# Vehicle Collision Prediction under Reduced Visibility Conditions
## 4 Experiments
### Experiment Results
#### 4.2.3 Collision Warning with Different Prediction Horizons  
Variations in collision warning levels for different prediction horizons are shown in Figure 10 for both non-congested traffic scenario (15 to 25 time slots) and congested scenario (1 to 15 time slots). We can observe that the predicted collision warning levels are higher when the prediction horizon is larger. Since we take the maximum of all collision warning levels in the prediction horizon, the final resulting collision warning level monotonically increases with prediction horizon. The maximum function is used so that the final result indicates the level of collision risk within the prediction horizon. When the final collision warning level is above a given threshold, 0.5 here, then a rear-end collision warning alert is activated. In the following, let us analyze the relation between the prediction horizon and PRT. As shown in Figure 11, if we want to avoid a predicted collision, an alert must be given at least \(T_{pr}+T_{br}\) time slots prior to the time of predicted collision. To avoid the collision in the time slot \(T_{br}\), a driver needs to react in the time slot \(T_{pr}\), brake with a deceleration \(a_{br}\), and decrease the SV's speed to the same as the \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & \\ \hline \multirow{2}{*}{**\# Input Nodes**} & \multicolumn{8}{c}{**Pracdiction Horizon (Time Slots)**} \\ \cline{2-11}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
4 & 0.32 & 1.08 & 2.16 & 3.39 & 4.61 & 5.73 & 6.57 & 7.26 & 8.03 & 8.90 \\
5 & 0.26 & 0.90 & 1.88 & 3.00 & 4.23 & 5.42 & 6.75 & 8.06 & 9.15 & 9.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Testing error of velocity prediction model. Figure 9: Comparison between actual velocity and predicted velocity with 4 inputs. (**a**) Comparison between actual velocity and predicted velocity in non-congested traffic scenario; (**b**) Comparison between actual velocity and predicted velocity in congested traffic scenario. [PERSON]'s speed. It is suggested that \(a_{br}\) is set to 6.56 ft/s\({}^{2}\) (2 m/s\({}^{2}\)) to ensure ride comfort [7]. Then, \(T_{br}\) can be computed as follows:  
\[T_{br}(t)=\frac{\Delta V}{a_{br}}=\frac{V_{SV}(t)-V_{LV}(T)}{a_{br}} \tag{8}\]  
where \(\Delta V\) is the difference between [PERSON]'s speed at time slot \(t\) and [PERSON]'s speed at time slot \(T\). It means after \(T_{br}\), [PERSON]'s speed is the same as [PERSON]'s speed. \(T\) is set the 71 st time slot in the non-congested traffic scenario, and the 115 th time slot in the traffic congestion scenario, respectively, in which we assume a collision occurred. By assuming that the target (\(LV\)) becomes visible at time slot \(t\) under a given target visibility (in feet or meter), \(T_{pr}\) can be calculated as follows:  
\[T_{pr}(t)=T_{re}-T_{br}(t) \tag{9}\]  
where \(T_{re}(t)\) is the time interval from time slot \(t\) to \(T\). For each time slot and we experimented with different prediction horizons and found the prediction that gives the earliest collision alert for that time slot. These prediction horizons were mapped to \(T_{pr}\), which we assume is the PRT of SV. We then used a third order polynomial to fit the mapping data between prediction horizon and PRT of SV. As shown in Figure 12a, the resulting function of best fitting in the non-congested traffic scenario is as follows:  
\[T_{ph}=0.932T_{pr}^{3}-4.6822T_{pr}^{2}+10.481T_{pr}+13.16 \tag{10}\]  
where \(T_{pr}\) is the SV's PRT, and \(T_{ph}\) is prediction horizon. As shown in Figure 12b, the resulting function of best fitting in the congested traffic scenario is as follows:  
\[T_{ph}=0.0207T_{pr}^{3}+0.3642T_{pr}^{2}+0.2078T_{pr}+0.6447 \tag{11}\]  
Note that the above are just two examples of 3 rd order polynomial fitting between PRT and prediction horizons.