# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 5 Empirical Applications
### Breast Cancer Wisconsin Data  
In this subsection an analysis of the Breast Cancer Wisconsin (Diagnostic) dataset is performed ([[https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+](https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+)]([https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+](https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+))(diagnostic), accessed on 1 February 2023). This data set was created by [39] and it has been very used for training of statistical methods (e.g., [40]). The endpoint of this statistical analysis is to unsupervised classify the kind of breast cancer (i.e., benign or malignant). The dataset consists of 569 patients: 357 with benign diagnosis and 212 with malignant status. Table 5 shows the features average by a breast cancer diagnosis. In order to include the variability/uncertainty that characterizes the dataset in the classification procedure, the radii have been fixed equal to the standard deviation of data \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{_Centers-Radii_} & \multicolumn{2}{c}{_Centers_} & \multicolumn{2}{c}{_Radii_} \\  & **Fuzzy** & **Fdist** & **w** & **Fdist** & **w** & **Fdist** & **w** \\ \hline \multicolumn{6}{c}{**Entropy-based fuzzy clustering for point data (EFC)**} \\ \hline  & 0\% & 0.000 & - & 0.000 & - & 9.387 & - \\ \(p=0.10\) & 1\% & 0.547 & - & 0.547 & - & 9.360 & - \\  & 5\% & 1.581 & - & 1.581 & - & 9.269 & - \\  & 10\% & 2.236 & - & 2.236 & - & 9.145 & - \\ \hline  & 0\% & 0.000 & - & 0.000 & - & 9.136 & - \\ \(p=0.20\) & 1\% & 0.100 & - & 0.100 & - & 9.108 & - \\  & 5\% & 1.581 & - & 1.581 & - & 9.015 & - \\  & 10\% & 2.236 & - & 2.236 & - & 8.886 & - \\ \hline  & 0\% & 0.000 & - & 0.000 & - & 8.480 & - \\ \(p=0.40\) & 1\% & 0.030 & - & 0.030 & - & 8.452 & - \\  & 5\% & 1.572 & - & 1.572 & - & 8.348 & - \\  & 10\% & 2.236 & - & 2.236 & - & 8.213 & - \\ \hline \multicolumn{6}{c}{**Entropy-based clustering for interval-valued data (EC-ID)**} \\ \hline  & 0\% & 0.000 & 0.490 & 0.000 & 0.200 & 0.566 & 0.500 \\  & 1\% & 0.706 & 0.500 & 0.707 & 0.254 & 0.625 & 0.500 \\  & 5\% & 1.581 & 0.500 & 1.581 & 0.380 & 1.571 & 0.500 \\  & 10\% & 2.236 & 0.500 & 2.236 & 0.445 & 2.236 & 0.500 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Benign** & **Malignant** & **_p-Value_** \\  & **(n = 357)** & **(n = 212)** & **\(p\)-Value** \\ \hline radius & \(12.1\pm 1.78\) & \(17.5\pm 3.20\) & \(<\)0.001 \\ texture & \(17.9\pm 4.00\) & \(21.6\pm 3.78\) & \(<\)0.001 \\ perimeter & \(78.1\pm 11.8\) & \(115\pm 22\) & \(<\)0.001 \\ area & \(463\pm 134\) & \(978\pm 368\) & \(<\)0.001 \\ smoothness & \(0.09\pm 0.01\) & \(0.10\pm 0.01\) & \(<\)0.001 \\ compactness & \(0.08\pm 0.03\) & \(0.15\pm 0.05\) & \(<\)0.001 \\ concavity & \(0.05\pm 0.04\) & \(0.16\pm 0.07\) & \(<\)0.001 \\ concave points & \(0.02\pm 0.01\) & \(0.09\pm 0.03\) & \(<\)0.001 \\ symmetry & \(0.17\pm 0.02\) & \(0.19\pm 0.03\) & \(<\)0.001 \\ fractal dimension & \(0.06\pm 0.01\) & \(0.06\pm 0.01\) & \(0.880\) \\ \hline \hline \end{tabular} For each feature we have the average \(\pm\) standard deviation.  
\end{table} 
Table 5: Breast Cancer Wisconsin: features average by diagnosis. features and then the EFC-ID model for malignant breast cancer diagnosis has been applied to the obtained interval-valued dataset. For comparison purposes, also the conventional fuzzy k-means clustering (FKM) for interval-valued data has been applied. This is because FKM is the major competitor model of EFC-ID, showing the best results in the simulation study. Figure 3a,b show the ROC curves obtained by the EFC-ID and FKM approaches, respectively. From Figure 3 we can see that EFC-ID outperforms the FKM with the AUC value equal to 0.9647 (CI 95% 0.9495-0.9778) against the 0.8770 (CI 95% 0.8621-0.9066) obtained by FKM. Table 6 shows the performance indexes computed on the results obtained by the two approaches.