Rough Similarity Degree And Research On Its Characteristics

In1982, Professor Z. Pawlak presented rough sets theory in which a undefinable set can be defined by the lower approximation and the upper approximation. In 2002, Professor Shi Kaiquan has 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 direction S-rough sets. S-rough sets make Z. Pawlak rough sets become static to dynamic and make rough sets theory have many applications. In 2005, Professor Shi Kaiquan presented function S-rough sets and it is founded based on function field. Because function is equal to law, function S-rough sets is applied in law-mining and recognition. This thesis give the research on characteristics of rough similarity degree based on Z. Pawlak rough sets, S-rough sets and function S-rough sets.The main research contents of this thesis are presentingλ-rough sets andλ-rough fuzzy sets, discussing the characteristics ofλ-rough sets andλ-rough fuzzy sets, presenting rough similarity degree, studying the characteristics and applications in rough sets theory and S-rough sets theory, studying function rough similarity degree and its applications in function rough space, defining rough class and class rough similarity degree, defining dynamic class and giving its characteristics about rough similarity degree.Chapter one firstly provides a brief introduction of the background and the resent situation of development and research and gives the definition and properties of Z. Pawlak rough sets. Secondly it gives the definition of S-rough sets and function S-rough sets which extend rough sets on theory.Chapter two firstly changes the basis of founding Z. Pawlak rough sets into fuzzy equivalence relation.λ-equivalence class andλ-rough sets are presented according to the conceptions ofλ-cut relation in fuzzy sets theory. By studying the general structure ofλ-rough sets we can obtain thatλ-rough sets is a general case of Z. Pawlak rough sets. Studying the basic properties ofλ-equivalence class andλ-rough sets theorem 2.2.7 and theorem 2.2.8 arc obtained according to the variation ofλvalue. Whenλ₁≤λ₂≤…≤λ_n, bothλ-equivalence class andλ-rough sets have the decomposition chains and there are separately [x]_λ1 (?) [x]_λ2 (?)…(?) [x]_λn and In this decomposition chain, (?)λ_i,λ_j∈[0,1],λ-equivalence class satisfies (1)[x]_λi∩[x]_λj≠φand (2) andλ-roughsets satisfies (1) and (2)Secondly because both rough sets and fuzzy sets are tools for solving the uncertain problem, the general form rough fuzzy sets was put forward by D. Dubois and H. Prade by combining rough sets and fuzzy sets. But the definition and structure ofλ-rough fuzzy sets presented in this thesis are based onλ-rough sets. Because there is the strongλ-cut relation in fuzzy sets theory, the conception of strongλ-rough fuzzy sets is presented. By the union and intersection decomposition theorem of fuzzy set, the union decomposition theorem 2.4.3 and the intersection decomposition theorem 2.4.7 ofλ-rough fuzzy sets and the union decomposition theorem 2.4.4 and the intersection decomposition theorem 2.4.8 of strongλ-rough fuzzy sets can be given. Because a fuzzy equivalence relation is a special fuzzy set it has the union and intersection decomposition form. The union and intersection decomposition theorem 2.5.1 of .(R|⁾-rough sets can be got.Whenλ-rough sets and strongλ-rough sets are got, (R|⁾-rough sets can be got by theorem 2.5.1. Afterλ-rough fuzzy sets of fuzzy sets can be obtained by the union decomposition theorem 2.4.3 or the intersection decomposition theorem 2.4.7 ofλ-rough fuzzy sets, Similarly (R|⁾-rough fuzzy sets can be got by the decomposition theorem 2.5.2.Chapter three firstly presents the definition of rough similarity degree〈X,Y〉_R and〈X,Y〉_R = min based on the axiomatic similarity degree. Further the definitions of lower rough similarity degree of lower approximation and upper rough similarity degree of upper approximation are given.Secondly the basic properties of rough similarity degree are given. Theorem 3.2.8 and theorem 3.2.9 are obtained and there are separately〈X,Y〉_R = 0 if and only if for arbitrary x∈X∩Y there is [x]∩(—|R)(X) =φor [x]∩(—|R)(Y) =φand〈X,Y〉_R = 1 if and only if for arbitrary x∈X∪Y－X∩Y there is [x] (?)(R|—)(X) and [x] (?) (R|—)(Y) and (—|R)(X) = (—|R)(Y). Definition 3.2.10 is presented andρ(X,Y) is defined by〈X,Y〉_R andρ(X,Y) = 1－〈X,Y〉_R. Theorem 3.2.11 proves thatρ(X,Y) satisfies the three formulas of distance (1)ρ(X,Y) = 0 (?) X≈_RY, (2)ρ(X,Y)=ρ(Y,X) and (3)ρ(X,Z)≤ρ(X,Y)+ρ(Y,Z). Soρ(X, Y) is a reasonable distance formula.Rough similarity degree has a special property that is the probability of the rough similarity degree between a determined set and an arbitrary set is 0 or 1 is larger than the probability of the rough similarity degree is between 0 and 1. This property do good to found clustering model and recognition model and the arithmetic of clustering is got based on rough similarity degree. After clustering n patterns are obtained. When a new sample is obtained it need to recognize that belongs to which pattern in n patterns and the arithmetic of rough pattern recognition is obtained. Using a simple example explains this model about rough pattern recognition in this thesis.Lastly the relations of between a set and its one direction S-sets and dual of one direction S-sets can be explained by rough similarity degree and theorem 3.5.5 and theorem 3.5.6 are obtained. The affection degree of single element migrate in S-rough sets can be analyzed and theorem 3.5.7 and theorem 3.5.8 are obtained.Chapter four firstly gives theorem 4.1.3 about the card variation of function equivalence class according to the attribute variation and theorem about the card variation of lower approximation and upper approximation based on the definition of function one direction S-rough sets. Similarly theorem 4.2.3 about the card variation of function equivalence class according to the attribute variation and theorem about the card variation of lower approximation and upper approximation are given based on the definition of dual of function one direction S-rough sets. Function in R-function equivalence class [u] is made discrete and rough law p(x) = a_nxⁿ + a_n－1x^n－1 +…+ a₁x + a₀ generated by R-function equivalence class [u] is given by polynomial formula 4.4. [u] accepts the attack of attribute incursion to generate f-generation [u]^f, the same as rough law p(x)^f = b_nxⁿ + b_n－1x^n－1 +…+ b₁x + b₀ generated by [u]^f is got. f-collision lawθ(x)^f is got by formula 4.8 that is p(x)^f +θ(x)^f = p(x) and the relation theorem of rough law generated by [u] and rough law generated by [u]^f and f-collision lawθ(x)^f is given.Secondly an operator of function rough similarity degree is defined in function field and function rough algebra space 4.4.5 is founded. The properties of function rough algebra space are given according to function one direction S-sets and dual of function one direction S-sets. For an arbitrary set Q (?), applying function rough similarity degree the relation theorem 4.4.6 between function one direction S-sets Q°and Q and the relation theorem 4.4.9 between dual of function one direction S-sets Q' and Q are given. Single element migrate in function S-rough sets is analyzed by function rough similarity degree and theorem 4.4.12 and theorem 4.4.13 are obtained.Lastly because a function is a law, the contraction and expansion of function space is according to the contraction and expansion of law space. The error defined by rough similarity degree measures the variation degree of law and the original law of system can be obtained by measuring the error and the law producing after system is invaded.Chapter five firstly defines rough class on class field by the definition of rough sets and gives the definition of class rough similarity degree. The properties of class rough similarity degree are given and two theorems are obtained which are theorem 5.2.3 in which for arbitrary subclassσandδand〈σ,δ〉_R = 1 if only if〈σ,δ〉_|R = 1 and〈σ,δ〉_R|- = 1 and theorem 5.2.4 in which for arbitrary subclassσandδand〈σ,δ〉_R = 0 if only if〈σ,δ〉_|R = 0 or〈σ,δ〉_R|- = 0.Secondly because class have two kinds of variable cases which are the number of set that is element of class is variable and the size of set. that is element of class is variable. When the number of element is variable one direction S-class I and dual of one direction S-class I of dynamic class are presented and when the size element is variable one direction S-class II and dual of one direction S-class II of dynamic class are presented. The definitions of one direction S-rough class I and II and dual of one direction S-rough class I and II are given based on the variation of class.Lastly the characteristics of rough similarity degree about one direction S-rough class I and dual of one direction S-rough class I are given based on the definition of one direction S-rough class I and dual of one direction S-rough class I and there are theorem 5.4.3 and theorem 5.4.4.The innovative viewpoints of this dissertation are as follows:Innovative point 1. Based on Z. Pawlak rough sets theory, rough sets is extended toλ-rough sets by the definition ofλ-cut relation in fuzzy theory. The basic properties ofλ-rough sets are listed and the decomposition chain ofλ-rough sets is obtained. Based onλ-rough sets,λ-rough fuzzy sets is founded. The decomposition forms ofλ-rough fuzzy sets are got by the decomposition theorem in fuzzy theory. Lastly the decomposition forms of (R|⁾-rough sets and (R|⁾-rough fuzzy sets are got.Innovative point 1 can be found in chapter 2.Innovative point 2. Based on the axiomatic similarity degree the conception and properties of rough similarity degree are given. A new definition of distance is got by the definition of rough similarity degree. Based on the special property of rough similarity degree which is the probability of the rough similarity degree between a determined set and an arbitrary set is 0 or 1 is larger than the probability of the rough similarity degree is between 0 and 1, the arithmetics of clustering and pattern recognition are given. Lastly the affection degree of single element migrate in S-rough sets on sets can be calculated by rough similarity degree.Innovative point 2 can be found in chapter 3. Innovative point 3. Based on function rough sets and rough similarity degree, function rough algebra space is founded. When sets vary, function algebra space has the corresponding variation and the contraction theorem and the expansion theorem are obtained. Because a function is a law, the error defined by function rough similarity degree can measure the variation degree of law and the original of system can be obtained by measuring the error and the law producing after system is invaded.Innovative point 3 can be found in chapter 4.Innovative point 4. The conception and properties of class rough similarity degree on class field are given and the conceptions of one direction S-rough class I and II and dual of one direction S-rough class I and II are presented. The characteristics of rough similarity degree about one direction S-rough class I and dual of one direction S-rough class I are given.Innovative point 4 can be found in chapter 5.