## 4 Experimental Results and Discussion
### Second Experimental Scenario: Real Data  
In this experiment, we conduct a comprehensive evaluation of imputation methods by applying our proposed approach using the WOA (We selected WOA as it performed best among the other optimizers in the previous experiment) alongside several state-of-the-art imputation techniques (single, KNN, MICE The code for the MICE algorithm is here: [[https://github.com/hjm9702/AL_with_missing_values_considering_imputation_uncertainty](https://github.com/hjm9702/AL_with_missing_values_considering_imputation_uncertainty)]([https://github.com/hjm9702/AL_with_missing_values_considering_imputation_uncertainty](https://github.com/hjm9702/AL_with_missing_values_considering_imputation_uncertainty)), accessed on 11 February 2025), and GAIN) on a set of well-known real-world datasets ([[https://archive.ics.uci.edu/datasets](https://archive.ics.uci.edu/datasets)]([https://archive.ics.uci.edu/datasets](https://archive.ics.uci.edu/datasets)), accessed on 11 February 2025) (see Table 5). We compare our missing-data imputation approach with various single and multiple imputation methods. Incorporating real-world datasets aims to bridge theoretical effectiveness with practical application. These datasets have varying numbers of features. In this experimental scenario, we use different missing rates to investigate the adaptability of each imputation method to varying degrees of data incompleteness. The results of this experiment are shown in Table 6. As illustrated:  
* **Missing Rate of 5%:** The proposed data imputation approach using WOA demonstrated exceptional performance across all datasets, achieving the best results in the majority. The KNN algorithm performed excellently in three out of ten datasets, showing promising results in others, and generally ranked second with an average rank of 2.1. MICE and Single methods performed satisfactorily, while GAIN struggled, yielding low performance across all datasets. Overall, FS-WOA had the best average ranking (1.4), whereas GAIN ranked the lowest. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\# Features** & **FS-PSO** & **FS-GWO** & **FS-SCA** & **FS-WOA** \\ \hline
5 features & 74.516 & 88.594 & 90.456 & 111.750 \\
8 features & 132.788 & 158.534 & 180.344 & 225.319 \\
12 features & 226.484 & 291.397 & 351.888 & 424.178 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: CPU time for different imputation algorithms applied to synthetic datasets. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **\# Data Points** & **\# Features** \\ \hline Iris & 150 & 4 \\ Breast cancer & 569 & 30 \\ Wine & 178 & 13 \\ Digits & 1797 & 64 \\ Diabetes & 767 & 8 \\ Ionosphere & 350 & 34 \\ Sonar & 207 & 60 \\ Vowel & 989 & 10 \\ Vehicle & 845 & 18 \\ Mnist & 3146 & 784 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Description of the real datasets that we used in our experiments. The datasets have (i) different numbers of features, ranging from four to 784, and (ii) different data sizes, ranging from 150 to 3146 instances. * **Missing Rate of 15%:** The FS-WOA approach again yielded the best results, outperforming other methods in over 50% of the datasets while also performing well in the rest. The KNN algorithm performed best on some datasets, and the MICE method showed remarkable performance on certain datasets, although it struggled on others. Overall, FS-WOA maintained the best average rank of 1.6, followed by KNN at 2.5, with GAIN at the bottom with an average rank of 4.2. * **Missing Rate of 25%:** The FS-WOA approach continued to outperform other methods, achieving optimal results in about half of the datasets. GAIN, with its more complex structure and higher data requirements, was significantly affected by the increased missing rates. The MICE and KNN methods showed promising performance in only a few datasets. In summary, the ranking was FS-WOA first, KNN second, MICE third, Single fourth, and GAIN last.