## 7 Results
### Dynamic All-Red Signal Control
In Figure 8b, red colored circular points represent those outliers identified from the dataset and the dashed line represents the quadratic polynomial curve fitted to the outliers. Hence, if we use the quadratic curve model on top of the Hougen-Watson model to predict the intersection passing time, then the predicted time will be long enough for _most_ RLRs so that they can completely clear an intersection within the time interval, which is much safer than using the Hougen-Watson model alone. However, as one can notice, it may be too conservative sometimes to use only one prediction model to predict intersection passing times for all types of RLRs. For a certain class of RLRs, the predicted intersection passing time predicted by the model may be unnecessarily longer than needed for such RLRs. Thus, for better traffic efficiency, we develop and use different prediction models for different RLR classes to predict the intersection passing time more precisely. Figure 9 shows the prediction model of each RLR class developed by the same framework of using Hougen-Watson model and quadratic polynomial curve fitting. Having these four different prediction models corresponding to each RLR types, it is now possible to determine the necessary all-red signal extension time more effectively than using only one prediction model once the type of RLR of an incoming vehicle is correctly classified by the MC-DCNN classifier. Table 4 shows the values of the Hougen-Watson model parameters determined by the Levenberg-Marquardt algorithm for each prediction model and also the values of the coefficients for the quadratic polynomial curve fitting of outliers. Regarding the values of the quadratic polynomial curve fitting shown in the table, \(p_{1}\) represents the second-order coefficient, \(p_{2}\) is the first-order coefficient, and \(p_{3}\) is the polynomial constant of a quadratic polynomial equation. Figure 8: All-red signal extension time for RLRs: Actual vs. Prediction. To evaluate the accuracy of the proposed multi-class intersection passing time prediction framework compared to the case of using only one prediction model, the _mixed_ RLR model in Table 4, we use the following standard deviation of residual \(\sigma_{est}\) defined as  
\[\sigma_{est}=\sqrt{\frac{\sum{(y-\hat{y})^{2}}}{N}} \tag{3}\]  
where \(N\) is the number of RLRs, \(y\) is the actual intersection passing time of an RLR, and \(\hat{y}\) is the predicted intersection passing time of the RLR. Table 5 is the result of the prediction accuracy of intersection passing time of RLRs measured by \(\sigma_{est}\) for the two prediction models in the case where the prediction time after yellow onset is 3 s. As shown in the result, the proposed multi-class model has \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Hougen-Watson Model**} & \multicolumn{6}{c}{**Polynomial Fitting**} \\ \hline  & \(\mathbf{\beta_{1}}\) & \(\mathbf{\beta_{2}}\) & \(\mathbf{\beta_{3}}\) & \(\mathbf{\beta_{4}}\) & \(\mathbf{\beta_{5}}\) & \(\mathbf{p_{1}}\) & \(\mathbf{p_{2}}\) & \(\mathbf{p_{3}}\) \\ \hline Type A RLR & \(2.83\times 10^{27}\) & \(-7.33\times 10^{26}\) & \(1.15\times 10^{27}\) & \(4.24\times 10^{27}\) & \(5.1\times 10^{6}\) & \(4.39\) & \(-25.03\) & \(39.06\) \\ Type B RLR & \(0.0642\) & \(0.1003\) & \(-0.0205\) & \(0.0382\) & \(-8.89\times 10^{5}\) & \(0.57\) & \(-1.45\) & \(2.83\) \\ Type C RLR & \(0.1625\) & \(0.2340\) & \(-0.0210\) & \(0.1941\) & \(-2.19\times 10^{5}\) & \(-0.21\) & \(2.1\) & \(-1.32\) \\ Type D RLR & \(2.8061\) & \(-0.6884\) & \(0.7655\) & \(-1.3043\) & \(0.0644\) & \(0\) & \(1\) & \(0\) \\ Mixed RLR & \(0.0733\) & \(0.0828\) & \(-0.0133\) & \(0.0510\) & \(-7.00\times 10^{5}\) & \(2.17\) & \(-8.48\) & \(10.39\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients of prediction models for all-red signal extension time. (Mixed RLR represents the prediction model shown in Figure 8 and others are corresponding to models shown in Figure 9).