## 4 Experimental Results
### Experimental Setup for Benchmark Problems  
The performance of the proposed method was evaluated by using nine synthetic datasets named \(DS_{1}\),..., \(DS_{9}\)[84]. Table 2 presents the details of each dataset:  
1. Column \(Instances\) indicates the total number of examples for each dataset, respectively. 2. Column \(Attributes\) indicates the number of features for each dataset, respectively. 3. Column \(Classes(offline)\) indicates the number of classes known at the offline phase. Figure 1: The general scheme of the HPFuzzNDA approach. Column _Classes_ indicates the minimum total number of classes for each dataset (the difference between this number and the corresponding value from the column \(Classes(offline)\) is the minimum number of novel classes that algorithms are expected to detect). Both the PFuzzND and the proposed HPFuzzNDA approaches use possibilistic fuzzy clustering algorithm PFCM during offline and online phases. The parameters of the PFCM algorithm were the same for all conducted experiments and are listed in Table 3. The following notations are used in the table: \(expo\) and \(nc\) are constants responsible for the membership and the typicality values, \(a\) and \(b\) are constants that show the influence of the membership and the typicality values during the decision making step, \(MaxIter\) is the maximum number of iterations for PFCM to generate a given number of clusters, and \(MinImp\) is the minimum difference between the objective function values calculated on two consequential iterations of the PFCM algorithm (if this difference was less than \(MinImp\) then calculations stopped). It should be noted that the PFCM clustering method uses a specific parameter \(\gamma_{i}\) for each \(i\)-th cluster. All these \(\gamma_{i}\) parameters influence the typicality values, which in turn are necessary to determine the clusters' centres. The \(\gamma_{i}\) parameters were introduced for the possibilistic clustering method PCM [73]. In this work, they were user-defined constants, while in [85] the authors proposed to calculate these values on each iteration of the clustering algorithm. In this study, experiments were conducted only for the iteratively calculated \(\gamma_{i}\). The same basic parameters were used during the offline and the online phases (including the novelty detection procedure) for both PFuzzND and HPFuzzNDA algorithms to make the comparison between them fair. As previously mentioned, \(c_{max}\) was a constant for the PFuzzND approach, and it changed over time for HPFuzzNDA. All parameters are listed in Table 4. In the table, \(k_{sm}\) is the number of clusters that should be designed for the short memory; it is a user-defined parameter (constant) for PFuzzND and automatically adjusted for HPFuzzNDA. \begin{table}
\begin{tabular}{c c c c c} \hline
**Datasets** & **Instances** & **Attributes** & **Classes (offline)** & **Classes** \\ \hline \(DS_{1}\) & 1000 & 2 & 3 & 4 \\ \(DS_{2}\) & 5500 & 2 & 2 & 3 \\ \(DS_{3}\) & 10,000 & 2 & 3 & 3 \\ \(DS_{4}\) & 11,000 & 2 & 2 & 3 \\ \(DS_{5}\) & 20,000 & 2 & 3 & 3 \\ \(DS_{6}\) & 10,000 & 2 & 4 & 4 \\ \(DS_{7}\) & 10,000 & 2 & 3 & 3 \\ \(DS_{8}\) & 10,000 & 3 & 3 & 3 \\ \(DS_{9}\) & 10,000 & 2 & 4 & 4 \\ \hline \end{tabular}
\end{table} 
Table 2: Benchmark datasets used in experiments. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(expo\) & 2 \\ \(MaxIter\) & 100 \\ \(MinImp\) & 0.0005 \\ \(a\) & 1 \\ \(b\) & 4 \\ \(nc\) & 3 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of the PFCM algorithm. Additionally, the NL-SHADE-RSP also contains its own parameters. Most of them were taken from [6] as it was shown that they allow NL-SHADE-RSP to achieve the best optimization results. However, the following parameters were set during our experiments:  
* \(NFE_{max}=c_{max}\times(p-1)\times 100\);
* \(P=c_{max}\times(p-1)+1\);
* initial \(NP\) = 25;
* \(NP_{max}=25\).