Exploratory Advanced Research Program

VASTO - Evolutionary Agent System for Transportation Outlook

Agent-Based Modeling and Simulation in The Dilemma Zone

5 Analysis Results

5.1 ABMS Scenario

5.1.1 DZ Model

The proposed driver behavior model was implemented using ABMS software. The simulation scenario was a segment of an urban arterial with three lanes in each direction containing two signalized intersections.

Figure 5-1: Image. Screenshot of simulation.

In the simulation (Figure 5-1), the nearest leading vehicle in the adjacent lane was considered the adjacent vehicle. The simulation time horizon was 30,000 hours and the number of replications was 30. Initial speed of vehicles was assumed to be uniformly distributed between 23.3 mi/h and 30 mi/h. Moreover, the coming flow rate was 1,200 vehicles/hour and perception-reaction time (δ) was 0.7 seconds. The number of sampled vehicles is 1,000. The free flow speed of each group was given by drivers' data collected by the HDS. Free flow speed of group A, group B, group C, and group D was Normal (38.89, 3.34) mi/h, Normal (42.38, 4.93) mi/h, Normal (52.67, 3.12) mi/h, and Normal (57.30, 3.34) mi/h, respectively. Parameters in the parenthesis represent mean and variance of the Normal distribution. Based on collected data from the HDS, decisionmaking behaviors of the drivers in the four groups were developed via the E-BDI framework (see section 5.2 for more details). The constructed simulation models were executed under the two different facility speed limit cases (40 mi/h and 55 mi/h), as mentioned in section 4.1.1.

5.1.2 E-BDI Framework

The E-BDI framework was used as an underlying model in ABMS to mimic participants' decisionmaking behaviors shown in the DZ experiments conducted with the HDS. Because E-BDI generates a multistage plan via probabilistic inference, it is able to represent the uncertain perception and reasoning processes of humans. ⁽²⁾

In the E-BDI framework, once an agent perceives the road condition (e.g., traffic), the perceptual processor in the Belief module creates Beliefs via the Bayesian Belief Network (BBN). The BBN uses conditional probabilities between variables to infer states of attributes (e.g., travel time) from the observed road conditions (e.g., free flow speed, traffic volume, and road length). Thus, each agent can have its own Beliefs about the attributes based on the BBN trained by its own experience. ⁽²⁾

Then, a real-time planner creates a multistage plan from the Beliefs. In this process, Extended Decision Field Theory (EDFT) is used to select one option from multiple alternatives regarding interdependencies between alternatives. ^{(17) (18)} It calculates the preference values of all alternatives based on perception and attention of a human on each attribute of each option (see section 0). Once the preference values converge, EDFT selects one option that has the highest preference value. By running EDFT multiple times with BBN, choice probabilities for all of the alternatives can be computed. More details on this work can be found in Lee, Son, and Jin (2010).

5.2 ABMS Model Calibration and Validation

5.2.1 Calibration of BBN

In the E-BDI framework, a driver's decisionmaking behavior in the DZ was represented by BBN with EDFT. To be more specific, the BBN provides the states of the attributes of each option from the environmental factors, and the EDFT computes the preferences on each option regarding the psychological aspect of driver deliberation. As mentioned in section 4.1.1, there were four groups classified by facility speed and degree of driving in a hurry. Each group had three environmental factors (i.e., RedLightCam, PedCountSig, and AdjVehBeh). In addition to these three factors, two more factors (distance to stop line of a vehicle (Distance) and approaching speed of a vehicle at the onset of the yellow phase (AppSpeed)), as included in the existing ITE DZ model, ⁽¹⁾ were considered to represent the status of an individual vehicle. To identify the impacts of the three environmental factors (i.e., external information) and the status of the individual vehicle (i.e., internal information) on the driver's decision in the DZ, external information and internal information were considered as attributes in the E-BDI framework.

The challenge of calibration is how to measure the correlations between the five observed factors and two attributes (i.e., two latent variables: internal information and external information). Although the status of the attributes is not observed by the HDS, the status can be inferred from the five observed factors and the drivers' decisions by an SEM approach. Structural equation modeling is a statistical technique for testing and estimating causal relations using a combination of statistical data and causal assumptions that has been widely used to find the relationships between observed variables and latent variables. ⁽¹⁹⁾ Figure 5-2 shows the hypothetical model for SEM analysis. ⁽⁹⁾

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Figure 5-2: Chart. Path model in SEM analysis. ⁽⁹⁾

Table 5-1 shows the standardized regression weights of each group. ⁽⁹⁾

Table 5-1: Standardized regression weights of driver groups. ⁽⁹⁾

To evaluate the validity of the developed models, a χ² goodness of fit test was conducted (H0: the data are consistent with the developed model; Ha: the data are not consistent with the developed model). The χ²/df value of the models was 1.846/7 (group A), 1.206/7 (group B), 2.274/7 (group C), and 3.926/7 (group D). In general, if χ²/df is less than 3, the model would be considered a desirable model for the given dataset. ⁽¹⁹⁾ Thus, the path model shown in Figure 5-2 was used as a fitted model to explain the collected data by HDS in this study (the model can be different based on a collected dataset). Regarding the standardized regression weight of each group, drivers in group A, group B, and group D were more sensitive to external information than internal information, i.e., Decision <- Internal information.

This sensitivity implies that the existing ITE DZ model, which mainly considers AppSpeed and Distance, would not be sufficient to explain the driver's decision when the driver also had external information. The gap in the standardized regression weights between internal information and external information in affecting a decision also shows that group B (i.e., |0.261-1.361| = 0.900) and group D (i.e., |0.270-1.032| = 0.762) cared more about external information than group A (i.e., |0.245-1.043| = 0.798) and that for group C, there was no impact of external information. In other words, external information was considered to a significant extent by drivers in the in-a-hurry situation.

From the SEM approach, the status of each attribute was estimated. Figure 5-3 shows the developed BBN. ⁽⁹⁾ In this process, the values of each attribute were discretized and normalized to use the same scale in the EDFT algorithm for each option. In fact, if there was no significant impact on the inference accuracy of the BBN, discretization of a continuous factor reduced the space complexity of the BBN so that inference process using the BBN could be improved.

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Figure 5-3: Chart. BBN for E-BDI framework. ⁽⁹⁾

The advantage of BBN is its ability to infer the status of attributes even though only partial observations of the parent variables (i.e., AppSpeed, Distance, RedLightCam, PedCountSig, and AdjVehBeh) were available. Additionally, because it is a probabilistic model using conditional probabilities between variables, BBN could handle the driver's uncertain perception process.

5.2.2 Calibration of EDFT

The calibration of EDFT is the process to find the parameters of EDFT to mimic real drivers' decisionmaking behaviors in the DZ.

Figure 5-4 shows the EDFT used in this study. ⁽⁹⁾

Figure 5-4: Equation. EDFT used in the DZ. ⁽⁹⁾

The first term on the right-hand side includes a stability matrix (S) to consider the effect of the preference at the previous state (the memory effect) and preference values (P) of two alternatives at time t. The second term on the right-hand side includes a contrast matrix (C) regarding the correlation between alternatives, a value matrix (M(t+h)) to represent the subjective perception of a human on each attribute of each option, and a weight vector (W(t)) to assign the weights of attention on each attribute. ^{(9) (17) (18)}

By fixing the values of the stability matrix S and contrast matrix C, the calibration process could be simplified as a searching process for the appropriate weight vector (W(t)) of each driver group under the given value matrix M(t). In this study, by comparing the decisions of real drivers with the inferred decisions given by the E-BDI framework, the weight vector W(t) with the highest prediction accuracy was found. Table 5-2 ⁽⁹⁾ reveals the selected weight vectors. ⁽⁹⁾

Table 5-2: Weight vector of driver groups.

Weight Vector	Group A	Group B	Group C	Group D

5.3 Modeling Experiments

To analyze the simulation results, a Speed-Distance (SD) diagram was used. The SD diagram shows drivers' decisions, together with the vehicles' approaching speed and distance to the stop line, at the onset of the yellow phase. In an SD diagram, there are two curves; one is called the stopping curve and the other is called the crossing curve. ⁽⁹⁾ The stopping curve represents the limited stopping distance under different vehicle speeds with respect to comfortable deceleration (i.e., -3.72 m/sec² in this research, as given by the Traffic Engineering Handbook). ⁽¹³⁾ Similarly, the crossing curve represents the limited crossing distance under different vehicle speeds within the length of the yellow phase.

By using these two curves, the space in the SD diagram can be separated into four zones: (1) cross zone-space beyond both curves, (2) stop zone-space under both curves, (3) option zone-space under stopping curve and above crossing curve, and (4) DZ-space under crossing curve and above stopping curve. A driver is able to clear the intersection with the current approaching speed before the traffic signal turns red in the cross zone and to stop comfortably at the stop line in the stop zone. Otherwise, a driver must choose either to go or to stop when the driver is in the DZ or the option zone.

5.3.1 Low-Facility Speed Limit

In this scenario, the simulation model consisted of 50 percent group A vehicles and 50 percent group B vehicles.

Figure 5-5 ⁽⁹⁾ and Figure 5-6 ⁽⁹⁾ show the experimental results under different configurations of RedLightCam and PedCountSig (i.e., presence or not). On the one hand, among the vehicles in Figure 5-5 (no RedLightCam and no PedCountSig), 36.31 percent cleared the intersection successfully (red stars), 60.71 percent stopped (blue circles), and 2.98 percent chose to cross but had to run a red light. On the other hand, among all vehicles in Figure 5-6 (RedLightCam and PedCountSig present), 36.37 percent cleared the intersection successfully (red stars), 61.22 percent stopped (blue circles), and 2.41 percent chose to cross but had to run a red light (black triangles). According to the definition of the option zone, there were no vehicles running a red light in the option zone because drivers could either stop their vehicles or go through an intersection before the traffic signal changed to a red light. However, because of traffic on the roadway, drivers could not suddenly change their vehicle speed. In other words, the actual maximum acceleration rate and deceleration rate were different from the theoretical values. Thus, vehicles that ran red lights were detected in the simulation model. In addition, comparing the proportion of red-light running vehicles in the two figures, when RedLightCam and PedCountSig were present, they were able to reduce the number of red-light running vehicles (i.e., from 2.98 percent to 2.41 percent).

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Figure 5-5: Chart. SD diagram of Group A (RedLightCam=no and PedCountSig=no).⁽⁹⁾

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Figure 5-6: Chart. SD diagram of Group A (RedLightCam=yes and PedCountSig=yes).⁽⁹⁾

When the red-light photo enforcement camera and pedestrian countdown signal do not exist at a signalized intersection (no RedLightCam and no PedCountSig), among all vehicles in group B, 36.60 percent cleared the intersection successfully (red stars), 60.85 percent stopped (blue circles), and 2.55 percent chose to cross but had to run a red light (black triangles) to do so.

On the other hand, when the RedLightCam and PedCountSig present, 39.38 percent cleared the intersection successfully (red stars), 58.94 percent stopped (blue circles), and 1.68 percent chose to cross but had to run a red light (black triangles). As found for group A, the presence of RedLightCam and PedCountSig reduced the number of vehicles running red lights. Significantly, group B had a higher reduction in the amount of red-light running vehicles (0.87 percent) than group A (0.57 percent) because drivers who were driving in the in-a-hurry situation cared more about the external information than other drivers who were driving in a normal situation.

Table 5-3: Summary of experiments (group A: 90 percent; group B: 10 percent).

Table 5-4: Summary of experiments (group A: 10 percent; group B: 90 percent).

Table 5-3 and Table 5-4 show the average numbers of crossed, stopped, and red-light running vehicles collected by 30 replications of each scenario. As previously mentioned, the presence of RedLightCam and PedCountSig reduced the number of red-light violations for group A and group B. In addition, the impact of RedLightCam and PedCountSig on drivers' decisions increased when the roadway had a greater number of group B vehicles, as shown in the tables on the reduction of red-light violations. Table 5-3 had a 0.47 percent reduction for group A and a 0.76 percent reduction for group B, and Table 5-4 had a 0.72 percent reduction for group A and 1.14 percent for group B because the drivers in group B considered the external information more significantly than those in group A. Moreover, group A had more red-light running vehicles than group B. This result meant that the approaching speed of group A was too slow to cross the intersection within the yellow phase (group A has Normal (38.89, 3.34) mi/h speed distribution and group B has Normal (42.38, 4.93) mi/h speed distribution).

5.3.2 High-Facility Speed Limit

Unlike group A and group B, drivers' decisions in group C were only related to the internal information mentioned in section 5.2.

Among all vehicles in group C, 56.24 percent cleared the intersection successfully (red stars), 42.26 percent stopped (blue circles), and 1.50 percent chose to cross but had to run the red light (black triangles). Comparing the distribution of decisions with group A, the number of red-light running vehicles in group C was smaller than the number of red-light running vehicles in group A. Noting that group C drivers had a 55 mile-per-hour speed limit, this smaller number of violations meant that drivers could make clear decisions under the high-facility speed limit (FacilitySpeed) under the experimental settings.

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Figure 5-7: Chart. SD diagram of group D (RedLightCam=no and PedCountSig=no).⁽⁹⁾

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Figure 5-8: Chart. SD diagram of group D (RedLightCam=yes and PedCountSig=yes).⁽⁹⁾

Group D, however, had different decisionmaking patterns from group C. Figure 5-7 ⁽⁹⁾ and Figure 5-8 ⁽⁹⁾ show the experimental results of group D. Among the vehicles in Figure 5-7, 57.72 percent cleared the intersection successfully (red stars), 40.68 percent stopped (blue circles), and 1.60 percent chose to cross but had to run the red light (black triangles).

Comparing the distribution of decisions in group D with group C, an urgent situation (i.e., driving in a hurry) made drivers have uncertain decisions. In other words, it increased the potential risk of a car crash at an intersection. If RedLightCam and PedCountSig were present at the intersection, the uncertainty was significantly mitigated. In Figure 5-8, 55.25 percent cleared the intersection successfully (red stars), 43.29 percent stopped (blue circles), and 1.46 percent chose to cross but had to run the red light (black triangles).

Similar to the low-facility speed limit case, experiments under different configurations of group C and group D were conducted. Table 5-5 and Table 5-6 show the average numbers of vehicles that crossed, stopped, red-light-running vehicles collected by 30 replications of each scenario. As indicated in Figure 5-7 and Figure 5-8, the presence of RedLightCam and PedCountSig reduced the number of red-light running vehicles. The number of red-light running vehicles decreased as the percentage of group D increased from 90 percent to 10 percent. The reduction in the amount of red-light running vehicles of group D was 0.06 percent in Table 5-5 and 0.43 percent in Table 5-6. Group C had a 0.01 percent reduction in Table 5-5 and 0.09 percent reduction in Table 5-6.

In fact, because the decision model of group C did not include external information as an attribute, the number of light violations of group C should not be affected by the presence of RedLightCam and PedCountSig. However, group C shared a roadway with group D, which was influenced by the external information so that group C was indirectly affected by the external information via physical interactions (i.e., actual movements of vehicles) with vehicles in group D. Therefore, in group C, there were small differences according to the presence of RedLightCam and PedCountSig.

Table 5-5: Summary of experiments (group C: 90 percent; group D: 10 percent).

Table 5-6: Summary of experiments (group C: 10 percent; group D: 90 percent).

5.4 Discussions

The E-BDI-based DZ model was proposed to predict drivers' decisions at an intersection. The proposed model can be used to assist in intersection design to reduce the potential risk of crashes as a result of drivers' responses. In addition to the two factors considered in the ITE DZ model (i.e., AppSpeed and Distance), various external factors in the roadway environment (i.e., RedLightCam, PedCountSig, and AdjVehBeh) were considered. The first two factors were defined as internal information, and the other three factors were defined as external information. The internal and external information were used as attributes in the E-BDI-based DZ model.

From the drivers' responses collected by the HDS, SEM analyzed the impact of unobserved attributes (i.e., internal information and external information) on drivers' decisions in the DZ.

According to the standardized regression weights given by SEM, most driver groups, except group C, used both external and internal information for their decisionmaking.

The SEM analysis results were utilized to calibrate BBN and EDFT in the E-BDI-based model to understand the impact of internal and external information on drivers' decisionmaking behaviors in the DZ.

According to the calibration results, when drivers were in a hurry situation (i.e., group B and group D), drivers' decisionmaking behaviors were significantly affected by the external information.

Agent-based modeling and simulation with the calibrated E-BDI-based DZ model showed the impact of external information via SD diagrams. It was executed with multiple vehicles to consider physical interactions between vehicles. The experimental results revealed that drivers tended to have a higher number of red-light running vehicles when a pedestrian countdown signal and a red-light photo enforcement camera were not installed at an intersection. The presence of a pedestrian countdown signal and red-light photo enforcement camera were able to reduce the number of red-light running vehicles significantly, particularly when drivers were in a situation that called for them to be in a hurry. The findings corresponded to the results from the E-BDI-based DZ model calibration. This is a good proof of concept for the potential development of a model to predict driver decisions in the DZ via ABMS with the E-BDI-based DZ model.