S-probability Rough Sets Containing Element Transfer With Stochastic Property

In 1982,Professor Z.Pawlak proposed rough sets theory which is a new mathematics tool of dealing with fuzzy and uncertain knowledge,and its main idea is finding the decision-making or classifying rules by knowledge reduction with keeping the classifying capability.Rough sets theory has been widely applied in the fields of knowledge discovery,data mining,artificial intelligence and pattern recognition,etc.Practical system(such as financial system,risk investment system,medical diagnosis system,etc) problems are often dynamic,classical Z.Pawlak rough sets are static and helpless for dealing with dynamic problems.In 2002,Professor Shi Kaiquan extended Z.Pawlak rough sets and presented singular rough sets which is shorted for S-rough sets.S-rough sets have two forms,one direction S-rough sets and two directions S-rough sets.S-rough sets developed Z.Pawlak rough sets, and provide a theoretical support for solving various dynamic problems(such as dynamic data mining,dynamic knowledge discover and system dynamic rough characteristic,etc) induced by element transfer or attribute transfer.The difference between S-rough sets and Z.Pawlak rough sets is that S-rough sets has dynamical properties,and these dynamical properties of S-rough sets are represented by element transfer.In practical applications,element transfer with stochastic properties often appear.This paper is concerned with the influences of element transfer with stochastic properties to S-rough sets.The main works of this paper are as follows:(1.)For one direction S-rough sets containing element transfer with stochastic properties,we introduce the definition of one direction S-probability rough sets,discuss the influences of element transfer with different probabilities to one direction S-probability rough sets,and give theorems and corollaries. (2.)We give the relation theorems among one direction S-probability rough sets,Z.Pawlak rough sets,one direction S-rough sets.One direction S-probability rough sets is the general form of Z.Pawlak rough sets and Z.Pawlak rough sets is the special case of one direction S-probability rough sets;one direction S-probability rough sets is the general form of one direction S-rough sets and one direction S-rough sets is the special case of one direction S-probability rough sets.(3.)We give the definition of two direction S-probability rough sets,discuss the influences of element transfer with different probabilities to two direction S-probability rough sets,and give theorems and corollaries.(4.)We establish the relation theorems among two direction S-probability rough sets,Z.Pawlak rough sets,one direction S-rough sets,two direction S-rough sets,dual of one direction S-rough sets.Two direction S-probability rough sets is the general form of Z.Pawlak rough sets and Z.Pawlak rough sets is the special case of two direction S-probability rough sets;two direction S-probability rough sets is the general form of one direction S-rough sets and one direction S-rough sets is the special case of two direction S-probability rough sets;two direction S-probability rough sets is the general form of two direction S-rough sets and two direction S-rough sets is the special case of two direction S-probability rough sets; two direction S-probability rough sets is the general form of dual of one direction S-rough sets and dual of one direction S-rough sets is the special case of two direction S-probability rough sets.(5.)We give the definition of dual of one direction S-probability rough sets, establish the relation theorems among dual of one direction S-probability rough sets,Z.Pawlak rough sets,dual of one direction S-rough sets.Dual of one direction S-probability rough sets is the general form of Z.Pawlak rough sets,Z.Pawlak rough sets is the special case of dual of one direction S-probability rough sets; dual of one direction S-probability rough sets is the general form of dual of one direction S-rough sets,dual of one direction S-rough sets is the special case of dual of one direction S-probability rough sets.