## 4 Analysis
### Imputation Performance  
To check and compare the performance of the proposed model, five test-case scenarios were set up as shown in Table 1. In this test, the performance measure used was the root mean squared error (RMSE). Where \(y_{e}(^{m})\) is the estimated value and \(y_{r}(^{m})\) is the observed value, respectively, whereas \(M\) denotes the total number of testing entries we used. \[RMSE=\sqrt{\frac{1}{M}\sum_{m=1}^{M}\left(y_{e}^{(m)}-y_{r}^{(m)}\right)^{2}} \tag{12}\]  
As stated, the missing target was detector number 6 with the MD strategy that assumed malfunctions for all observation days in the validation dataset. All speed data of the number 6 in the validation data were removed. The \(Ang_{t}\) was newly computed from the validation data, which did not include VDS 6 traffic-speed data. The comparison of the imputation performances especially in SBTSs II and III were of great interest for congestion management. The results of the performance comparisons when the training data and validation data were 80% and 20% respectively among the SVR-based cases, as shown in Table 2. There were some key findings. First, the model segmentation by using the \(Ang_{t}\) improved the performance and this was supported by the RMSE difference between Case 1 and Case 2. The rest of the comparisons were conducted to examine the effects of (i) the variable selection and (ii) the addition of the angle to the independent variables, and the standard model segmentation was applied afterwards. Second, the variable selection considering the factor loading improved the imputation performance. The evidence is visible from the RMSE differences between Case 2 and Case 4, and Case 3 and Case 5. Third, the use of the angle as an independent variable improved the imputation performance except when the variable selection was used in the SBTS II. This is supported by the RMSE differences between Case 2 and Case 3, and Case 4 and Case 5. The range of the computation time of the proposed approach for all cases was from 6 s to 27 s, and it is reasonable for actual implementations in the field for practitioners at DOTs. The imputation performances of the comparison method according to the missing rate of 10%, 20%, and 30% are shown in Table 3. This paper compared the performance of the proposed model against the linear interpolation method, ANN, and KNN. For the linear interpolation method, we used \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Variables** & **Segmentation of SBTS** & **Using \(Ang_{t}\)******* \\ \hline Case 1 & All & No & No \\ Case 2 & All & Yes & No \\ Case 3 & All & Yes & Yes \\ Case 4 & Selected & Yes & No \\ Case 5 & Selected & Yes & Yes \\ \hline \hline \end{tabular}  
* The angle between the first PC and the standard vector \(v_{s}\) at a time \(t\).  
\end{table} 
Table 1: Description of cases for empirical analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**RMSE***** (km/h) of Validation**} \\ \cline{2-4}  & **SSTS II** & **SBTS III** & **Aggregate** \\ \hline case 1 & 12.586 & 11.696 & 12.173 \\ case 2 & 11.607 & 11.961 & 11.785 \\ case 3 & 11.564 & 11.902 & 11.733 \\ case 4 & 11.019 & 11.222 & 11.120 \\ case 5 & 11.070 & 11.072 & 11.071 \\ \hline \hline \end{tabular}  
* Root mean squared error.  
\end{table} 
Table 2: Validation results of empirical analysis. a nearby detector's speed data at the same time. Before performing the ANN and KNN methods, it was necessary to determine some parameters; the number of hidden layers and the activation function for ANN and k value which is the number of neighbors for KNN. Here, we did not put much effort to apply a new or complex methodology to find optimal parameters. However, we iterated the above methods with changing parameters to find optimal parameters, which gave the minimum RMSE. In this paper, the following deterministic of the parameters was used; the number of hidden layers = 5, the activation function = linear function, and \(k\) value = 83.