摘要
1982年，Z.Pawlak教授提出了粗集理论，它是用下、上近似定义一个不可定义集合的理论。2002年史开泉教授将Z.Pawlak粗集进一步推广，提出了奇异粗集(singular rough sets)，简称S-粗集，它有三种形式：单向S-粗集，单向S-粗集对偶和双向S-粗集。S-粗集将Z.Pawlak粗集的静态形式扩展成粗集的动态形式，使粗集具有了更为广泛的应用。2005年史开泉教授再次提出函数S-粗集，它是建立在函数论域基础上的。由于函数与规律等价，因此函数S-粗集可以广泛地应用于规律挖掘，规律识别等。本文在Z.Pawlak粗集，S-粗集和函数S-粗集的基础上，提出了粗相似度以及给出粗相似度特性的研究。
本文的主要研究内容是：提出λ-粗集和λ-粗模糊集，讨论了λ-粗集和λ-粗模糊集的特征；提出了粗相似度，在粗集理论和S-粗集理论中研究了粗相似度的特征及应用；研究了函数粗相似度及其在函数粗空间中的应用；最后在类域上定义了粗类和类粗相似度，定义了动态类和给出了它的粗相似度特征。
第一章绪论，首先叙述了Z.Pawlak粗集理论的提出背景、发展和研究近况，给出了Z.Palwlak粗集的定义和性质；其次给出了理论上的扩展S-粗集和函数S-粗集的定义。
第二章首先将Z.Pawlak粗集的建立基础等价关系扩展到模糊等价关系。根据模糊集合理论中λ-截关系的概念，提出了λ-等价类和λ-粗集的概念。研究λ-粗集的一般结构得出λ-粗集是Z.Pawlak粗集的推广。研究λ-等价类和λ-粗集的基本性质，得到根据λ值的变化有定理2．2．7和定理2．2．8，即当λ_1≤λ_2≤…≤λ_n时，λ-等价类和λ-粗集都存在分解结构——分解链，分别是[x]_(λ_1)(?)[x]_(λ_2)(?)…(?)[x]_(λ_n)和((?)(X)，(?)(X))(?)((?)(X)，(?)(X))(?)…(?)((?)(X)，(?)(X))，而且在分解链中，对(?)λ_i，λ_j，∈[0，1]，λ-等价类满足条件(1)[x]_(λ_i)∩[x]_(λ_j)≠φ，(2)[x]_λ=∩[x]_(λ_i)和λ-粗集满足条件(1)((?)(X)，(?)(X))∩((?)(X)，(?)(X))≠φ，(2)((?)(X)，(?)(X))=(?)((?)(X)，(?)(X))。
其次，因为模糊集和粗集都是解决不确定性问题，所以D.Dubois和H.Prade将两者结合提出了粗模糊集的概念并给出了粗模糊集的一般形式。但本文是在λ-粗集的基础上给出了另一种粗模糊集的定义——λ-粗模糊集和它的一般结构。因为模糊关系存在强λ-截关系，所以提出了一种强λ-粗模糊集的概念。根据模糊集并分解定理和交分解定理，可以得到λ-粗模糊集的并分解定理2．4．3和交分解定理2．4．7和强λ-粗模糊集的并分解定理2．4．4和交分解定理2．4．8．最后根据模糊等价关系是特殊的模糊集，即模糊等价关系也存在并分解形式和交分解形式，就可以得到(?)-粗集的并交分解定理2．5．1。
在得到集合的λ-粗集和强λ-粗集后，利用定理2．5．1即可求得(?)-粗集。同样利用λ-粗模糊集的并分解定理2．4．3或交分解定理2．4．7求得模糊集合的λ-粗模糊集，类似利用定理2．5．2，进而求得(?)-粗模糊集。
第三章首先根据公理化相似度的定义提出了粗相似度_R的定义，即_R=min{|(?)(X)∩(?)(Y)|/|(?)(X)∪(?)(Y)|，|(?)(X)∩(?)(Y)|/|(?)(X)∪(?)(Y)|}，进一步定义了下近似的下粗相似度和上近似的上粗相似度。
其次给出了粗相似度的基本性质，得到了定理3．2．8和定理3．2．9，它们分别是_R=0的充要条件是对任意的x∈X∩Y，有[x]∩(?)(X)=φ或[x]∩(?)(Y)=φ和_R=1的充要条件是对任意的x∈X∪Y-X∩Y，有[x](?)(X)和[x](?)(Y)且(?)(X)=(?)(Y)。利用粗相似度_R给出了一种距离定义3．2．10即ρ(X，Y)=1-_R，定理3．2．11证明了ρ(X，Y)满足距离三公式(1)ρ(X，Y)=0(?)X≈_RY，(2)ρ(X，Y)=ρ(Y，X)和(3)ρ(X，Z)≤ρ(X，Y)+ρ(Y，Z)。即ρ(X，Y)是一种合理的距离形式。
粗相似度有一个特殊性质，即当一个集合确定时，其它集合与它的粗相似度的值是0或者1的可能性大于粗相似度值介于0和1之间的可能性。它的这个性质更有利于建立聚类模型和识别模型，因此在粗相似度的基础上给出一个聚类的算法。聚类后得到n类模式，当出现一个新样本时，需要用模式识别的方法去识别它属于哪个模式，因此接着就给出了粗模式识别的算法。文中应用一个简单的例子解释了这种粗模式识别模型。
最后用粗相似度对集合与其受干扰后形成单向S-集合和单向S-集合对偶的关系进行描述得到定理3．5．5和定理3．5．6；而且针对S-粗集中单元素迁移做了粗相似度的分析，得到定理3．5．7和定理3．5．8。
第四章首先在函数单向S-粗集定义的基础上，给出了属性变化时函数等价类基数变化定理4．1．3以及粗集的上下近似的基数变化定理；同样在函数单向S-粗集对偶定义的基础上，给出了属性变化时函数等价类基数变化定理4．2．3以及粗集的上下近似的基数变化定理。将R-函数等价类[u]中的函数进行离散，通过多项式公式4．4给出了R-函数等价类[u]的生成粗规律p(x)=α_nx~n+α_(n-1)x~(n-1)+…+α_1x+α_0；[u]受到属性入侵攻击的f-生成[u]~f，相似地可得[u]~f的生成粗规律p(x)~f=b_nx~n+b_(n-1)x~(n-1)+…+b_1x+b_0。由式4．8即p(x)~f+θ(x)~f=p(x)得到f-碰撞规律θ(x)~f，而且给出了[u]的生成粗规律和[u]~f的生成粗规律以及f-碰撞规律θ(x)~f的关系定理。
其次，在函数论域上定义一个算子函数粗相似度，就建立了一个函数粗代数空间4．4．5，在函数单向S-集合和函数单向S-集合对偶基础上给出这个函数粗代数空间的性质研究。任给集合Q(?)，应用函数粗相似度给出函数单向S-集合Q°和函数单向S-集合Q′与集合Q之间的关系定理4．4．6和定理4．4．9。针对函数S-粗集中单元素迁移做了粗相似度的分析，得到定理4．4．12和定理4．4．13．
最后，由于一个函数就是一个规律，因此函数空间的萎缩和扩张对应于规律空间的萎缩和扩张。利用粗相似度定义一个误差度量规律变化程度，进而通过测得误差和系统受侵后生成的规律就可以得到系统的原规律。
第五章首先在类论域上根据粗集定义给出了粗类的定义，而且给出了类粗相似度的定义。给出了类粗相似度的基本性质，而且得到了两个充要条件定理，即定理5．2．3对于任意子类σ和δ，<σ，δ>_R=1的充要条件是<σ，δ>_(?)=1和<σ，δ>_(?)=1；定理5．2．4对于任意子类σ和δ，<σ，δ>_R=0的充要条件是<σ，δ>_(?)=0或<σ，δ>_(?)=0。
其次由于类有两种变化形式，一种是类的元素集合个数的变化，一个是类的元素集合大小的变化，因此由元素个数的变化提出了动态类的单向S-类Ⅰ和单向S-类对偶Ⅰ，由元素大小的变化提出了动态类的单向S-类Ⅱ和单向S-类对偶Ⅱ。在类变化的基础上给出了动态类的单向S-粗类Ⅰ和Ⅱ以及单向S-粗类对偶Ⅰ和Ⅱ的定义。
最后根据类粗相似度和单向S-粗类Ⅰ以及单向S-粗类对偶Ⅰ的定义，给出了关于动态类的单向S-粗类Ⅰ和单向S-粗类对偶Ⅰ的粗相似度特性，即定理5．4．3和定理5．4．4．
本文的创新点：
创新点1，以Z.Pawlak粗集理论为基础，根据模糊集理论中的λ-截关系，将粗集拓展到λ-粗集，并给出λ-粗集的基本性质，得到了λ-粗集的分解结构——分解链；在λ-粗集的基础上建立了λ-粗模糊集，根据模糊集理论中的分解定理，得出了λ-粗模糊集的分解定理，最后得到了(?)-粗集和(?)-粗模糊集分解形式。
创新点1列于第二章中。
创新点2．在相似度公理化的基础上，给出了粗相似度的概念和基本性质，而且根据粗相似度的定义得到一种距离形式的定义方法；根据粗相似度的特殊性质即当一个集合确定时，其它集合与它的粗相似度的值是0或者1的可能性大于粗相似度值介于0和1之间的可能性，给出了聚类和粗模式识别的算法。最后用粗相似度计算S-粗集中单元素迁移对集合的影响程度。
创新点2列于第三章中。
创新点3．在函数粗集和粗相似度的基础上建立的函数粗代数空间，根据在集合变化时函数代数空间会出现萎缩或扩张，给出了相应的萎缩和扩张定理；一个函数就是一个规律，利用函数粗相似度定义一个误差度量规律变化程度，进而通过测得误差和入侵后的规律就可以得到系统的原规律。
创新点3列于第四章中。
创新点4．在类论域上给出了类粗相似度的概念和性质，提出了单向S-粗类Ⅰ和Ⅱ以及单向S-粗类对偶Ⅰ和Ⅱ的概念；最后给出了单向S-粗类Ⅰ和单向S-粗类对偶Ⅰ的粗相似度特性。
创新点4列于第五章中。
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. Whenλ_1≤λ_2≤…≤λ_n, bothλ-equivalence class andλ-rough sets have the decomposition chains and there are separately [x]_(λ_1) (?) [x]_(λ_2) (?)…(?) [x]_(λ_n) and In this decomposition chain, (?)λ_i,λ_j∈[0,1],λ-equivalence class satisfies (1)[x]_(λ_i)∩[x]_(λ_j)≠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_nx~n + a_(n－1)x~(n－1) +…+ a_1x + a_0 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_nx~n + b_(n－1)x~(n－1) +…+ b_1x + b_0 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.
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