Covering-based Rough Set Models Research Articles

Neighborhood-related q-rung orthopair fuzzy covering-based rough set models and their applications for multi-attribute decision making

Neighborhood-related q-rung orthopair fuzzy covering-based rough set models and their applications for multi-attribute decision making

L-valued covering-based rough sets and corresponding decision-making applications

Considering L to be a complete residuated lattice, by introducing the notion of L-valued covering on an L-set (as a universe), and then an L-valued neighborhood based on it, we present the concept of L-valued covering-based rough sets. We mainly address the following issues in this paper: Firstly, we present four types of L-valued neighborhood operators and study some of their respective properties. Secondly, we construct L-valued lower (resp., upper) approximation operators and then discuss some of their properties. Finally, we try to propose a kind of MADM problem based on an L-valued covering-based rough set model (for L=[0,1]).

Two new fuzzy covering-based rough approximation operators via inclusion degree

Approximate accuracy is an important concept in rough set theory, which is defined by upper and lower approximations. Generally speaking, the higher precision means the better application performance. The approximation accuracy can be improved by minimizing the upper approximation and maximizing the lower approximation. Recently, Zhou [52] introduced two types of fuzzy-covering based rough set models by using inclusion relation between fuzzy sets. In this paper, by replacing inclusion relation with implication degree, we investigate two new fuzzy covering-based rough set models. Compared with inclusion relationship, the inclusion degree can describe the contained relation between fuzzy sets in more detail. This makes our upper approximation smaller than Zhou’s upper approximation, while the lower approximation is larger than Zhou’s. Therefore, the approximate accuracy of our model is higher than that of Zhou. Furthermore, we apply the new model to the study of multi-attribute decision-making (MADM). Combined with the car buying problem, we demonstrate the effectiveness of our model and compare it with other methods. The results show that we can get the same optimal choice as other methods. However, according to Zhou’s model, we cannot get the optimal choice.

Single-Valued Neutrosophic Covering-Based Rough Set Model Over Two Universes and Its Application in MCDM

Single-Valued Neutrosophic Covering-Based Rough Set Model Over Two Universes and Its Application in MCDM

Some neighborhood-related fuzzy covering-based rough set models and their applications for decision making

Fuzzy rough set (FRS) has a great effect on data mining processes and the fuzzy logical operators play a key role in the development of FRS theory. In order to generalize the FRS theory to more complicated data environments, we firstly propose four types of fuzzy neighborhood operators based on fuzzy covering by overlap functions and their implicators in this paper. Meanwhile, the derived fuzzy coverings from an original fuzzy covering are defined and the equalities among overlap function-based fuzzy neighborhood operators on a finite fuzzy covering are also investigated. Secondly, we prove that new operators can be divided into seventeen groups according to equivalence relations, and the partial order relations among these seventeen classes of operators are discussed, as well. And then, two types of neighborhood-related fuzzy covering-based rough set models are proposed via the overlap function-based fuzzy neighborhood operators and the t-norm-based fuzzy neighborhood operators. Furthermore, the groups and partially order relations are also discussed. Finally, a novel fuzzy TOPSIS methodology is put forward to solve a biosynthetic nanomaterials select issue, and the rationality and enforceability of our new approach is verified by comparing its results with nine different methods.

Boundary region-based variable precision covering rough set models

As an essential way of knowledge representation and knowledge acquisition in uncertainty theory, variable precision rough set (VPRS) has been widely considered by researchers. Firstly, the corresponding CVPRS model via boundary region is investigated based on an existing covering-based rough set model. Its algebraic structures and properties are systematically studied. This model mainly depicts a pair of boundary operators and a pair of approximation operators with a certain threshold from the perspective of single knowledge granularity. And it not only relaxes the complete inclusion relation but also retains the basic properties of the classical rough set. Secondly, an attribute reduction approach is given for a covering-based decision information systemusing the proposed CVPRS model, which keeps the lower or upper approximation unchanged. The different performances of boundary operators and their related three indices in these reduction methods are compared. The results show that boundary operators and distribution functions have outstanding advantages in these applications. Finally, an extraction method of necessity rules and possibility rules corresponding to decision classes are established. The validity and security of the above methods are verified theoretically. Numerical experiments further show that the proposed models and methods have a certain application value.

Attribute reduction based on observational consistency in intuitionistic fuzzy multi-covering decision systems

Attribute reduction is an important issue in data mining, machine learning and other applications of big data processing. Covering-based rough set and intuitionistic fuzzy (IF) set models are both the effective theoretical tools of uncertainty or imprecise computation, and thus IF covering rough set model has been acknowledged as a positive approach to attribute reduction. Based on IF covering rough set model, this study explores a kind of parameterized IF observational consistency in IF multi-covering decision system, and proposes an attribute reduction method. This article firstly defines the concepts of regular IF β-covering, parameterized IF observational sets on the regular IF β-covering approximation space. Secondly, the parameterized IF observational consistency is defined to be the principal of attribute reduction in the IF multi-covering decision system, and the related IF discernibility matrix is developed to provide a way of attribute reduction. For multi-observational consistency corresponding to an observational parameters set, an unified multi-observational discernibility matrix is constructed, which avoids the disadvantage of needing to construct multiple corresponding discernibility matrices separately. Furthermore, an attribute reduction algorithm based on iterative dissolving of unified multi-observational discernibility matrix is proposed, and the experiment to demonstrate effectiveness of algorithm is presented. Experiments with UCI datasets shows that, the proposed method is a good way for improving both the rates of attribute-reduced and the classification accuracy of reduced datasets.

Fuzzy covering-based rough set on two different universes and its application

In this paper, we propose a new type of fuzzy covering-based rough set model over two different universes by using Zadeh’s extension principle. We mainly address the following issues in this paper. First, we present the definition of fuzzy \(\beta\)-neighborhood, which can be seen as a fuzzy mapping from a universe to the set of fuzzy sets on another universe and study its properties. Then we define a new type of fuzzy covering-based rough set model on two different universes and investigate the properties of this model. Meanwhile, we give a necessary and sufficient condition under which two fuzzy \(\beta\)-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Moreover, matrix representations of the fuzzy covering lower and fuzzy covering upper approximation operators are investigated. Finally, we propose a new approach to a kind of multiple criteria decision making problem based on fuzzy covering-based rough set model over two universes. The proposed models not only enrich the theory of fuzzy covering-based rough set but also provide a new perspective for multiple criteria decision making with uncertainty.

Decision self-information based on parameterized fuzzy β neighborhood and its application in three-way multi-attribute group decision-making

<abstract><p>As a special kind of entropy, decision self-information effectively considers the uncertainty information of both the lower and upper approximations. However, it is limited to rough binary relations, which limits its application to complex problems. In addition, parameterized fuzzy β covering, as an extension of the covering-based rough set model, can effectively characterize the similarity between samples. We combine decision self-information with a parameterized fuzzy β neighborhood to propose decision self-information in fuzzy environments, and we study its important properties. On this basis, a three-way multi-attribute group decision-making algorithm is established, and a practical problem is solved. The effectiveness of the proposed method is verified by experimental analysis.</p></abstract>

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.

A commentary on B. Yang, B.Q. Hu, “On some types of fuzzy covering-based rough sets” [Fuzzy Sets Syst. 312 (2017) 36–65

This paper shows that the eight propositions and the four corollaries about four types of fuzzy covering-based rough set models in Yang and Hu (2017) [1] are incorrect through counter-examples, and the eight propositions are corrected.

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.

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 fitting model for attribute reduction with fuzzy β-covering

As an intuitive extension of covering-based rough set models, fuzzy β covering provides an effective means to deal with uncertain information. However, classical fuzzy β covering can not fit a given data set well, which limits its popularization and application. On the one hand, this model can not guarantee that the lower approximation is contained in the upper approximation. It does not ideally describe the differences in sample classification. On the other hand, fuzzy β covering is sensitive to noise hidden in data, which leads to unstable classification performance. To address these issues, we set forth a new fitting model for attribute reduction based on fuzzy β covering. To this end, parameterized fuzzy β neighborhoods in terms of a family of fuzzy coverings are introduced to characterize the similarity between samples, using which the granular structures of fuzzy lower and upper approximations of a decision are presented. Fuzzy β covering decision tables are then formalized to deal with knowledge reduction from the viewpoint of maintaining the discriminative ability of a fuzzy covering family. Furthermore, a heuristic attribute reduction algorithm is developed to reduce redundant fuzzy coverings. Extensive numerical experiments are further conducted to examine the effectiveness and feasibility of the proposed model.

PF-TOPSIS method based on CPFRS models: An application to unconventional emergency events

To improve the utilization of rough sets, some novel uncertain models via rough sets have been appeared. Although many rough set models can only handle some intuitionistic fuzzy data, they can’t handle the data in the Pythagorean environment. For effectively solving the complicated problems in the Pythagorean environment, by applying the notion of Pythagorean fuzzy set (PFS) theory which is a natural generalization of intuitionistic fuzzy set (IFS) theory together with combining the covering-based rough set models and fuzzy rough set models, we introduce the concept of covering-based Pythagorean fuzzy rough set (CPFRS) models via Pythagorean fuzzy β-neighborhoods. Particularly, two kinds of uncertain degrees of our extended models are investigated. We find that our proposed models can effectively handle the complex data in the Pythagoras environment for the theoretical analysis with CPFRS models. We set forth two different Pythagorean fuzzy TOPSIS methodologies to deal with the multi-attribute decision-making (MADM) problem by taking the advantage of the CPFRS models. Finally, we compare and analyze the results of the two methods with two existing through a practical example.

Covering-based soft fuzzy rough theory and its application to multiple criteria decision making

Covering-based rough set model is a vital topic as a generalization of rough sets which is a tool for AI and data mining. For the purpose, we establish two classes of hybrid uncertain systems: soft rough fuzzy-covering models and soft fuzzy rough covering models by means of (fuzzy) soft neighborhoods. By means of fuzzy soft measure degrees, we deduce \(\alpha \)-soft rough fuzzy coverings, D-soft rough fuzzy coverings, \(\alpha \)-soft fuzzy rough coverings, and D-soft fuzzy rough coverings by these two hybrid uncertain systems, respectively. The relationships among the proposed covering-based rough set models are given. Based on the theoretical discussion for the combination of covering fuzzy rough sets and soft sets, we set forth a new method to multiple criteria decision-making problem. By comparative analysis, we obtain that the optimal results are the same between the aggregation operator method and our proposed method, which means that our method is reasonable and effective.

Intuitionistic fuzzy $$\beta $$-covering-based rough sets

Covering-based rough set is an important extended type of classical rough set model. In this model, concepts are approximated through substitution of a partition in classical rough set theory with a covering in covering-based rough set theory. Various generalized covering-based rough sets have been investigated, however, little work has been done on extending four classical covering-based rough set to intuitionistic fuzzy (IF) settings. In this study, four novel IF covering-based rough set models are developed by combining an IF $$\beta $$-covering with four classical covering-based rough set models. First, we present the concept of IF $$\beta $$-minimal description, and then construct four order relations on IF $$\beta $$ approximation space. Second, we propose four IF $$\beta $$-covering-based rough set models and derive that they are generalizations of four existing covering-based rough sets in IF settings. We also discuss the properties of these IF $$\beta $$-covering-based rough sets and reveal their relationships. We use the existing distance between two IF sets to characterize the uncertainty of the presented IF $$\beta $$-covering-based rough sets. Third, we define the reducts of IF $$\beta $$-covering decision systems and examine their discernibility-function-based reduction methods for these IF $$\beta $$-covering-based rough sets. Fourth, we present four optimistic and pessimistic multi-granulation IF $$\beta $$-covering-based rough sets and analyze their properties and uncertainty measures from multi-granulation perspective. Fifth, we study the discernibility-function-based reduction methods for the presented multi-granulation IF $$\beta $$-covering-based rough sets. Finally, we discuss another two neighborhood-based IF covering-based rough sets. This study can provide a covering-based rough set method for acquiring knowledge from IF decision systems.

Communication between fuzzy information systems using fuzzy covering-based rough sets

Information acquisition and communication play an important role in the field of information technology. Meanwhile, communication between fuzzy information systems is a basic problem in fuzzy systems. In this paper, we discuss the communication between fuzzy information systems by using fuzzy covering mappings and fuzzy covering-based rough sets, respectively. First, some properties of fuzzy covering mappings are investigated by Zadeh's extension principle. Then we address the communication between fuzzy information systems by using fuzzy covering mappings. Especially, by introducing some notions in fuzzy information system, we study the relationship between two fuzzy information systems which are defined in the same universe. Second, we propose a novel type of fuzzy covering-based rough set model in fuzzy information system. Some properties of this model are investigated. Finally, we study the communication between fuzzy information systems using fuzzy covering-based rough sets.

A comprehensive study of fuzzy covering-based rough set models: Definitions, properties and interrelationships

Fuzzy covering-based rough set models are hybrid models using both rough set and fuzzy set theory. The former is often used to deal with uncertain and incomplete information, while the latter is used to describe vague concepts. The study of fuzzy rough set models has provided very good tools for machine learning algorithms such as feature and instance selection. In this article, we discuss different types of dual fuzzy rough set models which all consider fuzzy coverings. In particular, we study two models using non-nested level-based representation of fuzziness. In addition to the study of the theoretical properties for each model, interrelationships between the different models are discussed, resulting in a Hasse diagram of fuzzy covering-based rough set models for a finite fuzzy covering, an IMTL-t-norm and its residual implicator.

On some types of fuzzy covering-based rough sets

Fuzzy coverings are a natural extension of the coverings by replacing crisp sets with fuzzy sets. Recently, an excellent introduction to the definition of a fuzzy β-covering is due to Ma and two fuzzy covering-based rough set models are presented. In this paper, by introducing some new definitions of fuzzy β-covering approximation spaces, the properties of fuzzy β-covering approximation spaces and Ma's fuzzy covering-based rough set models are studied. Furthermore, three new types of fuzzy covering-based rough set models as generalizations of Ma's models are first proposed in this paper. First, some properties of fuzzy β-covering and its fuzzy β-neighborhood family are proposed. We present a necessary and sufficient condition for fuzzy β-neighborhood family induced by a fuzzy β-covering to be equal to the fuzzy β-covering itself. Then we study the characterizations of Ma's fuzzy covering-based rough set models and give a necessary and sufficient condition for two fuzzy β-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Finally, this paper proposes three new types of fuzzy covering-based rough set models by introducing a new notion of a fuzzy complementary β-neighborhood.