## 3 Results
### Graphical Pre-Screening of the Candidate Distance Measures  
#### 3.3.1 The Ultrafiltration Process Multi-Characteristic Distance Measure Selection  
The candidate distance measures that were indicatively selected to be tested were (1) the Euclidean distance, (2) the maximum/Chebyshev distance, (3) the Manhattan distance, (4) the Canberra distance and (5) the Minkowski distance. For the Minkowski distance, the model parameter \(p\) was fitted at a value of 4. Two-cluster and three-cluster \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**95\% Confidence Intervals**} \\  & **Spearman’s** & **Significance** & \multicolumn{2}{c}{**(Two-Tailed) a,b**} \\ \cline{3-5}  & **Rho** & **(Two-Tailed)** & **Lower** & **Upper** \\ \hline j—EC & \(-\)0.703 & 0.035 & \(-\)0.935 & \(-\)0.049 \\ j—Turb & \(-\)0.151 & 0.699 & \(-\)0.751 & 0.586 \\ j—TN & \(-\)0.377 & 0.318 & \(-\)0.840 & 0.403 \\ j—TP & \(-\)0.469 & 0.203 & \(-\)0.870 & 0.305 \\ j—NO\({}_{3}\) & 0.468 & 0.204 & \(-\)0.306 & 0.870 \\ EC—Turb & 0.500 & 0.170 & \(-\)0.268 & 0.879 \\ EC—TN & 0.667 & 0.050 & \(-\)0.019 & 0.926 \\ EC—TP & 0.667 & 0.050 & \(-\)0.019 & 0.926 \\ EC—NO\({}_{3}\) & \(-\)0.881 & 0.002 & \(-\)0.976 & \(-\)0.507 \\ Turb—TN & 0.217 & 0.576 & \(-\)0.540 & 0.779 \\ Turb—TP & 0.333 & 0.381 & \(-\)0.444 & 0.824 \\ Turb—NO\({}_{3}\) & \(-\)0.390 & 0.300 & \(-\)0.844 & 0.390 \\ TN—TP & 0.133 & 0.732 & \(-\)0.598 & 0.743 \\ TN—NO\({}_{3}\) & \(-\)0.746 & 0.021 & \(-\)0.945 & \(-\)0.139 \\ TP—NO\({}_{3}\) & \(-\)0.339 & 0.372 & \(-\)0.826 & 0.439 \\ \hline \hline \end{tabular}  
* Estimation is based on [PERSON]’s r-to-z transformation. b Estimation of standard error is based on the formula proposed by [PERSON], [PERSON].  
\end{table} 
Table 4: Confidence intervals for Spearman’s \(\rho\) correlation coefficients of the six nanofiltration process characteristics. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{2}{l}{**Characteristics**} & \multirow{2}{*}{**QCD**} & \multirow{2}{*}{**Efficiency**} & \multicolumn{2}{c}{**Relative**} & \multicolumn{2}{c}{**Cumulative Relative**} \\ \cline{3-5} \multicolumn{2}{l}{**(Nanofiltration Process)**} & & & **Efficiency** & **Efficiency** & **Efficiency** \\ \hline \hline \multicolumn{2}{l}{**NO\({}_{3}\)**} & 0.614 & 0.377 & 0.389 & 0.389 \\ \hline \hline \multicolumn{2}{l}{**EC**} & 0.613 & 0.376 & 0.387 & 0.776 \\ \hline \multicolumn{2}{l}{**TN**} & 0.36 & 0.130 & 0.134 & 0.910 \\ \hline \hline \multicolumn{2}{l}{**J**} & 0.24 & 0.0576 & 0.0594 & 0.969 \\ \hline \multicolumn{2}{l}{**TP**} & 0.17 & 0.0289 & 0.0298 & 0.999 \\ \hline \multicolumn{2}{l}{**Turbidity**} & 0.032 & 0.00102 & 0.00106 & 1 \\ \hline \multicolumn{2}{l}{Total} & 0.970 & 1 \\ \hline \hline \end{tabular}  
* Estimation is based on [PERSON]’s r-to-z transformation. b Estimation of standard error is based on the formula proposed by [PERSON], [PERSON] and [PERSON].  
\end{table} 
Table 5: Relative importance of the six nanofiltration process characteristics based on the efficiency (QCD2). configurations were evaluated to examine the linear and non-linear dependencies on the four controlling factors. From Section 3.2.1, the participating ultrafiltration process characteristics in the clustering assignment were (1) the turbidity, (2) the total nitrogen content and (3) the NO\({}_{3}\) content. Shepard graphs were prepared and are contrasted in Figure 8 to provide quick visual evidence. It appeared that for both preset cluster sizes, the Euclidean distance measure provided the most satisfying representation of the relationship between the configuration distances and the dissimilarities. However, it was not easily discerned whether the two- or three-cluster setting was more accurate in delineating this behavior. Therefore, the Kruskal stress estimations for all five distance measures and the two cluster sizes are tabulated in Table 6. Ostensibly, the three-cluster setting was favored by the fitting performances of all five distance measures.