# Representing and Inferring Massive Network Traffic Condition: A Case Study in Nashville, Tennessee
## 3 Results
### Performance of the Proposed Weighted Average Method  
Figure 5 shows the performance of the proposed weighted method compared with the L2 norm, SVD-based leverage, and random sampling methods. The red line in the graph indicates that the weighted method outperforms all other three options when the compression rate (CR) is lesser than 32 (\(2^{5}=32\)). The proposed weighted method resulted in lower PRD than other methods when the CR does not exceed 32, whereas the drawback of the weighted sampling method is relatively longer computational time. When the CR surpasses 32, the proposed weighted method is only better than the SVD-based leverage method, partly because the presetting default value of the weight (w) is 0.5, which provides too much weight for SVD-based leverage. Random sampling is even better than the L2 norm option when the CR is less than 64, which might be due to randomness and fluctuation. It is worth noting that SVD-based leverage and proposed weighted sampling perform best with a CR less than 8, whereas the error for the SVD-based leverage option increases dramatically as the CR becomes greater than 32.  
Figure 5: Performance of the proposed method without clustering.  
Table 1 presents the summary statistics of the performance of the proposed method (weighted) compared to other three models (L2 norm, SVD-based leverage, and random sampling). It should be noted that random sampling is carried out to show the minimal time needed to generate the decomposition results, since we assume that the time consumed by column selection could be ignored under a random sampling scenario. However, random sampling is impractical because the performance fluctuates, and the selected columns are not fixed each time the sample is obtained. That is also why we repeated the process five times and used the average performance. Among the three methods, the L2 norm-based method is the most computation-friendly, and the proposed weighted method uses more computational resources than the other two methods. Overall, this is a trade-off between accuracy and computing cost.  
Taking CR = 16 with the proposed weighted average method as an example, Figure 6 depicts the absolute error of the original matrix and inferred matrix for the testing period (23 July 2023, to 29 July 2023). The horizontal axis represents the dates and the vertical axis is the detectors. There are 31 links selected, and the PRD = 9.5%. The absolute error is less than 10 mph for 94% of time and detectors. Some recurring patterns can be found, probably due to the recurring peak-hour congestion at some detector links.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Compression Rate (CR)**} & \multicolumn{3}{c}{**PRD (\%)**} & \multicolumn{3}{c}{**Computing Time (Milliseconds)**} \\ \cline{2-9}  & **L2** & **SVD** & **Weighted** & **Random** & **L2** & **SVD** & **Weighted** & **Random** \\ \hline
2 & 7.8\% & 3.8\% & 3.6\% & 5.8\% & 793.1 & 982.5 & 1075.5 & 645.9 \\
4 & 9.7\% & 6.3\% & 6.1\% & 7.7\% & 692.0 & 870.9 & 964.9 & 562.6 \\
8 & 10.1\% & 8.4\% & 8.3\% & 8.9\% & 705.0 & 899.3 & 939.8 & 534.3 \\
16 & 10.8\% & 10.0\% & 9.5\% & 9.6\% & 648.2 & 860.0 & 973.0 & 512.7 \\
32 & 11.4\% & 12.0\% & 10.8\% & 10.5\% & 691.7 & 915.1 & 939.8 & 512.0 \\
64 & 11.7\% & 16.4\% & 12.5\% & 11.4\% & 631.2 & 835.1 & 940.4 & 567.0 \\
128 & 11.7\% & 20.6\% & 13.5\% & 12.4\% & 624.3 & 882.4 & 905.8 & 522.4 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the performance of different methods without clustering.  
Figure 6: Performance of the proposed weighted method when CR = 16.