# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 4 Simulation Study  
To investigate the performance of the entropy-based fuzzy clustering (EFC-ID) model, a simulation study has been carried out. The aim of this simulation study is to study the behavior of EFC-ID in different cases that could be occurred in empirical applications (e.g., well-separated and not well-separated clusters, presence of fuzzy points, groups structure applied on centers, on radii, or on both, etc.). In particular, is very interesting to study the EFC-ID model in terms of weights (\(w\)) estimate and identification of the natural partition for different _degrees of entropy_. The proposed model has also been compared with other fuzzy clustering models for interval-valued data, i.e., fuzzy \(k\)-means clustering (FKM) proposed by [35] and fuzzy relational clustering (FRC), and with other crisp models, i.e., hierarchical clustering (HC). Moreover, EFC-ID has been also compared with another version of the entropy-based clustering model, i.e., the entropy-based fuzzy clustering model for point data (here called EFC) and the entropy-based clustering for interval-valued data (EC-ID). For FRC and HC a dissimilarity measure for interval-valued data based on OR proposed by [37] has been used. Regarding the simulation scheme, three data generation scenarios have been considered. In each scenario, the simulated dataset is constructed in such a way that two well-separated clusters (\(k=2\)) with the same size are generated (i.e., cluster 1: \(1,\ldots,n/2\), cluster 2: \(n/2+1,\ldots,n\)). Following the simulation line proposed by [8; 19; 21], we have:  
* _centers-radii scenario_, where the centers and the radii of the interval-valued data generated have a group structure. * _centers scenario_, where the radii of the interval-valued data are all randomly generated, while the centers of the data generated have a group structure. * _radii scenario_, where the centers of the interval-valued data are all randomly generated, while the radii of the data generated have a group structure. In Table 2, the details on the simulated scheme are shown. Each simulated dataset is composed of one hundred objects (\(n=100\)) and two interval-valued variables (\(J=2\)). Moreover, for the purpose of evaluating the fuzzy clustering performances of the proposed model in presence of fuzzy points (i.e., data points with memberships degree for each cluster equal to 0.5), three different percentages of fuzzy points (1, 5, and 10%) have been included in the 100 objects. In this way, we have 4 different datasets for each scenario. Note that for each scenario the data-generating process has been replicated 300 times. Finally, we have also set three values of the fuzziness parameter \(m\) (1.1, 1.5, and 2, respectively) and three values of the fuzzy entropy parameter \(p\) (0.10, 0.20, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Centers** & **Radii** \\ \hline _centers-radii_ & Cluster 1: U[0, 1] & Cluster 1: U[0, 1] \\  & Cluster 2: U[3.5, 4.5] & Cluster 2: U[1.5, 2.5] \\ _centers_ & Cluster 1: U[0, 1] & Cluster 1: U[0, 2] \\  & Cluster 2: U[3.5, 4.5] & Cluster 2: U[0, 2] \\ _radii_ & Cluster 1: U[0, 2] & Cluster 1: U[0, 1] \\  & Cluster 2: U[0, 2] & Cluster 2: U[1.5, 2.5] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data generation in the simulation scheme. and 0.40), to detect how the clustering performance is affected by these parameters. For the hierarchical clustering model, HC (i.e., hard clustering), the _single linkage_ and the _complete linkage_ approaches have been considered. For evaluating the performance of the model the Frobenius distance (Fdist) computed between the natural (generated) memberships matrix \(\mathbf{U}_{c}\) and the memberships matrix \(\mathbf{\hat{U}}\) obtained by the model, has been used. This approach is often used as a stopping rule in some fuzzy clustering algorithms [38]. The Frobenius distance has been then averaged over the 300 simulation runs.