## 4 Experimentation, Performance Evaluation, and Discussion
### Metrics for Performance Evaluation  
At the first instance, experiments were conducted coupling seven other metaheuristic algorithms (ABC, IWO, SOS, TLBO, DE, PSO and genetic algorithm (GA)) with K-means using the same framework as the proposed HFA-K-means algorithm. Their performances were evaluated and compared with that of the proposed HFA-K-means algorithm over the 20 high-dimensional datasets using the DBI as the cluster validity index. This was to investigate the performance of the proposed HFA-K-means algorithm in comparison with these metaheuristic-based K-means hybrid algorithms in order to further justify its selection as the best option before its comparison with results from the literature. The experimental results of the performances of these seven metaheuristic-based K-means hybrid algorithms compared with the proposed HFA-K-means algorithm are presented in Table 3. The non-parametric Friedman mean rank test was also conducted to further validate the clustering solution of the various metaheuristic-based K-means hybrid algorithms. The results from the Friedman mean rank test are presented in Table 4.  
The proposed HFA-K-means algorithm was compared with conventional metaheuristic search methods and other FA-based hybrid algorithms for a comprehensive and objective investigation of the clustering performance. The proposed algorithm was evaluated against classical FA, K-means, GA, DE, PSO, and IWO with four hybridized metaheuristics algorithms (FADE, PSDDE, IWODE, Improved SOSK-means) and other FA variants (FAPSO, FATLBO, and FADE). The DBI was used to evaluate the proposed algorithm's performance over 20 high-dimensional datasets with dimensions between 9 and 256 and dataset sizes between 1000 and 100,000, as shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Datasets** & **Dataset Types** & **Dimension of** & **Number of Data** & **Number of** & **References** \\  & **Data** & **Objects** & **Clusters** & \\ \hline A1 & Synthetically generated & 2 & 3000 & 20 & [58; 59; 60] \\ A2 & Synthetically generated & 2 & 5250 & 35 & [58; 59; 60] \\ A3 & Synthetically generated & 2 & 7500 & 50 & [58; 59; 60] \\ Birch1 & Synthetically generated & 2 & 100,000 & 100 & [38; 40; 61] \\ Birch2 & Synthetically generated & 2 & 100,000 & 100 & [38; 40; 61] \\ Birch3 & Synthetically generated & 2 & 100,000 & 100 & [58; 60; 61] \\ Bridge & Grey-scale image blocks & 16 & 4096 & 256 & [60; 62] \\ D31 & Shape sets & 2 & 3100 & 31 & [60; 63] \\ Dim002 & Synthetically generated & 2–15 & 1351–10,126 & 9 & [58; 60; 64] \\ Dim1024 & High-dimensional & 1024 & 1024 & 16 & [58; 60; 65] \\ Housec5 & RGB image & 3 & 34,112 & 256 & [60; 62] \\ Housec8 & RGB image & 3 & 34,112 & 256 & [60; 66] \\ Letter & UCI dataset & 16 & 20,000 & 26 & [60; 62] \\ Finland & Mopsi locations & 2 & 13,467 & 4 & [60; 67] \\ S1 & Synthetically generated & 2 & 5000 & 15 & [58; 60; 68] \\ S2 & Synthetically generated & 2 & 5000 & 15 & [58; 60; 68] \\ S3 & Synthetically generated & 2 & 5000 & 15 & [58; 60; 68] \\ S4 & Synthetically generated & 2 & 5000 & 15 & [7; 58; 60; 68] \\ T4.8k & Miscellaneous & 2 & 8000 & 3 & [60; 69] \\ Yeast & UCI dataset & 8 & 1484 & 10 & [60; 62] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The characteristics of twenty high-dimensional datasets. The summary of the results obtained by the proposed HFA-K-means for the clustering problem over 20 high-dimensional datasets is presented in Table 5. The results are presented to show the best, worst, average, and standard deviation values over 20 independent runs of each of the algorithms on 20 datasets. This signifies the optimal, the worst, the mean, and the standard deviation of the clustering solutions. The average computation time taken to achieve the optimal solution for the HFA-K-means is shown in Figure 2. To investigate the performance gain in terms of the computation time, the HFA-K-means computation time is compared with the computation time of a standard hybrid of FA and K-means.