# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
###### Abstract
An interval-valued data can be formalized as \(x_{ij}=[\underline{x}_{ij},\bar{x}_{ij}]\), \(i=1,\ldots,n\) and \(j=1,\ldots,J\), where \(x_{ij}\) is the \(j\)-th interval-valued variable observed on the \(i\)-th observation, \(\underline{x}_{ij}\) and \(\bar{x}_{ij}\) denote the lower and upper bounds of the interval, respectively, (i.e., the extreme values registered for the \(j\)-th interval-valued variable on the \(i\)-th observation). Then, in a \(n\times J\) interval-valued data matrix, each observation is represented as a hyperrectangle (in \(\mathbb{R}^{J}\)) having \(2^{J}\) vertices [10]. However, in this work, we use a simpler notation of interval-valued data, consisting to consider centers and radii, separately. In particular \(i\) for the centers we indicate \(\mathbf{C}\) the \(n\times J\)_centers matrix_ whose generic element \(c_{ij}=2^{-1}(\underline{x}_{ij}+\bar{x}_{ij})\) is the center (i.e., the midpoint) of the associated interval; \(ii\) for the radii we define \(\mathbf{R}\) the \(n\times J\)_radii matrix_ whose generic element \(r_{ij}=2^{-1}(\bar{x}_{ij}-\underline{x}_{ij})\) is the radius of the associated interval. Then, by considering this reformulation of the interval-valued data, the complete interval-valued matrix can be formalized as follows:  
\[\mathbf{X}\equiv\big{\{}x_{ij}=[c_{ij},r_{ij}]:i=1,\ldots,n;\,j=1,\ldots,J \big{\}}. \tag{1}\]  
In Figure 1, a bi-dimensional artificial dataset is represented. In this dataset two groups of 50 subjects classified as normotensive (black color with \(\mu^{\prime}=[75,140]\)) and hypertensive (red color with \(\mu^{\prime}=[100,210]\)) have been simulated. In the left plot of Figure 1, the dataset is represented in ordinary form (i.e., with a radius equal to zero), while in the right one, the dataset is represented in interval-valued form (i.e., with a radius bigger than zero). In the literature on data analysis, a great deal of attention is paid to statistical methods to treat interval-valued data, in different research areas [7; 8; 9; 11; 12; 13]. In a classical cluster analysis framework different interesting methods have been suggested. In particular, Ref. [14] proposed a clustering method for symbolic data; Ref. [15] proposed a similarity measure for comparing interval-valued data and a modified agglomerative method for clustering symbolic data. Ref. [16] proposed a partitional dynamic clustering method for interval data based on adaptive Hausdorff distances; [PERSON]. [17] suggested clustering methods for interval data based on single adaptive distances. However, an interesting line of research has focused on clustering of interval-valued data based on fuzzy approaches, where the weighting exponent \(m\) controls the extent of membership sharing between fuzzy clusters [7; 18; 19; 20; 21]. [PERSON]. [22] remarked that this \"strange\" parameter is unnatural and has no physical meaning. Then, in the above objective function, we may remove \(m\), but in this case, the procedure cannot generate the membership update Equations [23]. To this purpose, [PERSON]. [22; 24] suggested a new approach to fuzzy clustering by proposing the so-called Maximum Entropy Inference Method. The idea underlies the paper by [25] where the trade-off between fuzziness and compactness is dealt with by introducing a unique objective function reformulating the maximum entropy method in terms of regularization of the fuzzy c-means (FCM) function. In the literature, many authors proposed the entropy-based approach as regularization in fuzzy clustering modeling. In particular, [PERSON]. [26] proposed an entropy-based fuzzy clustering method that automatically identifies the number and initial locations of cluster centers. Successively, it removes all data points having a similarity larger than a threshold with the chosen cluster center. The procedure is repeated till all data points are removed; [PERSON]. [27; 28] suggested a generalized objective function with additional variables.