Quantity Characteristics Of Rough Information Matrix And Its Applications

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:Chapterâ… introduces 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.Chapterâ…¡mainly 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? Chapterâ…¢firstly 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=(α₁(X₁₁),α₂(X₁₂),…,α_n(X_1n)) be precision matrix of rough information vector((X_11-,X₁₁^-),(X_12-,X₁₂^-),…,(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 groupε₁,ε₂,…,ε_n,and the combination coefficient is 1.Chapterâ…£further 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.Chapterâ…¤introduces 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 chapterâ…¥,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 Chapterâ…¢are 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.