Rough Set And Research On Its Several Characteristics

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. In 2002. Professor Shi Kaiquan extended Z. Pawlak rough sets and presented singular rough sets which is shorted for S-rough sets. It has three forms, one direction S-rough sets, dual of one direction S-rough sets and two directions S-rough sets. S-rough sets makes Z. Pawlak rough sets change from static to dynamic and makes rough sets theory have more applications . In 2005, Professor Shi Kaiquan presented function S-rough sets which is based on function universe. Since a function is equal to a law. function S-rough sets can be applied in law mining and law recognition, etc.The main research contents of this thesis are: Giving the axiomatized definition of rough similarity degree; presenting the concepts of rough similarity degree and rough close degree in rough fuzzy sets and discussing their applications in rough recognition of fuzzy pattern; proposing the definition of condition rough entropy of sets which is used to measure the uncertainty degree of rough sets; discussing the method to deal with knowledge reduction based on rough entropy in inconsistent systems: presenting a definition form of variable precision rough fuzzy sets:θ-rough fuzzy sets; studying the dynamic structure characteristics of S-rough sets and discussing the relationship between single element transfer and the structure of S-rough sets.Chapter one firstly provides a brief introduction of the background and the resent situation of development and research about rough sets, and introduces the definition and properties of Z. Pawlak rough sets. Secondly it gives the definitions of S-rough sets and function S-rough sets which extend rough sets on theory.Chapter two gives the axiomatized definition of rough similarity degree. Suppose there is the binary function on F(U) which is the power set of U, and F(U)×F(U)→[0,1], (X,Y) (?)S(X,Y).If S satisfies: (1) S(X,Y) = S(Y,X); (2) S(X,Y) = 1 (?) X≈_R Y; (3) (?)X∩(?)Y =φ(?) S(X,Y) = 0; (4) X (?) Y (?) Z (?) S(X, Z)≤{S(X, Y)(?)S(Y, Z)}, then we say S(X. Y) is the rough similarity degree between X and Y. Based on the upper rough similarity degree and lower rough similarity degree, several familiar forms of rough similarity degree are proposed and their properties are given. The rough similarity degree of rough sets is the generalization of the similarity degree of crisp sets.In Chapter three, the definition of rough similarity degree in rough fuzzy sets isproposed, that is _R = min{(?)}. The basic properties of roughsimilarity degree in rough fuzzy sets are given, and Theorem 3.2.10 and Theorem 3.2.11 are gotten, which are: _R = 0 if and only if for any x∈U, we have (?)_R(x) = 0 or (?)_R(x) = 0; and _R = 1 if and only if for any x∈U, we have (?)_R(x) = (?)_R(x) and (?)_R(x)=(?)_R(x).In fuzzy sets theory, we usually use the close degree to measure the similarity degree between two fuzzy sets. In this chapter, by providing the concepts of meanvalue fuzzy sets and rough membership measure, the close degree is introduced into rough fuzzy sets, which is called rough close degree. The Hamming rough close degree, Euclid rough close degree, minimal-maximal rough close degree and minimal-average rough close degree are defined. Their properties are discussed and Theorem 3.3.8 and Theorem 3.3.9 are introduced. Theorem 3.3.10 discusses the relationship between rough similarity degree and minimal-maximal rough close degree.Based on the two mew measurements, we can cluster the rough fuzzy sets in the approximation space, then according to the cluster result, we can deal with the rough pattern recognition of new fuzzy examples. At last, a simple example is used to explain the rough pattern recognition model in this chapter.Chapter four gives the research on uncertainty measure of rough sets. Firstly, the roughness of knowledge is described, and the concept of rough entropy of knowledge as well as the relationship between knowledge and rough entropy in information systems are introduced. To decision information systems, the concept of conditional rough entropy is introduced, and based on it a new measurement which can be used to measure the uncertainty degree in rough sets, the condition entropy of sets, is proposed. In an approximation space, the uncertainty of knowledge decreases monotonously as the granularity of information becomes smaller through finer partitions. Theorem 4.5.4 indicates that the conditional entropy has the same characteristic, so using it to measure the roughness of rough sets is reasonable. An example shows that the condition entropy H_R(X) is better than the rough degreeρ_R(X) in measuring the uncertainty degree. Theorem 4.5.10 states that in an approximation space, uniting X and Y makes the rough entropy diminish. And in a decision system, since every decision set is a rough set. uniting different decision sets makes the uncertainty lower.Moreover, in the decision system (U,A∪D,f), let the partition educed by the decision set D be U/IND(D) = {D₁,D₂,... ,D_m} and the conditional entropy ofknowledge D given B be H(D/B). then H{D/B) = (?) H_b(D_j). which is Theorem4.6.2. It reveals the relationship in essence between the conditional entropy of knowledge and the one of sets. For the decision system (U, A∪D, f), the conditional entropy of D measures the whole uncertainty degree of the decision system, and the uncertainty decreases monotonously as the partition educed by conditional attributes A becomes smaller. Each decision set D_i measures the partial uncertainty degree, and the uncertainty of each D_i all decreases monotonously as the partition educed by A becomes smaller, which explains the fact that the whole uncertainty degree becomes smaller.Chapter five researches the approaches to knowledge reduction in information systems based on rough entropy. Firstly the approach to knowledge reduction in nondecision information system is introduced, and then the decision information system is studied. Since the consistent system can be regarded as an special form of inconsistent system, we only discuss the method of knowledge reduction in inconsistent systems. The concepts of rough entropy of element and decision set in decision systems are given. The relations between rough entropy and alternative types of knowledge reduction in inconsistent systems are discussed, which is stated by Theorem 5.3.8. In an inconsistent system (U, A∪D, f), B (?) A is a distribution consistent set if and only if for any u_i∈U, H_B(D/u_i) = H_A(D/u_i); B (?) A is a distribution consistent set if and only if for any D_j, 1≤j≤m, we have H_B(D_j) = H_A(D_j): B (?) A is a possible consistent set(upper approximation consistent set) if and only if(?); B (?) A is a lower approximation consistent set if and onlyif |{k : H_B(D/u_k) = 0}| = |{k : H_A(D/u_k) = 0}|, 1≤k≤n. The approach to knowledge reduction based on conditional entropy is that first we find out the core of attribute set, then we add the most important attribute to the core, and finally we can get the smallest reduct of the information system. The algorithm for distribution reduct, possible reduct (upper approximation reduct) and lower approximation reduct is given. Finally, an instance is solved, which verifies the validity of the approaches.Chapter six discusses the variable precision rough fuzzy sets. A new definition form of variable precision rough fuzzy sets,θ- rough fuzzy sets, is presented. Theorem 6.2.7 states that as the precisionθbecomes smaller, the positive region and negative region will be larger whereas the borderline region will be smaller, so the approximation precision will be larger and the roughness will be smaller. Specialty, ifθ= 1, then (?)^θand (?) will respectively degenerate into (?) and A under rough fuzzy sets meaning; and (?) and (?) will respectively degenerate into (?) and (?). Therefore,θ-rough fuzzy sets is the generalization of rough fuzzy sets under variable precision meaning. Theorem 6.2.9 is that theθ-lower approximation (?) andθ-upper approximation (?) of rough fuzzy set A with the parameter 0 <β≤α≤1 respectively equal theθ-lower approximation ofα-cut set andθ-upper approximation ofβ-cut set of A. Specially, if A is a classical set, then for anyα,β∈(0,1], (?) and (?) respectively degenerate into the lower approximation (?)(A) and the upper approximation (?)(A) under Pawlak meaning; and (?) and (?) respectively degenerate intoθ- (?)(A) andθ- (?)(A). Therefore,θ-rough fuzzy sets is the generalization of variable precision rough sets under fuzzy meaning. Finally, the conclusions in this chapter are verified by an example.Chapter seven introduces the single element transfer in S-rough sets and the dynamic structure characteristic of S-rough sets. Theorem 7.2.2 states that for X (?) U. if the element u transfers into X, then RX largens and RX unchanges if and only if [u] (?) Bn(X), [u] - u (?) X; (?)X unchanges and RX largens if and only if [u] (?) Neg(X), card([u])≥2; (?)X unchanges and (?)X both largen if and only if [u] (?) Neg(X), card([u]) = 1; (?)X unchanges and (?)X both unchange if and only if [u] (?) Bn(X), ([u] - u)∩X≠φ. Similarly, Theorem 7.3.2 discusses the case of single element transfer out of X.The dynamic structure characteristic of two direction S-rough sets:X^f = {u|u∈U, u(?)X, f(u) = x∈X} is the f-expansion of X, and X^? = {x|x∈X,(?)(x) = u (?) X} is the (?)-contraction of X. Let X^f = {u₁,u₂,... ,u_p}, X^? = {u₁, u₂,..., u_q}. According to X^* = X∪X^f - X^?, we can assume X^* is from first expanding then contracting of X, and X expands into X^°through p times of transfer in, then X^°contracts into X^* through q times of transfer out. Every single element transfer will affect the structure of rough sets, which can be judged by Theorem 7.2.2 and Theorem 7.3.2.The innovative viewpoints of this dissertation are as follows:Innovative point 1. Giving the axiomatized definition of rough similarity degree. Based on the upper rough similarity degree and lower rough similarity degree, several new forms of rough similarity degree are given. Proposing the concepts of rough similarity degree and rough close degree in rough fuzzy sets and discussing their applications in rough recognition of fuzzy patterns.Innovative point 1 can be found in Chapter 2 and Chapter 3.Innovative point 2. Based on the conditional entropy theory, defining the conditional entropy of sets which can be used to measure the uncertainty degree of rough sets. It is applied in decision systems. For a decision system, the conditional entropy of each decision set measures the partial roughness and the sum conditional entropy of all decision sets measures the uncertainty degree of the whole decision system.Innovative point 2 can be found in chapter 4.Innovative point 3. The concepts of rough entropy of elements and decision sets in decision information systems are given. The relationships between conditional rough entropy and alternative types of knowledge reduction in inconsistent systems are investigated . The approaches to look for distribution reduction, possible reduction, upper approximation reduction and lower approximation reduction are given.Innovative point 3 can be found in chapter 5.Innovative point 4. Presenting a new definition form of variable precision rough fuzzy sets:θ-rough fuzzy sets, and giving its properties.Innovative point 4 can be found in chapter 6.Innovative point 5. Introducing the single element transfer in S-rough sets, and the changes of S-rough sets structure caused by it. Explaining the dynamic structure characteristic of S-rough sets by the single element transfer.Innovative point 5 can be found in chapter 7.