Structured probabilistic rough set approximations

Probabilistic rough set approximations are proposed based on a conditional probability to describe the desired levels of precision between the equivalence classes and an approximated set. This definition shows the detailed information on individuals satisfying some conditions but ignores the structural information. In this paper, applying the structured granules in a coarsened-grained universe, we introduce structured probabilistic rough set approximations between the power sets of the original universe and the coarsened-grained universe. By using the zooming-in and structured probabilistic rough approximation operators, two pairs of probabilistic rough lower and upper approximations on the same universe are investigated. Related properties and relationships of them are investigated. Furthermore, applying the Bayesian decision procedure, conditional probability and loss functions, three-way classifications in structured probabilistic rough set approximations are then proposed to classify the structured granules of the coarsened-grained universe into three disjoint structured probabilistic regions. This method gives the values of the pair of thresholds. Meanwhile, by using the minimum-risk decision rules, we also can construct the structured probabilistic rough lower and upper approximations. Finally, we discuss the monotonicity of structured probabilistic positive and negative regions.