## 4 Experimentation, Performance Evaluation, and Discussion
### Results and Discussion  
The experimental results presented in Table 3 indicate that the proposed HFA-K-means algorithm had the lowest mean clustering value of 0.5522 for all the 20 high-dimensional datasets. This indicates that the proposed algorithm averagely performed better compared to the other seven metaheuristic-based K-means algorithms. The proposed HFA-K-means algorithm and PSO-based K-means recorded best results in five of the datasets each while the best performance records for the other ten datasets were shared among the other competing K-means-based hybrid algorithms. The Friedman mean rank test result of the eight competing algorithms presented in Table 4 also shows that the proposed HFAK-means \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Datasets** & **FA** & **K-Means** & **HFA-K-Means** \\ \hline A1 & 2.00 & 3.00 & 1.00 \\ A2 & 2.00 & 3.00 & 1.00 \\ A3 & 1.50 & 3.00 & 1.50 \\ Birch1 & 1.53 & 3.00 & 1.48 \\ Birch2 & 2.00 & 3.00 & 1.00 \\ Birch3 & 1.85 & 3.00 & 1.15 \\ Bridge & 2.00 & 3.00 & 1.00 \\ D31 & 1.95 & 3.00 & 1.05 \\ Dim002 & 1.50 & 3.00 & 1.50 \\ Dim1024 & 1.55 & 3.00 & 1.45 \\ Housec5 & 1.55 & 3.00 & 1.45 \\ Housec8 & 1.95 & 3.00 & 1.05 \\ Letter & 1.50 & 3.00 & 1.50 \\ Finland & 1.83 & 3.00 & 1.18 \\ S1 & 1.53 & 3.00 & 1.48 \\ S2 & 1.60 & 3.00 & 1.40 \\ S3 & 2.00 & 3.00 & 1.00 \\ S4 & 1.53 & 3.00 & 1.48 \\ T4.8k & 2.00 & 3.00 & 1.00 \\ Yeast & 1.50 & 3.00 & 1.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: [PERSON] rank-sum test for the HFA-K-means and the two classical algorithms tested across the twenty datasets for the twenty algorithm runs. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Datasets** & **HFA-K-Means vs.** & **HFA-K-Means vs.** & **FA vs. K-Means** \\ \hline A1 & 0.001 & 0.001 & 0.001 \\ A2 & 0.001 & 0.001 & 0.001 \\ A3 & 1.000 & 0.001 & 0.001 \\ Birch1 & 0.317 & 0.001 & 0.001 \\ Birch2 & 0.001 & 0.001 & 0.001 \\ Birch3 & 0.002 & 0.001 & 0.001 \\ Bridge & 0.001 & 0.001 & 0.001 \\ D31 & 0.001 & 0.001 & 0.001 \\ Dim002 & 1.000 & 0.001 & 0.001 \\ Dim1024 & 0.180 & 0.001 & 0.001 \\ Housec5 & 0.180 & 0.001 & 0.001 \\ Housec8 & 0.001 & 0.001 & 0.001 \\ Letter & 1.000 & 0.001 & 0.001 \\ Finland & 0.001 & 0.001 & 0.001 \\ S1 & 0.317 & 0.001 & 0.001 \\ S2 & 0.638 & 0.001 & 0.001 \\ S3 & 0.001 & 0.001 & 0.001 \\ S4 & 0.317 & 0.001 & 0.001 \\ T4.8k & 0.001 & 0.001 & 0.001 \\ Yeast & 1.000 & 0.001 & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Wilcoxon signed-rank test \(p\) values for equal medians on DB Index. algorithm ranked better with the lowest mean rank of 2.48 to further justify its superior performance over the PSO-based K-means that recorded a mean rank of 3.67. The computation results shown in Table 5 show that the proposed HFA-K-means can cluster high-dimensional datasets efficiently. The high convergence rate of the HFA-K-means is reflected in Figure 2, showing the average computation time required to achieve convergence. In comparison with the classical FA and K-means algorithms, as shown in Table 4, the proposed HFA-K-means recorded the best performance in 16 of the 20 datasets, namely, A1, A2, Birch1, Birch2, Birch3, Bridge, D31, DIM1024, Housec5, Housec8, Finland, S1, S2, S3, S4, and T4.8k. HFA-K-means recorded ties with FA in the remaining ones, namely A3, DIM002, Letter, and Yeast. In terms of average performance across all the 20 datasets, the HFA-K-means recorded the lowest value, followed by the FA. The lesser values of HFA-K-means compared with FA and K-means in most datasets show better performance. Based on this, we can infer that HFA-K-means can effectively and efficiently automatically cluster high-dimensional datasets. Table 7 compares the proposed algorithm with other classical metaheuristic algorithms. The results show that HFA-K-means demonstrated the best performance in f14 datasets, namely, A1, A2, Birch2, Birch3, Bridge, D31, DIM1024, Housec5, Housec8, Letter, Finland, S3, T4.8k, and Yeast. PSO outperformed the HFA-K-means in three datasets, namely A3, S2, and S4.