## 5 Experimental Results
### Noise Removal  
This section specifically evaluates the performance of NBR clustering in removing noise from noisy synthetic datasets, in contrast to the previous section, which presented percentage accuracy results on NBR-Clust's effectiveness in clustering noisy datasets. The Arbelaitz point set data is utilized, which contains 10% noise that is randomly generated following a \"uniform distribution over the sampling window\" [63]. A visualization of an unequal density dataset with noise is illustrated in Figure 7. The cases of primary interest are those where NBR-Clust successfully removes noise without compromising clustering accuracy, and it is preferred if this is accomplished in one iteration. The measures precision and recall are used to quantify NBR-Clust performance when identifying noise nodes. Let \(N_{D}\) denote the number of noise nodes in the dataset; \(|S|\), the number of nodes in the critical attack set; and \(N_{S}\), the number of noise nodes contained in that attack set. Precision is defined  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**VAT**} & \multicolumn{3}{c}{**Integrity**} & \multicolumn{3}{c}{**Tenacity**} \\ \hline  & & \multicolumn{3}{c}{**no Noise**} & \multicolumn{3}{c}{**with Noise**} & \multicolumn{3}{c}{**no Noise**} & \multicolumn{3}{c}{**with Noise**} & \multicolumn{3}{c}{**no Noise**} & \multicolumn{3}{c}{**with Noise**} \\ \hline \multirow{2}{*}{**D**} & \multirow{2}{*}{**K**} & **eq** & **uneq** & **eq** & **uneq** & **eq** & **uneq** & **eq** & **uneq** & **eq** & **uneq** & **eq** & **uneq** \\  & & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** & **dens** \\ \hline
2 & 2 & **100** & **100** & **100** & **96** & **100** & **100** & **49** & **80** & **100** & **100** & 52 & 81 \\
2 & 4 & 99 & 100 & 100 & 100 & **99** & **99** & **36** & 69 & **99** &  
as \(\mathrm{P}=\frac{N_{5}}{|S|}\), and recall is defined as \(\mathrm{RC}=\frac{N_{5}}{N_{D}}\). Precision indicates how successful the algorithm was in identifying the noise nodes as the attack set nodes, while recall is the percentage of the overall noise nodes that were contained in the attack set. Table 8 shows the precision and recall of NBR-Clust using VAT, integrity and tenacity when removing noise from noisy datasets. Integrity gives the most balanced mix of precision and recall. VAT generally has a high precision--the nodes it finds are quite likely to be noise nodes, but due to the smaller attack set sizes, it does not find a large percentage of noise nodes. Tenacity has the opposite problem, often finding over 90% of noise nodes, but also with the high probability that many of them are not noise, implying low precision but high recall. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**VAT**} & \multicolumn{4}{c}{**Integrity**} & \multicolumn{4}{c}{**Tenacity**} \\ \hline  & & **eq dens** & **uneq dens** & **eq dens** & **uneq dens** & **eq dens** & **uneq dens** & **eq dens** & **uneq dens** \\ \hline
**D** & **K** & **Pre** & **Rec** & **Pre** & **Rec** & **Pre** & **Rec** & **Pre** & **Rec** & **Pre** & **Rec** & **Pre** & **Rec** \\ \hline
2 & 2 & 63 & 6 & 65 & 3 & 25 & 40 & 40 & 57 & 17 & 93 & 19 & 96 \\
2 & 4 & 58 & 7 & 68 & 5 & 22 & 24 & 36 & 46 & 16 & 91 & 15 & 91 \\
2 & 8 & 74 & 14 & 74 & 10 & 53 & 28 & 34 & 21 & 15 & 81 & 15 & 84 \\
4 & 2 & 80 & 8 & 89 & 8 & 26 & 68 & 30 & 77 & 17 & 94 & 17 & 98 \\
4 & 4 & 83 & 13 & 93 & 11 & 30 & 57 & 17 & 96 & 37 & 72 & 17 & 96 \\
4 & 8 & 82 & 16 & 85 & 13 & 96 & 49 & 47 & 51 & 15 & 95 & 16 & 95 \\
8 & 2 & 80 & 10 & 75 & 5 & 26 & 75 & 27 & 75 & 18 & 94 & 19 & 95 \\
8 & 4 & 85 & 18 & 90 & 16 & 38 & 45 & 32 & 69 & 17 & 88 & 17 & 95 \\
8 & 8 & 92 & 22 & 94 & 23 & 96 & 51 & 44 & 63 & 16 & 93 & 17 & 93 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Precision and Recall (in Percentage) of Hierarchical NBR Clustering Noise Removal on Noisy Synthetic Datasets.