## 3 Risk Perception and Fuzzy Decision-Based Driving Behavior Model
### Fuzzy Decision  
Fuzzy decision-making theory is intended to handle decision-making problems characterized by uncertainty, vagueness, and subjectivity. It has been effectively applied to the modeling of driving behavior [54; 55; 56]. When the leading vehicle approaches a signalized intersection during the yellow light phase, a decision must be made whether to proceed or stop. In our previously developed risk-perception-based model, each driver's desired risk was assumed to be a fixed value. However, this assumption overlooks the inherent uncertainty influenced by various factors, such as individual personality traits and road environment conditions. Therefore, in this study, we improve the model by incorporating fuzzy decision-making based on the distance to the stop line, current speed, and perceived risk of the traffic signal. For following vehicles, which are not responsible for leading decisions, their motion decisions are simplified by considering only the perceived risk induced by the preceding vehicle. We used 2303 yellow phase-approaching cases extracted from high-resolution event-based data collected at six intersections along Trunk Highway 55 in the Twin Cities area, as shown in Figure 3. The data were gathered using the SMART-SIGNAL system developed by the University of Minnesota, which continuously records vehicle detector actuations and signal phase changes [57; 58]. The cases were investigated from west to east in the direction of the particular phase, in which the green phase time is 110 s, yellow phase time is 5.5 s, and the total cycle length is 210 s, including detailed timestamps of vehicle arrivals at the stop line after yellow onset. The data did not include information on the speed limit at the intersection entrance or on the upstream road segment. However, analysis of the 2303 cases shows that the approach speeds follow a normal distribution, with a mean of 22.94 m/s and a standard deviation of 3.57 m/s and values ranging from 6.40 m/s to 29.92 m/s. The acceleration data also follow a normal distribution, with a mean of 0.61 m/s\({}^{2}\), a standard deviation of 1.41 m/s\({}^{2}\), and a range from \(-\)7.92 m/s\({}^{2}\) to 3.93 m/s\({}^{2}\). Among the cases, 1833 involved accelerations through the intersection, while 470 involved decelerations. As observed from the cases, even when a vehicle is very close to the stop line, there are instances where a deceleration decision is still made due to the predicted signal risk  
Figure 3: Data collection site. being high and approaching 1. This occurs even though the remaining distance to the stop line may be insufficient to meet the braking requirements, potentially resulting in a yellow-running outcome despite the driver's intention to stop. Therefore, in addition to the commonly considered factors, i.e., distance to the stop line and vehicle speed at the onset of the yellow phase, we introduced the perceived signal risk calculated using Equation (9) as the third factor in the decision-making process. The marginal histograms illustrating the distributional relationships among these three factors are shown in Figure 4.  
Based on the 2303 yellow phase-approaching cases, membership functions for the three input factors were constructed, as shown in Figure 5. Subsequently, through fuzzy inference and defuzzification (the principles and procedures for creating fuzzy rules are not elaborated here), a fuzzy decision tree representing driver behavior during the yellow phase was developed, as illustrated in Figure 6. The output of the model is the probability of passing or not passing the stop line. An overall prediction accuracy of 80% was achieved for classifying pass and no-pass decisions. Accordingly, we characterize the driver's behavior approaching intersections during the yellow phase through the lens of risk perception and fuzzy decision-making, without limiting the analysis to whether the vehicle was located within the dilemma zone.