Data granulation by the principles of uncertainty
  Researches in granular modeling produced a variety of mathematical models,
such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets,
which are all suitable to characterize the so-called information granules.
Modeling of the input data uncertainty is recognized as a crucial aspect in
information granulation. Moreover, the uncertainty is a well-studied concept in
many mathematical settings, such as those of probability theory, fuzzy set
theory, and possibility theory. This fact suggests that an appropriate
quantification of the uncertainty expressed by the information granule model
could be used to define an invariant property, to be exploited in practical
situations of information granulation. In this perspective, a procedure of
information granulation is effective if the uncertainty conveyed by the
synthesized information granule is in a monotonically increasing relation with
the uncertainty of the input data. In this paper, we present a data granulation
framework that elaborates over the principles of uncertainty introduced by
Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is
possible to apply such principles regardless of the input data type and the
specific mathematical setting adopted for the information granules. The
proposed framework is conceived (i) to offer a guideline for the synthesis of
information granules and (ii) to build a groundwork to compare and
quantitatively judge over different data granulation procedures. To provide a
suitable case study, we introduce a new data granulation technique based on the
minimum sum of distances, which is designed to generate type-2 fuzzy sets. We
analyze the procedure by performing different experiments on two distinct data
types: feature vectors and labeled graphs. Results show that the uncertainty of
the input data is suitably conveyed by the generated type-2 fuzzy set models.
