## 7 Results
### Clustering  
As described in Section 4.1, we use the DTW algorithm to measure the similarity between a pair of speed profile time-series. Figure 6 shows several examples of speed profile time-series data, selected from different clusters which are determined later through the HCA clustering process, to illustrate the effect of the DTW algorithm for optimal alignment of two time-series data and the similarity measure calculated between them. Figure 6a shows a comparison between speed profiles from Type A RLR and Type B RLR clusters. The similarity measure between these two time-series data, calculated as the accumulated pairwise Euclidean distance, is 131.26 in this case. Similarly, Figure 6b,c also show the similarity results from different RLR clusters where similarity measures calculated from the DTW algorithm are 87.85 and 71.23, respectively. On the other hand, Figure 6d shows the similarity result between a pair of speed profile time-series selected from the same cluster, which is Type B RLR in this case. For these speed profile time-series data, the similarity measure from the DTW algorithm is less than 25, which is substantially lower than the other three cases in the figure and hence clearly indicates that these two time-series are quite similar to each other in terms of their shapes while they may be in slightly different phases. Next, to determine the number of clusters via HCA based on the similarity measures, it is necessary to choose a threshold appropriately for the value of a similarity measure. If the threshold for cluster separation is too low, then there will be too many clusters formed and the driving characteristics of RLRs between clusters are not clearly distinguishable. Therefore, we investigate the hierarchical structure of clusters generated from HCA for all RLR traffic datasets and choose to separate clusters when the similarity measure suddenly increases more than 50 in the HCA process since clusters formed \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Decision Model** & **Behavior** & **Mean** & **Standard Deviation** \\ \hline \multirow{4}{*}{Speed (m/s)} & \multirow{4}{*}{Continuous} & RLR & 14.1337 & 1.5924 \\  & & Go & 13.6342 & 1.3388 \\  & & Stop & 12.2642 & 4.4733 \\ \cline{2-5}  & \multirow{4}{*}{One} & RLR & 15.2914 & 1.3388 \\  & & Go & 13.8057 & 2.7491 \\  & & Stop & 12.3939 & 4.3449 \\ \hline \multirow{4}{*}{Acceleration (m/s\({}^{2}\))} & \multirow{4}{*}{Continuous} & RLR & 0.3404 & 0.7978 \\  & & Go & 0.2910 & 0.6586 \\  & & Stop & 0.1817 & 0.8693 \\ \cline{2-5}  & \multirow{4}{*}{One} & RLR & -0.0500 & 0.4520 \\  & & Go & 0.2076 & 0.5830 \\  & & Stop & 0.1709 & 0.7996 \\ \hline \multirow{4}{*}{DTI (m)} & \multirow{4}{*}{Continuous} & RLR & 37.6579 & 3.7382 \\  & & Go & 16.0680 & 10.6932 \\  & & Stop & 63.8649 & 21.3053 \\ \cline{2-5}  & \multirow{4}{*}{One} & RLR & 51.3914 & 6.3918 \\  & & Go & 19.1971 & 12.2729 \\  & & Stop & 65.6994 & 21.5267 \\ \hline \multirow{4}{*}{Headway (m)} & \multirow{4}{*}{Continuous} & RLR & 88.8779 & 76.6408 \\  & & Go & 88.9833 & 81.1717 \\ \cline{1-1}  & & Stop & 86.4493 & 78.2994 \\ \cline{1-1} \cline{2-5}  & \multirow{4}{*}{One} & RLR & 106.6986 & 77.8763 \\ \cline{1-1}  & & Go & 95.3744 & 83.9599 \\ \cline{1-1}  & & Stop & 89.2601 & 79.3263 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical results of traffic simulation data collected at the time of yellow onset. from this are most reasonably distinguishable in terms of their driving characteristics. As a result, there are four different clusters formed for RLRs, as described in Section 4.1. Figure 7 shows the result of clustering generated through the HCA process for all RLR traffic data collected from the Vissim simulation. Figure 7a-d shows RLR speed profiles of each cluster.