# A Maximum-Entropy Fuzzy Clustering Approach for Cancer Detection When Data Are Uncertain
## 6 Concluding Remarks  
Following a fuzzy approach, in this paper, a new fuzzy clustering technique for interval-valued data is suggested. In particular, by considering a suitable weighted distance, we propose a fuzzy clustering model with entropy regularization (i.e., the EFC-ID model). Moreover, an approach has been proposed where the uncertainty that characterizes the data has been considered in the classification procedure using the degree of variability to estimate it.  
The principal advantages of this approach consist of _(i)_ the use of entropy regularization approach in a fuzzy clustering framework is the maximum entropy principle that provides a new perspective to facing the problem of fuzzifying the clusterization of the units while ensuring the maximum of compactness of the obtained clusters; _(ii)_ including the uncertainty of the data in the classification procedure leads more homogeneous and separated groups partitions; _(iii)_ the multi-group approach facilitates the use of this approach for other purposes as stages detection, response classes identification, prognosis classification, etc.); _(iv)_ the external procedure of uncertainty recognition leads to fix a different kind of interval measures (e.g., specific percentile differences); _(v)_ the weighted distance guarantees that the point data has a bigger weight than the radii. In this way, the risk to associate the biggest relevance to the uncertainty in the classification procedure is reduced.  
Conversely, the principal disadvantages consist of _(i)_ the unknown membership function of the imprecise data; _(ii)_ the approach could suffer in the case of a small sample size which inflates the radii.  
To investigate the performance and effectiveness of the proposed model a simulation study has been carried out. In particular, the aim of the simulation study has been to study the behavior of the EFC-ID model in terms of weights (_w_) estimate and identification of the natural partition for different degrees of entropy in different cases that could be occurred in empirical applications: _(i)_ well-separated clusters, _(ii)_ not well-separated clusters (i.e., when fuzzy points there are in the data structure), _(iii)_ groups structure applied on centers only, on the radii only, or on both. Results have shown that the proposed approach is more able to distinguish the natural clusters as well as to identify prototypes with respect to other methodologies. The proposed model has been compared with crisp and fuzzy models for interval-valued data. We have also analyzed two real case studies. In all cases, the proposed approach has shown good performance in identifying the natural partition and the advantages of the use of EFC-ID have been detailed illustrated.  
For future research could be interesting to embed a cross-validation approach in EFC-ID to select different uncertainty measures with respect to the standard deviation (e.g., inter-quartile range) in order to obtain also non-symmetrical imprecise data (e.g., trapezoidal data [43].  
Conceptualization, M.F.; methodology, M.F., I.D.B. and P.C.; software, M.F. and I.D.B.; validation, D.B. and P.C.; formal analysis, M.F., D.B. and P.C.; writing--original draft preparation, P.C.; writing--review and editing, P.C.; visualization, M.F. and I.D.B.; supervision, P.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.