## 3 Results
### Graphical Pre-Screening of the Candidate Distance Measures
#### 3.4.2 The Nanofiltration Process Parameter-Free Projection Self-Organized Clustering  
The dendrogram for the three nominated nanofiltration process characteristics (electrical conductivity, total nitrogen content and NO\({}_{3}\)\({}^{-}\) content) is drawn in Figure 11. The Databionic classifier was adjusted to the Euclidean distance metric. The tracking of position points was carried out by projection in a compact structure. Nevertheless, two cluster sizes were examined this time, at k = 2 and 3. The length of the branches was evaluated on the ultrametric portion of the distance. The tree configuration was surprisingly identical for both of the tested cluster sizes. It was determined that there were at least two clusters that exhibited substantial differences between them. It might be said that the behavior of the data entries that were associated with runs # 1, 2 and 3 were separable from the rest of the dataset. The implementation of the Davies-Bouldin self-validator index, set at (\(p\) = 2, q = 1), greatly simplified the comparison process in four aspects in this case (Table 9):  
1. The self-validation of the tri-characteristic clustering was in agreement regardless of the cluster size; the Davies-Bouldin index was confined to values between 0.34 and 0.41 for all four estimations.
2. The Davies-Bouldin index estimations, within a preset cluster size, was in agreement regardless of the selection of the two centrotypes. This might imply a more reliable \"internal standard\" with respect to the ultrafiltration process outcomes.
3. It was factor A that mimicked the behavior of the self-validator estimations, thus delivering the maximum information by ensuring that the similarity between the factor-A vectoring and the inherent internal clustering pattern were almost indistinguishable; the membership identification entries from the Databionic classifier and the fractional factorial setting vector for factor A matched. Particularly, for the case in which the cluster size was set at k = 3, the centroid- and medoid-based Davies-Bouldin index estimates were also identical; their computed values were 0.41 and 0.40, respectively. From this behavior, it was inferred that factor A should be assigned a simpler linear model.
4. The remaining three factors may be deemed weak since their Davies-Bouldin index magnitudes were substantially larger.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & & & **Davies–Bouldin Index Estimation For Fractional Factorial Vectors** \\ \hline
**Centrotypes** & **Self-validation** & **A** & **B** & **C** & **D** \\ \hline
**Centroids** & 0.74 (0.80) \({}^{*}\) & 1.89 & 6.79 & 1.88 & 4.7 \\ \hline
**Medoids** & 0.76 (1.07) \({}^{*}\) & 1.27 & 2.08 & 3.55 & 4.44 \\ \hline \hline \end{tabular}  
* In parenthesis the estimation was obtained using the parameters (\(p\) = 2, q = 2).  
\end{table}
Table 8: Davies–Bouldin index estimation of ultrafiltration process tri-characteristic dataset for self-validation and fractional factorial clustering performance (\(p\) = 2, q = 1).  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & & & & **Davies–Bouldin Index Estimation for Fractional Factorial Vectors** \\ \hline
**Cluster Size** & **Centrotypes** & **Self-validation** & **A** & **B** & **C** & **D** \\ \hline
**k = 2** & **Centroids** & 0.39 & 0.41 & 12.1 & 19.97 & 22.87 \\ \hline  & **Medoids** & 0.34 & 0.4 & 5.63 & 5.61 & 5.6 \\ \hline
**k = 3** & **Centrotypes** & **Self-validation** & **A** & **B** & **C** & **D** \\ \hline  & **Centroids** & 0.41 & 0.41 & 12.1 & 19.97 & 22.87 \\ \hline  & **Medoids** & 0.4 & 0.4 & 5.63 & 5.61 & 5.6 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Davies–Bouldin index estimation of nanofiltration process tri-characteristic dataset for self-validation and fractional factorial clustering performance (\(p\) = 2, q = 1).