## 3 Results
### Graphical Pre-Screening of the Candidate Distance Measures
#### 3.4.1 The Ultrafiltration Process Parameter-Free-Projection Self-Organized Clustering  
The dendrogram for the three nominated ultrafiltration process characteristics (turbidity, total nitrogen content and NO\({}_{3}\)\({}^{-}\) content) is drawn in Figure 9. The Databionic auto-classifier was propped to take advantage of the Euclidean distance metric, adjusting the position type for the projection points to complement the compact structure type, while a three-cluster model was assigned. The length of the branches was scaled on the ultrametric portion of the distance. It was observed that the first branch (runs # 1, 2) was bifolious and well separated from the rest of the runs in the L\({}_{9}\)(3\({}^{4}\)) OA formulation. The second branch retained information from runs # 2-6, and the last branch was more balanced as the  
Figure 9: Shepard graphs of the five tested distance measures for two-cluster (**A**) and three-cluster configurations (**B**) (nanofiltration process characteristics).  
information was retrieved by the last three OA runs (# 7-9). To further examine the relationship of the tri-characteristic micro-dataset, the \"fitting\" performance was internalized. By using a popular algorithmic internal validator (the Davies-Bouldin index), the similarity evaluation process was not only restricted to quantifying a clustering effectiveness between the predicted cluster memberships and the inherent dataset. In fact, the Davies-Bouldin index could assume an extended role, i.e., that of an internal standard by which the four fractional factorial \"membership\" vectors were also assessed against the dataset. In Table 8, list the Davies-Bouldin index scores are listed for the two commonly implemented types of centrotypes: the cluster centroids and the medoids. The Davies-Bouldin Index was adjusted to the ordinary metric parameters (p, q) = (2, 1). From Table 8, it can be seen that the centroid- and medoid-based index estimations were in agreement. Additionally, the index parameter settings set at (p, q) = (2, 2) were attempted and estimated to verify that the initial (2, 1) setting offered a finer resolution of the clustering self-validation performance. There was as much as a 40% reduction in the internal validation prediction when selecting the Davies-Bouldin medoid metric, which was set at the parameter pair (2, 1) over the (2, 2) option. A corresponding difference also existed but was smaller for the centroid centrotype option. The Davies-Bouldin index estimations for the fractional factorial vectors revealed that there was a consensus in pointing at a factor that minimized the index estimate the most, and it was rather independent of the centrotype choice.  
In both cases, it was the factor A (membrane type) that stood out, and its fractional factorial setting vector emulated the cluster membership vector closer to that which was derived from the Databonic classifier solution (Figure 10). Actually, the medoid metric version substantially reduced the Davies-Bouldin index estimate to a value of 1.27 with respect to the centroid-based prediction value of 1.89 (Table 8). However, there was a split decision on the activeness of factor C (temperature), which returned a comparable similarity prediction to factor A when the solution was based on the centroid metric, but it was not verifiable by the alternative medoid estimate. Of course, validating the micro-dataset using the medoid-mediated fractional factorial vector sequences caused the similarity index estimations to be more evenly moderated in compiling the profiled hierarchy landscape. The rest of the factors could be contemplated to be inactive. Factor A was more likely to be approximated with a linear model.  
Figure 10: Dendrogram for emergent self-organized classification of the ultrafiltration process characteristics (Euclidean distance measure, cluster size =3, structure type = compact, position type = projected points).