Modeling Crossing Behavior of Drivers at Unsignalized Intersections with Consideration of Risk Perception

. Drivers ’ risk perception is vital to driving behavior and traffic safety. In the dynamic interaction of a driver-vehicle-environment system, drivers’ risk perception changes dynamically. This study focused on drivers’ risk perception at unsignalized intersections in China and ana lyzed drivers’ crossing behavior. Based on cognitive psychology theory and an adaptive neuro-fuzzy inference system, quantitative models of drivers’ risk perception were established for the crossing processes between two straight-moving vehicles from the orthogonal direction. The acceptable risk perception levels of drivers were identified using a self-developed data analysis method. Based on game theory, the relationship among the quantitative value of drivers’ risk perception, acceptable risk perception l evel, and vehicle motion state was analyzed. The models of drivers’ crossing behavior were then established. Finally, the behavior models were validated using data collected from real-world vehicle movements and driver decisions. The results showed that the developed behavior models had both high accuracy and good applicability. This study would provide theoretical and algorithmic references for the microscopic simulation and active safety control system of vehicles.


Introduction
Traffic safety in intersections has attracted increasing attention. According to the National Highway Traffic Safety Administration (NHTSA) [1], approximately 47% of the total 10,064,000 crashes in the United States in 2013 occurred at intersections and nearby areas. The American Institute of Transportation Engineers have declared that road intersection safety was an important subject that required careful solutions [2]. Moreover, previous researches showed that drivers' behavior at intersections could significantly affect the capacity and safety of intersections [3,4]. Thus, conducting research on drivers' behavior at intersections is necessary.
Unlike signalized intersections, unsignalized intersections have no positive indication to inform drivers when it is appropriate to enter the intersection. Drivers' behavior at unsignalized intersections is more complex than that at signalized intersections. Each driver makes decisions about when, where and how to complete a required maneuver based on the elements of a decision context, including his/her perceptions of distance, velocity, and the performance of his/her vehicle [5]. Accordingly, the number of vehicular conflicts and accidents at unsignalized intersections is higher than that at other intersections. According to NHTSA [1] statistical data, approximately 46% of intersection crashes in the United States in 2013 occurred at unsignalized intersections. In China, crashes also occurred at unsignalized intersections in 2013, accounting for approximately 60% of total intersection crashes [6]. Therefore, we mainly focused on drivers' behavior at unsignalized intersections and established drivers' behavior models in this study. The information analysis of drivers, particularly risk perception, is notably a key problem in understanding driving behavior and improving traffic safety [7]. The level of drivers' risk perception changes dynamically in the dynamic interaction of a driver-vehicle-environment system. Once a perceived risk is out of an accepted region (i.e., an acceptable risk perception level), a driver will accordingly speed up or slow down to adjust his/her perceived risk. That is, drivers' risk perception significantly influences driving behavior. Thus, establishing drivers' risk perception models and determining acceptable risk perception levels are important to analyze drivers' behavior at unsignalized intersections.
In addition, the lack of stop signs and roundabouts at unsignalized intersections in China causes difficulty in controlling or guiding traffic. In most cases, when one vehicle encounters another vehicle at an unsignalized intersection in China, neither of the drivers will completely stop their vehicle. Instead, one driver will gradually approach the intersection and adjust his/her driving behavior by gaming with the other vehicle. That is, the encounter between the two vehicles is actually a game process between the two drivers, and thus, can be analyzed using game theory.
In this study, we selected typical crossing cases at unsignalized intersections in China (in each case, one straight-moving vehicle encountered another straightmoving vehicle from the orthogonal direction) and developed drivers' behavior models based on risk perception and game theory. The models were expected to possibly reflect drivers' psychological characteristics and describe drivers' behavior at unsignalized intersections with improved accuracy.

Quantifying risk perception of drivers
Drivers' risk perception exhibits nonlinear characteristics because of the effects of various factors. Thus, describing the risk perception of drivers using one particular formula is difficult. According to recent literature [8], an adaptive neuro-fuzzy inference system (ANFIS) is a combination of neural network and fuzzy logic approaches; hence, it inherently has the advantages of both, such as having a good learning mechanism and reasoning capability. Accordingly, we adopted ANFIS to model drivers' risk perception at unsignalized intersections in China in this study.
A typical unsignalized intersection, shown in Fig. 1(a), where pedestrians as well as non-motor and motor vehicles traveled together, was selected for this study.  We chose a tall building on the southwest corner of the intersection and used a video-based system to record traffic conditions. Then, we selected 150 typical crossing cases, including two straight-moving vehicles from the orthogonal direction, as shown in Fig. 1(b). To avoid interference from other vehicles to drivers' crossing behavior, we selected only the simplest crossing cases. In each case, one straight-moving vehicle encountered another straight-moving vehicle and no other object. Furthermore, the vehicle from the right side was labeled as "right vehicle," that is, ‫ܥ‬ ோ in Fig. 1(b) in this study, whereas the vehicle from the left side was labeled as "left vehicle," that is, ‫ܥ‬ in Fig. 1(b). Point C in Fig. 1(b) indicates the crossing point of the trajectories of the two vehicles.

Risk perception parameters
In our previous study [9], we identified the main factors that influenced the decision of straight-moving drivers when they encountered another vehicle moving at unsignalized intersections in China. For right vehicle drivers, SPEED_DIFFERENCE (i.e., the relative speed between the right vehicle and the left vehicle), DISTANCE_DIFFERENCE (i.e., the relative distance between the right vehicle to the crossing point and the left vehicle to the crossing point), and SPEED_L (i.e., the speed of the left vehicle) were main factors that affected drivers' risk perception and their yielding/preemptive decisions. For left vehicle drivers, the main factors that influenced their risk perception and decisions were SPEED_DIFFERENCE, CROSS_TIME_R (i.e., the travel time of the right vehicle to the crossing point), CROSS_DISTANCE_R (i.e., the distance between the right vehicle and the crossing point). In this study, we used these parameters to model drivers' risk perception in driving processes.

Quantitative method
In general, ANFIS has a six-layer feedforward neural network structure. For a detailed structure of ANFIS, see Jang et al. [10]. According to our previous study [9], both right and left vehicle drivers completed their yielding/preemptive decisions at 0.9 s before reaching the crossing point. To understand and model drivers' risk perception conveniently, we used the values of the input and output variables at the drivers' decision moment to train ANFIS of drivers' risk perception in this study. All the input variables were fuzzy variables, which should be described and measured using linguistic rather than precise numerical values. The membership functions of all the input variables were initially supplied exogenously. The output variable was the quantitative value of drivers' risk perception. For two drivers in a crossing process, if one driver decelerates or stops the vehicle before reaching the crossing point and makes a yielding decision, then this condition generally represents a more serious perceived risk of this driver than the other driver. Under this condition, we considered the risk perception value of this driver as "1." By contrast, if one driver makes a preemptive decision, then the risk perception value of this driver is smaller than that of the other driver. We considered the risk perception value of this driver as "0." The fuzzy logic toolbox in MATLAB was used to develop and test the drivers' risk perception models. The specific steps were described as follows.
Step 1: Generating and inputting training and testing data. We chose 100 crossing cases as training samples and extracted the training data for this study. The parameter data obtained from the other 50 cases were used as testing data. Step 2: Identifying the type of membership functions. In this study, a Gaussian membership function was selected to fuzzify all the input variables.
Step 3: Using the "genfis1" function to generate the original fuzzy inference structure (FIS).
Step 4: Setting the training parameters of ANFIS.
Step 6: Using the "evalfis" function to test the obtained performance of FIS.

Model results
Drivers' risk perception models can be obtained using the quantitative method described earlier. Fig. 2 shows the input-output risk perception values of drivers.   Fig. 2, it was known that the model result could possibly reflect the changes in drivers' risk perception within a small model error, which indicated that the model could satisfy practical requirements. The comparison between the observed and predicted values for drivers' risk perception also showed that the structure and algorithm of the established models were reasonable.

Indentifying acceptable risk perception level
Previous studies have indicated that the risk perception abilities of drivers varied under different characteristics of drivers during the actual driving process. However, each driver has a fixed level of acceptable risk perception that can guide his/her behavior. According to risk compensation theory (RCT), individuals tend to adjust their behavior in response to perceived changes in risk. After training, drivers should have a similar acceptable risk level in similar driving situations. In this study, we considered this level as a "range," which was assumed as [α, β]. According to risk homeostasis theory (RHT), drivers would adjust their driving procedures based on this range to maintain a stable and constant condition. When the perceived risk level of drivers was less than α, they would speed up to achieve their driving purpose. During this time, these drivers generally made preemptive decisions in the crossing process at unsignalized intersections. If the risk perception value was larger than β, then drivers would slow down and make yielding decisions to ensure safety. If the risk perception value of drivers was within the range of [α, β], then they would make either yielding or preemptive decisions when encountering another vehicle.
The acceptable risk perception level is related to the risk perception abilities of drivers and their driving operation. Thus, identifying the acceptable risk perception level is critical in this study. According to the established quantitative method, we can obtain the ranges of drivers' risk perception under preemptive and yielding conditions. Then, the acceptable risk level can be determined according to these two ranges. As shown in Fig. 3

Modeling method
The lack of control measures for right-of-way at unsignalized intersections in China makes gap-forcing behavior common [9]. The previous studies and models may not be applicable in China. Moreover, they cannot explicitly consider the dynamic interactions among drivers and cognitive decision features. To address such shortcomings, an approach based on game theory inspired by the early works of Kita et al. [11] was adopted to model drivers' behavior in the present study.
In general, various factors influence drivers' crossing behavior. Among these, the velocity change of vehicles can effectively reflect the change in drivers' risk perception and their decision behavior. To simplify the model, in this study, we analyzed the game process of drivers and established game behavior models based on the following factors: vehicle speed and drivers' risk perception. In addition, a previous study [11] showed that the interaction process between right and left drivers could be considered a zero-sum non-cooperative game Risk  According to game theory, the player set for the crossing process in this study was ‫ܥ‬ = ‫ܥ{‬ ோ , ‫ܥ‬ }, as shown in Fig. 1(c). ‫ܥ‬ ோ and ‫ܥ‬ might take different measures, such as acceleration, deceleration, or uniform motion, according to perceived risk. Thus, the strategy set for drivers could be considered ܵ = {ܵ ଵ , ܵ ଶ , ܵ ଷ } , where ܵ ଵ , ܵ ଶ, and ܵ ଷ denoted acceleration, uniform motion, and deceleration, respectively.
Assuming that the velocity of ‫ܥ‬ ோ and ‫ܥ‬ at the beginning of time step i were ܸ ோ and ܸ , their acceleration (or deceleration) were ܽ ோ and ܽ , and the distance from both vehicles to the crossing point were ‫ܮ‬ ோ and ‫ܮ‬ , respectively. Assuming that the risk perception values of ‫ܥ‬ ோ and ‫ܥ‬ drivers at time step i were ܲ ோ and ܲ , respectively. As mentioned earlier, , DISTANCE_DIFFERENCE (i.e., ‫ܮ‬ ோ − ‫ܮ‬ ), and SPEED_L (i.e., ܸ ) were the main factors that affected the risk perception of right vehicle drivers. Meanwhile, the main factors that affected the risk perception of left vehicle drivers were SPEED_DIFFERENCE, CROSS_TIME_R (i.e.,ܶ ோ ), and CROSS_DISTANCE_R (i.e., ‫ܮ‬ ோ ). We assumed that all drivers would continue driving with the same acceleration at time step i to reach the crossing point. Thus, ܶ ோ can be calculated using Equation 1: Then, ܲ ோ and ܲ could be obtained according to the established risk perception models.
At time step i+1, the risk perception value of ‫ܥ‬ driver (ܲ ାଵ ) could be determined by considering parameters (ܸ ோ ାଵ − ܸ ାଵ , ܶ ோ ାଵ , and ‫ܮ‬ ோ ାଵ ) in the risk perception model of left vehicle drivers. Where, Similarly, the risk perception values of both drivers at time steps ݅ + 2, ݅ + 3 to N could be calculated.
For both drivers, when the increase in velocity change was high and the risk perception value was low, the payoff of the drivers was high. Thus, we defined the payoff function of the drivers as follows: where ݆ = ܴ, ‫;ܮ‬ ݅ = 1,2, … , ܰ. ‫ܨ‬൫∆ܸ ൯ and ‫1(ܩ‬ − ܲ ) denoted the normalization for ∆ܸ and (1 − ܲ ) , respectively. ݉ and ݊ represented the expectation coefficients of the drivers to vehicle speed and risk perception, and ݉ ≥ 0, ݊ ≥ 0, ݉ + ݊ = 1. In actual driving processes, the speed and risk perception factors had different influences on drivers' payoff. If the risk perception value of a driver was lower than his/her acceptable risk perception level, then he/she would focus on improving vehicle speed. If the perceived risk value of a driver was higher than his/her acceptable risk perception level, then he/she would focus on changing risk perception. However, if the risk perception value of a driver was within the range of the acceptable risk level, then we could conclude that the speed and risk perception factors were equally important to the driver's payoff. Therefore, based on the acceptable risk perception level of drivers [ߙ , ߚ ], the speed coefficient ݉ can be defined as follows: To analyze the game behavior between ‫ܥ‬ ோ and ‫ܥ‬ drivers, solving the game model at each time DOI: 10.1051/ 0 (2016) matecconf/201 MATEC Web of Conferences step was necessary. The method for this process was as follows.
Step 1: According to the established motion state and risk perception model, the velocity changes of the two drivers and their risk perception values at the first time step (i.e., ∆V ୖ ଵ , ∆V ଵ , P ୖ ଵ , and P Step 2: The payoff values of the two drivers under different strategy combinations were evaluated based on Equation 5.
Step 3: The Nash equilibrium strategy at the first time step was obtained according to the payoff value, and consequently, the behavior strategies of the two drivers were determined. Then, the motion information of the two vehicles at the beginning of the second time step was calculated based on the determined behavior strategies.
Step 4: According to the motion state and the established risk perception models, the velocity changes of the two drivers and their risk perception values at the second time step under different strategy combinations were calculated. By repeating steps 2 and 3, the behavior strategies of the two drivers at the second time step and the motion state of the two vehicles at the beginning of the third time step were obtained.
Step 5: Similarly, the behavior strategies of the two drivers at time steps 3 to N were calculated.

Model results
To facilitate understanding, we analyzed the game behavior of the drivers within a certain period as follows: 2 s before the preemptive driver reached the crossing point. Then, we divided this period into 10 time steps. Each time step was 0.2 s. According to related literature, we assumed that if C ୖ and C drivers chose acceleration strategy (S ଵ ), then acceleration was considered 1 . 0 m/s ଶ (i.e., a ୖ = 1.0 m/s ଶ , a = 1.0 m/s ଶ ). If the two drivers adopted deceleration strategy ( S ଷ ), then deceleration was determined as 1.5 m/s ଶ (i.e., a ୖ = −1.5m/s ଶ , a = −1.5 m/s ଶ ). If the drivers chose uniform motion (S ଶ ), then a ୖ = a = 0. Based on Equation 5, the payoff values of the two drivers under different strategy combinations at the first time step were calculated. The result showed that at the first time step, the Nash equilibrium strategy was both ‫ܥ‬ ோ and ‫ܥ‬ decelerated when the payoffs of ‫ܥ‬ ோ and ‫ܥ‬ drivers were the largest, that is, 0.47 and 0.54, respectively. Then, we could determine the motion information of the two vehicles at the beginning of the second time step according to the estimated strategies: ‫ܮ‬ ோ ଶ = 11.27 m , ‫ܮ‬ ଶ = 7.97 m , ܸ ோ ଶ = 4.7 m/s , and ܸ ଶ = 4.5 m/s . Similarly, the Nash equilibrium at the second time step could be inferred, when both C ୖ and C also decelerated.
Based on this analogy, the Nash equilibrium strategy at each time step could be determined. Table 1 shows the estimated drivers' strategies when reach each Nash equilibrium.

Model test
As shown in Table 1, both ‫ܥ‬ ோ and ‫ܥ‬ drivers decelerated at the beginning of the crossing game, and then ‫ܥ‬ continued to decelerate; however, ‫ܥ‬ ோ chose to accelerate. According to the estimated strategies, vehicle speed and distance to the crossing point at the beginning of each time step could be calculated. Fig. 4 shows the comparison between the estimated and the actual results.   Table 2 and Equation 2, the distance from both vehicles to the crossing point at the beginning of each time step could be estimated. Fig. 4(b) showed the estimated distance from the two vehicles to the crossing point could accurately reflect the actual distance. In addition, ‫ܥ‬ ோ was estimated to be closer to the crossing point than ‫ܥ‬ at the last time step. That was, ‫ܥ‬ ோ driver would make a preemptive decision and pass the crossing point first. This result was also consistent with the observed data. In this study, we also established the game models for another 50 crossing cases, and then calculated the game strategies of the drivers. We estimated vehicles' speed, distance from the vehicles to the crossing point, and final preemptive or yielding decisions of the drivers. Table 2 shows the error of the estimated speed and distance. As shown in Table 2, the error between the estimated and the actual results was minimal. It indicated the established models could efficiently describe drivers' behavior. By comparing the estimated and the actual preemptive/yielding decision, 48 cases were estimated accurately. Accuracy was 96%.

Discussion and conclusions
In this study, we focused on typical crossing cases at unsignalized intersections. In each case, one straightmoving vehicle encountered another straight-moving vehicle. Based on one of our previous studies on typical crossing cases, the main parameters that affected the risk perception of right and left vehicle drivers were determined. Then, we used these parameters to establish the quantitative models of drivers' risk perception based on ANFIS. By comparing the actual data and the model prediction results, we verified the practicality of the proposed quantitative risk perception models. Then, according to a data analysis method that we developed, the acceptable risk perception levels for right and left vehicle drivers were identified.
Based on game theory, the relationship among the quantitative value of risk perception, the acceptable risk perception level, and vehicle motion state was analyzed. By analyzing the Nash equilibrium strategy, the acceleration/deceleration strategies of the drivers at each time step could be identified. Then, we could calculate vehicle speed and the distance from a vehicle to the crossing point at the beginning of each time step. It was shown that the estimated results were consistent with the fact. By analyzing a test intersection, the established game models were proven to have good applicability.
In summary, this study could promote the development of risk analysis theory from qualitative analysis to quantitative calculation, as well as expand the application of this theory to traffic safety. Moreover, we considered drivers' risk perception factor and used game theory to model their behavior. Such efforts made the model accurate and reasonable. Furthermore, the results of this study could provide a good theory and algorithm for the microscopic simulation and active safety control system of vehicles.