# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 3 Model and Algorithm
### Optimization Problem
 \tag{6}\]  
\[\frac{\partial L_{m}(u_{ig},\lambda,w)}{\partial\lambda}=0\Leftrightarrow \sum_{g=1}^{k}u_{ig}-1=0. \tag{7}\]  
From Equation (6), we obtain  
\[log(u_{ig})=\frac{1}{p}\Big{[}\lambda-\Big{[}(1-w)^{2}d^{2}(\mathbf{c}_{i}, \mathbf{c}_{g})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})\Big{]}\Big{]}-1, \tag{8}\]  
and then  
\[u_{ig}=\exp\biggl{[}\frac{\lambda}{p}-\frac{1}{p}\Big{[}(1-w)^{2}d^{2}(\mathbf{ c}_{i},\mathbf{c}_{g})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})\Big{]}-1 \biggr{]}. \tag{9}\]  
By considering Equation (7):  
\[\exp\biggl{(}\frac{\lambda}{p}-1\biggr{)}=\frac{1}{\sum_{g=1}^{k}\biggl{[} \frac{1}{\exp\bigl{[}(1/p)[(1-w)^{2}d^{2}(\mathbf{c}_{i},\mathbf{c}_{g})+w^{2} d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})]\bigr{]}}\biggr{]}}, \tag{10}\]  
and by replacing Equation (10) in Equation (9), we obtain  
\[u_{ig}=\frac{1}{\sum_{g^{\prime}=1}^{k}\left[\frac{\exp\bigl{[}(1/p)[(1-w)^{2} d^{2}(\mathbf{c}_{i},\mathbf{c}_{g})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})] \bigr{]}}{\exp\bigl{[}(1/p)[(1-w)^{2}d^{2}(\mathbf{c}_{i},\mathbf{c}_{g^{\prime }})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g^{\prime}})]\bigr{]}}\right]}. \tag{11}\]  
The normalization condition for \(w\) is implicitly satisfied. To take into account the _coherence condition_, observing that Equation (5) is a parabola with respect to \(w\), the optimum value of \(w\) results in the minimum between the abscissa of its vertex and 0.5 [8], i.e.,  
\[w=\min\Biggl{\{}\frac{\sum_{i=1}^{n}\sum_{g=1}^{k}u_{ig}\bigl{[}d^{2}(\mathbf{c} _{i},\mathbf{c}_{g})\bigr{]}}{\sum_{i=1}^{n}\sum_{g=1}^{k}u_{ig}\bigl{[}d^{2}( \mathbf{c}_{i},\mathbf{c}_{g})+d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})\bigr{]}}, 0.5\Biggr{\}}. \tag{12}\]  
Finally, we compute the centroids for the centers and radii through the steps shown in Equations (13) and (14), respectively. \[ \begin{split}&\frac{\partial J_{\text{EFC}-ID}(\mathbf{U},\mathbf{ \tilde{X}},w)}{\partial\mathbf{c}_{g}}=0\Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}(1-w)^{2}d^{2}(\mathbf{c}_{i}, \mathbf{c}_{g})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})\Big{]}+p\sum_{i=1}^{ n}u_{ig}log(u_{ig})=0\Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}(1-w)^{2}(\mathbf{c}_{i}^{2}+2 \mathbf{c}_{i}\mathbf{c}_{g}+\mathbf{c}_{g}^{2})+w^{2}(\mathbf{r}_{i}^{2}+2 \mathbf{r}_{i}\mathbf{r}_{g}+\mathbf{r}_{g}^{2})\Big{]}\\ &+p\sum_{i=1}^{n}u_{ig}log(u_{ig})=0\Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}(1-w)^{2}(\mathbf{c}_{i}+\mathbf{c}_ {g})\Big{]}=0\Leftrightarrow\\ &\mathbf{c}_{g}=\frac{\sum_{i=1}^{n}u_{ig}\mathbf{c}_{i}}{\sum_{ i=1}^{n}u_{ig}}.\end{split}  \tag{13}\]  
\[ \begin{split}&\frac{\partial J_{\text{EFC}-ID}(\mathbf{U}, \mathbf{\tilde{X}},w)}{\partial\mathbf{r}_{g}}=0\Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}(1-w)^{2}d^{2}(\mathbf{c}_{i},\mathbf{ c}_{g})+w^{2}d^{2}(\mathbf{r}_{i},\mathbf{r}_{g})\Big{]}+p\sum_{i=1}^{n}u_{ig}log(u_{ig})=0 \Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}(1-w)^{2}(\mathbf{c}_{i}^{2}+2 \mathbf{c}_{i}\mathbf{c}_{g}+\mathbf{c}_{g}^{2})+w^{2}(\mathbf{r}_{i}^{2}+2 \mathbf{r}_{i}\mathbf{r}_{g}+\mathbf{r}_{g}^{2})\Big{]}\\ &+p\sum_{i=1}^{n}u_{ig}log(u_{ig})=0\Leftrightarrow\\ &\sum_{i=1}^{n}u_{ig}\Big{[}w^{2}(\mathbf{r}_{i}+\mathbf{r}_{g}) \Big{]}=0\Leftrightarrow\\ &\mathbf{r}_{g}=\frac{\sum_{i=1}^{n}u_{ig}\mathbf{r}_{i}}{\sum_{ i=1}^{n}u_{ig}}.\end{split}  \tag{14}\]  
In order to show an example of application we consider the bi-dimensional interval-valued dataset described in Introduction. In Table 1 are shown the mean and variance of centers and radii used to generate 300 observations of a bi-dimensional interval-valued data with a structure of three groups (i.e., 100 observations for each cluster). By applying the EFC-ID model on this dataset, we have the results shown in Figure 2. Then, the centroids of centers and radii have been correctly identified with the Adjusted Rand index (ARI) value equal to \(1\).