Presumably correct decision sets

The paper presents a granulation method termed presumably correct decision sets as a data analysis tool to handle uncertainty in the form of inconsistency. In this approach, each decision class in a decision system will be associated with three regions containing weak members, borderline members, and strong members. Such sets are defined based on the membership degree of each instance to its neighborhood without considering their actual decision classes. As a second step, we derive the presumably correct and incorrect sets by contrasting the decision classes determined by a neighborhood function with the actual class labels. We extract these sets from either the regions containing strong members or the whole universe. This defines the strict and relaxed versions of our theoretical formalism. In that way, we can isolate those instances that will be difficult to handle by pattern classification algorithms as they are responsible for the inconsistent patterns. The numerical simulations using synthetic and real-world datasets illustrate the advantages of our model compared to rough sets, which is deemed a solid state-of-the-art approach to cope with inconsistency. Towards the end, we introduce an explanation method that allows determining which problem features make the presumably incorrect instances difficult.