## 3 Results
### Selecting Optimal Cluster Number for Sewage Pollution Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(p\) & \(d\)/2 & \(df\)1 & \(F\) & **Wilks\({}^{\prime}\) Lambda** & \begin{tabular}{c} **Independent** \\ **Variables** \\ \end{tabular} \\ \hline \(<\)0.001 & 256 & 1 & 2119 & 0.10 & \begin{tabular}{c} Mechanically treated \\ sewage \\ \end{tabular} \\ \(<\)0.001 & 256 & 1 & 117.4 & 0.68 & Totally treated sewage \\ \hline \multicolumn{5}{c}{b} \\ \hline \(p\) & \(R^{2}\) & \(t\) & **Standard Error** & **Coefficient** & \
\begin{tabular}{c} **Independent** \\ **Variables** \\ \end{tabular} & **Dependent Variable** \\ \hline \(0.051\) & \(-\) & 1.956 & 2.30 & 4.51 & Constant \\ \hline \(<\)0.001 & 0.88 & 46.03 & 0.028 & 1.30 &
\begin{tabular}{c} Mechanically treated \\ sewage \\ \end{tabular} & Raw sewage \\ \hline \(<\)0.001 & 0.24 & \(-\)10.83 & 0.029 & \(-\)0.318 & Totally treated sewage \\ \hline \hline \end{tabular}
\end{table} 
Table 4: (a) Multivariate regression analysis of mechanically treated sewage and totally treated sewage. Tests on independent variables. (b) Multivariate regression analysis of raw sewage, mechanically treated sewage and totally treated sewage. Tests on dependent variable as raw sewage. other extract levels might need more investigation, while the consistent behaviour of phenols and chlorides suggests predictable patterns for more efficient methods. The dataset in Figure 6a consists of the ether extract data analysed using a clustering method. The analysis yielded the following metrics: an average Silhouette width of 0.4184, a Within-Cluster Sum of Squares (WCSS) of 12.564, an \(F\)-statistic of 0.96961 and a variance explained of 49.23%. The Silhouette plot for phenols (Figure 6b) revealed two distinct clusters (Cluster 1 and Cluster 2) with varying degrees of internal cohesion. Cluster 1 exhibited consistently higher Silhouette coefficients, indicating strong intra-cluster similarity. Cluster 2 showed lower Silhouette coefficients, suggesting weaker internal cohesion and potential overlap with other clusters. A clustering analysis for phenols yielded the following metrics: an average Silhouette width of 0.49331, a Within-Cluster Sum of Squares of 0.0036858, an \(F\)-statistic of 1.3887 and a variance explained of 58.14%. For chlorides (Figure 6c), the average Silhouette coefficient was 0.738, indicating strong cluster cohesion and good separation. Two clusters were identified. Cluster 2 was larger with a broad range of Silhouette scores, while Cluster 1 had lower Silhouette values, suggesting less cohesion. An \(F\)-statistic of 1.42 and variance explained (58.65%) suggest  
Figure 4: Principal Component Analysis (PCA) of sewage pollutants including the ether extract, phenols, chlorides and sulfates. Figure 5: Detrended Correspondence Analysis (DCA) of sewage pollutants including ether extract, phenols, chlorides and sulfates. Phenols were related to sulfates. The ether extract was closer to chlorides. that while the clusters capture a significant portion of variability, there may be room for improvement by testing other numbers of clusters or adjusting the clustering algorithm. In the case of sulfates (Figure 6d), Cluster 1 had higher Silhouette coefficients, indicating strong cohesion and separation. Cluster 2 had more variation in Silhouette values, with some points close to 0 or negative, suggesting less cohesion. Overall, the clustering with \(k=2\) was reasonable, but Cluster 2 may need further exploration. WCSS was 51.268, \(F\)-statistic was 0.963, variance explained was 49.06% and the average Silhouette coefficient was 0.431.