# Dilemma Zone: Modeling Drivers’ Decision at Signalized Intersections against Aggressiveness and Other Factors Using UAV Technology

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Defining Dilemma Zone

_{c}) and the maximum yellow passing distance (X

_{0}). For the calculation of X

_{c}and X

_{0}, the following equations are used, while constant parameter values are usually applied for perception/reaction times and acceleration/deceleration rates [2,5,10].

- V
_{0}is the vehicle’s approaching speed - δ
_{stop}is the perception/reaction time for safe stopping - α
_{stop}is the maximum deceleration rate for safe stopping - δ
_{pass}is the perception/reaction time for safe passing - α
_{pass}is the maximum acceleration rate for passing - τ is the yellow time interval
- W is the width of the crossing road
- L is the typical vehicle length

_{c}> X

_{0}, the distance between X

_{c}and X

_{0}is defined as the type I dilemma zone. When X

_{c}< X

_{0}, the distance between X

_{c}and X

_{0}is defined as the option zone. While a driver caught in the dilemma zone can neither comfortably stop before the stop line nor clear the intersection successfully, the option zone is an area where either stopping or crossing can be performed successfully [2,7].

## 3. Literature Review

## 4. Materials and Methods

#### 4.1. Study Area

#### 4.2. Data Collection

#### 4.3. Data Analysis

- Approaching speed (from the onset of the yellow signal until the moment the vehicle stopped or passed the stop line)
- Distance to stop line (from the onset of the yellow signal until the moment the vehicle stopped or passed the stop line)
- Acceleration/deceleration (from the onset of the yellow signal until the moment the vehicle stopped or passed the stop line)
- Driver’s decision to stop or clear the intersection
- Type of vehicle
- The position of the vehicle in case a platoon is formed (Platoon leader, 1st or 2nd follower)

- Approaching speed (at the onset of the yellow signal)
- Average speed (between the initiation of the yellow signal and the moment the vehicle stopped or passed the stop line)
- Distance to stop line (at the onset of the yellow signal)
- Acceleration/deceleration (at the onset of the yellow signal and more precisely 0.5 s after the initiation of the yellow signal, for ensuring that perception/reaction time, assumed 1.5 s, has not elapsed)
- Average acceleration/deceleration (between the initiation of the yellow signal and the moment the vehicle stopped or passed the stop line)
- Existence of an approaching speed greater than the posted speed limit
- Categorization of drivers based on their behavior (if a driver stopped before or after the stop line, or if he/she crossed the intersection with yellow or red signal)

- Calculation of safe stopping distance (SSD) and critical crossing distance (CCD) for all vehicles (based on type I dilemma zone Equations (1) and (2), and assuming constant values for perception/reaction time = 1.5 m/s
^{2}and maximum acceleration and deceleration rates = 3.5 m/s^{2}) - Calculation of vehicle’s relative position (based on the safe stopping distance (SSD), critical crossing distance (CCD) and the actual distance to stop line)

#### 4.4. Modeling Drivers’ Behavior

#### 4.4.1. Formulation of Initial Binary Logistic Model

- Approaching speed (both at the onset of the yellow signal and average)
- Distance to stop line
- Acceleration/deceleration (both at the onset of the yellow signal and average)
- Other potential explanatory variables, including drivers’ position in the platoon (platoon leader, 1st and 2nd follower), lane change, etc.

- P
_{i}is the probability of the ith case to stop - Z
_{i}is the result of a linear function of the various factors (explanatory variables)

- The Nagelkerke R Square index, which gives an indication of the size of the sample variance that is ultimately interpreted by the regression. The closer to 1 is the value of this indicator, the better the model adapts to the sample data.
- Hosmer and Lemeshow test has been also used to check the proper adaptation of the sample data. Values of sig.> 0.05 at significance level a = 95% indicate that the model is well adapted to the data.
- Another measure of the good adaptation of the model is the SPSS Classification Table, which compares the observed probabilities with those provided for by the model. The higher the percentage of cases of the dependent variable correctly predicted based on the model, the better the model adjustment [57].

#### 4.4.2. Formulation of Latent Class Model

- y
_{n}is the nth observation of the manifest variables - S is the number of classes
- π
_{j}is the prior probability of membership in class j - P
_{j}is the class specific probability of y_{n}given the class specific parameters θ_{j} - θ
_{j}are the class specific parameters

^{2}and +0.9 m/s

^{2}. The thresholds used for the approaching speed and acceleration/deceleration recoding are presented in Table 1.

#### 4.4.3. Formulation of Final Binary Logistic Model

## 5. Results

#### 5.1. Sample Statistics

#### 5.2. Initial Binary Logistic Regression Model Results

^{st}or 2

^{nd}followers only if the lead driver crossed the intersection and, therefore, the following drivers had a choice to do the same).

#### 5.3. Latent Class Analysis Results: Driver Classification according to Aggressiveness

#### 5.4. Final Binary Logistic Regression Model Results

_{s}= 0.230).

^{2}are almost 60% more likely to cross the intersection. The corresponding acceleration/deceleration value for conservative drivers is over 0.50 m/s2. For all relevant positions that drivers can be found when they face the yellow signal, the acceleration/deceleration value that increases the chance of drivers crossing the intersection is relatively lower for aggressive drivers than for conservative ones. This practically means that aggressive drivers are more willing to cross the intersection even if they are decelerating on the start of the yellow phase, in contrast to the more conservative drivers for whom the probability of passing is associated with higher acceleration values.

## 6. Discussion–Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Variable | Variable Coding | Description | Min | Max | Mean | Std. Deviation |
---|---|---|---|---|---|---|

Acceleration Deceleration Yellow (m/s^{2}) | Acceleratio_Deceleration_Yellow | Measure of acceleration or deceleration on the yellow signal | −2.79 | 10.06 | 0.82 | 1.10 |

Average Speed (m/s) | Average_Speed | Average speed between yellow signal and stop line | 2.81 | 34.73 | 19.23 | 7.07 |

Speed Yellow (m/s) | Speed_Yellow | Measure of speed on the yellow signal | 3.92 | 32.95 | 20.60 | 4.21 |

Distance Yellow (m) | Distance_Yellow | Distance from stop line on the yellow signal | 0.85 | 129.90 | 67.09 | 34.86 |

Average Acceleration Deceleration Rate (m/s^{2}) | Average_Acceleration_Deceleration_Rate | Average acceleration or deceleration rate between yellow signal and stop line | −5.09 | 5.49 | 0.16 | 1.73 |

SSD (m) | SSD | Safe Stopping Distance | 8.06 | 204.53 | 95.76 | 30.39 |

CCD (m) | CCD | Critical Crossing Distance | 13.30 | 129.44 | 81.01 | 16.49 |

Variable | Variable Coding | Description | Range | Frequency |
---|---|---|---|---|

Category | Category | Classification of drivers based on their behavior | A: Stop | A: 32.10% |

B: Pass after stop line | B: 2.60% | |||

C: Pass with yellow | C: 57.70% | |||

D: Pass with red | D: 7.60% | |||

Platoon Leader | Platoon_Leader | The first car in case of platoon on the onset of yellow | 0: No | 0: 28.80% |

1: Yes | 1: 71.20% | |||

1st Follower | 1_Follower | The second car in case of platoon on the onset of yellow | 0: No | 0: 77.30% |

1: Yes | 1: 22.70% | |||

2nd Follower | 2_Follower | The third car in case of platoon on the onset of yellow | 0: No | 0: 93.80% |

1: Yes | 1: 6.20% | |||

Decision | Decision | Decision to stop or pass the stop line | 0: Stop | 0: 34.70% |

1: Pass | 1: 65.30% | |||

Decision Previous | Decision_Previous | Decision to stop or pass the stop line of previous car (for 1st and 2nd followers only) | 0: Stop | 0: 12.40% |

1: Pass | 1: 87.60% | |||

Change Lane | Change_Lane | Change lane between the onset of the yellow signal and the stop line | 0: No | 0: 98.10% |

1: Yes | 1: 1.90% | |||

Greater than Speed Limit | Greater_than_Speed Limit | The speed on yellow signal is above 70 km/h | 0: No | 0: 33.70% |

1: Yes | 1: 66.30% | |||

Acceleration | Acceleration | Acceleration or deceleration after the onset of the yellow signal | 0: No | 0: 35.40% |

1: Yes | 1: 64.60% | |||

Relative Position | Rel_Position | Vehicle’s relative position, speed and acceleration on the onset of the yellow signal | 1: Obvious Decision Stop | 1: 31.20% |

2: Option Zone | 2: 0.60% | |||

3: Dilemma Zone | 3: 9.30% | |||

4: Obvious Decision Pass | 4: 58.90% | |||

Type of vehicle | Type_of_vehicle | Type of vehicle | 1: Car | 1: 91.40% |

2: Truck | 2: 5.30% | |||

3: Moto | 3: 3.20% |

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**Figure 1.**Types of dilemma zone: (

**a**) Type I dilemma zone, (

**b**) Option zone, (

**c**) Type II dilemma zone.

**Figure 8.**Distribution of decision probability depending on changes in acceleration and the relative position of vehicles (aggressive drivers).

**Figure 9.**Distribution of decision probability depending on changes in acceleration and the relative position of vehicles (non-aggressive drivers).

Variable Initial Coding | New Recoded Variable | Recoding Thresholds | Recoding Values | Interpretation |
---|---|---|---|---|

Speed_Yellow | Speed_Yellow_Recoded | Speed_Yellow ≤ 60 km/h | 1 | Low Approaching Speed |

60 km/h < Speed_Yellow ≤ 70 km/h | 2 | Medium Approaching Speed | ||

70 km/h < Speed_Yellow ≤ 80 km/h | 3 | High Approaching Speed | ||

Speed_Yellow > 80 km/h | 4 | Very High Approaching Speed | ||

Acceleration_Deceleration_ Yellow | Acceleration_Deceleration_Yellow_Recoded | Acceleration_Deceleration_Yellow < −0.9 m/s^{2} | 1 | High Deceleration |

0.9 m/s^{2} ≥ Acceleration/Deceleration_Yellow ≥ −0.9 m/s^{2} | 2 | Medium Deceleration/Acceleration | ||

Acceleration_Deceleration_Yellow > 0.9 m/s^{2} | 3 | High Acceleration |

Number of Latent Classes | BIC |
---|---|

2 | 3714.978 |

3 | 3441.026 |

4 | 3332.407 |

5 | 3290.169 |

6 | 3312.328 |

Variable | Description | Behavior | Mean | Std. Deviation | Minimum | Maximum |
---|---|---|---|---|---|---|

Distance (m) | Distance from stop line at the onset of the yellow signal | Stop | 97.14 | 21.78 | 39.06 | 128.90 |

Go | 48.39 | 23.89 | 6.55 | 104.10 | ||

Passed with red | 95.80 | 16.81 | 61.15 | 127.90 | ||

Stop after stop line | 86.54 | 18.69 | 61.32 | 125.70 | ||

Speed (m/s) | Approaching speed at the onset of the yellow signal | Stop | 17.66 | 3.59 | 3.92 | 27.37 |

Go | 22.65 | 3.40 | 11.81 | 32.95 | ||

Passed with red | 20.59 | 3.07 | 12.81 | 26.34 | ||

Stop after stop line | 18.91 | 3.26 | 13.35 | 25.10 | ||

Acceleration/Deceleration (m/s^{2}) | Acceleration/Deceleration at the onset of the yellow signal | Stop | 0.27 | 0.87 | −2.52 | 2.72 |

Go | 1.13 | 1.13 | −2.00 | 10.06 | ||

Passed with red | 0.94 | 0.70 | −0.18 | 2.92 | ||

Stop after stop line | 0.14 | 1.24 | −2.79 | 1.34 |

Decision | Dilemma Zone | Obvious Decision Pass | Obvious Decision Stop | Option Zone |
---|---|---|---|---|

Stop | 36.73% | 4.85% | 86.59% | 66.67% |

Pass | 63.27% | 95.15% | 13.41% | 33.33% |

Variable | Estimate | Std. Error | p-Value | OR |
---|---|---|---|---|

Speed_Yellow | 0.72 | 0.09 | 0.00 | 2.06 |

Distance_Yellow | −0.11 | 0.01 | 0.00 | 0.89 |

Acceleration_Deceleration_Yellow | 1.92 | 0.28 | 0.00 | 6.79 |

Constant | −5.59 | 1.26 | 0.00 | 0.00 |

Goodness of Fit Metrics | ||||

Nagelkerke R Square | 0.83 | |||

Hosmer and Lemeshow Test | 0.81 | |||

Classification (overall percentage) | 91.40% |

Speed_Yellow_Recoded | ||||

Pr(1) | Pr(2) | Pr(3) | Pr(4) | |

Class_1 | 0.0000 | 0.0000 | 0.4454 | 0.5546 |

Class_2 | 0.4011 | 0.5989 | 0.0000 | 0.0000 |

Acceleration_Deceleration_Yellow_Recoded | ||||

Pr(1) | Pr(2) | Pr(3) | ||

Class_1 | 0.0287 | 0.5057 | 0.4655 | |

Class_2 | 0.0282 | 0.7119 | 0.2599 | |

Platoon_Leader | ||||

Pr(1) | Pr(2) | |||

Class_1 | 0.7414 | 0.2586 | ||

Class_2 | 0.6610 | 0.3390 | ||

1st_Follower | ||||

Pr(1) | Pr(2) | |||

Class_1 | 0.2184 | 0.7816 | ||

Class_2 | 0.2542 | 0.7458 | ||

2nd_Follower | ||||

Pr(1) | Pr(2) | |||

Class_1 | 0.0374 | 0.9626 | ||

Class_2 | 0.0847 | 0.9153 | ||

Greater_than_Speed_Limit (Approaching Speed) | ||||

Pr(1) | Pr(2) | |||

Class_1 | 1 | 0 | ||

Class_2 | 0 | 1 |

**Table 7.**Estimated class population shares and predicted class memberships (by modal posterior prob.).

Estimated Class Population Shares | |

Class_1 | Class_2 |

0.6629 | 0.3371 |

Predicted Class Memberships (by Modal Posterior Prob.) | |

Class_1 | Class_2 |

0.6629 | 0.3371 |

Fit for 2 Latent Classes: |

number of observations: 525 |

number of estimated parameters: 19 |

residual degrees of freedom: 172 |

maximum log-likelihood: −1797.987 |

AIC(2): 3633.973 |

BIC(2): 3714.978 |

G^2(2): 669.3934 (Likelihood ratio/deviance statistic) |

X^2(2): 725.9775 (Chi-square goodness of fit) |

Variable | Estimate | Std. Error | p-Value | OR |
---|---|---|---|---|

Acceleration_Deceleration_Yellow | 1.72 | 0.25 | 0.00 | 5.56 |

Distance_Yellow | −0.05 | 0.01 | 0.00 | 0.95 |

Obvious_Decision_Stop | −3.09 | 0.60 | 0.00 | 0.05 |

Option_Zone | −3.46 | 1.38 | 0.01 | 0.03 |

Dilemma_Zone | −1.10 | 0.56 | 0.05 | 0.33 |

Observed_LCA_Classification | 1.92 | 0.46 | 0.00 | 6.80 |

Constant | 3.68 | 0.70 | 0.00 | 39.70 |

Goodness of Fit Metrics | ||||

Nagelkerke R Square | 0.81 | |||

Hosmer and Lemeshow Test | 0.34 | |||

Classification (overall percentage) | 91.00% |

**Table 10.**Odds ratio for aggressive and non-aggressive drivers based on the relative position of vehicles.

Obvious Decision Pass | Obvious Decision Stop | Option Zone | Dilemma Zone | |
---|---|---|---|---|

Odds Ratio | 36.93 | 1.68 | 1.16 | 12.29 |

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## Share and Cite

**MDPI and ACS Style**

Papaioannou, P.; Papadopoulos, E.; Nikolaidou, A.; Politis, I.; Basbas, S.; Kountouri, E.
Dilemma Zone: Modeling Drivers’ Decision at Signalized Intersections against Aggressiveness and Other Factors Using UAV Technology. *Safety* **2021**, *7*, 11.
https://doi.org/10.3390/safety7010011

**AMA Style**

Papaioannou P, Papadopoulos E, Nikolaidou A, Politis I, Basbas S, Kountouri E.
Dilemma Zone: Modeling Drivers’ Decision at Signalized Intersections against Aggressiveness and Other Factors Using UAV Technology. *Safety*. 2021; 7(1):11.
https://doi.org/10.3390/safety7010011

**Chicago/Turabian Style**

Papaioannou, Panagiotis, Efthymis Papadopoulos, Anastasia Nikolaidou, Ioannis Politis, Socrates Basbas, and Eleni Kountouri.
2021. "Dilemma Zone: Modeling Drivers’ Decision at Signalized Intersections against Aggressiveness and Other Factors Using UAV Technology" *Safety* 7, no. 1: 11.
https://doi.org/10.3390/safety7010011