## 4 Analysis  
### SBTS Identification  
The \(Ang_{t}\) was calculated using the speed data and the results are shown in Figure 3. The figure shows the speed-contour plot on 3 rd March 2016 in the test site and the plot of the \(Ang_{t}\) from the same day. The bottleneck was normally activated at number 3 and the queue tail occurred near number 9. As described in the methods section, the \(Ang_{t}\) plot had a similar pattern with the speed-contour plot in the real traffic data. The \(Ang_{t}\) represented a whole SBTS as opposed to the prediction approach based on spot-based traffic states which had a major limitation of being unable to operate when the data was missing. The \(Ang_{t}\) indicated the degree of the congested area in the section. The low \(Ang_{t}\) value represented the spread of congestion over a larger range of road sections in the study area. In addition, the speed of the evolution of traffic congestion can be interpreted by the angle. When the evolution of traffic congestion started rapidly, \(Ang_{t}\)'s inclination was steep. On the other hand, when the evolution of traffic congestion started smoothly, \(Ang_{t}\)'s inclination was also smooth. Many researchers have divided traffic flows into several different transitional states with their own criteria to identify traffic flow characteristics. [PERSON] [26] proposed finer classifications of SBTSs based on the fundamental diagrams. The free flow was further divided into the SBTSs of free fluid traffic and bunched fluid traffic. The congested section state was further classified into bunched congested traffic and standing congested traffic. In the 2010 edition of the Highway Capacity Manual [28], the traffic flow of the freeway was classified into six different SBTSs based on traffic densities. [PERSON] and [PERSON] [29] divide SBTSs using a speed-occupancy graph. They proposed the boundary lines on the graph, which can divide the SBTS into two free flow, congestion and transition SBTSs. The existing methods on SBTS identification were suitable when complete data were available because they considered spot-based traffic states. However, the identification of the SBTSs was needed for the missing value imputation due to the absence of the target data. This is the reason why identifying the SBTS using \(Ang_{t}\) is critical for missing value imputations. In this study, three traffic states were considered. The SBTS I contained the traffic state of all spots that were in the free flow state. The SBTS II was the traffic condition of the section that the queue length in the analyzed site was near the empirical maximum and did not expand anymore. SBTS III  
Figure 2: Study area in Korean freeway. (**a**) Geographical information. (**b**) Vehicle detection system (VDS) information. was between I and II that the traffic condition of the section was either queue build-up or diminishing, e.g., backward-forming and forward-recovery shockwaves. In the empirical case, the SBTS III may contain the traffic conditions of the section where the queue is not maximized, yet maintained. The boundaries between these states can be identified easily from diagrams as shown in Figure 3. In the study section, the SBTSs were classified into three SBTSs based on the \(Ang_{t}\) value measured using 12 loop detector stations. The data from 3 March 2016 showed a significant difference in terms of peak patterns between morning and evening, and the \(Ang_{t}\) certainly captured the difference in this degree of congestion. The results of t-tests show that the three SBTSs are all statistically different with 95% confidence levels. This verifies that \(Ang_{t}\) can be used as a classification index of traffic SBTS. The characteristics of the three SBTSs are defined as in the following. * SBTS I: free flow state at all spot (\(Ang_{t}\geq 173\))
* SBTS II: maximized queue (\(Ang_{t}\leq 165\))
* SBTS III: between I and II (\(173\geq Ang_{t}\geq 165\)). The classification performance of the SBTSs using the \(Ang_{t}\) was compared with that of using the average speed.