## 3 Results
### Graphical Pre-Screening of the Candidate Distance Measures
#### 3.3.2 The Nanofliltration Process Characteristics Multi-Characteristic Distance Measure Selection  
The same procedure was repeated for the nanofiltration process as in the preceding sub-section. The same five distance measures and the two cluster sizes were assessed for the nanofiltration process characteristics that were nominated from Section 3.2.2, i.e., the electrical conductivity, the NO\({}_{3}\) content and the total nitrogen content. The two- and three-cluster configurations of the dataset in terms of the Shepard graphs are depicted in Figure 9. While the Euclidean distance measure still provided a tighter fit between the configuration distances and the dissimilarities for the two-cluster case, the Manhattan, Canberra and Minkowski (\(p\) = 4) distance measures competed fairly well with each other. To make this distinction more accurate, the Kruskal stress estimates for all five distance measures and the two cluster sizes are tabulated in Table 7. Again, it was observed that the three-cluster setting option led to lower Kruskal stress values for the Manhattan, Canberra and Minkowski (\(p\) = 4) distance measures. However, the minimum estimated Kruskal stress value was 5.42 \(\times\) 10\({}^{-14}\). It was obtained in the case of a two-cluster setting when a Euclidean distance measure was considered; it was not distinguishably different from the three-cluster size setting.  
\begin{table}
\begin{tabular}{l c c} \hline
**Distance Measure** & **Cluster Number** & **Kruskal Stress Value** \\ \hline Euclidean & 2 & 6.89 \(\times\) 10\({}^{-14}\) \\ Euclidean & 3 & 4.66 \(\times\) 10\({}^{-14}\) \\ Maximum & 2 & 3.13 \\ Maximum & 3 & 8.51 \(\times\) 10\({}^{-3}\) \\ Manhattan & 2 & 2.63 \\ Manhattan & 3 & 5.12 \(\times\) 10\({}^{-3}\) \\ Canberra & 2 & 9.36 \\ Canberra & 3 & 2.14 \\ Minkowski (\(p\) = 4) & 2 & 4.19 \(\times\) 10\({}^{-3}\) \\ Minkowski (\(p\) = 4) & 3 & 2.22 \(\times\) 10\({}^{-3}\) \\ \hline \end{tabular}
\end{table}
Table 6: [PERSON] stresses pre-screening for the distance measures at two cluster number settings (ultrafiltration process characteristics).  
Figure 8: Shepard graphs of the five tested distance measures for two-cluster (**A**) and three-cluster configurations (**B**) (ultrafiltration process characteristics).  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Distance Measure** & **Cluster Number** & **Kruskal Stress Value** \\ \hline Euclidean & 2 & \(5.42\times 10^{-14}\) \\ Euclidean & 3 & \(6.01\times 10^{-14}\) \\ Maximum & 2 & \(2.32\times 10^{-3}\) \\ Maximum & 3 & \(5.15\times 10^{-3}\) \\ Manhattan & 2 & \(9.03\times 10^{-3}\) \\ Manhattan & 3 & \(5.58\times 10^{-14}\) \\ Canberra & 2 & \(5.98\times 10^{-3}\) \\ Canberra & 3 & \(6.58\times 10^{-14}\) \\ Minkowski (\(p=4\)) & 2 & \(7.91\times 10^{-3}\) \\ Minkowski (\(p=4\)) & 3 & \(5.78\times 10^{-14}\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: [PERSON] stresses pre-screening for the distance measures at two-cluster number settings (nanofiltration process characteristics).  
The two-cluster Euclidean-distance-measured Kruskal stress estimate was comparable to the Kruskal stress values that were obtained for the three-cluster Manhattan, Canberra and Minkowski (\(p=4\)) distance measure models. For practical purposes, the Euclidean distance measure was retained in the tuning of the Databionic unsupervised classifier, but it was tested on both cluster sizes.  
Ultrametric Self-Organizing Clustering and Validating Metric Comparison to Fractional Factorial Setting Vectors