are very complicated and are affected by a variety of factors. To the best of our knowledge, no simulation model is well developed so far that can accurately describing the behaviors of pedestrians crossing and vehicle yielding simultaneously. Even the widely used simulation software systems such as the Vissim and Paramics are notfully embedded with the capability to simulate these behaviors. For instance, Vissim only allows users to define the priority rule for pedestrians and make the vehicle fully yielding to pedestrians, which is too idealistic (all drivers comply with the traffic rules) to completely capture the real cases.

Some limited efforts had been made previously to describe vehicle yielding and pedestrian crossing behaviors in simulation environment. Sun and Benekohal (2003) developed a simulator called PMIS (Pedestrian and Motorist Interaction Simulator) to reflect the decision-making process of driver yielding and pedestrian crossing at uncontrolled mid-block crosswalks (Sun and Benekohal, 2003). This work provided a general approach to incorporate the interactions between pedestrians and vehicles into a microscopic simulation model. Meanwhile, Chae extended the well-known microscopic simulation software Vissim to convey the yield-crossing gap-crossing process by redefining priority rules for vehicles and pedestrians, respectively (Chae, 2005). Schroeder and Bastian Rouphail (2007) discussed a framework for modeling and implementing pedestrian-vehicle interaction in microsimulation (Schroeder and Bastian Rouphail, 2007). Further, a more explicit framework was proposed by Schroeder and Rouphail in 2010 (Sun and Ukkusuri Benekohal, 2003). They thought that the driver yielding behavior described as logit models could be inserted into the core algorithms in the simulation. Thus, the simulation model should include algorithms of driver yielding, pedestrian crossing, vehicle following, lane-changing etc. Recently, a simplified model for pedestrian-vehicle interactions at road crossings based on a discrete-event system was developed by Ottomanelli and Iannucci Sassanelli (2012). This simulation model concentrated on pedestrian crossing behaviors and characterized it with a logic process. But vehicle yielding behaviors are not taken into consideration in this latest work.

Micro-simulation is very suitable for describing vehicular and pedestrian traffic due to its flexibility in the representation of their behaviors. Cellular automaton (CA) models have achieved meaningful success in simulating complex scenarios in recent a few decades. They are able to characterize complex traffic behaviors with simple and flexible transition rules accurately. Moreover, with the development of computer technology, CA models have the potential to achieve high performance in simulating traffic dynamics. Over the years, a large number of cellular based models for traffic simulation have been put forward by researchers. Nagel and Schreckenberg (1992) proposed a stochastic discrete automaton model to simulate freeway traffic(Nagel and Schreckenberg, 1992). Their work provided solid foundation for traffic flow simulation with the CA approach and demonstrated the reliability of the CA model in representing traffic dynamics. Liang et al. (2010) proposed an improved CA model for traffic flow based on the analysis of driver behavior (Liang et al., 2010). CA model for pedestrian flow simulation was first introduced by Blue and Adler (2002). The pedestrian behaviors of lane movement, forward movement were well captured by only a set of simple local rules in the CA model. Meanwhile, the pedestrian flow pattern generated with CA microsimulation is consistent with the well-established fundamental properties. Numerous modified CA models were developed based on the work of Blue and Adler. These models were demonstrated to be highly capable of replicating selforganization phenomena emerging in pedestrian flows (Burstedde et al., 2001; Kirchner and Schadschneider, 2002; Lu et al., 2012). However, the vehicle and pedestrian movements are considered separately in most of previous work about CA microsimulations, and few existing studies have incorporated pedestrian and vehicle interactions at unsignalized crosswalks into such models. Xin et al. (2014) proposes a pedestrian-vehicle cellular automata model to investigate the dynamic of mixed traffic around the unsignalized midblock crosswalk. But their proposed model focuses on pedestrian gap acceptance behavior while the drive yielding behavior is not discussed in detail (Xin et al., 2014).

This study attempts to develop a reliable microscopic model for simulating vehicle and pedestrian behaviors at unsignalized crosswalks realistically. To make up the gap of previous study, we propose a CA model that incorporates the interactions between pedestrian and vehicle. The basic concept about the development of this microscopic model is to integrate the behavioral models for vehicle yielding and pedestrian crossing into the simulation algorithms. A detailed vehicle and pedestrian behavioral data from real-world observations are adopted to calibrate the model, which is subsequently used to predict the pedestrian and vehicle movement at unsignalized crosswalks.

The remainder of this paper is organized as follows. In the next section, a detailed description about the methodology to establish the simulation model is presented. In section three, the procedures regarding how the model is calibrated with field data are detailed discussed. Following that,the simulation results are shown to validate the capability of the proposed model in predicting the interaction events, pedestrian delay and vehicular traffic dynamic. Finally, the benefits of utilizing the proposed model are discussed, followed by some suggestions for future work.

2. Model

In this study, the discrete CA model is used as the benchmark model to simulate for vehicle and pedestrian interaction behaviors at unsignalized mid-block crosswalks, in which the pedestrians and vehicles are allowed to move between cells at every time step. Meanwhile, to represent the heterogeneity of traffic behaviors, pedestrians and vehicles are regarded as automaton identities with specific attributes. In the following sub-sections, we will discuss the rational and the conceptual framework of the proposed model in detail.

2.1. Model framework

The simulation system consists of vehicles, pedestrians, road, crosswalk and waiting areas for pedestrians as shown in Fig. 1.The exemplar mid-block crossing area is depicted in yellow. The crosswalk is discretized into cells, each with a size of 0.5m × 0.5m. Each pedestrian occupies one cell while a vehicle occupies multiple cells. For example, a vehicle with a size of 3.5m × 7.0m occupies 7 × 14 = 98 cells.

The overall framework of this simulation model is shown in Fig. 2, which can be divided into five phases. Phase 1 is to generate pedestrians and vehicles and place them into the geometric boundary of the model. Unlike the general CA models, both the pedestrians and vehicles are populated with various attributes such as age, gender, attitude, vehicle type etc. Moreover, the arrival patterns of pedestrians and vehicles following certain statistical distributions are obtained and confirmed from field data. In Phase 2, all the parties start to move. The pedestrians in the waiting area begin to walk on the crosswalk and vehicles entering the system begin to move on the road. Phase 3 describes the dynamic interaction process when the vehicles and pedestrians approach the conflict area simultaneously. In this circumstance,the vehicles may or may not yield to the pedestrians while the pedestrians may decide to cross or wait. After the decision dilemma process, the pedestrians walk out of the interaction area and the vehicles leave the crosswalk, one before the other in either order in Phase 4. In Phase 5, when the pedestrians reach the opposite side of the crosswalk and the vehicles arrive at the boundary at the other end of the road, they are removed from the system.

2.2. Model development

Based on the overall framework described above, the model is designedas anintegratedsimulationsystemthatinvolves the walking and crossing behaviors of pedestrians as well as the moving and yielding behaviors for vehicles. These behaviors in every phase are modeled with the specific rules described below.

2.2.1. Phase 1: vehicle and pedestrian generation and entry

The vehicles and pedestrians are generated with various attributes. Each vehicle is assigned with certain vehicular type. The arriving distributions of vehicles of all vehicular types are based on field data collected from real-world observations. Besides, once the vehicles are generated, they are also assigned an initial speed randomly using a normal distribution. Ifthe randomly generated speed is less than or equal to the maximum allowable speed, the speed is valid and would be used; otherwise, another speed is generated for the vehicle. Similarly, the attributes of gender, age, attitude etc. are assigned to pedestrians.

Before a new vehicle enters the system, the simulation model would check whether the number of empty cells between front vehicle and the entry boundary is sufficient for the newly entering vehicle with given speed. Ifthe gap is large enough,the new vehicle is allowed to enter the system; Otherwise it will be held to enter until the gap is sufficient. For a newly generated pedestrian, the simulation model also checks whether there are empty cells in the waiting area. If there are, the pedestrian is allowed to enter into the system and positioned in the available cells randomly.

2.2.2. Phase 2,4,5: vehicle and pedestrian movement

The well-known Nagel-Schreckenberg model (Nagel and Schreckenberg, 1992) is applied to mimic driving maneuvers and actions of vehicles in our model. Note, however, the NaSch model used a single large cell for a vehicle while we allow a vehicle occupies multiple small cells. This distinction fosters a higher resolution andfidelity for themovement of a vehicle inourmodel.Inthe NaSch model, the velocity of vehicle varies from 0 to Vmax (cells/time step), where Vmax is the maximum allowable speed, i.e. the specified speed limit for the modeled roadway segment. In the NaSch model, the positions of vehicles are updated at each time step with the following four consecutive steps.

• 1)Acceleration. If the vehicle velocity is below Vmax, the vehicle would speeds up with acceleration rate a,. Vn(t + 1) → min [Vn (t) + a, Vmax]. The magnitude of acceleration rate given for the vehicle is explained in the next section.

Deceleration. If there is a vehicle in front and the gap to the vehicle in front is smaller than allowed (or not safe enough) for the vehicle velocity, the vehicle velocity is decreased to a speed that matches the size of that gap. Vn(t + 1) → min [Vn (t + 1), dn(t)].

2)Randomization. With probability r, the velocity of vehicle is decreased by a, if the vehicle is not already standing still. r ⇒ Vn(t + 1) → max [Vn(t + 1) − a, 0]
3)Position update. Once the speed of vehicles are determined from the three steps above, all the vehicles move forward with the current velocity. Xn (t + 1) → Xn (t) + Vn (t + 1)

where Vn (t) is the velocity of the vehicle at time step t, dn (t) represents the empty cells in front of the vehicle, a is the acceleration rate of the vehicle and Xn(t) is the position of the vehicle at time step t.

The pedestrian walking behavior is described by an SFCA model that was developed by Lu et al. in 2013 (Ren et al., 2014). SFCA is a microscopic CA model for pedestrian flow simulation at the crosswalk. It uses the principle of social forces to model the interactions among pedestrians. At each time step, the positions of pedestrians are updated with the rules described below.

Each pedestrian moves to one of the neighboring cells or stay still in the same cell according to the transition probability at every time step. The possible neighboring cells that a pedestrian may move into are depicted in Fig. 3. Each movement is made with a transition probability Pt which depends on the state of the neighboring cells and the social forces. Driving forces act between pedestrians and the destinations, whereas the boundaries and pedestrians induce the repulsive forces. Moreover, interaction forces are used to represent interactions among pedestrians. These forces, as illustrated in Fig. 1 of last section, are expressed through suitable scalar functions. With these formulations, the pedestrians are more likely to move towards an empty cell in the direction where the force is smaller. This model has been calibrated and validated with real-world data and empirical analysis. A more detailed

Fig. 1. Schematic diagram of the simulation model.

Fig. 2. The overall framework for the simulation model.

description about this model is given in our previous study Ren et al., 2014).

2.2.3. Phase 3:Vehicle yielding and pedestrian crossing behaviors t is reasonable to assume that when a driver initially see some pedestrians in the crosswalk within the vision distance Dv, his/her decision regarding whether to yield for pedestrians isn't made immediately but a dynamic process which rely on the varied situation. The algorithm to characterize the dynamic process of driver's yielding decisions is summarized in Fig. 4. The detailed steps of this algorithm are presented as follows:

Fig. 3. The neighborhood adopted in the model.

Fig. 4. The algorithm for modeling of vehicle yielding behavior.

Ifthe subject vehicle is within the vision distance,then check first whether the vehicle is already moving past the crosswalk.
Ifthe subject vehicle is still approaching the crosswalk,thencheck whether there is a leading vehicle in front of the subject vehicle.
If such leading vehicle exists, then calculate the headway between the leading vehicle and the subject vehicle to determine whether the subject vehicle needs to accelerate or decelerate in the next time step.
In the case the subject vehicle itself is the leading vehicle approaching the crosswalk, the existence of pedestrians in the conflict area will be examined.
If there are pedestrians in the conflict area, the subject vehicle begins to decelerate to avoid the collision with pedestrians. Otherwise, the vehicle would keep moving forward according to the NaSch model rules.
However, if there are some pedestrians in the waiting area, the vehicle still needs to decide whether to yield for them even if there is no pedestrian in the conflict zone. The probability for

vehicle to yield under these conditions is provided by a vehicle yielding model, which we will describe explicitly in the following section.

For the pedestrians waiting to cross the street,they will observe the gaps between vehicles before crossing. If the gap is acceptable, the pedestrians will decide to cross. However, if a pedestrian considers the observed gaps are not enough to cross, he or she will wait until the available gap occurs between the vehicles. This process is summarized in Fig. 5. The probability for pedestrians to cross is determined by the pedestrian gap acceptance model, which will also be presented in detail in the following section.

In addition to the vehicle yielding and pedestrian crossing behaviors explained in Figs. 4 and 5, there are additional pedestrian-vehicle interaction scenarios that need to be captured in the overall model.

Itis possible that a vehicle driver may not yield to the pedestrian even if the pedestrian attempts to enter the conflict area. To mimic this phenomenon in the simulation model, we assume the drivers originally decide to keep moving would have probability to yield to pedestrian finally due to the changed situation.With this respect, one can find that the vehicles, still with probability1 − -, that would keep moving and not yield to pedestrians even if the pedestrians attempt to cross the street. In this case, the pedestrians have no choice but to return to the original cell and wait for another gap to cross.

Moreover, a deadlock is also likely to occur when a vehicle yields to the pedestrian and at the same time a pedestrian waits for the vehicle to cross. To break the deadlock, we assume the driver changes the yielding behavior and decides to move forward with probabilityı, while the pedestrian proceeds ahead and moves into the conflict area with probability1 − ı.

3. Model calibrations and simulation

3.1. Parameters in the model

Theparameters intheproposedmodeldescribedabove are summarized in Table 1. It is worth mentioning that the probability for vehicle yielding and pedestrian crossing under various situations are calculated based on the following binary logit model which is consistent with previous study(Schroeder and Rouphail, 2011).:

$$P{Y = 1} = \frac{e^{\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k + \varepsilon}}{1 + e^{\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k + \varepsilon}}$$

where ˇ⁰ is a constant term andˇi is the estimable parameter for the independent variableXi(for arbitrary i = 1, 2,...k).P is the probability of the independent variable Yequals to 1.

Before calibrating the binary logit model, it is critical to identify the potential pivotal variables that affect the driver yielding and pedestrian crossing behaviors. The parameters of these variables in this model would be calibrated with field data using an event-based methodology that has been suggested in previous work (Schroeder and Rouphail, 2011). Specifically, the pedestrianvehicle interaction is defined as the driver's decision process and corresponding vehicular movement regarding whether or not the vehicle approaching crosswalk should yield to pedestrian in the conflict area. With this definition,the variables describing the characteristics of pedestrians, vehicles and traffic dynamics as well as the discrete outcome of driver yielding and pedestrian crossing are extracted. Following the same definitions presented in previous work (Schroeder and Rouphail, 2011), the relevant variables are listed in Table 2. Logistics regression analysis is then conducted to find the best fitting models that capture the relationship between independent variable (outcome of vehicular yielding or pedestrians crossing behavior) and a set of dependent variables that describe the attributes of pedestrians, vehicles and traffic.

3.2. Calibration process

The data collected on two unsignalized crosswalks located on urban streets in Nanjing, China (see Fig. 6) is used to demonstrate the calibration process of the propose model. These data were collected on workdays during morning and evening peak hours (7:30–9:30am, 16:30–18:30) from April 21 to April 25 of 2014. As mentioned in 3.1, the data to be collected should include the site configuration, the vehicle and pedestrian motion, and the vehicle and pedestrian interaction events.

3.2.1. Calibration of configuration parameters

The value of independent variables related to the configurations of urban roadways and unsignalized crosswalks is obtained by actual measurements. The lengths of the roadway segments in site A and site B are 500 m and 400 m respectively according to Google map. The widths of these lanes are uniformly set as 3.75m. There are 4 lanes in total for both sites. The width of pedestrian crosswalk marks in both side are all 6 m and corresponding length are24m, 26 m respectively.

3.2.2. Calibration of pedestrian and vehicle motion model

Various probability distributions are applied to determine the best-fitting arrival patterns of vehicles and pedestrians. It turns out that the Poisson distribution properly fits the field data, where the probability function is given by

Vehicle: $$P\nu(x) = \frac{(\lambda_1 t)e^{(-\lambda_1 t)}}{x!}$$

Pedestrian: $Pp(x) = \frac{(\lambda_2 t)e^{(-\lambda_2 t)}}{x!}$

where P(x) is the probability of x vehicles or pedestrians arriving during every time interval; and t is the time interval; ¹ and ² is the average arriving rate, for pedestrians and vehicles respectively. Fig. 7 shows the calibrated result for the pedestrian arrival pattern in site A. The number of pedestrians is counted in every 10 s, and the empirical probability density function is shown in Fig. 7a. It turns out that the Poisson distribution model fits the field data well with parameter A ¹ = 2.70 and the Kolmogorov-Smirnov test indicates that the test statistic is 0.249 which is less than the critical value 0.294. Thus,the null hypothesis thatthe observed data follows Poisson distribution is not rejected at 95% confidence level. The other vehicle and pedestrian arrival patterns are obtained in the similar way and their parameters are shown in Table 5.

The formulation for pedestrian transition probability at each time step has been calibrated in our previous publications and the value of related parameters can be found in (Ren et al., 2014). Meanwhile, according to the empirical data, the average walking speed of pedestrians is about 1.25m/s. Since the pedestrian is allowed to move one cell in each time step in the simulation model, the equivalent time step in the simulation is set as ⁰.5m ¹.25m/s = 0.4s. In the two sites, the maximum speed of vehicles is 40 km/h, approximately 11m/s. Scaling the speed by the size of the cells and the time step, the maximum speed of vehicle in our simulation is about 11×0.4m ⁰.5m/cell = 8.8cells ≈ 9cells per time step. Moreover, the acceleration rate of the vehicle can also be estimated using this approach. The acceleration rate is around ⁰.50m (0.4s) ² = 3.12m/s2when the vehicle speed decreases or increases by one cell per time step. Recall that the vision distance is defined as the distance at which the driver begins to pay attention to the presence of pedestrian in the conflicting area. The vision distance is estimated as the distance which the vehicle requires to stop completely from maximum speed by the

Fig. 5. The algorithm for modeling of pedestrian crossing behavior.

Table 1 Associated parameters in the simulation model.

	Parameters	Variable	Unit (in reality)	Unit (in simulation)
Configuration	Vehicle Lane length	L1	m	cells
	Vehicle Lane Width	W1	m	cells
	Number of vehicle lanes		–	–
	Crosswalk width	W2	m	cells
	Crosswalk length	L2	m	cells
Vehicle	Attributes	A = {type, speed}
	Arrival rate	1	pcu/s	pcu/time step
	Maximum speed	Vmax	m/s	cells/time step
	Randomization probability	r	–	–
	Yielding probability	ˇ0+ˇ1X1+ˇ2X2++ˇkXk+ε e Py = ˇ0+ˇ1X1+ˇ2X2++ˇkXk+ε	–	–
	Vehicle accelerate rate	1+e a	m/s2	cells/(time step)2
	Vehicle vision distance	Dv	m	cells
Pedestrian	Attributes	B = {Gender, Age,Group Clothecolor, Attitude}	–	–
	Walk speed	V	m/s	cells/time step
	Arrival rate	2	ped/s	ped/time step
	Transition probability	pi exp(−kwf˛B) × exp(−kf f˛ˇ) Pi t = 3 pj exp(−kwf˛B) × exp(−kf f˛ˇ) j=1 i = 1, 2, 3	–	–
	Crossing probability	ˇ0+ˇ1X1+ˇ2X2++ˇkXk+ε e Pc = ˇ0+ˇ1X1+ˇ2X2++ˇkXk+ε 1+e	–	–
Others	Parameter 1		–	–
	Parameter 2	ı	–	–

TABLE 2 Definition of variables in vehicle yielding behavior and pedestrian crossing models.

Category	Variables	Type
Pedestrian characteristics GE-Gender
	0-Female; 1-Male;	Categorical
	AG-Age
	0 ≤ 18; 1–19–39; 2–40–59;3 ≥ 60;	Categorical
	GR-Group
	0-Walk alone; 1-Walk in group;	Categorical
	CC-Clothes color
	0-Bright; 1- Normal;2- Grey;	Categorical
	AT-Attitude
	0-Assertive;1-Hesitant	Categorical
	PO-Position
	0-Near side; 1-Far side;	Categorical
Vehicle characteristic	VT-Vehicle Type	Categorical
	0-Car (including Taxi); 1-Heavy vehicle(including Bus);
	DP-Distance to conflicting pedestrians(m)	Continuous
	VS-Vehicle Speed(km/h)	Continuous
	TH-Time headway(s)	Continuous
	FO-Following
	0-Following other cars 1-not following other cars	Categorical
	IQ-In queue
	0-part of the queue 1-not in queue	Categorical
Traffic condition	PR-Previous vehicle
	0-Previous car passed with yielding
	1-Previous car passed without yielding	Categorical

Fig. 6. Locations and photos of the sites for field data collection.

Fig. 7. Fitting Result of Pedestrian Arrival Pattern in Site A.

specified deceleration rate. Thus, the vision distance is calculated by (11m/s)2 2×(3.12m/s2) , which equals to proximately 20m.

Note in above calculations, the vehicle speed, acceleration, and vision distance are all covert to number of cells. These values can easily be adjusted to reflect the movement of the vehicles in fractions of a cell instead of one full cell in each time step. This simplification of increment by one cell is found to be satisfactory for the purpose of the simulation, as we will show with our simulation results in the next section. However, these parameters can be revised to accommodate a desired resolution of the movement if a higher fidelity of vehicle movements is considered necessary.

TABLE 3 Descriptive Statistics for Sites.

	Site A		Site B
	Mean	Std. Dev.	Mean	Std. Dev.
Yield	0.54	0.496	0.58	0.497
Gender	0.55	0.431	0.57	0.429
Age	1.43	0.423	1.75	0.464
Group	0.58	0.665	0.68	1.509
Clothes color	0.78	0.823	0.94	0.810
Attitude	0.41	0.378	0.35	0.410
Position	0.40	0.489	0.44	0.502
Vehicle Type	0.24	0.667	0.30	0.634
Distance to conflicting pedestrians	38.7	19.728	41.7	23.531
Vehicle Speed	24.8	6.127	26.4	6.800
Time headway	4.30	0.891	4.42	1.023
Following	0.27	0.389	0.21	0.415
In queue	0.80	0.376	0.76	0.431
Previous vehicle	0.49	0.497	0.59	0.474

TABLE 4 Results of Logistic Regression for Site A and B.

	Variable	Estimate	exp(ˇi)	Std Error	Pr > 2
Site A	intercept Attitude Group Time headway Vehicle Speed AIC SC -2Log L	−2.42 1.87 0.38 1.58 −0.10 579.02 583.06 99.92	0.09 6.49 1.46 3.74 0.90 Percent Concordant Somers' D Gamma C	0.23 0.38 0.19 0.27 0.04 92.3 0.906 0.955 0.973	< .0001 < .0001 0.043 < .0001 0.0065
Site B	Intercept Attitude Time headway Vehicle Speed AIC SC -2Log L	−2.47 1.59 2.12 −0.11 543.02 561.04 95.23	0.08 4.90 3.06 0.89 Percent Concordant Somers' D Gamma C	0.53 0.23 0.11 0.02 88.4 0.769 0.770 0.884	< .0001 < .0001 < .0001 < .0001

3.2.3. Calibration of vehicle-pedestrian interaction model

1375 and 1479 pedestrian-vehicle interaction events are observed in site A and site B respectively (2854 events in total) for model calibration. The explanatory variables regarding characteristics of pedestrians, vehicles and traffic conditions are extracted from approximately 40 h of videos.

The attributes of pedestrian gender, age, group etc. are observable and obvious in a great majority of cases while the attributes of pedestrian attitude towards the vehicle are determined subjectively. If the pedestrian moves forth and back before crossing, his or her attitude is considered hesitant, otherwise it is considered assertive. In order to capture the relevant variables, the camera is placed carefully as shown in Fig. 1. Moreover, another video camera placed on a high building is used to record the vehicle trajectories when approaching to the crosswalk. The inputs of these variables are shown in Table 3.

The statistical software SAS, is used to conduct the logistic regression analysis for both site A and site B. The parameters of the resultant models are shown in Table 4. It is found that the independent variables including pedestrian attitude, group, time headway between vehicles, and vehicle speed all significantly impact drivers' yielding behaviors at Site A, while the pedestrian attitude, time headway between vehicles, and vehicle speed are salient variables for drivers' yielding behaviors at Site B. Moreover, the data from both sites suggest that drivers are more likely to yield to a pedestrian with assertive attitude to cross. Itis also found thatthe vehicle at a higher speed is less likely to yield. These conclusions are consistent with those of previous work (Schroeder and Rouphail, 2011). According to the binary logit model, the effects of each variable on the probability of yielding can be described by odds ratio (exp(ˇi)). The odds ratio is interpreted as an increase in the odds of the response when the variable increases from 0 to 1. For example, the odds ratio value for pedestrian attitude is 1.87 for Site A, and 1.59 for Site B, which can be interpreted that the driver are 6.49 and 4.90 times more likely to yield to the pedestrian with assertive attitude than the one with hesitant attitude. Similarly, increase time headway between vehicles also result in higher probability of driver yielding. Table 4 demonstrates that the yield probability would increase by 3.74 times and 3.06 times for Site A and Site B respectively if the time headway between vehicles is increased by one second. It worth mentioning that the values of indicators (i.e., the percentage of concordant observations, Somers' D, Gamma and C) in Table 4 suggest the models fit very well to the data.

Similarly, we also use binary logit model to estimate pedestrian crossing behavior. For simplicity, this model only incorporates the most significant variable. The results from the goodness-of-fit test suggest that the estimated models fit the observed data quite well as the Chi-square statistic (Site A 48.01, Site B 45.21) and p-values for Pr >ChiSq is <0.00001.

3.3. Simulations

The simulation experiments are performed based on the calibrated model. Still using the aforementioned cases, the validity of the proposed model is investigated. Each simulation is conducted for 20 h to be consistent with the duration of the data collection in both site A and site B. The total number of interaction events as well as the vehicle yielding rate in an hour is outputted. Table 6 compares the simulation results with the observed data. Recall to alleviate in 2014. It shows that the two indicators generated by the developed simulation model fit well with the observed data from both Site A and Site B. The differences between the observed and simulated numbers of pedestrian-vehicle interaction events for Site A and Site B are 13.2% and 9.43%, respectively. For the yielding rate, the differences are only 10.67% for site A and 3.77% for Site B. Since a difference within 15% is acceptable in most studies (Choe and Skabardonis Varaiya, 2002), the performance of the simulated result confirms the validity of the proposed simulation model.

Pedestrian delay is a very important indicator to determine pedestrian level of service. At unsignalized crosswalks, it not only depends on the pedestrian volume and vehicle volume but also the driver yielding pattern since they all together determine the number of opportunities for pedestrian crossing (National Research Council Transportation Research Board, 2010). To compare the difference between the simulated results and the actual observations, the pedestrian delay data are also collected from the video. The average pedestrian delays from the simulation are 19.35s and 26.85s at Site A and at Site B, respectively. The distributions of simulated pedestrian delay at Site A and Site B are shown in Fig. 8. It shows that the pedestrian delay ranges from 0 to 120s and most of the pedestrians (around 70%) experience the delays less than 80s, while the percentage of pedestrians experienced more than 100s delay is quite low. Fig. 8 alsopresents the distributions of pedestrian delay collected from field observations, which are remarkably similar to the patterns obtained from simulations. Since the mean value of pedestrian delay estimated from field data is about 21.34s and 28.63s,the difference between the observed and simulated resultis only 9.32% and 6.21%. To test whether the benchmark distribution formfor estimatingdelay inthe simulationmodel differ fromthat of observations, the two-sample Kolmogorov-Smirnov test is applied. The p-values of Kolmogorov–Smirnov are 0.872 and 0.805 for site A and site B, respectively, which indicates that the distributions of the pedestrian delay estimated from field data and simulations comply with the same distribution at a confidence level of 0.95.

Table 5 Associated parameters in the simulation model.

Category	Parameters	Site A Value	Site B Value
Configuration	Vehicle lane lengthL1	500m	400m
	Vehicle Lane Width	3.75m	3.75m
	Number of lanesN	4	4
	Crosswalk widthW	6m	6m
	Crosswalk lengthL2	24m	26m
Vehicle	Attributes A	A= {0.24,V0} V0∼N(5.5, 2.0)	A= {0.30,V0},V0∼N(5.5, 2.0)
	Average arrival rate 1	0.09pcu/s	0.22pcu/s
	Maximum speedVmax	11m/s	11m/s
	Randomization probabilityr	[0, 1]	[0, 1]
	Yielding probabilityPy	e−2.42+1.87AT+0.38GR+1.58TH−0.10VS Py =	e−2.47+1.59AT+2.12TH−0.11VS Py =
	accelerate rate a	1+e−2.42+1.87AT+0.38GR+1.58TH−0.10VS 3.12m/s2	1+e−2.47+1.59AT+2.12TH−0.11VS 3.12m/s2
	vision distanceDv	20m	20m
Pedestrian	Attributes B	B = {0.55,1.43, 0.58, 0.78, 0.41}	B = {0.57,1.75, 0.68, 0.94, 0.35}
	Walk speed V	1.25m/s	1.25m/s
	Average arrival rate 2	0.27ped/s	0.15ped/s
	Transition probability Pi t	pi exp(−kwf˛B) × exp(−kf f˛ˇ) Pi t = 3 pj exp(−kwf˛B) × exp(−kf f˛ˇ) j=1	pi exp(−kwf˛B) × exp(−kf f˛ˇ) Pi t = 3 pj exp(−kwf˛B) × exp(−kf f˛ˇ) j=1
	Crossing probabilityPc	i = 1, 2, 3 e−3.07+0.81Gap Pc =	i = 1, 2, 3 e−3.67+0.98Gap Pc =
Others	Parameter 1	1+e−3.07+0.81Gap [0, 1]	1+e−3.67+0.98Gap [0, 1]
	Parameter 2 ı	[0, 1]	[0, 1]

TABLE 6 Comparison results of the simulated and observed.

	Site A				Site B
	# of Interaction events		Yielding rate		# of Interaction events		Yielding rate
	Mean	StdDev	Mean	StdDev	Mean	StdDev	Mean	StdDev
Observed	68	15.8	0.53	0.51	75	17.6	0.53	0.49
Simulated	59	–	0.48	–	67	–	0.51	–
Difference	9	6.9	0.05	0.16	8	8.5	0.02	0.07
Difference (%)	13.2	–	9.43	v	10.67	–	3.77	–

Fig. 8. Distributions of pedestrian delay in site A and site B.

These results reveal that the proposed simulation model is capable of providing reasonably good predictions on pedestrian delay.

Furthermore, the proposed simulation model can also provide representations of traffic flow by creating vehicular trajectories graphically. Fig. 9 shows some sample data of the evolution of vehicular trajectory at Site A and Site B. It should be noted that Fig. 9(a and b) only represent the traffic flow on one of the lanes. From Fig. 9, it can be seen that some vehicles make a temporary stop to yield for pedestrian at the unsignalized crosswalk. As the traffic volume in site A is not high, the congestion only exists at a small segment and dissipates very quickly. In site B, however, the traffic volume is very high and the vehicles form a longer queue due to pedestrian crossing. Thereby, one can see that the developed simulation model accurately replicate the phe-

Fig. 9. Spatiotemporal diagram of vehicle flow (a) on Site A (b) on Site B in simulation.

nomenon of interruption of the traffic flow by crossing pedestrians at the unsignalized crosswalk. However, since the real vehicular trajectory data is not available, in this study, we cannot provide a quantitative comparison of the simulated vehicular trajectory with that of field data. Thereby, a more detailed study is required to explore the relationship among the traffic flow, the vehicle and pedestrian interaction, and driver yielding rate in the future.

3.4. Safety evaluation

The calibrated simulation model is then applied to evaluate pedestrian safety at unsignalized crosswalk. When a pedestrian arrives atthe crosswalk, he or she is exposed to either safe or unsafe gap and need to make a decision to cross or wait. In some cases, the time lag also plays very important role in pedestrian crossing decision, which is defined as the difference between the time at which the pedestrian's arrival at the crosswalk and the time when next conflicting vehicle arrivals (Schroeder and Bastian Rouphail, 2007). The time lag is considered safe if it is larger than the critical value, otherwise it is defined as unsafe. The occurrences of the safe lag time (OSL) reflects the chances for pedestrian safe crossing. A variety of factors such as the vehicle volume, speed, yielding behaviors etc. affect this indicator. Moreover, there are cases that the pedestrians make a risky decision to cross with an unsafe time lag, which result in a vehicle-pedestrian conflict. The percentage of the risky crossing (PRC), namely the conflict rate (CR) is thus an important index to evaluate pedestrian safety and can be identified with the proposed simulation model. Moreover, based on the definition of the time to collision (TTC), it is described as difference between the time at which the pedestrian entering the conflicting area and the time that next conflicting vehicle arriving at the crosswalk in the simulation (Gettman and Head, 2003).

Based on the definitions above, the simulation model is performed to evaluate the pedestrian safety at both site A and site B. The critical time lag is set at 4s which is considered safe as the pedestrian generally need only 3 s to cross the conflict area at the speed of 1.25m/s. The occurrence of safety time lag, the percentage of the risky crossing as well as the average time to collision are obtained with simulation. The results indicate that the pedestrian safety levels are relatively low as the conflict rates are as large as 42.1% and 44.3% for site A and site B respectively. For mean TTC, they are just 2.52s and 2.43s at sites A and site B, respectively. About 40.3% and 38.6% crossable lag occurred during the investigated periods at these two unsignalized crosswalk. This implies that the pedestrians may cross with risky lag which induces high pedestrian-vehicle conflicts rates. According to these results, to improve the safety of pedestrian crossing, it is quite necessary to introduce some treatments (e.g. the yielding sign before the crosswalk, the signal control) in the two sites to improve the conditions for pedestrian crossing.

To gain more insights about the pedestrian safety, we also simulate different scenarios in Site B by varying the driver yielding rates. The results are shown in Fig. 10. It demonstrates thatthe conflict rate decreases dramatically as the yielding rate increases. This implies that the higher the yielding rate, the safer the pedestrian would beat the unsignalized crosswalk. In addition, the pedestrian delay reduces with respect to (w.r.t.) increased the yielding rate. However, the variation of vehicle delay w.r.t. yielding rate shows a different trend. When the yielding rate varies between 0.1 and 0.5, the vehicle delay decreases w.r.t. increased yielding rate. It can be interpreted that the lower yielding rate would result in the higher conflict rate, to avoiding an accident, the frequencies of vehicle's stopping and decelerating behavior may also increases. Thereby, the vehicle delay increases as yielding rate decreases when it is in [0.1, 0.5]. The vehicle delay reaches the minimum value when the yielding rate is 0.60, but it increases when the yielding rate is over 0.60. In this case, the vehicle delay is mainly induced by the driver yielding behavior. These results suggestthatthe driver should yield properly to achieve a better trade-off between the traffic safety and efficiency at the unsignalized crosswalk.

4. Conclusions and discussion

In this paper, a CA based simulation model for pedestrianvehicle interactions at unsignalized mid-block crosswalks is proposed and developed. The model is shown to be capable of

Fig. 10. Simulation results under different yielding rate.

capturing the behaviors of both vehicles (e.g., moving and yielding) and pedestrians (e.g., waiting and crossing) when approaching at the unsignalized interactions. The associated parameters in the model are calibrated with large amount of observations from two unsignalized intersections in Nanjing, China. Through comparison of the average interaction events and driver yielding rates obtained from simulation model and field data, we demonstrate that the simulation model can replicate pedestrian-vehicle interaction with high accuracy and reliability. In addition, the proposed model also makes a good prediction of pedestrian delay.

With the merits discussed in this study, the proposed simulation model is a useful tool for either assessing pedestrian crossing safety or for evaluation of effectiveness of countermeasures and treatments. It should be pointed out while the pedestrians are assigned withdifferent attributes intheproposedmodelthey are all assumed to be homogeneous. Thus,the walking speeds of all pedestrians (including the old male, the old female, the young male and young female) are the same. This is one of the deficiencies of the proposed model and will be improved in further study. However, as the micro simulation model has high transferability, our model can be extended to replicate other crossing situations as well, and can be further developed into an easy-to-use software platform for pedestrian crossing facility design. For future study, we would like to address all of the pre-mentioned gaps of the proposed simulation model and to quantitatively investigate the effects of various factors on the pedestrian safety as well as traffic performance at the unsignalized mid-block crosswalks.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No.51578149, 51408190, 51308311, 51408321) the Natural Science Foundation of Zhejiang Province (No. LY15E080013) and Sponsored by K.C.Wong Magna Fund in Ningbo University. The authors are very grateful to the anonymous reviewers for their valuable suggestions and comments.

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