## 4 Results
### Algorithms Comparison
From the comparison of the algorithms of the rest category, CLA is recommended. Figure 1:0 Comparison of the rest algorithms using: (**a**) DBI, (**b**) MDI, (**c**) IAI, (**d**) IEI, (**e**) CH, (**f**) SI. Table 8 presents the algorithms ranking per validity indicator. The minCEntropy ranks 1 st according to J, IAI and CDI. K-medoids ranks 1 st in DBI and 4 th in CDI. The Modified K-means#1 is present in 10 indices. According to Criterion#3, in general terms, it is the most efficient algorithm. IWFA K-means is the 2 nd best. Also, minCEntropy ranks high in the lists. The results indicate that the partitional algorithms present better performance followed by hierarchical ones. Among the latter, SL is the most robust while MVM and UPGMC have satisfactory performance. No fuzzy and neural-network based algorithms are present. Among the algorithms of the rest category, IRC and CLA provide adequate clusterings. Criterion#4 is important in applications with vast amount of data. Table 9 presents the execution time per algorithm as measured in 2.20 GHz Pentium(r) B960 Dual Core(tm) with 8 GB RAM system. The time refers only to the execution of the algorithm for 2 to 30 clusters excluding the calculation of the validity indicators value. Furthermore, the time of parametric analyses for proper parameter calibration is not included. The last column in the Table corresponds to the ratio of the required time of an algorithm to the required time of the K-means. All algorithms are executed considering their optimal parameters. It can be observed that hierarchical algorithms are the fastest. SOM and AVQ have considerably longer execution times, making these algorithms not appropriate for real-time applications. The required time for FDL and ISODATA cannot actually take part in the comparison since many different executions are needed by changing their parameters in order to provide clusterings with specific number of clusters. Criterion#5 is application dependent. It refers to the potential of an algorithm to fit into the special requirements of an application. In the present paper, Criterion#5 considers the following attributes: Empty clusters generation and outlier detection. The results are presented in Figure 1:1 The concept of \"empty clusters\" refers to the fact that an algorithm leads to lower number of clusters than the one it is requested. The concept \"outlier detection\" refers to the fact that an algorithm has the potential to track and isolate atypical patterns. Partitional algorithms do not result in empty clusters formation. However, they tend to produce clusters with almost similar number of members. On the contrary, hierarchical algorithms such as the SL and the CL algorithms can identify outliers. \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Validity Indicator** & **Algorithms’ Ranking** & **Validity Indicator** & **Algorithms’ Ranking** \\ \hline \multirow{4}{*}{J} & 1. minCEntropy & & 1. IRC \\  & 2. Modified K-means\#1 & & 2. IWFA K-means \\  & 3. MVM & & 3. Modified K-means\#1 \\  & 4. K-means\_A & & 4. BCEC2 \\ \hline \multirow{4}{*}{MIA} & 1. SL & & 1. minCEntropy \\  & 2. IWFA K-means & & 2. Modified K-means\#1 \\  & 3. UPGMC & & 3. MVM \\  & 4. UPGMA & & 4. K-means\_A \\ \hline \multirow{4}{*}{CDI} & 1. minCEntropy & & 1. IWFA K-means \\  & 2. Modified K-means\#1 & & 2. Modified K-means\#1 \\  & 3. MVM & & 3. SVC \\  & 4. K-medoids & & 4. IRC \\ \hline \multirow{4}{*}{WCBCR} & 1. SL & & 1. Modified K-means\#1 \\  & 2. UPGMC & & 2. IWFA K-means \\  & 3. Modified K-means\#1 & & 3. minCEntropy \\  & 4. UPGMA & & 4. MVM \\ \hline \multirow{4}{*}{SMI} & 1. Modified K-means\#1 & & 1. IWFA K-means \\  & 2. CLA (N3) & & 2. Modified K-means\#1 \\  & 3. MVM & & 3. SL \\  & 4. CLA (N2) & & 4. UPGMC \\ \hline \multirow{4}{*}{SMI2} & 1. Modified K-means\#1 & & 1. K-medoids \\  & 2. CLA (N3) & & 2. SL \\ \cline{1-1}  & 3. MVM & & 3. UPGMC \\ \cline{1-1}  & 4. CLA (N2) & & 4. UPGMA \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Algorithms ranking per validity indicator.