## 4 Discussion  
### Bar Plots-Feature-To-Cluster Assignment  
In the following, typical results of the mathematical modeling are presented. The selected types of illustration are typical examples of how those results can be presented. Figure 4 shows the proportion as calculated according to the above-derived equation for each characteristic element. The frequency of occurrence is normalized to one, which means that one represents 100% of examples in the cluster having the considered characteristic, whereas 0 means 0% of the examples show this specific characteristic. Thus, the values, and especially the shape, of the graphs indicate the importance of specific characteristics for each cluster. It is straightforward to create illustrations, as shown in Figure 4 for each cluster, to compare the significance of each group of characteristics with each other. These illustrations help the experts who provide the initial data to neutrally evaluate the results.  
In Figure 5, the significance of each characteristic in a cluster in relation to the so-called parent cluster is shown. Based on the investigated cluster, a specific number 'i' of sub-clusters results from the calculations. For simplicity, again, only two sub-clusters are used in the explanatory illustrations. A significance of '1' means that the specific characteristic is only present in one sub-cluster of all sub-clusters extracted out of one parent cluster. A significance of '0' means that this characteristic is not existing in the sub-cluster 'i'. Again, this type of illustration helps to group the classifications and allows a graphical evaluation of the results based on the mathematical modeling. This allows a dispassionate evaluation of the results of the initial classification of the example cases. In comparison to the classical evaluation of large groups of objects, these findings and the selected methods of presentation are big advantages to avoid misjudgments by the experts.  
Figure 3: Complete-link dendrogram for final PUE-objects (\(n=54\)).  
Figure 6 gives a graphical overview of the whole assessment of data, including the conciseness, given as proportion, and significances of each characteristic number for several layers of classes (parent cluster down to each layer of sub-clusters). Again, the values are normalized to '1', representing all cases in the corresponding cluster, and '0', meaning no proportion/significance. The visual presentation of the results of mathematical modeling can be used to distinguish relevance. If both criteria (proportion and significance) are high  
Figure 4: The illustration shows how present a feature is in the determining cluster.  
Figure 5: The significance of each characteristic to the clusters.  
for a specific characteristic within a specific cluster, it can be stated that this characteristic is relevant for the assignment of objects into the cluster. If only the proportion is high, this means that the characteristic criteria are relevant for the cluster but should not be exclusively assigned. In other words, many objects within the observed cluster possess specific characteristics, but other objects in other clusters possess this characteristic as well. Vice versa, it can be stated if only the significance is high, this means that the characteristic is the reason for assigning the object to the cluster, but many other objects in the same cluster do not possess this characteristic.  
Summarizing, the combination of proportion and significance, and especially their gradient between the different layers of clusters (parent cluster, sub-clusters, sub-sub-clusters,... ), allows a description for each specific cluster. Based on the graphical presentation, which is created automatically out of the results of the mathematical modeling, the experts can easily determine the relations between the list of objects.