# Vehicle Collision Prediction under Reduced Visibility Conditions
## 2 Related Work
### Research on Collision Warning Models  
In parametric models, the Time-To-Collision (TTC) model is one of the most common models. [PERSON] [8] defined TTC as \"the time required for two vehicles to collide if they continue at their present speeds and on the same path\". The TTC model is computed as shown below:  
\[\text{TTC}=D/\Delta V \tag{1}\]  
where \(D\) is the distance between the center of a subject vehicle (SV) to the center of the lead vehicle (LV) and \(\Delta V\) is the relative speed. However, there are some limitations in the TTC model. First, it cannot be computed when the relative speed is negative or zero. Second, several modified TTC algorithms have been proposed, but they do not suggest the best timing for warning because they do not consider human factors under different traffic conditions [6]. The stopping distance algorithm (SDA) with kinematics analysis is one of the most common models. [PERSON] et al. [9] proposed the SDA model for 3 scenarios, namely lead vehicle stopped (LVS), lead vehicle decelerating (LVD), and lead vehicle moving (LVM). In LVS and LVM scenarios, the SDA model is computed as follows:  
\[D_{W}=\frac{\left(V_{F}-V_{L}\right)^{2}}{-2a_{F}}+\left(V_{F}-V_{L}\right) \times\left(T_{R}+t_{sys}\right)+D_{S} \tag{2}\]In a LVD scenario, the SDA model is computed as follows:  
\[D_{W}=\frac{V_{F}^{2}}{-2a_{F}}+V_{F}\times\left(T_{R}+t_{sys}\right)-\frac{(V_{ L})^{2}}{2a_{L}}+D_{S} \tag{3}\]  
where \(V_{F}\) and \(V_{L}\) are the velocities of the following vehicle and the lead vehicle, respectively, \(T_{R}\) is the reaction time of a driver in the vehicle, and \(a_{F}\) and \(a_{L}\) are the braking intensities of the two vehicles, respectively, \(t_{sys}\) is system delay, and \(D_{S}\) is a desired safety gap when the velocity of the following vehicle is the same as that of the lead vehicle. [PERSON] et al. introduced a vehicle kinematics (VK) model, based on the DSRC technology. The authors proposed the model to solve information delay and transmission delay from DSRC. However, problems still exist in the kinematics rear-end CW model, that is, either high false alert or high missing alert occurs in the model due to the delay [10].  
In non-parametric models, artificial neural networks (ANN) are one of the most common models. Figure 1 shows a basic architecture of a multi-layer back-propagation neural network (BPNN). [PERSON] et al. used a neural network model to address the problem of high frequency of false alerts and missing alerts occurring in the VK model [10]. [PERSON] et al. also proposed a framework for real-time CWS based on a Multi-Layer Perceptron Neural Network (MLPNN) in V2I environment [6]. The authors tried to solve the impact of different driver reaction times in a rear-end CWS. But a case study done in St. Petersburg, Russia in 2014 pointed out that the costs of V2I technology are very high due to network deployment of Road Side Units (RSU). It is more feasible to use DSRC for V2V applications [11]. [PERSON] et al. used fuzzy logic for coarse assessment, and the ratio of collision occurrence for fine assessment. Then, the authors used the neural network ensembles to calculate collision probabilities [12]. [PERSON] et al. proposed a Crash Risk Increase Indicator (CRII) for traffic in fog conditions. It included not only local traffic flow characteristics but also traffic flow relationships. It was shown using a logistic model that crash risk is prone to increase at ramp vicinities and locations with heavy traffic in fog conditions [13]. [PERSON] et al. also proposed a Rear-End Collision Risk Index (RCRI) which used the safe stopping distances of two consecutive vehicles in the same lane and the clearance distance between them (so that they do not collide into each other). Different vehicle types, different lanes, and driving maneuver can affect the accuracy of RCRI [14].