# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 5 Empirical Applications
### Multiple Myeloma Data  
In this subsection, an analysis of the Multiple Myeloma Data available in survminer R-package is performed ([[https://cran.r-project.org/web/packages/survminer/survminer.pdf](https://cran.r-project.org/web/packages/survminer/survminer.pdf)]([https://cran.r-project.org/web/packages/survminer/survminer.pdf](https://cran.r-project.org/web/packages/survminer/survminer.pdf)), accessed on 1 February 2023). Multiple Myeloma data is extracted from publicly available gene expression data (GEO Id: GSE4581). The endpoint of this statistical analysis is to unsupervised classify the molecular groups of Multiple Myeloma. The original dataset consists of 256 patients with the IMWG molecular cytogenetic classification as shown in Table 7. However, since there is a large number of groups, we have selected a sub-sample including in the analysis only three molecular groups: Hyperdiploid, MAF, and MMSET (i.e., 131 patients). In this dataset, we have six features that correspond to six different gene expressions (i.e., CCND1, CRIM1, DEPDC1, IRF4, TP53, and WHSC1). Table 8 shows the gene expression average by molecular groups of Multiple Myeloma.  
Furthermore, in this case, 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-features and then the EFC-ID model has been applied for molecular group identification. Such as the previous subsection, the conventional fuzzy \(k\)-means (FKM) for interval-valued data has been applied for comparison purposes. Notice that, in this application, the optimal entropy parameter \(p\) has been fixed to 0.25. In order to evaluate the performance of both approaches, three different evaluation criteria have been used (i.e., the Chi-Squared Index [41], the Accuracy rate and the Adjusted Rand Index [42]). The results obtained are shown in Table 9.  
Furthermore, in this case, the three evaluation criteria show an EFC-ID performance gain than the conventional FKM. In particular, both the empirical applications show that FKM, unlike EFC-ID, unbalances the weights in favor of centers (\(wc=0.97\)), providing a result similar to the simple FCM for point data (i.e., when \(wc=1\)), and then neglecting the information included in the uncertainty of data. Therefore, the important contribution  
\begin{table}
\begin{tabular}{c c} \hline
**Molecular Group** & **\# Patients** \\ \hline Cyclin D-1 & 22 \\ Cyclin D-2 & 43 \\ Hyperdiploid & 66 \\ Low bone disease & 31 \\ MAF & 21 \\ MMSET & 44 \\ Proliferation & 29 \\ \hline \end{tabular}
\end{table}
Table 7: Multiple Myeloma Data: IMWG molecular cytogenetic classification.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Hyperdiploid** & **MAF** & **MMSET** & **\(p\)-Value** \\
**(n = 66)** & **(n = 21)** & **(n = 44)** & **\(p\)-Value** \\ \hline CCND1 & \(1280\pm 1500\) & \(434\pm 765\) & \(427\pm 602\) & \(<\)0.001 \\ CRIM1 & \(73\pm 174\) & \(59\pm 91\) & \(482\pm 392\) & \(<\)0.001 \\ DEPDC1 & \(169\pm 111\) & \(382\pm 356\) & \(311\pm 236\) & 0.006 \\ IRF4 & \(14500\pm 3850\) & \(12700\pm 4930\) & \(12300\pm 3950\) & 0.013 \\ TPS3 & \(1880\pm 669\) & \(1350\pm 1040\) & \(1240\pm 475\) & \(<\)0.001 \\ WHSC1 & \(139\pm 136\) & \(481\pm 454\) & \(8100\pm 4560\) & \(<\)0.001 \\ \hline \end{tabular} For each feature we have the average \(\pm\) standard deviation.  
\end{table}
Table 8: Multiple Myeloma Data: gene expression average by molecular groups.  
\begin{table}
\begin{tabular}{c c c} \hline
**EFC-ID** & **FKM** \\ \hline Chi-Squared & 91.636 & 88.257 \\ Accuracy & 0.710 & 0.601 \\ Adjusted Rand Index & 0.329 & 0.211 \\ \hline \end{tabular}
\end{table}
Table 9: Performance indexes obtained by EFC-ID and FKM for Multiple Myeloma classification.  
of this proposal is the use of entropy regularization which improves the separability of groups and their homogeneity without neglecting the degree of uncertainty characterized by the data.