粗信息矩阵的数量特征及其应用
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摘要
在科学技术和经济管理领域中,人们经常会接收到各种各样的信息,但有的信息是粗糙的,这类信息我们称之为粗信息.该类信息是不能用精确的集合来表示的,这样就对人们认识和开发使用该类信息造成了严重的影响,从而丧失了可利用的重要信息资源.为了处理具有信息不确定性、不精确、不完善的系统,波兰数学家Z.Pawlak教授于1982年提出了一种新的数学理论——粗糙集理论,粗糙集是把一个不可定义的集合X用上、下近似来定义,Z.Pawlak粗糙集的一般性研究工作为系统理论奠定了理论基础并获得了广泛的应用与研究.但在实际问题中往往遇到的信息由于受到多种因素的影响而变动.因此Z.Pawlak粗集在解决某些问题时受到了一定的限制.2002年史开泉教授改进了Z.Pawlak粗糙集,提出奇异粗集(Singular rough sets),简称S-粗集,2005年改进了S-粗集,提出函数S-粗集,并分别将它用于知识、规律挖掘中.本文利用已有的研究,将粗集理论、S-粗集理论、矩阵理论与粗系统信息挖掘、识别相嫁接,给出一系列新的研究与应用.
本文所讨论的内容是基于Z.Pawlak粗糙集、S-粗集理论、矩阵理论、模糊集理论的框架体系下展开讨论的.一方面,提出粗信息矩阵、粗粒度矩阵,在此基础上进行广泛而较深入的研究,所得到的内容都是新的;另一方面,借助已有的粗相似度,提出了粗相似度矩阵,讨论了粗相似度矩阵的有关性质及定理.同时,论文又介绍并讨论了函数S-粗集的结构特征并给出函数S-粗集对偶这种单规律的属性特征及控制准则,并给出基于该粗集下的金融风险投资决策规律分析.最后提出了属性模糊集的概念,讨论了属性模糊集的特征,给出了属性模糊集的分解定理.本论文各章的具体内容概括如下:
第一章介绍了粗糙集理论的发展概况、研究现状与前景展望,根据粗糙集的目前研究现状与前景的预测方向分析,重点选择对粗糙集的理论进行完善这一方向作为本文的重点研究内容,兼顾到后续论文研究内容的需要,简要介绍了Z.Pawlak粗集与它的结构特征、知识的数值特征等基本概念与理论,为后面章节的研究奠定了必要的基础.
第二章主要研究知识挖掘的数量特征及挖掘的一系列准则与定理.给出了k阶f知识、k阶(?)知识、知识挖掘度的概念;讨论了知识挖掘的链式特征并给出了f、(?)知识挖掘基数链定理,f、(?)知识挖掘粒度链定理,f、(?)知识挖掘过滤度链定理,f、(?)知识挖掘度链定理,并给出了知识粒度与知识挖掘度关系原理的应用.给出f、(?)知识最小、最大挖掘度定理。另外,本文根据单向S-粗集中的知识具有阶梯特征,利用知识的阶梯特性,能够找到人们事先不知道的知识;从知识依赖的另一个角度挖掘所需要的知识,给出了知识的阶梯度,利用阶梯度,直接挖掘所需要的知识。本文给出了F-阶梯知识对的F-阶梯度依赖-分辨定理、F-阶梯知识对的F-阶梯度依赖-不可分辨定理、最大F-阶梯知识的挖掘-发现定理、阶梯知识第一、第二挖掘-发现准则等,如定理2.4.3和定理2.4.4,2.4.5,并给出具体应用。讨论最小F-阶梯知识的挖掘-发现定理:若{([X°]_(F,K),[X°]_K~F|k=1,2,…,m)是F-阶梯知识[X°]_(F,k)与F-阶梯知识[X°]_K~F构成的F-阶梯知识对集合;则存在([X°]_(F,p),[X°]_P~F),p∈(1,2,…,m),满足LAD([X°]_(F,p))=(?){LAD([X°]_(F,i));LAD([X°]_p~F)=(?){LAD([X°]_i~F)};GRD([X°]_(F,p)≤GRD([X°]_(F,i))_(i=1,2,…m);GRD([X°]_p~F)≤GRD([X°]_i~F)_(i=1,2,…m)。
基于在经典系统中的一个系统具有的特征可以从研究系统的系统矩阵中获取,因此提出如下问题:由粗集生成的粗系统,粗系统的特征是否也可以从研究粗系统的系统粗矩阵中获取?第三章正是基于这种考虑提出粗信息矩阵,即由M_α(X)_-,M_α(X)~-构成的矩阵对(M_α(X)_-,M_α(X)~-),称作粗集(X_(ij-,X_(ij)~-)生成的粗信息矩阵。讨论了信息矩阵存在性定理、粗信息矩阵存在性定理。
在定义了信息矩阵的包含、相等以及和粗信息矩阵、积粗信息矩阵(∪,∩,～)运算的基础上,研究了粗信息矩阵的性质(性质3.2.1-性质3.2.8)。得到了粗信息矩阵的粗集生成原理。从讨论我们可以看出,粗信息矩阵中的每个元素是一个序对,是满足二元属性的粗集,从而不仅推广了普通数域上的矩阵理论,而且推广了经典的Z.Pawlak粗集,同时得到了一系列满足不同二元属性的粗糙集。也就是说,不仅从粗集个数方面得到推广,而且从维数方面得到更一般的粗集,为以后人们处理经济、管理问题、从大量的数据中提出有用的知识提供了一个很好的理论依据。
在静态粗信息矩阵的基础上,受S-粗集研究问题的思路启发,本文给出粗信息矩阵的三种形式。提出并讨论了粗信息矩阵的粒度矩阵的特征,即由G(X)_-=(g_(ij-))_(m×n)G(X)~-=(g_(ij)~-)_(m×n)构成的矩阵对(G(X)_~-,G(X)~-)称作X(?)U的粗信息粒度矩阵(rough information granulation matrix),记作G=(G(X)_-,G(X)~-)。定义了下粗粒度和(积、余)矩阵、上粗粒度和(积、余)矩阵、粗粒度和(积、余)矩阵的运算,研究了其运算的性质。给出了粗粒度矩阵满足的运算律,详细内容参见§3.5。
其次,研究了粗信息矩阵与粗粒度矩阵的关系定理,即粗信息矩阵的包含对应着粗粒度矩阵的包含;定义了粗信息粒度矩阵的和与积运算,讨论了粗信息粒度矩阵、单向S-粗信息粒度矩阵、单向S-对偶粗信息粒度矩阵的关系,得到了一系列重要的定理。
最后给出了粗信息精度向量,得到了信息向量可定义与不可定义的充要条件。如定理3.8.6:设β=(α_1(X_(11)),α_2(X_(12)),…,α_n(X_(1n)))为粗信息向量((X_(11-),X_(11)~-),(X_(12-),X_(12)~-),…,(X_(1n-),X_(1n)~-))的精度矩阵,则所有的X_(1j),是可定义的充要条件是向量β为基本单位向量组ε_1,ε_2,…,ε_n的线性组合,且组合系数为1。
第四章进一步讨论粗信息矩阵的数量特征,提出粗信息向量的粗相似度矩阵,并研究了它的基本性质及结构。讨论了粗相似度矩阵与实二次型、正定矩阵、半正定矩阵的关系定理。
第五章介绍了函数单向S-粗集对偶与函数粗集,讨论了规律属性的萎缩特征,研究了规律的属性控制与识别相关定理及准则,给出了函数单向S-粗集对偶在金融风险识别中的应用。
第六章模糊集(Fuzzy Sets)理论是Zadeh教授1965年提出的,它是研究不确定性理论的一种精确的数学方法。联系模糊集与经典集合的桥梁是模糊分解定理,该定理揭示了模糊集的结构,一个模糊集是由若干个子模糊集叠加而成,而每个子模糊集就是λ∈[0,1]与经典集A_λ(λ的截集)的数积,如果我们引入属性模糊集,那么属性模糊集的属性分解情况如何?它具有那些特性?与我们熟知的模糊分解定理的关系怎样?这些问题,在文献中很少见。基于此,本章提出了属性模糊集的概念,给出它的属性分解定理,并且,当存在属性迁移时,给出了属性模糊集的属性链式定理,回答了属性分解定理与分解定理的一致性。
论文的主要创新点:
创新点1.建立了.f、(?)不同阶知识的最小、最大挖掘度及一系列挖掘准则与定理.给出F-阶梯知识对的挖掘发现准则,(?)隐藏知识依赖、发现的原理.
创新点2.提出粗信息矩阵,建立了静态、动态粗信息矩阵,粗粒度矩阵的相关理论与结构特征的讨论,给出了一系列重要的性质与定理.为粗系统理论的深入细致的研究奠定了很好的基础.整个论文的第三章内容与结论都是新的.
创新点3.提出粗相似矩阵的概念,并对该部分内容进行了详细细致的研究.完善了相似度理论,能够融合知识内容,将粗相似度矩阵与实二型结合进行讨论,有一定的理论意义与实用价值.
创新点4.研究了属性规律的特征,结合函数S-粗集讨论了属性控制与识别相关定理与准则,为规律挖掘提供了理论保证.
创新点5.提出属性模糊集的概念,将模糊集的重要定理与粗糙集的属性进行一定程度上嫁接渗透研究.
In science,technology and economic management fields,people often receive a variety of information.But some information is rough,so they are called rough information.Rough information can't be expressed as an accurate set,which hampers people to understand and use this kind of and make the available important information resources be lost.In order to deal with the systems which have uncertainty,imprecise and imperfect characteristic,the Polish mathematician Professor Z.Pawlak proposed a new mathematical theory,rough sets theory in 1982. Rough sets define a set of X with upper and lower approximation which can't be defined by set.The general research of Z.Pawlak's rough sets has laid a theoretical foundation for the system theory and make it get a wide range of applications and research.Some information encountered at the practical problems often change because of the effects of many factors,therefore Z.Pawlak's rough sets would have a certain amount of restrictions when to resolve certain problems.In 2002 Professor Shi Kaiquan improved Z.Pawlak's rough sets,and proposed the singular rough sets (Singular rough sets),called S-rough sets in brief.In 2005,he further proposed the function S-rough sets,and used it for the mining of knowledge and the law.In this paper,a series of new research and discussion are combined with the rough sets theory, S-rough sets theory,matrix theory and the mining of rough information system, identification and a series of new research and applications will be given.
The contents discussed in this article are based on the Z.Pawlak's rough sets, S-rough sets theory,matrix theory and fuzzy sets theory.On the one hand,it proposes rough information matrix and rough granulation matrix,then does wide-ranging and more in depth studies on the basis of them,thus the contents are new;On the other hand,it proposes rough similarity matrix through by using rough similarity,and discusses the coarse nature and theorem of the similarity matrix.Meanwhile,the paper also introduces and discusses the structural characteristics of the function S-rough sets and gives single law attribute characteristics and control criteria of the function S-rough sets,and then gives investment decision-making laws analysis of the financial risks based on rough sets.Finally,it proposes the concept of attribute fuzzy sets,discusses the characteristics of attribute fuzzy sets and gives the decomposition theorem of attribute fuzzy sets.In this paper,the specific content of each chapter are summarized as following:
ChapterⅠintroduces the development,research status and prospect about rough sets theory,then according to the current research directions and prospects of rough sets,it chooses improving rough sets theory as the focus of this article's study.For the needs of the follow-up papers' study,it briefly introduces the basic concepts and theories about the Z.Pawlak's rough sets and its structural characteristics,the numerical characteristics of knowledge,which laid the necessary foundation for the following section of the research.
ChapterⅡmainly studies the number characteristics of knowledge mining and a series of criterions and theorems about mining.It gives f,(?) knowledge of the k order,and the concept of knowledge mining degrees;it discusses the chain characteristics of knowledge mining and gives base chain theorem of f and (?) knowledge mining,granulation chain theorem of f and(?) knowledge mining, filter chain theorem of f and(?) knowledge mining and mining degrees chain theorem of f and(?) knowledge and then gives the application of the relationship principle between granularity and mining degree of knowledge,the minimum and maximum mining degree theorem of f and(?) knowledge.In addition,this paper introduces one direction S-rough sets with the ladder characteristics of knowledge, which can help people find knowledge unknown in advance;it also mine the required knowledge from another angle-dependent knowledge mining,and gives the ladder of knowledge,then makes the use of the ladder to mine the required knowledge directly. In this paper it gives F-ladder dependence degree discernibility of F-ladder knowledge pair,F-ladder dependence degree indiscernibility theorem of F-ladder knowledge pair,the largest F- ladder knowledge mining discovery theorem,the first criteria on F-ladder knowledge,such as theorem 2.4.3,theorem 2.4.4 and theorem 2.4.5,and gives specific applications.To discuss the smallest F-ladder knowledge mining-discovery theorem:If {([X°]_(F,k),[X°]_k~F)|k=1,2,…,m} is the collection of F-ladder knowledge pair composed by F-ladder knowledge[X°]_(F,k) and F-ladder knowledge[X°]_k~F;then there must be([X°]_(F,p),[X°]_p~F),p∈(1,2,…,m), which meets LAD([X°]_(F,p))=(?){LAD([X°]_(F,i))};LAD([X°]_p~F)=(?){LAD([X°]_i~F)}; GRD([X°]_(F,p)≤GRD([X°]_(F,i))_(i=1,2,…m);GRD([X°]_p~F)≤GRD([X°]_i~F)_(i=1,2,…m).
In the classical system the characteristics of a system can obtain from studying the system matrix of the system,so the following question is proposed based on that: is the features of rough system that is made up with rough sets also available from rough system matrix of the research rough systems? ChapterⅢfirstly raises rough information matrix based on this consider,that is,Definition 3.1.3,a pair of the matrix (M_α(X)_-,M_α(X)~-) which is composed of M_α(X)_- and M_α(X)~- is called rough information matrix generating from rough sets(X_(ij-),X_(ij)~-).This paper discusses existence theorem of the information matrix and the existence theorem of rough information matrix,namely Theorem 2.1.1,Theorem 2.1.2.
Based on the definitions of the information matrix inclusion,equivalent and sum rough information matrix,product rough information matrix(∪,∩,～)calculation, some properties of rough information matrix are given.And it gets rough set generation principle of the rough information matrix.From the discussion we can see that each element of rough information matrix is a ordered pair,which is rough sets with the dual attribute,thus it promotes not only the matrix theory on the general number field but also the classical Pawlak's rough sets,meantime it gets a series of rough sets that meet different binary attribute.In other words,it not only promotes the number of the rough sets,but also gets general rough sets from the aspect of the dimension.It provides a good theoretical basis for people to deal with economic, management issues,and propose useful knowledge from the large amount of data.
On the base of static rough information matrix,getting inspiration from the research questions of Professor Shi Kaiquan's S-rough sets,this paper gives the three forms of rough information matrix,and it presents and discusses characteristics of rough information granulation matrix,that is,matrix pair(G(X)_-,G(X)~-)consisting of G(X)_-=(g_(ij-))_(m×n) and G(X)~-=(g_(ij)~-)_(m×n) is called rough information granulation matrix of X(?)U,denoted as G=(G(X)_-,G(X)~-).This paper defines calculations of the lower rough granulation sum(product,complementary operations) matrix,the upper rough granulation sum(product,complementary operations) matrix,rough granulation sum(product,complementary operations) matrix,and research the nature of its operations.It gives some calculation laws which is contented with rough granulation matrix,see the details§3.5.
Secondly,it studies the relationship theorem between rough information matrix and rough granulation matrix,gives sum and product of rough information granulation matrix,and discusses one direction S- rough information granulation matrix,the dual of one direction S- rough information granulation matrix.At the same time,a series of important theorems are discussed.
Finally,it gives accuracy vector of rough information,and obtaines necessary and sufficient conditions about information vector defined or not defined.Theorem 3.8.6:supposed thatβ=(α_1(X_(11)),α_2(X_(12)),…,α_n(X_(1n))) be precision matrix of rough information vector((X_(11-),X_(11)~-),(X_(12-),X_(12)~-),…,(X_(1n-),X_(1n)~-)),then all X_(ij) are able to be defined if and only if the vectorβis a linear combination of the basic unit vector groupε_1,ε_2,…,ε_n,and the combination coefficient is 1.
ChapterⅣfurther discusses quantity characteristics of rough information matrix, proposes rough similarity matrix of rough information vector,and studies its basic characteristics and structural,and then discusses the theorem of the relationship among rough similarity matrix,the true quadratic model,positive definite matrix and half-positive definite matrix.
ChapterⅤintroduces dual of function one direction S-rough sets and function rough set,discusses the shrinking characteristics of the law and property,and studies the relevant theorems and criteria about attribute control and identification of the law. It gives the application of dual of function one direction S-rough sets at financial risk identification.
In chapterⅥ,Fuzzy Sets theory proposed by Professor Zadeh in 1965 is a precise mathematical method to study the uncertain theories.Fuzzy decomposition theorem is the bridge associated fuzzy sets and the classic set,and the theorem reveals the structure of fuzzy sets,that is,a fuzzy set is the overlay form of a number of sub-fuzzy sets,and each sub-fuzzy sets are the number product ofλ∈[0,1]and the classical set A_λ(cut-off set ofλ).If we introduce an attribute fuzzy set,then how is the attribute decomposition situation of the attribute fuzzy sets? What characteristics does it have? What is the relationship between it and our well-known fuzzy decomposition theorem? These problems are very rare in literatures.Based on this,this chapter proposes the concept of attribute fuzzy sets,gives its attribute decomposition theorem,gives the attribute chain theorem of the attribute fuzzy sets when the attribute transfer exist,and answers the consistency of property decomposition theorem and the decomposition theorem.
The main innovative viewpoints of this thesis are as follows:
Innovation point 1.Set up the minimum,the maximum mining degree and a series of mining criterions and theorems about different order knowledge of f,(?).The dependence -indiscernibility theorem about F-ladder knowledge pair,The mining-discovery criterion of F-ladder knowledge pair are discussed.
Innovation point 2.Firstly propose rough information matrix,set up the static and dynamic rough information matrix,discuss the relevant theory and structural features of rough granulation matrix,and give a series of nature and important theorems.It has laid a good foundation for the rough system theory-depth and meticulous research.All the contents and conclusions in ChapterⅢare new.
Innovation point 3.Put forward the concept of rough similarity matrix,and do in detail and meticulous research for the part of the contents,give criterion and theorems of rough system clustering.Perfected the similarity theory,be able to integrate knowledge of the content,discuss rough similarity Matrix combined with real quadratic,and have a certain theoretical significance and practical value.
Innovation point 4.Study the characteristics of attribute law,discuss the relevant theorems and criterion of attribute control and identification combined with function S-rough sets,and provide a theoretical guarantee for the law mining.
Innovation point 5.Propose the concept of attribute fuzzy sets,study important theorems of fuzzy sets and the attribute of rough sets to a certain extent graft infiltration.
本文所讨论的内容是基于Z.Pawlak粗糙集、S-粗集理论、矩阵理论、模糊集理论的框架体系下展开讨论的.一方面,提出粗信息矩阵、粗粒度矩阵,在此基础上进行广泛而较深入的研究,所得到的内容都是新的;另一方面,借助已有的粗相似度,提出了粗相似度矩阵,讨论了粗相似度矩阵的有关性质及定理.同时,论文又介绍并讨论了函数S-粗集的结构特征并给出函数S-粗集对偶这种单规律的属性特征及控制准则,并给出基于该粗集下的金融风险投资决策规律分析.最后提出了属性模糊集的概念,讨论了属性模糊集的特征,给出了属性模糊集的分解定理.本论文各章的具体内容概括如下:
第一章介绍了粗糙集理论的发展概况、研究现状与前景展望,根据粗糙集的目前研究现状与前景的预测方向分析,重点选择对粗糙集的理论进行完善这一方向作为本文的重点研究内容,兼顾到后续论文研究内容的需要,简要介绍了Z.Pawlak粗集与它的结构特征、知识的数值特征等基本概念与理论,为后面章节的研究奠定了必要的基础.
第二章主要研究知识挖掘的数量特征及挖掘的一系列准则与定理.给出了k阶f知识、k阶(?)知识、知识挖掘度的概念;讨论了知识挖掘的链式特征并给出了f、(?)知识挖掘基数链定理,f、(?)知识挖掘粒度链定理,f、(?)知识挖掘过滤度链定理,f、(?)知识挖掘度链定理,并给出了知识粒度与知识挖掘度关系原理的应用.给出f、(?)知识最小、最大挖掘度定理。另外,本文根据单向S-粗集中的知识具有阶梯特征,利用知识的阶梯特性,能够找到人们事先不知道的知识;从知识依赖的另一个角度挖掘所需要的知识,给出了知识的阶梯度,利用阶梯度,直接挖掘所需要的知识。本文给出了F-阶梯知识对的F-阶梯度依赖-分辨定理、F-阶梯知识对的F-阶梯度依赖-不可分辨定理、最大F-阶梯知识的挖掘-发现定理、阶梯知识第一、第二挖掘-发现准则等,如定理2.4.3和定理2.4.4,2.4.5,并给出具体应用。讨论最小F-阶梯知识的挖掘-发现定理:若{([X°]_(F,K),[X°]_K~F|k=1,2,…,m)是F-阶梯知识[X°]_(F,k)与F-阶梯知识[X°]_K~F构成的F-阶梯知识对集合;则存在([X°]_(F,p),[X°]_P~F),p∈(1,2,…,m),满足LAD([X°]_(F,p))=(?){LAD([X°]_(F,i));LAD([X°]_p~F)=(?){LAD([X°]_i~F)};GRD([X°]_(F,p)≤GRD([X°]_(F,i))_(i=1,2,…m);GRD([X°]_p~F)≤GRD([X°]_i~F)_(i=1,2,…m)。
基于在经典系统中的一个系统具有的特征可以从研究系统的系统矩阵中获取,因此提出如下问题:由粗集生成的粗系统,粗系统的特征是否也可以从研究粗系统的系统粗矩阵中获取?第三章正是基于这种考虑提出粗信息矩阵,即由M_α(X)_-,M_α(X)~-构成的矩阵对(M_α(X)_-,M_α(X)~-),称作粗集(X_(ij-,X_(ij)~-)生成的粗信息矩阵。讨论了信息矩阵存在性定理、粗信息矩阵存在性定理。
在定义了信息矩阵的包含、相等以及和粗信息矩阵、积粗信息矩阵(∪,∩,～)运算的基础上,研究了粗信息矩阵的性质(性质3.2.1-性质3.2.8)。得到了粗信息矩阵的粗集生成原理。从讨论我们可以看出,粗信息矩阵中的每个元素是一个序对,是满足二元属性的粗集,从而不仅推广了普通数域上的矩阵理论,而且推广了经典的Z.Pawlak粗集,同时得到了一系列满足不同二元属性的粗糙集。也就是说,不仅从粗集个数方面得到推广,而且从维数方面得到更一般的粗集,为以后人们处理经济、管理问题、从大量的数据中提出有用的知识提供了一个很好的理论依据。
在静态粗信息矩阵的基础上,受S-粗集研究问题的思路启发,本文给出粗信息矩阵的三种形式。提出并讨论了粗信息矩阵的粒度矩阵的特征,即由G(X)_-=(g_(ij-))_(m×n)G(X)~-=(g_(ij)~-)_(m×n)构成的矩阵对(G(X)_~-,G(X)~-)称作X(?)U的粗信息粒度矩阵(rough information granulation matrix),记作G=(G(X)_-,G(X)~-)。定义了下粗粒度和(积、余)矩阵、上粗粒度和(积、余)矩阵、粗粒度和(积、余)矩阵的运算,研究了其运算的性质。给出了粗粒度矩阵满足的运算律,详细内容参见§3.5。
其次,研究了粗信息矩阵与粗粒度矩阵的关系定理,即粗信息矩阵的包含对应着粗粒度矩阵的包含;定义了粗信息粒度矩阵的和与积运算,讨论了粗信息粒度矩阵、单向S-粗信息粒度矩阵、单向S-对偶粗信息粒度矩阵的关系,得到了一系列重要的定理。
最后给出了粗信息精度向量,得到了信息向量可定义与不可定义的充要条件。如定理3.8.6:设β=(α_1(X_(11)),α_2(X_(12)),…,α_n(X_(1n)))为粗信息向量((X_(11-),X_(11)~-),(X_(12-),X_(12)~-),…,(X_(1n-),X_(1n)~-))的精度矩阵,则所有的X_(1j),是可定义的充要条件是向量β为基本单位向量组ε_1,ε_2,…,ε_n的线性组合,且组合系数为1。
第四章进一步讨论粗信息矩阵的数量特征,提出粗信息向量的粗相似度矩阵,并研究了它的基本性质及结构。讨论了粗相似度矩阵与实二次型、正定矩阵、半正定矩阵的关系定理。
第五章介绍了函数单向S-粗集对偶与函数粗集,讨论了规律属性的萎缩特征,研究了规律的属性控制与识别相关定理及准则,给出了函数单向S-粗集对偶在金融风险识别中的应用。
第六章模糊集(Fuzzy Sets)理论是Zadeh教授1965年提出的,它是研究不确定性理论的一种精确的数学方法。联系模糊集与经典集合的桥梁是模糊分解定理,该定理揭示了模糊集的结构,一个模糊集是由若干个子模糊集叠加而成,而每个子模糊集就是λ∈[0,1]与经典集A_λ(λ的截集)的数积,如果我们引入属性模糊集,那么属性模糊集的属性分解情况如何?它具有那些特性?与我们熟知的模糊分解定理的关系怎样?这些问题,在文献中很少见。基于此,本章提出了属性模糊集的概念,给出它的属性分解定理,并且,当存在属性迁移时,给出了属性模糊集的属性链式定理,回答了属性分解定理与分解定理的一致性。
论文的主要创新点:
创新点1.建立了.f、(?)不同阶知识的最小、最大挖掘度及一系列挖掘准则与定理.给出F-阶梯知识对的挖掘发现准则,(?)隐藏知识依赖、发现的原理.
创新点2.提出粗信息矩阵,建立了静态、动态粗信息矩阵,粗粒度矩阵的相关理论与结构特征的讨论,给出了一系列重要的性质与定理.为粗系统理论的深入细致的研究奠定了很好的基础.整个论文的第三章内容与结论都是新的.
创新点3.提出粗相似矩阵的概念,并对该部分内容进行了详细细致的研究.完善了相似度理论,能够融合知识内容,将粗相似度矩阵与实二型结合进行讨论,有一定的理论意义与实用价值.
创新点4.研究了属性规律的特征,结合函数S-粗集讨论了属性控制与识别相关定理与准则,为规律挖掘提供了理论保证.
创新点5.提出属性模糊集的概念,将模糊集的重要定理与粗糙集的属性进行一定程度上嫁接渗透研究.
In science,technology and economic management fields,people often receive a variety of information.But some information is rough,so they are called rough information.Rough information can't be expressed as an accurate set,which hampers people to understand and use this kind of and make the available important information resources be lost.In order to deal with the systems which have uncertainty,imprecise and imperfect characteristic,the Polish mathematician Professor Z.Pawlak proposed a new mathematical theory,rough sets theory in 1982. Rough sets define a set of X with upper and lower approximation which can't be defined by set.The general research of Z.Pawlak's rough sets has laid a theoretical foundation for the system theory and make it get a wide range of applications and research.Some information encountered at the practical problems often change because of the effects of many factors,therefore Z.Pawlak's rough sets would have a certain amount of restrictions when to resolve certain problems.In 2002 Professor Shi Kaiquan improved Z.Pawlak's rough sets,and proposed the singular rough sets (Singular rough sets),called S-rough sets in brief.In 2005,he further proposed the function S-rough sets,and used it for the mining of knowledge and the law.In this paper,a series of new research and discussion are combined with the rough sets theory, S-rough sets theory,matrix theory and the mining of rough information system, identification and a series of new research and applications will be given.
The contents discussed in this article are based on the Z.Pawlak's rough sets, S-rough sets theory,matrix theory and fuzzy sets theory.On the one hand,it proposes rough information matrix and rough granulation matrix,then does wide-ranging and more in depth studies on the basis of them,thus the contents are new;On the other hand,it proposes rough similarity matrix through by using rough similarity,and discusses the coarse nature and theorem of the similarity matrix.Meanwhile,the paper also introduces and discusses the structural characteristics of the function S-rough sets and gives single law attribute characteristics and control criteria of the function S-rough sets,and then gives investment decision-making laws analysis of the financial risks based on rough sets.Finally,it proposes the concept of attribute fuzzy sets,discusses the characteristics of attribute fuzzy sets and gives the decomposition theorem of attribute fuzzy sets.In this paper,the specific content of each chapter are summarized as following:
ChapterⅠintroduces the development,research status and prospect about rough sets theory,then according to the current research directions and prospects of rough sets,it chooses improving rough sets theory as the focus of this article's study.For the needs of the follow-up papers' study,it briefly introduces the basic concepts and theories about the Z.Pawlak's rough sets and its structural characteristics,the numerical characteristics of knowledge,which laid the necessary foundation for the following section of the research.
ChapterⅡmainly studies the number characteristics of knowledge mining and a series of criterions and theorems about mining.It gives f,(?) knowledge of the k order,and the concept of knowledge mining degrees;it discusses the chain characteristics of knowledge mining and gives base chain theorem of f and (?) knowledge mining,granulation chain theorem of f and(?) knowledge mining, filter chain theorem of f and(?) knowledge mining and mining degrees chain theorem of f and(?) knowledge and then gives the application of the relationship principle between granularity and mining degree of knowledge,the minimum and maximum mining degree theorem of f and(?) knowledge.In addition,this paper introduces one direction S-rough sets with the ladder characteristics of knowledge, which can help people find knowledge unknown in advance;it also mine the required knowledge from another angle-dependent knowledge mining,and gives the ladder of knowledge,then makes the use of the ladder to mine the required knowledge directly. In this paper it gives F-ladder dependence degree discernibility of F-ladder knowledge pair,F-ladder dependence degree indiscernibility theorem of F-ladder knowledge pair,the largest F- ladder knowledge mining discovery theorem,the first criteria on F-ladder knowledge,such as theorem 2.4.3,theorem 2.4.4 and theorem 2.4.5,and gives specific applications.To discuss the smallest F-ladder knowledge mining-discovery theorem:If {([X°]_(F,k),[X°]_k~F)|k=1,2,…,m} is the collection of F-ladder knowledge pair composed by F-ladder knowledge[X°]_(F,k) and F-ladder knowledge[X°]_k~F;then there must be([X°]_(F,p),[X°]_p~F),p∈(1,2,…,m), which meets LAD([X°]_(F,p))=(?){LAD([X°]_(F,i))};LAD([X°]_p~F)=(?){LAD([X°]_i~F)}; GRD([X°]_(F,p)≤GRD([X°]_(F,i))_(i=1,2,…m);GRD([X°]_p~F)≤GRD([X°]_i~F)_(i=1,2,…m).
In the classical system the characteristics of a system can obtain from studying the system matrix of the system,so the following question is proposed based on that: is the features of rough system that is made up with rough sets also available from rough system matrix of the research rough systems? ChapterⅢfirstly raises rough information matrix based on this consider,that is,Definition 3.1.3,a pair of the matrix (M_α(X)_-,M_α(X)~-) which is composed of M_α(X)_- and M_α(X)~- is called rough information matrix generating from rough sets(X_(ij-),X_(ij)~-).This paper discusses existence theorem of the information matrix and the existence theorem of rough information matrix,namely Theorem 2.1.1,Theorem 2.1.2.
Based on the definitions of the information matrix inclusion,equivalent and sum rough information matrix,product rough information matrix(∪,∩,～)calculation, some properties of rough information matrix are given.And it gets rough set generation principle of the rough information matrix.From the discussion we can see that each element of rough information matrix is a ordered pair,which is rough sets with the dual attribute,thus it promotes not only the matrix theory on the general number field but also the classical Pawlak's rough sets,meantime it gets a series of rough sets that meet different binary attribute.In other words,it not only promotes the number of the rough sets,but also gets general rough sets from the aspect of the dimension.It provides a good theoretical basis for people to deal with economic, management issues,and propose useful knowledge from the large amount of data.
On the base of static rough information matrix,getting inspiration from the research questions of Professor Shi Kaiquan's S-rough sets,this paper gives the three forms of rough information matrix,and it presents and discusses characteristics of rough information granulation matrix,that is,matrix pair(G(X)_-,G(X)~-)consisting of G(X)_-=(g_(ij-))_(m×n) and G(X)~-=(g_(ij)~-)_(m×n) is called rough information granulation matrix of X(?)U,denoted as G=(G(X)_-,G(X)~-).This paper defines calculations of the lower rough granulation sum(product,complementary operations) matrix,the upper rough granulation sum(product,complementary operations) matrix,rough granulation sum(product,complementary operations) matrix,and research the nature of its operations.It gives some calculation laws which is contented with rough granulation matrix,see the details§3.5.
Secondly,it studies the relationship theorem between rough information matrix and rough granulation matrix,gives sum and product of rough information granulation matrix,and discusses one direction S- rough information granulation matrix,the dual of one direction S- rough information granulation matrix.At the same time,a series of important theorems are discussed.
Finally,it gives accuracy vector of rough information,and obtaines necessary and sufficient conditions about information vector defined or not defined.Theorem 3.8.6:supposed thatβ=(α_1(X_(11)),α_2(X_(12)),…,α_n(X_(1n))) be precision matrix of rough information vector((X_(11-),X_(11)~-),(X_(12-),X_(12)~-),…,(X_(1n-),X_(1n)~-)),then all X_(ij) are able to be defined if and only if the vectorβis a linear combination of the basic unit vector groupε_1,ε_2,…,ε_n,and the combination coefficient is 1.
ChapterⅣfurther discusses quantity characteristics of rough information matrix, proposes rough similarity matrix of rough information vector,and studies its basic characteristics and structural,and then discusses the theorem of the relationship among rough similarity matrix,the true quadratic model,positive definite matrix and half-positive definite matrix.
ChapterⅤintroduces dual of function one direction S-rough sets and function rough set,discusses the shrinking characteristics of the law and property,and studies the relevant theorems and criteria about attribute control and identification of the law. It gives the application of dual of function one direction S-rough sets at financial risk identification.
In chapterⅥ,Fuzzy Sets theory proposed by Professor Zadeh in 1965 is a precise mathematical method to study the uncertain theories.Fuzzy decomposition theorem is the bridge associated fuzzy sets and the classic set,and the theorem reveals the structure of fuzzy sets,that is,a fuzzy set is the overlay form of a number of sub-fuzzy sets,and each sub-fuzzy sets are the number product ofλ∈[0,1]and the classical set A_λ(cut-off set ofλ).If we introduce an attribute fuzzy set,then how is the attribute decomposition situation of the attribute fuzzy sets? What characteristics does it have? What is the relationship between it and our well-known fuzzy decomposition theorem? These problems are very rare in literatures.Based on this,this chapter proposes the concept of attribute fuzzy sets,gives its attribute decomposition theorem,gives the attribute chain theorem of the attribute fuzzy sets when the attribute transfer exist,and answers the consistency of property decomposition theorem and the decomposition theorem.
The main innovative viewpoints of this thesis are as follows:
Innovation point 1.Set up the minimum,the maximum mining degree and a series of mining criterions and theorems about different order knowledge of f,(?).The dependence -indiscernibility theorem about F-ladder knowledge pair,The mining-discovery criterion of F-ladder knowledge pair are discussed.
Innovation point 2.Firstly propose rough information matrix,set up the static and dynamic rough information matrix,discuss the relevant theory and structural features of rough granulation matrix,and give a series of nature and important theorems.It has laid a good foundation for the rough system theory-depth and meticulous research.All the contents and conclusions in ChapterⅢare new.
Innovation point 3.Put forward the concept of rough similarity matrix,and do in detail and meticulous research for the part of the contents,give criterion and theorems of rough system clustering.Perfected the similarity theory,be able to integrate knowledge of the content,discuss rough similarity Matrix combined with real quadratic,and have a certain theoretical significance and practical value.
Innovation point 4.Study the characteristics of attribute law,discuss the relevant theorems and criterion of attribute control and identification combined with function S-rough sets,and provide a theoretical guarantee for the law mining.
Innovation point 5.Propose the concept of attribute fuzzy sets,study important theorems of fuzzy sets and the attribute of rough sets to a certain extent graft infiltration.
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