## 7 Results
### Dynamic All-Red Signal Control  
For the safety of intersection traffic under the threat of RLRs, an approach of all-red signal extension has been proposed to extend the all-red signal to a _pre-fixed_ time duration, which is typically less than 5 s, in order to prevent vehicles from other directions entering the intersection when an RLR is detected. However, the fixed-time all-red signal extension may not be effective as drivers can adapt easily to the fixed extension time. In addition, it may reduce the intersection traffic efficiency in case the all-red signal extension time is chosen too conservatively and it may also reduce the traffic safety in case the all-red signal extension time is too short. To address such issues related to the fixed-time all-red signal extension approach, we incorporate the driving characteristics of RLR to determine the necessary all-red extension time. For this purpose, we adopt a nonlinear regression model, called the Hougen-Watson model, to develop an all-red extension time prediction model based on the traffic data collected from the Vissim simulation. The Hougen-Watson model performs nonlinear fitting through multivariate input of speed, DTI and headway, and has the advantage of being easily usable because it is provided as a Matlab function. Figure 8 shows the comparison between the actual intersection passing time calculated from the traffic data and the predicted intersection passing time by the Hougen-Watson prediction model for all RLRs traffic data. In the figure, circular points represent RLRs. For each RLR, the actual and the predicted intersection passing times for the vehicle can be compared between the values in the vertical and horizontal axis. The diagonal line, called the _Base line_ in the figure, represents when the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline  & 0 & 56.6\% & 57.5\% & 58.6\% & 57.1\% & 57.4\% & 57.7\% & 59\% \\  & 0.5 & 79.8\% & 85.4\% & 85\% & 85.2\% & 84.6\% & 85.4\% & 85.1\% \\ Window & 1 & 84.3\% & 86.5\% & 87.7\% & 87.2\% & 87.8\% & 87.3\% & 87.5\% \\ size & 1.5 & 88.4\% & 88.6\% & 89.9\% & 90.2\% & 90.3\% & 90.0\% & 90\% \\ (sec) & 2 & 90.6\% & 92.2\% & 92.6\% & 93.3\% & 93.3\% & 93.4\% & 93.3\% \\  & 2.5 & 94\% & 94.7\% & 95.6\% & 95.9\% & 95.8\% & 95.8\% & 96\% \\  & 3 & 98.9\% & 98.9\% & 99.2\% & 99.3\% & 99.3\% & 99.4\% & 99.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification accuracy of Support Vector Machine (SVM) classifier. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline  & 0 & 87.3\% & 92.5\% & 92.9\% & 90.7\% & 89.2\% & 95.4\% & 99.5\% \\ Window & 0.5 & 90.7\% & 95.2\% & 96.4\% & 97.4\% & 98.1\% & 98\% & 99.6\% \\ size & 1 & 93.5\% & 96.4\% & 97.1\% & 97.8\% & 98.3\% & 98.4\% & 99.9\% \\ (sec) & 1.5 & 93.5\% & 96.9\% & 97.4\% & 98.4\% & 98.6\% & 99.1\% & 99.9\% \\  & 2 & 93.8\% & 96.2\% & 97.1\% & 99\% & 98.9\% & 99.3\% & 99.9\% \\  & 2.5 & 94.3\% & 96.7\% & 97.2\% & 98.7\% & 98.2\% & 98.9\% & 99.9\% \\  & 3 & 94.5\% & 97.3\% & 97.5\% & 98.6\% & 98.6\% & 99.3\% & 99.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification accuracy of the Multi-Channel Deep Convolutional Neural Networks (MC-DCNN) classifier. actual and predicted time matches. Thus RLRs above the base line actually take longer time than the predicted intersection passing time to completely cross an intersection. As shown in the figure, a large number of RLRs are shown above the base line. Therefore, for such RLRs, the Hougen-Watson prediction model alone is not enough to predict the necessary all-red signal extension time for all types of RLRs. To address this issue, we identified RLRs, called the _Outliers_, from the dataset in which actual intersection passing times are larger and also maximally deviated from their predicted intersection passing times.