A Vector Is a Granule: A Novel Extension of the Variable Precision Rough Set Model

In the current extended variable precision rough set models (VPRS models), a granule is a set of objects. Examples of such models include the variable precision neighbourhood rough set model and the variable precision covering-based rough set model. These objects can be considered as real vectors when dealing with real-valued data, and, therefore, a granule is effectively a set of real vectors. In this paper, we show that a single vector can be considered as a granule, too. More precisely, we introduce the granule vector and the approximation vector, and show that the negativity of the inner product of these two vectors is equivalent to a granule being contained the lower (or upper) approximation. Based on this equivalence, we propose a novel extension of VPRS model which treats a single vector as a granule. In particular, this generalised model can deal with applications not covered by the existing models.