## 4 Experimental Results
### Numerical Results for Benchmark Problems  
The aim of the conducted experiments was to compare the HPFuzzNDA approach with the original PFuzzND algorithm. Thus, both algorithms were executed on all 9 datasets and repeated 10 times. Results obtained on all datasets are presented in Table 5. In these tables, results averaged by the number of program runs are shown, and \(Acc\) means \begin{table}
\begin{tabular}{c c c} \hline \hline  & \
\begin{tabular}{c} **PFuzzND** \\ **Parameters** \\ \end{tabular} &
\begin{tabular}{c} **HPFuzzNDA Parameters** \\ \end{tabular} \\ \hline \multirow{4}{*}{Offline Phase} & \(m=2\) & \(m=2\) \\  & \(c_{max}=8\) & initial \(c_{max}=2\) \\  & \(p_{offline}=10\%\) & \(p_{offline}=10\%\) \\  & initial \(k_{class}=4\) & \\ \hline \multirow{4}{*}{Online Phase} & \(\theta_{init}=0.3\) & \(\theta_{init}=0.3\) \\  & \(\theta_{class}=0.3\) & \(\theta_{class}=0.3\) \\  & \(\theta_{adapt}=0.1\) & \(\theta_{adapt}=0.1\) \\  & & \(p_{in}=0.05\) \\  & & \(p_{cmax}=0.02\) \\  & & \(B=40\%\) (of currently examined instances) \\ \hline \multirow{4}{*}{Novelty Detection} & \(m=2\) & \(m=2\) \\  & \(N_{sm}=40\) & \(N_{sm}=40\) \\ \cline{1-1}  & \(ts=500\) & \(ts=500\) \\ \cline{1-1}  & \(phi=0.45\) & \(phi=0.45\) \\ \cline{1-1}  & \(k_{sm}=4\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Basic and newly introduced parameters of the PFuzzND and HPFuzzNDA approaches. accuracy in % while \(MF\) denotes the macro \(F\)-score metric values (they are also given in %). Additionally, the \(UnkR\) values here should be multiplied by \(10^{-3}\). Overall, the proposed method was able to detect class extensions at the classification step and adapt the model during the online phase. Moreover, the HPFuzzNDA algorithm could also detect the new classes which emerged along the online phase for datasets that had unlabelled instances (the latter is shown in the column with the unknown measure rate values). Generally, compared to the PFuzzND approach, the new HPFuzzNDA algorithm correctly classified instances more frequently. Additionally, to compare the efficiency of HPFuzzNDA and PFuzzND the Mann-Whitney rank sum statistical test with significance level \(p_{MW}=0.01\) was used. Statistical tests were applied to the accuracy and macro \(F\)-score values, and obtained results are demonstrated in Table 6. It should be noted that in Table 6, the following notations are used:  
1. \"+\" means that HPFuzzNDA showed significantly better results than PFuzzND on a given problem;
2. \"-\" means that HPFuzzNDA showed significantly poorer results than PFuzzND on a given problem;
3. \"=\" means that there were no significant differences between results of the HPFuzzNDA and PFuzzND approaches for a given problem;  
Thus, the HPFuzzNDA algorithm achieved significantly better results for four benchmark datasets and performed worse on only one dataset. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(DS_{1}\) & \(DS_{2}\) & \(DS_{3}\) \\ \hline \multirow{3}{*}{Accuracy} & \(1.758\) & \(-3.250\) & \(3.704\) \\  & \(=\) & \(-\) & \(+\) \\ \cline{1-1} \cline{3-5} \multirow{3}{*}{Macro \(F\)-score} & \(1.914\) & \(-3.326\) & \(3.704\) \\  & \(=\) & \(-\) & \(+\) \\ \hline \multirow{3}{*}{Accuracy} & \(DS_{4}\) & \(DS_{5}\) & \(DS_{6}\) \\ \hline \multirow{3}{*}{Accuracy} & \(-0.265\) & \(3.780\) & \(0.529\) \\  & \(=\) & \(+\) & \(=\) \\ \cline{1-1} \cline{3-5} \multirow{3}{*}{Macro \(F\)-score} & \(-0.227\) & \(3.780\) & \(0.529\) \\  & \(=\) & \(+\) & \(=\) \\ \hline \multirow{3}{*}{Accuracy} & \(DS_{7}\) & \(DS_{8}\) & \(DS_{9}\) \\ \cline{1-1} \cline{3-5} \multirow{3}{*}{Macro \(F\)-score} & \(-1.022\) & \(2.685\) & \(3.780\) \\  & \(=\) & \(+\) & \(+\) \\ \cline{1-1} \cline{3-5} \multirow{3}{*}{Macro \(F\)-score} & \(-0.983\) & \(2.646\) & \(3.779\) \\  & \(=\) & \(+\) & \(+\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the Mann–Whitney rank sum statistical test.