## 4 Experimental Results and Discussion
### First Experimental Scenario: Synthetic Data  
This experiment aims to create a controlled environment to evaluate the effectiveness of various optimization algorithms in the informative feature selection step of our proposed imputation method. Specifically, we will examine their performance under different conditions. To identify the best optimization algorithm among the PSO, GWO, WOA, and SCA, we utilize synthetic datasets with varying rates of missing values. To do this, we generated the synthesized data using the \"sklearn\" Python toolbox, which allows for the creation of regression datasets with variable correlation between features. This approach ensures that the data are suitable for our analysis and reflects realistic relationships among the variables. This approach enables us to assess the proposed imputation method alongside different optimization algorithms and conduct a comparative analysis that highlights their strengths and limitations. Table 2 presents the details of these datasets. In this experiment, we imputed the data with 5%, 15%, and 25% missing rates using our proposed imputation method with different optimization algorithms. The results are shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline
**Dataset** & **\# Data Points** & **\# Features** \\ \hline SynD\({}_{1}\) & 750 & 11 \\ SynD\({}_{2}\) & 900 & 7 \\ SynD\({}_{3}\) & 800 & 9 \\ SynD\({}_{4}\) & 800 & 8 \\ SynD\({}_{5}\) & 1200 & 10 \\ SynD\({}_{6}\) & 1000 & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: Description of the Synthetic datasets. The table has six synthetic datasets with a size of 750 to 1200 data points and a size of seven to eleven features. The first three datasets with uncorrelated features, and the remaining three datasets with dependent features. \begin{table}
\begin{tabular}{c c c c c} \hline
**Dataset** & **FS-PSO** & **FS-GWO** & **FS-SCA** & **FS-WOA** \\ \hline \multicolumn{5}{c}{**Missing rate 5\%**} \\ \hline SynD\({}_{1}\) & 0.0586 (4.0) & 0.0567 (3.0) & 0.0567 (2.0) & **0.0554 (1.0)** \\ SynD\({}_{2}\) & **0.0378 (1.0)** & 0.0409 (4.0) & 0.0394 (3.0) & 0.0384 (2.0) \\ SynD\({}_{3}\) & 0.0448 (4.0) & 0.0432 (3.0) & **0.0418 (1.0)** & 0.0430 (2.0) \\ SynD\({}_{4}\) & 0.0067 (3.0) & **0.0062 (1.0)** & 0.0066 (2.0) & 0.0070 (4.0) \\ SynD\({}_{5}\) & 0.0038 (2.5) & 0.0041 (4.0) & 0.0038 (2.5) & **0.0036 (1.0)** \\ SynD\({}_{6}\) & 0.0091 (4.0) & **0.0086 (1.0)** & 0.0087 (2.0) & 0.0088 (3.0) \\ \hline Average ranks & 3.08 & 2.67 & **2.08** & 2.17 \\ \hline \multicolumn{5}{c}{**Missing rate 15\%**} \\ \hline SynD\({}_{1}\) & 0.1625 (3.0) & 0.1647 (4.0) & **0.1585 (1.0)** & 0.1603 (2.0) \\ SynD\({}_{2}\) & **0.1226 (1.0)** & 0.1253 (4.0) & 0.1239 (2.0) & 0.1240 (3.0) \\ SynD\({}_{3}\) & 0.1328 (4.0) & 0.1293 (3.0) & 0.1275 (2.0) & **0.1237 (1.0)** \\ SynD\({}_{4}\) & 0.0227 (4.0) & 0.0218 (3.0) & **0.0200 (1.0)** & 0.0203 (2.0) \\ SynD\({}_{5}\) & **0.0197 (1.0)** & 0.0214 (4.0) & 0.0208 (3.0) & 0.0205 (2.0) \\ SynD\({}_{6}\) & 0.0329 (4.0) & 0.0295 (2.0) & 0.0325 (3.0) & **0.0292 (1.0)** \\ \hline Average ranks & 2.83 & 3.33 & 2.00 & **1.83** \\ \hline \multicolumn{5}{c}{**Missing rate 25\%**} \\ \hline SynD\({}_{1}\) & 0.2689 (4.0) & 0.2608 (2.0) & 0.2631 (3.0) & **0.2568 (1.0)** \\ SynD\({}_{2}\) & 0.1869 (4.0) & 0.1825 (2.0) & 0.1859 (3.0) & **0.1799 (1.0)** \\ SynD\({}_{3}\) & 0.2060 (4.0) & 0.2034 (3.0) & 0.2013 (2.0) & **0.2008 (1.0)** \\ SynD\({}_{4}\) & 0.0431 (4.0) & 0.0428 (3.0) & 0.0399 (2.0) & **0.0396 (1.0)** \\ SynD\({}_{5}\) & 0.0527 (4.0) & 0.0482 (2.0) & **0.0481 (1.0)** & 0.0500 (3.0) \\ SynD\({}_{6}\) & **0.0546 (1.0)** & 0.0559 (3.0) & 0.0561 (4.0) & 0.0557 (2.0) \\ \hline Average ranks & 3.50 & 2.50 & 2.50 & **1.50** \\ \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of imputation methods on different synthetic datasets with a rate of 5%, 15%, and 25% missing data (in the form of MSE \((mk)\)). * **With a missing rate of 5%:** The WOA produced the best results on two datasets, secured second place on two others, and ranked third and fourth once.