Models Of Multi-granulation Interval-set Probabilistic Rough Sets

Rough set theory,as a mathematical tool for dealing with uncertainty,incomplete and fuzziness,has been widely applied in many fields such as machine learning,artificial intelligence,and so on.With the deeply study,different generalizations of rough sets are constructed in order to solve various practical problems.Among them,the probabilistic rough set shows the approximate description that satisfies certain degree of inclusion between the approximated set and equivalence classes by using the conditional probability and two threshold values.The approximate characterizations for any interval-set are given by the interval-set rough set.Multi-granulation rough set is used to describe an approximated set in the view of multiple granular space(based on a family of equivalence relations).On the basis of probabilistic rough sets,interval-set rough sets and multi-granulation rough sets,we mainly study interval-set probabilistic rough sets,multi-granulation interval-set probabilistic rough sets and variable multi-granulation interval-set probabilistic rough sets in this paper.The main research works of this paper are shown as follows:The first part introduces the idea of probabilistic rough set into interval-set rough set,and proposes the interval-set probabilistic rough set.The properties of them are studied.The monotonicity of the corresponding interval-set probabilistic rough lower and upper approximations is given when the two thresholds are changed.Finally,the rough measurements of the interval-set probabilistic rough set are discussed.In the second part,the multi-granulation interval-set probabilistic rough set is studied.It is well-known that Pawlak's rough lower and upper approximations of an approximated set are given in a single granular space(based on an equivalence relation),while multi-granulation rough set is described in multi-granulation spaces(based on a family of equivalence relations).For the multi-granulation space,multi-granulation interval-set rough sets are proposed,optimistic and pessimistic multi-granulation interval-set rough sets are also discussed.Their properties are investigated.The relationships among the various interval-set rough sets above are given.Moreover,the multi-granulation interval-set probabilistic rough sets are then defined in multi-granulation spaces.Their properties of optimistic and pessimistic multi-granulation interval-set probabilistic rough sets are shown.An example is used to explain the superiority for the optimistic and pessimistic multi-granulation interval-set probabilistic rough sets to approximate an interval set.The third part presents the variable multi-granulation interval-set probabilistic rough set.For any threshold,the variable multi-granulation interval-set rough sets are proposed partially satisfying the inclusion or non-empty overlapping between the equivalent classes and the approximated interval-set in a multi-granulation space.Their properties are given.Furthermore,based on the conditional probability and variable multi-granulation interval-set rough sets,the variable multi-granulation interval-set probabilistic rough sets are investigated.The properties and measurements of them are discussed.