Concept Analysis Using Quantitative Structured Three-Way Rough Set Approximations

One important topic of concept analysis is to learn an intension of a concept through a given extension. In the case where an exact intension cannot be formulated due to limited information, rough set theory introduces approximations to roughly learn the intension. Pawlak originally proposes a qualitative formulation of approximations which allows no error in the learned intension. Various quantitative formulations have been studied as generalizations, most of which use probabilistic measures. In contrast, non-probabilistic formulations have not been fully investigated. On the other hand, three-way approximations and structured approximations have been proposed to emphasize the semantics of approximations for the purpose of learning and interpreting intension. To combine the benefits of these two directions of generalizations, this paper investigates quantitative structured three-way approximations based on both probabilistic and non-probabilistic measures in the context of both complete and incomplete information.