Pessimistic multigranulation reduction of partially labeled generalized neighborhood decision information systems based on related family and matrix

The multigranulation rough set model is an important rough set model that approximates the target concept using a multigranularity structure. The multigranulation reductions of the generalized neighborhood decision information systems (GNDISs) based on multigranularity rough sets are general models for the multigranulation reductions of decision information systems (DISs) with no missing decision attribute values. In practical applications, missing labels exist in many datasets. Unfortunately, the theory of multigranulation reduction of GNDISs is not suitable for attribute reduction of partially labeled data. For this reason, the concept of partially labeled generalized neighborhood decision information systems (p-GNDISs) is proposed in this paper, and pessimistic multigranulation reduction of p-GNDISs is discussed. Moreover, the related family-based approach is provided for getting all the partially labeled, pessimistic reducts (PLP-reducts) of a p-GNDIS. Meanwhile, the matrix operations of the generalized neighborhood pessimistic lower approximation and the pessimistic multigranulation positive region on a p-GNDIS are presented. Relationships between the Boolean matrix of the related family and the matrices for computing pessimistic lower approximations are explored. Then, a logic algorithm to get a PLP-reduct of a p-GNDIS by matrix operations is presented. The pessimistic multigranulation reduction of p-GNDISs by related families method and matrix operations provides a theoretical foundation for designing algorithms of multigranulation reduction for partially labeled data.