## 4 Clustering Ordered Permutations
### Empirical Investigations  
A comprehensive empirical investigation of the proposed binary conversion methods was performed, considering the ART-1 and Improved-ART-1 algorithms. The two binary conversion techniques are compared in the following subsections based on misclassification, homogeneity, and average distance. In this investigation, the number of seeds is 5, the number of solutions per seed is 500, the half-width is 3, the vigilance value is 0.5, and the number of objects is 40. Since the analysis provides slightly different solutions during each run, a total of 10 solutions was considered, and then the average of these ten solutions was taken. Table 1 presents a comparative study between binary conversion method-1 and method-2, adopting the ART-1 and Improved-ART-1 neural networks. The initial result indicates that the binary conversion method-1 (M1) outperforms method-2 using ART-1 and Improved-ART-1 in terms of all the clustering performance criteria (i.e., less misclassification, high homogeneity, and shorter average distance). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**ART-1**} & \multicolumn{2}{c}{**Imp. ART-1**} \\ \cline{2-5}  & **M1** & **M2** & **M1** & **M2** \\ \hline Misclassification & 1834 & 1900 & 22 & 183 \\ Homogeneity (\%) & 42 & 35 & 99 & 72 \\ Average Distance & 47 & 55 & 8 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance analysis between binary conversion method-1 and method-2 considering ART-1 and Improved-ART-1 algorithms. The previous analysis shows that the binary conversion method-1 provides satisfactory results. Now, the investigation is further extended to different problem sizes considering 10, 20, 30 and 40 objects, where the other parameters mentioned at the beginning of this section are kept constant. Table 2 shows the performance of binary conversion method-1 on both ART-1 and Improved-ART-1 neural networks, considering four different problem sizes. The results indicate that method-1 performs better than method-2, providing less misclassification, higher homogeneity, and a shorter average distance. Figure 4 presents a comparative study between ART-1 and Improved-ART-1, in terms of misclassification, homogeneity, and average distance using binary conversion M1, since M1 outperforms compared to M2. It is seen that the number of ART-1 misclassifications is nearly 99% higher than when using Improved ART-1. Therefore, Improved-ART-1 with M1 leads to better-quality solutions than ART-1. Furthermore, a projection between the expected cluster numbers and the actual cluster generated by both neural networks using binary conversion method-1 is illustrated in Figure 5. It is reported that the Improved-ART-1 with method-1 matches and follows the trend of the anticipated number of clusters with an increment in seed number until 6, where the vigilance value is 0.5, degree of perturbation is 4, half-width is 3, and the \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**P1 (10 Objects)**} & \multicolumn{4}{c}{**P2 (20 Objects)**} & \multicolumn{4}{c}{**P3 (30 Objects)**} & \multicolumn{4}{c}{**P4 (40 Objects)**} \\ \cline{2-13}  & \multicolumn{2}{c}{**ART-1**} & \multicolumn{2}{c}{**Imp. ART-1**} & \multicolumn{2}{c}{**ART-1**} & \multicolumn{2}{c}{**Imp. ART-1**} & \multicolumn{2}{c}{**ART-1**} & \multicolumn{2}{c}{**ART-1**} & \multicolumn{2}{c}{**Imp. ART-1**} & \multicolumn{2}{c}{**Imp. ART-1**} \\ \cline{2-13}  & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** & **M1** & **M2** \\ \hline M & 2059 & 2215 & 941 & 1407 & 1829 & 1984 & 58 & 170 & 1921 & 1726 & 46 & 176 & 1834 & 1900 & 22 & 183 \\ H & 26 & 26 & 39 & 52 & 44 & 34 & 96 & 85 & 34 & 47 & 98 & 75 & 42 & 35 & 99 & 72 \\ A. D. & 8 & 8 & 5 & 7 & 18 & 20 & 11 & 12 & 33 & 33 & 17 & 18 & 47 & 55 & 8 & 21 \\ \hline \hline \end{tabular}
\end{table}