Covering Approximation Space Research Articles

Covering rough set models, fuzzy rough set models and soft rough set models induced by covering similarity

In order to study the rule of quantity characteristics of rough set theory, a novel notion of covering similarity in a covering approximation space is introduced. Based on this new notion, novel covering rough set models, novel fuzzy rough set models and novel soft rough set models are defined. Further, the concept of similarity matrix is presented, and then the matrix calculation formulas of all the new concepts and new models are deduced. Moreover, aim to better solve the practical problems by comparing the similarity, a concept of similarity interval is introduced, then the covering classification is carried out and a partial order of all the classes is derived. Finally, a few practical examples are given to illustrate the simple applications of all the new concepts and models in this paper.

Soft rough sets based on covering and their applications

Soft rough based on covering SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document} theory has become a useful and well-known area of research in theories of uncertainty. The present work follows up on this flourishing research topic. We introduce a new model of SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document} in a fusion of soft set theory SST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}ST$\\end{document} and rough set depending on covering CRS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {C}RS$\\end{document}. We put forth a definition of soft rough covering approximation space SCAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}CAS$\\end{document} via neighborhood concept. Some axiomatic systems of our model of SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document} are discussed. We study the relationship between our model of SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document} and three other SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document}-models. An algorithm for reduction of the attributes of the information systems using SCAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}CAS$\\end{document} is established. Based on the theoretical discussion, we set forth some applications of our model which will be helpful in decision making process via SRC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {S}RC$\\end{document} theory.

Covering Approximations Approach to Interval Ordered Information Systems

It is been proved that the theory of rough set is very beneficial in working with conflict problems induced by the information endearment The original idea of the rough set is not accurate, however, when preference orders of characteristics domains (standard) are to be considered. Seeking the issue of covering approximation relative to a control relation in interval-ordered information systems is a new mathematical tool to set up a control-based rough set approach. These Mathematical tools apply some topological generalizations, and control relations, covering approximations, and interval information systems. These results using our approach are more accurate than using a classical approach such as the approach of Pawlak. Topologies reproduce types of information systems with one value. By defining a control relation to interval information systems, we generalized the Pawlak approximation space to a covering approximation space and then we involve this approach to work with the interval-ordered information systems. Results gained by the proposed approach to generate two different rough approximations called j-lower and j-upper approximations. We applied j-rough notions such as j-rough membership, j-rough equality, and j-rough inclusion relations using topological generalizations. Our proposed method has improved the results gained from interval information systems. In this Research, there are many improved approaches to investing generalized approximation relative to a control relation in interval information systems, and we use the covering approximation in the rough set approach. This is a generalization of the Pawlak approach applied to interval information systems. This approach opens the way for other generalizations by making new algorithms, which simplify the calculations on it, and we can add more topological concepts to support decision-making in real-life applications.

Reductions of a covering approximation space from algebraic points of view

We first make a great improvement on the early definition of a reducible element of a covering approximation space and formulate the definition of a reduction of the covering. Then the crucial links between the two concepts are established and a method for obtaining an optimal reduction of the covering is proposed. More importantly, we introduce the representation matrix for a finite covering approximation space and define a new type of operation on matrices. To obtain a reduction of the covering, we need only to deal with the representation matrix, so that the reduction of the covering can be executed on computers. The concept of the product of two covering approximation spaces is defined in this paper and we explore the connections between the reductions of the covering of the product space and those of the coverings of the factor spaces. Furthermore, we deal with the connections from the standpoint of algebra. A new type of matrix product is defined and the property of the product is explored. Drawing on the product, we can research the reduction of the product space.

Grained matrix and complementary matrix: Novel methods for computing information descriptions in covering approximation spaces

As two important tools of information descriptions, minimal description (MinD) and maximal description (MaxD) are used to eliminate redundant information when describing an object in covering-based rough sets (CbRSs). Hence, there are many key issues (such as reductions, neighborhoods and approximations) have to do with them in CbRSs. However, it is complicated and time-consuming to use set approaches to compute them in a large cardinal covering approximation space. So the matrix approaches have been used to calculate them by which computers can implement the corresponding calculations easily. Based on the previous research work and inspired by the demand of knowledge discovery under large scale covering information systems, we propose two new matrix methods to calculate MinD and MaxD in CbRSs in this paper. Firstly, a note on the existing methods (matrix approach-1 and approach-2) is presented. Secondly, a new matrix, called grained matrix, is presented. After combining grained matrices and the mean of “sum”, the new matrix approach-1 is presented to calculate MinD and MaxD. Thirdly, three new matrix operations and the notion of complementary matrix are given which can simplify the calculation of computers. Based on new operations, we present the new matrix approach-2 to compute MinD and MaxD. Finally, the computational efficiency of the presented approaches are shown and compared with other methods by experiments on several UCI data sets.

Characteristic numbers and approximation operators in generalized rough approximation system

Covering rough set theory and binary relation based rough set theory are regarded as two important different generalizations of Z. Pawlak's classical rough set theory. In fact, when serial relations are considered, they are the same thing. If R is a serial relation from a universe U to another universe V, then {{R−1(y)}:y∈V} is a covering of U and {{R(x)}:x∈U} is a covering of V. In a covering approximation space (U,C), the relation R defined as R(x)={C∈C:x∈C} is just a serial relation from U to V=C. So U and V have the same status and (U,V,R) is indeed a whole, then (U,V,R) is called an Approximation System in order to take a comprehensive study from a new and higher perspective. By introducing nthl neighborhoods and kthr neighborhoods recursively, a new set of upper and lower approximation operators is established, and the Approximation System is further referred to as the Generalized Rough Approximation System (GRAS). In order to describe some stability of the Approximate System, characteristic numbers such as l-cycle number, r-cycle number, link number and connect number are also introduced. From there, partitions of GRAS can be constructed and sub-GRAS can be defined. This makes the whole of GRAS modularized, which is a big simplification when it comes to solving real problems. In addition, some applications of characteristic numbers and approximation operators are illustrated by examples, and the Boolean matrix calculation formulas of all approximations are proved. Finally, the important special GRAS (U,U,R) is studied in detail and an application example is given.

A novel notion in rough set theory: Invariant subspace

All current work on calculating approximate sets of a rough set inevitably requires the participation of all elements in the universe. However, it is cumbersome and causes a huge waste of resources when the universe is quite big and the computed set is small enough. By introducing the notion of invariant subspace in rough set theory, we skillfully reduce the range of elements involved in the computations of approximations of a rough set from the whole universe U to a suitable invariant subspace V of U, and then give the modular method within the reduced range. Moreover, we present the modular Boolean matrix method, such that the calculation of upper and lower approximations of rough sets can be converted into operations on modular matrices. Finally, the results in covering approximation space are generalized to fuzzy β-covering approximation space to calculate the upper and lower approximations of the crisp sets. In particular, an algorithm for calculating the value of β is proposed to make the involved information has the highest distinction degree. This β is more reliable and meaningful than the empirical one, and an example about COVID-19 is put forward to simply illustrate its application.

Interdependency of Complex Fuzzy Neighborhood Operators and Derived Complex Fuzzy Coverings

Complex fuzzy coverings (CFCs) are the natural mixture of the complex fuzzy sets (CFSs) and coverings, which are the modified versions of the coverings by replacing crisp sets with CFSs. This manuscript aims to explore the complex fuzzy neighborhood operators (CFNOs) by introducing the notions such as $\beta $ -neighborhood system ( $\beta $ -NO), complex fuzzy $\beta $ -minimal description (CF $\beta $ -MND), and complex fuzzy $\beta $ -maximal description (CF $\beta $ -MXD). First, we explore the complex fuzzy $\beta $ -covering approximation space (CF $\beta $ -CAS) and then we propose the above notions and investigate their properties. Additionally, we construct the CFNOs based on the complex fuzzy $\beta $ -coverings (CF $\beta $ -Cs). Finally, the CF $\beta $ -Cs were derived by using CFNOs, and their properties are considered. These all notions are also verified with the help of suitable examples to show that the presented approaches are extensive, reliable, and proficient techniques.

On local multigranulation covering decision-theoretic rough sets

Multi-granulation decision-theoretic rough sets uses the granular structures induced by multiple binary relations to approximate the target concept, which can get a more accurate description of the approximate space. However, Multi-granulation decision-theoretic rough sets is very time-consuming to calculate the approximate value of the target set. Local rough sets not only inherits the advantages of classical rough set in dealing with imprecise, fuzzy and uncertain data, but also breaks through the limitation that classical rough set needs a lot of labeled data. In this paper, in order to make full use of the advantage of computational efficiency of local rough sets and the ability of more accurate approximation space description of multi-granulation decision-theoretic rough sets, we propose to combine the local rough sets and the multigranulation decision-theoretic rough sets in the covering approximation space to obtain the local multigranulation covering decision-theoretic rough sets model. This provides an effective tool for discovering knowledge and making decisions in relation to large data sets. We first propose four types of local multigranulation covering decision-theoretic rough sets models in covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects. Moreover, some important properties and decision rules are studied. Meanwhile, we explore the reduction among the four types of models. Furthermore, we discuss the relationships of the proposed models and other representative models. Finally, illustrative case of medical diagnosis is given to explain and evaluate the advantage of local multigranulation covering decision-theoretic rough sets model.

INCREMENTALLY UPDATING APPROXIMATION IN INCOMPLETE INFORMATION SYSTEMS UNDER THE VARIATION OF OBJECTS

Rough membership functions in covering approximation space not only give numerical characterizations of covering-based rough set approximations, but also establish the relationship between covering-based rough sets and fuzzy covering-based rough sets. In this paper, we introduce a new method to update the approximation sets with rough membership functions in covering approximation space. Firstly, we present the third types of rough membership functions and study their properties. And then, we consider the change of them while simultaneously adding and removing objects in the information system. Based on that change, we propose a method for updating the approximation sets when the objects vary over time.

Three types of fuzzy covering-based rough set models

In this paper, three new types of fuzzy covering-based rough set models are proposed. First, the membership-degree-based fuzzy covering rough set is defined in the fuzzy covering approximation space by separately comparing the membership degree of each object belonging to the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering for the same object. Second, the membership-function-based fuzzy covering rough set is obtained by globally comparing the membership function of the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering. Finally, the membership-function-based fuzzy covering rough set model is generalized to the fuzzy β-covering approximation space. The properties of these new rough set models are discussed. Furthermore, the relationship between these new rough sets and the relationship between the new rough set and Pawlak rough set are described.

Multigranulation single valued neutrosophic covering-based rough sets and their applications to multi-criteria group decision making

In this paper, three types of (philosophical, optimistic and pessimistic) multigranulation single valued neutrosophic (SVN) covering-based rough set models are presented, and these three models are applied to the problem of multi-criteria group decision making (MCGDM).Firstly, a type of SVN covering-based rough set model is proposed.Based on this rough set model, three types of multigranulation SVN covering-based rough set models under the concept of multigranulation SVN $beta$-covering approximation space are proposed, where $beta$ is a SVN number.Moreover, the connections among these four models are investigated.Secondly, some conditions under which different multigranulation SVN $beta$-covering approximation spaces induced the same multigranulation SVN covering approximation operators are presented.Finally, three novel methods are presented to MCGDM problems under the multigranulation SVN covering-based rough set models.Furthermore, the proposed MCGDM methods are compared with other methods through a problem about paper defect diagnosis.

Couple fuzzy covering rough set models and their generalizations to CCD lattices

We make gradual generalizations in this paper, from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy covering rough set models in fuzzy rough set theory, and further to the concepts of couple L-fuzzy covering rough set models in L-fuzzy rough set theory.Given a fuzzy covering approximation space (U,C˜) and β∈(0,1], for each x∈U, we divide C˜ into two parts ⊤x={C˜i∈C˜:C˜i(x)≥β} and ⊥x={C˜i∈C˜:C˜i(x)<β}. To fully describe x from positive aspect and negative aspect, both the two parts ⊤x and ⊥x are important, especially the combination of them. So, in this paper, based on the two parts, we define 1st couple β-fuzzy covering rough set models [(P−˜,P+˜),(Q−˜,Q+˜)] and 2nd couple fuzzy β-covering rough set models [(P−‾,P+‾),(Q−‾,Q+‾)]. Both the two types of couple fuzzy β-covering rough set models are generalizations of the twin approximations defined in covering rough set theory. Since each pair of operators in these models are not only closely related but complementary, they can be used to analyze and solve practical problems from positive and negative aspects so as to make a crucial decision. So we then give some examples to show their practical value. The relationships between our models and some other models introduced in previous literature are investigated, and the matrix methods are given to calculate the related approximations and to describe the relationships between every couple models.To generalize the couple fuzzy β-covering rough set models to the CCD lattice are of a bit complicated, because any two elements in the lattice cannot be compared with each other generally. After some effort we successfully construct the ideal couple L-fuzzy β-covering rough set models in L-fuzzy rough set theory which are just the generalizations of the two types of couple fuzzy β-covering rough set models. We also obtain the lattice matrix representations to calculate the related approximations.

Fuzzy β-covering approximation spaces

All fuzzy covering-based rough set models are constructed under a corresponding fuzzy covering approximation space (FCAS). The fuzzy β-covering approximation space (β-FCAS) is a generalization of the FCAS through replacing the value 1 with a parameter β. In other words, the β-FCAS is the basis of studying fuzzy covering-based rough sets and their applications. Therefore, it is necessary to study some questions in the fuzzy covering approximation space, such as problems of reduction, relationships among some basic concepts and relationships between two fuzzy covering approximation spaces. In this article, we investigate the questions in the β-FCAS further. Firstly, the definitions of I-irreducible element and I-reduct are presented, which can be seen as the complement of the existing notions. Then, relationships among some concepts in the β-FCAS are investigated, such as the relationship between fuzzy β-minimal description and β-reduct, and the relationship between fuzzy β-covering and its I-reduct. Thirdly, inspired by some concepts in the β-FCAS, we present some new concepts between two β-FCASs and their properties. Based on these new concepts, we present some conditions such that two fuzzy β-coverings have the same reduct (or I-reduct). Finally, based on the results above, seven derived β-FCASs are investigated further and a corresponding lattice of them is proposed.

A new rough set model based on multi-scale covering

Multi-scale rough set has become one of the hot research topics in rough set. In this article, from the point of view of granule, we first give the definition of the scale relationship between two coverings. Then some properties about neighborhood and approximation operators are explored based on multi-scale covering. Further, we investigate the multi-scale covering approximation space and thus construct a rough set model based on multi-scale covering. In the multi-scale covering decision approximation space, several optimal scale coverings are studied. Finally, we show the connection between multi-scale covering rough sets and multi-scale information systems, and generalize the multi-scale covering approximation space. Rough sets based on multi-scale covering are more extensive than multi-scale information systems. In addition, we design couple of algorithms and several experiments are conducted to verify the efficiency of the algorithms.

TOPSIS-WAA method based on a covering-based fuzzy rough set: An application to rating problem

In this paper, we firstly study two pairs of covering-based fuzzy rough set models and propose the TOPSIS-WAA method based on a covering-based fuzzy rough set. Subsequently, we design a rating scheme based on a multi-criteria decision-making method in a finite fuzzy covering approximation space. The rating scheme is implemented based on the preset subjective ratio and the calculated objective ranking of all alternatives. Furthermore, we use the relevant data of some customers of Industrial and Commercial Bank of China (ICBC) to illustrate the feasibility of our method. At the same time, two test rules are used to verify the validity of the proposed method. Moreover, we compare five different decision-making methods with our method to demonstrate the superiority of our method. Finally, the performance of our method is verified from the perspectives of the best alternative and the optimal ranking.

Novel matrix-based approaches to computing minimal and maximal descriptions in covering-based rough sets

Minimal and maximal descriptions of concepts are two important notions in covering-based rough sets. Many issues in covering-based rough sets (e.g., reducts, approximations, etc.) are related to them. It is well known that, it is time-consuming and error-prone when set representations are used to compute minimal and maximal descriptions in a large scale covering approximation space. To address this problem, matrix-based methods have been proposed in which calculations can be conveniently implemented by computers. In this paper, motivated by the need for knowledge discovery from large scale covering information systems and inspired by the previous research work, we present two novel matrix-based approaches to compute minimal and maximal descriptions in covering-based rough sets, which can reduce the computational complexity of traditional methods. First, by introducing the operation “sum” into the calculation of matrix instead of the operation “⊕”, we propose a new matrix-based approach, called approach-1, to compute minimal and maximal descriptions, which does not need to compare the elements in two matrices. Second, by using the binary relation of inclusion between elements in a covering, we propose another approach to compute minimal and maximal descriptions. Finally, we present experimental comparisons showing the computational efficiency of the proposed approaches on six UCI datasets. Experimental results show that the proposed approaches are promising and comparable with other tested methods.

Updating reducts in fuzzy $$\beta $$-covering via matrix approaches while coarsening and refining a covering element

In a dynamic environment, knowledge reduction of information systems with variations of object sets, attribute sets and attribute values is an important topic, and related covering reduction of dynamic fuzzy $$\beta $$ -covering approximation space when coarsening and refining a covering element has attracted little attention. How to update the reducts based on the original information is an important problem since it can help to improve the efficiency of knowledge discovery. This paper provides the definitions of the coarsening and refinement of a covering element in the dynamic fuzzy $$\beta $$ -covering approximation space. Then approaches for updating the reducts by the relation character matrix are discussed when coarsening and refining a covering element. And the corresponding algorithms for computing new reducts are derived. Finally, several examples were given to illustrate the validity of the proposed approaches.

An MADM approach to covering-based variable precision fuzzy rough sets: an application to medical diagnosis

In medical diagnosis, how to select an optimal medicine from some medicines with similar efficacy values to treat diseases has become common problems between doctors and patients. To solve this problem, we describe it as a multi-attribute decision-making (MADM) in a finite fuzzy covering approximation space. This paper aims to propose two pairs of covering-based variable precision fuzzy rough sets. By combining the proposed rough set model with the VIKOR method, we construct a novel method to medicine selection MADM problems in the context of medical diagnosis. A real-life case study of selecting a proper medicine to treat Alzheimer’s disease is given to demonstrate the practicality of our proposed method. Through a comparative analysis and an experimental analysis, we further explore the effectiveness and stability of the established method.

Knowledge reduction methods of covering approximate spaces based on concept lattice

Both rough sets and concept lattices, which are two complementary tools in data analysis, are analyzed based on binary relations. The relations between rough sets and concept lattices are important research topic. In this paper, the methods of union reduction and intersection reduction in covering approximation spaces based on concept lattice are discussed, and the relations between union reduction of covering approximation spaces and concept lattices reduction are investigated. We also discuss the relations of element characteristics between covering approximation spaces and the concept lattices. Meanwhile, the connections between reduction of a covering approximation space and that of its compliment space are revealed. The research results establish a bridge between the rough sets and concept lattices and help one to gain much more insights into the two theories.