# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 4 Simulation Study
The results are presented in Table 3 with respect to different percentages of fuzzy points, different fuzziness/fuzzy entropy parameters, and different linkage methods. Table 3 shows that the average values of Fdist recorded for EFC-ID and HC are exactly equal to zero in the case of well-separated clusters (i.e., the natural partition is correctly identified by the model), whereas FKM shows Fdist values slightly higher. Another remarkable finding is that the clustering performance of our proposed model is slightly affected by the percentage of fuzzy points with respect to the other models, especially when the _degree of fuzzy entropy_ increases. Moreover, EFC-ID shows better performance than all the other approaches especially in the _radii scenario_. Notice that all the results of the not fuzzy approaches can be compared with EFC-ID when \(p=0.2\) (i.e., the medium _fuzziness degree_). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{_Centers-Radii_} & \multicolumn{3}{c}{_Centers_} & \multicolumn{3}{c}{_Radii_} \\  & **Fuzzy Points** & **Fdist** & **w** & **Fdist** & **w** & **Fdist** & **w** \\ \hline \multicolumn{6}{c}{**Entropy-based fuzzy clustering for interval-valued data (EFC-ID)**} \\ \hline \multirow{4}{*}{\(p=0.10\)} & 0\% & 0.000 & 0.490 & 0.000 & 0.200 & 0.023 & 0.500 \\  & 1\% & 0.000 & 0.500 & 0.158 & 0.254 & 0.140 & 0.500 \\  & 5\% & 1.578 & 0.500 & 1.581 & 0.380 & 0.665 & 0.500 \\  & 10\% & 2.236 & 0.500 & 2.236 & 0.445 & 1.664 & 0.500 \\ \hline \multirow{4}{*}{\(p=0.20\)} & 0\% & 0.000 & 0.490 & 0.000 & 0.200 & 0.522 & 0.492 \\  & 1\% & 0.014 & 0.500 & 0.043 & 0.254 & 0.513 & 0.493 \\  & 5\% & 1.480 & 0.500 & 1.544 & 0.381 & 0.580 & 0.496 \\  & 10\% & 2.231 & 0.500 & 2.232 & 0.445 & 0.902 & 0.500 \\ \hline \multirow{4}{*}{\(p=0.40\)} & 0\% & 0.000 & 0.490 & 0.000 & 0.200 & 0.702 & 0.388 \\  & 1\% & 0.006 & 0.500 & 0.017 & 0.254 & 7.028 & 0.340 \\  & 5\% & 0.996 & 0.500 & 1.205 & 0.388 & 6.887 & 0.350 \\  & 10\% & 2.092 & 0.500 & 2.097 & 0.452 & 6.707 & 0.364 \\ \hline \multicolumn{6}{c}{**Fuzzy \(k\)-means clustering (FKM)**} \\ \hline \multirow{4}{*}{\(m=1.10\)} & 0\% & 0.000 & 0.490 & 0.000 & 0.200 & 0.021 & 0.500 \\  & 1\% & 0.016 & 0.500 & 0.171 & 0.254 & 0.284 & 0.500 \\  & 5\% & 1.593 & 0.500 & 1.623 & 0.382 & 1.275 & 0.500 \\  & 10\% & 2.236 & 0.500 & 2.239 & 0.445 & 2.178 & 0.500 \\ \hline \multirow{4}{*}{\(m=1.50\)} & 0\% & 0.002 & 0.492 & 0.005 & 0.202 & 0.632 & 0.500 \\  & 1\% & 0.014 & 0.500 & 0.048 & 0.257 & 0.641 & 0.500 \\  & 5\% & 1.556 & 0.500 & 1.729 & 0.407 & 0.730 & 0.500 \\  & 10\% & 2.475 & 0.500 & 2.527 & 0.490 & 1.112 & 0.500 \\ \hline \multirow{4}{*}{\(m=2.00\)} & 0\% & 0.176 & 0.500 & 0.533 & 0.500 & 7.071 & 0.338 \\  & 1\% & 0.178 & 0.500 & 0.540 & 0.500 & 7.036 & 0.340 \\ \cline{1-1}  & 5\% & 0.995 & 0.500 & 1.603 & 0.500 & 6.892 & 0.350 \\ \cline{1-1}  & 10\% & 2.245 & 0.500 & 2.574 & 0.500 & 6.708 & 0.364 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Clustering models performance with well-separated clusters (0% of fuzzy points) and not well-separated clusters (1%, 5%, 10% of fuzzy points). Concerning the weights, we can note that \(wc=ws=0.5\) in the _centers-radii scenario_ (i.e., when the variability of data is balanced between centers and radii) and in the _radii scenario_ (i.e., when the variability of radii is higher than the variability of centers and then, _ws_ assume the maximum value). Finally, \(wc\geq ws\) when the variability of centers is higher than the variability of radii (i.e., in the _centers scenario_). In order to complete the evaluation of the proposed model, we have compared the EFC-ID results with the entropy-based fuzzy clustering model for point data (EFC) and the entropy-based clustering (EC-ID) model (i.e., the crisp version for interval-valued data). In Table 4, all the results are reported. The results in Table 4 show that the EFC-ID performance is better also than other entropy-based clustering models.