A fuzzy functional k-means approach for monitoring Italian regions according to health evolution over time

In recent decades, functional data analysis has attracted the attention of many researchers in mathematics and statistics. For this reason, both the theory and applications have proliferated in the literature. Much of classical statistics has been rewritten in functional terms to handle data that are or can be represented by functions through appropriate smoothing operations. Within the framework of supervised and unsupervised classification, numerous techniques have been proposed to identify homogeneous groups of functional data based on different possible metrics and semi-metrics depending on the specific context. A limitation of these techniques is that they always lead to crisp-type groupings. Recently, in fuzzy set theory, many classification methods have been proposed to obtain non-crisp groupings so that the researcher is not forced to assign a statistical unit to a single group in a unique way. Following this approach, it is possible to carry out a classification that contemplates the possibility that a statistical unit belongs to different groups at the same time with different degrees of membership. The objective of this article is to propose a fuzzy functional unsupervised classification algorithm that takes into account both the functional and the fuzzy approach in order to identify similar patterns of functional data. After presenting the method, a possible application is proposed using the health composite indicator concerning the Italian regions in the period 2010–2015. The final aim of this work is to provide professionals with a tool capable of monitoring the risks of health imbalances at the national level, identifying similar behaviours at the local level but embracing the uncertainty that the fuzzy functional classification preserves in results.