# Representing and Inferring Massive Network Traffic Condition: A Case Study in Nashville, Tennessee
## 4 Discussion  
As shown in Figure 5 and Table 1, the proposed weighted approach demonstrated lower PRD compared to alternative methods when the CR remained below 32. However, the downside of employing the weighted sampling method is the comparatively longer computational time. When the CR threshold surpasses 32, the proposed weighted  
Figure 8: Comparison of three methods with and without clustering: (**a**) L2 norm-based importance; (**b**) SVD-based leverage importance; (**c**) weighted average of L2 norm and leverage. method only outperforms the SVD-based leverage method, partially because the default weight value (w) is set at 0.5, assigning too much weight to the SVD-based leverage approach. A possible future direction could be tuning the hyperparameter w to achieve better performance over the L2 norm-based and SVD-based leverage methods on training data, followed by an evaluation of the method on the testing period. The PRD of the L2 norm-based method remains relatively stable (between 8% and 12%) as the CR increases from 2 to 128, while the SVD-based leverage method yields good performance with a CR lesser than 16, but its error rises very quickly as the CR becomes greater than 16. It is worth noting that the performance of the random sampling method is even better than that of the other three methods when the CR changes between 16 and 64. Previous researchers either failed to consider testing their method within this range or forgot to compare with the performance of the random sampling method. Figure 6 displays the heatmap of the absolute error between the original and inferred test data. A compression rate of 16 yields a PRD of 9.5%, and the absolute error is within 10 mph for 94% of the time and detectors. Additionally, the largest error always occurred recurrently during the peak hour at some specific detectors, which implies that the CUR decomposition method does not perform well during congestion periods. There also exist some non-recurring large errors, which might be due to the occurrence of some atypical incidents such as severe weather, special events, and scheduled road work. Another interesting finding is that clustering almost has no room for improvement for SVD-based leverage and the proposed weighted average method. As for the L2 norm-based method, clustering helps reduce the error to as low as 1% to 2%. Integrated with the clustering method, all three methods even demonstrated an increase in error when the CR was greater than 16, except for the SVD method with CR = 128. There might be various reasons accounting for this phenomenon. First, if the data do not exhibit clear clusters or if the clusters are not well defined, clustering algorithms may not yield meaningful or useful groupings. CUR decomposition relies on selecting representative columns and rows, and if the data do not naturally cluster, selecting clusters may not provide an advantage. Second, determining the optimal number of clusters (k) is challenging. Choosing an incorrect value for k can lead to poor cluster quality and, consequently, suboptimal CUR selection. Another reason could be the choice of the clustering algorithm. If the clustering algorithm is not well-suited to the data or the specific goals of CUR decomposition, it may not lead to improvements. Since we are clustering time series in the context of traffic speed, a dynamic time warping distance could be considered as the similarity metric to perform the clustering analysis. In addition, k-means clustering is simple and basic, so it might be beneficial to try other advanced clustering algorithms, such as hierarchical clustering or DBSCAN, to yield better clustering results and compare the CUR performance with different clustering methods. This study opens the door to various intriguing possibilities for further investigation of how to improve clustering results. The detectors used in this study are all from interstates without disruption by signal timing.