## 5 Discussion  
For the first problem, both algorithms were able to detect the novel 4-th class. However, a portion of the instances originally belonging to the novel 4-th class, were considered as anomalies and discarded by both algorithms. The number of clusters, which were designed by HPFuzzNDA for normal classes, varied from five to nine, while for the novel class this number changed from two to six. In addition, all classes were divided into different numbers of clusters. Meanwhile, all normal and novel classes were divided into eight clusters by the PFuzzND approach, thus, the number of clusters increased to the maximum value and did not change thereafter. Figure 2 shows the example of results obtained on this dataset by both algorithms. Generally, PFuzzND achieves a lower accuracy and higher value of the unknown rate measure on \(DS_{1}\). In Figure 2, it is evident that PFuzzND found more anomalies than HPFuzzNDA, but the latter was more successful while determining instances from normal classes. PFuzzND showed significantly better results compared to HPFuzzNDA only on the second dataset according to the Mann-Whitney test. Both algorithms were able to determine that there are only two normal classes and anomalies (novel classes were not detected, which is correct in the case of this dataset). The number of clusters that were designed by HPFuzzNDA for normal classes varied from 3 to 5, while it stayed at 8 by the end of PFuzzND work. Best results obtained for this dataset by both algorithms are presented in Figure 3. To be more specific, for the example demonstrated in Figure 3, PFuzzND's accuracy was 94.49%, while HPFuzzNDA correctly classified 93.53% of the labelled data. Nevertheless, from this figure, it is clear that HPFuzzNDA was able to determine class extensions better than the PFuzzND approach. This might present a sign that for the second dataset it would perhaps be better to use HPFuzzNDA but with an initially increased number of clusters for classes. According to the numerical results presented in the previous section, there is no significant difference between results obtained by both algorithms on \(DS_{4}\), \(DS_{6}\) and \(DS_{7}\). The fourth dataset is similar to the second as there are only two normal classes and anomalies. HPFuzzNDA mostly designed models with two clusters for each class, but PFuzzND still increased the number of clusters up to the maximum value causing an unnecessary partition of instances. Example of results obtained for \(DS_{4}\) is demonstrated in Figure 4.  
Figure 2: Example of instances from the first dataset, classified by HPFuzzNDA (the **left** graph) and by PFuzzND (the **right** graph) algorithms. Red points belong to the novel class. The 6-th and the 7-th datasets contain neither anomalies nor novel classes. HPFuzzNDA mostly forms four or five clusters for each normal class, while PFuzzND shows the same behaviour as it did for previously mentioned problems. It should be noted that both algorithms mistakenly marked some instances as anomalies for \(DS_{6}\), but the number of \"anomalies\" found by the PFuzzND approach was much bigger compared to HPFuzzNDA. This fact is highlighted in Table 5, as the value of the unknown rate measure obtained by PFuzzND for \(DS_{6}\) is much greater than that obtained by HPFuzzNDA. Furthermore, slightly better accuracy was achieved by HPFuzzNDA on both datasets, \(DS_{3}6\) and \(DS_{7}\), due to the fact that PFuzzND was not always able to determine the \"borders\" between present classes. The latter can be clearly seen in Figures 5 and 6. HPFuzzNDA significantly outperformed PFuzzND on \(DS_{3}\), \(DS_{5}\), \(DS_{8}\) and \(DS_{9}\) datasets. For \(DS_{3}\) the number of clusters varied from 4 to 9 for each class; for \(DS_{5}\), \(DS_{8}\) and \(DS_{9}\) it changed from 2 to 4. It should be noted that PFuzzND constantly increased the number of clusters for each class until it reached the maximum value. Thus, it can be concluded that a lower number of clusters leads to better results for these datasets.