# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
###### Abstract
These authors consider a covariance matrix and show an equivalence between their Kullback-Leibler (KL) fuzzy clustering and the Gaussian mixture model. The method of fuzzy clustering using the KL information is called the entropy-based method of FCM; [PERSON]. [29] suggested an axiomatic derivation of the Maximum Entropy Inference (and also of the possibilistic) clustering approach, based on a unifying principle of physics, that of Extreme Physical Information (EPI) defined by [30]; Ref. [23] suggest fuzzy unsupervised clustering models based on Shannon entropy regularization in order to classify time-varying data; Ref. [31] proposed a new fuzzy clustering method based on FCM and the relative entropy is added to its objective function as a regularization function to maximize the dissimilarity between clusters; Ref. [32] presented an entropy-based FCM segmentation method that incorporates the uncertainty of classification of individual pixels within the classical framework of FCM; Ref. [33] showed a novel method considering noise intelligently based on the existing FCM approach, called adaptive-FCM and its extended version (adaptive-REFCM) in combination with relative entropy; more recently, Ref. [34] proposed an entropy-based regularization approach to fuzzify the partition and to weight features, enabling the method to capture more complex patterns, identify significant features, and yield better performance facing high-dimensional data. Notice that the here-cited proposals on the models with entropy-based regularization, regard applications on ordinary point data. Following this research line, in this work, an entropy-based fuzzy clustering technique for interval-valued data for cancer detection is suggested. The novelty of this statistical approach is to consider the uncertainty of the data in the classification procedure using the standard deviation of data variables as a measure of the uncertainty. Moreover, the presence of an entropy-based regularization redresses the uncertainty among the statistical units, especially in the boundary region guarantying a more precise classification with respect to the other competitor models. The model is named entropy-based fuzzy clustering for interval-valued data (EFC-ID). Since for all kinds of cancer, it is particularly important to improve the accuracy of early diagnosis, and that conventional early diagnosis mainly relies on human experience, an automatic classification procedure can be improved the cancer detection in screening stages. The paper is organized as follows: in Section 2 the principal ingredients of EFC-ID  
Figure 1: Artificial data generated by two bi-variate Normal distributions. To the left we have a dataset in an ordinary form; to the right, we have an interval-valued dataset. approach are provided; in Section 3 the mathematical structure and the algorithm of the model are described; in Section 4 a detailed simulation study and comparison with other fuzzy and not fuzzy clustering models for interval-valued data is proposed; in Section 5 the results obtained by the EFC-ID application on empirical data are shown; finally, in Section 6 some concluding remarks and the lines for future research in this field are provided.