Rough Approximation Operators Research Articles

Rough sets models inspired by supra-topology structures.

Our aim of writing this manuscript is to found novel rough-approximation operators inspired by an abstract structure called "supra-topology". This approach is more relaxed than topological ones and extends the scope of applications because an intersection condition of topology is dispensed. Firstly, we generate eight types of supra-topologies using -neighborhood systems induced from any arbitrary relation. We elucidate the relationships between them and investigate the conditions under which some of them are identical. Then, we create new rough sets models from these supra-topologies and present the main characterizations of their lower and upper approximations. We apply these approximations to classify the regions of the subset and compute its accuracy measures. The master merits of the current approach are to produce the highest accuracy values compared with all approaches given in the published literature under a reflexive relation as well as preserve the monotonicity property of accuracy and roughness measures. Moreover, we demonstrate the good performance of the followed technique through analysis of some data of dengue fever disease. Ultimately, we debate the advantages and disadvantages of the followed approach and make a plan for some upcoming work.

Approximation operators and accuracy measures of rough sets from an infra-topology view.

In this work, our target point is to focus on rough approximation operators generated from infra-topology spaces and examine their features. First, we show how infra-topology spaces are constructed from -neighborhood systems under an arbitrary relation. Then, we exploit these infra-topology spaces to form new rough set models and scrutinize their master characterizations. The main advantages of these models are to preserve all properties of Pawlak approximation operators and produce accuracy values higher than those given in several methods published in the literature. One of the unique characterizations of the current approach is that all the approximation operators and accuracy measures produced by the current approach are identical under a symmetric relation. Finally, we present two medical applications of the current methods regarding Dengue fever and COVID-19 pandemic. Some debates regarding the pros and cons of the followed technique are given as well as some upcoming work are proposed.

On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called IO-implications) in this paper. More specifically, we first give the definition of the 1st type of FRSs and study some essential properties. Subsequently, the definition of the 2nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1st type of FRSs and the 2nd type of FRSs is described. That is, in a fuzzy approximation space, the 2nd type of lower approximation is contained in the 1st type of lower approximation, while the 1st type of upper approximation is included in the 2nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses.

Notes on “On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices”

Jiang and Hu characterized (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices with L-fuzzy negations, where L denotes a complete lattice. However, there are some faults in the study of (O,G)-fuzzy rough sets. As there exist complete lattices without strict L-fuzzy negations or L-fuzzy involutive negations, the properties of G-lower L-fuzzy rough approximation operators need be further discussed. Meanwhile, it is wrong that the G-lower L-fuzzy rough approximation operators preserve the arbitrary union of L-fuzzy sets and O-upper L-fuzzy rough approximation operators preserve the arbitrary intersection of L-fuzzy sets. Hence, the characterizations of (O,G)-fuzzy rough sets and multigranulation (O,G)-fuzzy rough sets need be rectifies. For the convenience of readers, a table is provided to show the correspondences among the conclusions.

A new approach to generalized neighborhood system-based rough sets via convex structures and convex matroids

Generalized neighborhood system-based rough sets (GNSs) play a key part in rough set theory. In this paper, we introduce a new approach to GNSs from the aspects of convex structures and convex matroids. Firstly, we propose the concepts of matroidal exterior operators and matroidal interior operators and investigate their relationship with convex matroids, respectively. We prove that there is a one-to-one correspondence among matroidal exterior operators, matroidal interior operators and convex matroids. Secondly, we introduce two new types of GNSs which are called co-directed intersection closed and exchangeable. Then we establish the compatible relationships between convex structures (resp. convex matroids) and co-directed intersection closed (resp. exchangeable) GNSs by using matroidal exterior operators and matroidal interior operators as the linkages. Finally, we present axiomatic characterizations of rough approximation operators corresponding to a co-directed intersection closed generalized neighborhood system operator (gns operator) as well as its combinations with serial, reflexive, weak-transitive and weak-unary gns operators.

Notes on “On (IO,O)-fuzzy rough sets based on overlap functions”

Qiao investigated the properties and topological structures of (IO,O)-fuzzy rough sets, which extended the classical conjunction operator in rough approximation operator to an overlap function O. However, there are some faults in the characterizations of (IO,O)-fuzzy rough sets, such as wrong conclusions and strict condition, even if the overlap function O is assumed to be a continuous t-norm with no non-trivial zero divisors. This paper further discusses (IO,O)-fuzzy rough sets and rectifies those faults. Moreover, some examples are presented to show those faults.

Real-valued hemimetric-based fuzzy rough sets and an application to contour extraction of digital surfaces

Binary relations, coverings and neighborhood systems/operators are useful tools to study fuzzy rough set theory. In this paper, we use the notion of real-valued hemimetric, a weak version of the standard metric, as the basic structure to define and study fuzzy rough sets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper and lower rough approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the Lukasiewicz logical system. It is shown that upper definable fuzzy subsets, lower definable fuzzy subsets and Lipschitz fuzzy subsets are the same thing in this model. Definable fuzzy subsets are exactly the upper sets with respect to the induced fuzzy preorder, which form a stratified Alexandrov fuzzy topology. A comparison between hemimetric-based fuzzy rough sets and fuzzy preorder-based fuzzy rough sets has been made. Results show that the former can be considered as a real-valued extension of the latter. At the end, an application of the hemimetric-based rough set model to contour extraction of digital surfaces is proposed.

On (GO, O)-fuzzy rough sets derived from overlap and grouping functions

Rough sets, as a powerful tool to deal with uncertainties and inaccuracies in data analysis, have been continuously concerned and studied by many scholars since it was put forward, especially the research on various rough set models. On the other hand, overlap and grouping functions, as two newly aggregation operators and mathematical model to handle the problems involving in information fusion, have been successfully applied in many real-life problems. In this paper, based on overlap and grouping functions, we propose a new fuzzy rough set model named (GO, O)-fuzzy rough sets and consider its characterizations along with topological properties. Properly speaking, firstly, we utilize QL-operators (and also QL-implications) constructed from overlap and grouping functions and fuzzy negations to define the lower approximation operator in (GO, O)-fuzzy rough set model named GO-lower fuzzy rough approximation operator and the upper approximation operator in (GO, O)-fuzzy rough set model is considered as the O-upper fuzzy rough approximation operator in (IO, O)-fuzzy rough set model proposed by Qiao recently. Secondly, we discuss lots of basic properties of (GO, O)-fuzzy rough sets, especially for the properties of GO-lower fuzzy rough approximation operator. Thirdly, we focus on the relationship between (GO, O)-fuzzy rough sets and concrete fuzzy relations. Finally, we give the topological properties of the upper and lower approximation operators in (GO, O)-fuzzy rough set model.

A rough set model based on (L, M)-fuzzy generalized neighborhood systems: a constructive approach

Considering L and M being different commutative quantales, this paper mainly proposes a many-level version of L-fuzzy generalized neighborhood systems and then a pair of many level upper and lower rough approximation operators based on it are defined and discussed. It is proved that these approximation operators include L-fuzzy generalized neighborhood-based approximation operators and M-level L-fuzzy relation-based approximation operators as their special circumstances. Moreover, we define the measure of many level rough approximations that in a certain sense characterizes the quality of the obtained approximation. Furthermore, when the M-level L-fuzzy generalized neighborhood system is non-increasing, serial, reflexive, transitive, and unary, then the corresponding M-level L-rough approximation operators are discussed and characterized, respectively. Moreover, the relationship between M-level L-rough approximation operators and -quasi-fuzzy topologies is presented.

On Topological Properties of Generalized Rough Approximation Operators

On Topological Properties of Generalized Rough Approximation Operators

Interval-Valued Intuitionistic Fuzzy Soft Rough Approximation Operators and Their Applications in Decision Making Problem.

It has been found that fuzzy sets, rough sets and soft sets are closely related concepts. Many complicated problems in economics, engineering, social sciences, medical science and many other fields involve uncertain data. These problems, which one comes in real life, cannot be solved using classical mathematical methods. There are several well-known theories to describe uncertainty, for instance, fuzzy set theory, rough set theory, and other mathematical tools. But all of these theories have their inherit difficulties as pointed out by D. Molodtsov. In 1999, D. Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. The concept of rough sets, proposed by Z. Pawlak as a framework for the construction of approximations of concepts. It is a formal tool for modeling and processing insufficient and incomplete information. Zhou and Wu first proposed the concept of intuitionistic fuzzy rough sets (IFrough sets). The aim of this paper is to introduce the concept of interval-valued intuitionistic fuzzy soft rough sets (IVIFS rough sets). We also investigate some properties of IVIFS rough approximation operators. Some basic operations and properties are studied. Lastly applications have been shown in decision making problems.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

Axiomatic characterizations of L-fuzzy rough sets by L-fuzzy unions and L-fuzzy intersections

In this paper, considering L being a complete residuated lattice, we first introduce the concept of mediate L-fuzzy relations and give its characterizations by L-fuzzy upper and lower rough approximation operators. Then we provide an axiomatic approach to characterize L-fuzzy rough sets by L-fuzzy unions and L-fuzzy intersections. Concretely, we present axiomatic characterizations of L-fuzzy rough sets with respect to serial, reflexive, symmetric, transitive, mediate and Euclidean L-fuzzy relations. Finally, we give characterizations of L-fuzzy upper (lower) rough approximation operators by L-closure (interior) operators from a categorical aspect.

Axiomatic characterizations of [formula omitted]-valued rough sets using a single axiom

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory. There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach. Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L-valued rough sets with an L-set as the basic universe of defining L-valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets, we will present the axiomatic characterizations of L-valued upper and lower rough approximation operators on an L-set with respect to reflexive, symmetric, transitive L-valued relations on an L-set as well as their compositions. Choosing an L-set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

Single axiomatic characterization of a hesitant fuzzy generalization of rough approximation operators

Single axiomatic characterization of a hesitant fuzzy generalization of rough approximation operators

L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations

For L a complete co-residuated lattice and R an L-fuzzy relation, an L-fuzzy upper approximation operator based on co-implication adjoint with L is constructed and discussed. It is proved that, when L is regular, the new approximation operator is the dual operator of the Qiao–Hu L-fuzzy lower approximation operator defined in 2018. Then, the new approximation operator is characterized by using an axiom set (in particular, by single axiom). Furthermore, the L-fuzzy upper approximation operators generated by serial, symmetric, reflexive, mediate, transitive, and Euclidean L-fuzzy relations and their compositions are characterize through axiom set (single axiom), respectively.

On (IO,O)-fuzzy rough sets based on overlap functions

In the last past years, as a class of continuous binary aggregation functions, there are many scholars keeping a watchful focus on overlap functions for their widely applicability in various actual problems. On the other side, after that rough sets were presented as a formal tool to handle indetermination and inaccuracy in the analysis of data, there arise many works concerning the extension study of rough approximation operators in rough set model. This paper mainly discusses the so-called (IO,O)-fuzzy rough set model obtained through extending the classical conjunction operator in rough approximation operator to overlap functions. Firstly, we investigate a number of essential properties of (IO,O)-fuzzy rough sets, especially for the connections between the obtained new fuzzy rough approximation operators and fuzzy relations. Secondly, we give the topological properties of (IO,O)-fuzzy rough sets. Finally, we show a brief comparison of the (IO,O)-fuzzy rough sets with other common fuzzy rough set models.

Hesitant fuzzy β covering rough sets and applications in multi-attribute decision making

In present paper, we put forward four types of hesitant fuzzy β covering rough sets (HFβCRSs) by uniting covering based rough sets (CBRSs) and hesitant fuzzy sets (HFSs). We firstly originate hesitant fuzzy β covering of the universe, which can induce two types of neighborhood to produce four types of HFβCRSs. We then make further efforts to probe into the properties of each type of HFβCRSs. Particularly, the relationships of each type of rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped. Moreover, the relationships between our proposed models and some other existing related models are established. Finally, we give an application model, an algorithm, and an illustrative example to elaborate the applications of HFβCRSs in multi-attribute decision making (MADM) problems. By making comparative analysis, the HFβCRSs models proposed by us are more general than the existing models of Ma and Yang and are more applicable than the existing models of Ma and Yang when handling hesitant fuzzy information.

A note on the relationships between generalized rough sets and topologies

Quite recently, Wu and Liu (2020) [11] raised an open problem when they discussed the relationships between generalized rough sets and topologies. Said precisely, each binary relation generates a topology through the lower rough approximation operator, then for two binary relations on the same set, is there a sufficient and necessary condition such that the union of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. In this note, we give a positive answer to this problem and provide a union-join condition. Furthermore, note that the main reason that the expected identity hold not consists in the fact that the union of two topologies is not a topology, in general. Then replacing the union of two topologies with the supremum of them, we give a sufficient and necessary condition (supremum-join condition) such that the supremum of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. We verify that the supremum-join condition is much simpler than the union-join condition. At last, we give a characterization on discrete topology generated by a binary relation using the sober separation.

Picture Fuzzy Rough Set and Rough Picture Fuzzy Set on Two Different Universes and Their Applications

The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on δ , ζ , ϑ -cut of picture fuzzy relation ℛ on two different universes (i.e., by combining picture fuzzy sets (PFSs) with rough sets (RSs)). Then, we discuss several interesting properties and related results on the PFRSs. Furthermore, we define some notions related to PFRSs such as (Type-I/Type-II) graded PFRSs, the degree α and β with respect to ℛ δ , ζ , ϑ on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree α and β with respect to ℛ δ , ζ , ϑ and investigate the basic properties of above notions. Finally, an approach based on the rough picture fuzzy approximation operators on two different universes in decision-making problem is introduced, and we give an example to show the validity of this approach.