Fuzzy Approximation Operators Research Articles

Fundamental properties of fuzzy rough sets based on triangular norms and fuzzy implications: the properties characterized by fuzzy neighborhood and fuzzy topology

Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics.

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.

Novel variable precision fuzzy rough sets and three-way decision model with three strategies

Variable precision (fuzzy) rough sets are interesting generalizations of Pawlak rough sets and can handle uncertain and imprecise information well due to their error tolerance capability. The comparable property (CP for short), i.e., the lower approximation is included in the upper approximation, is fundamental in Pawlak rough sets since many theories and applications rely on this property. However, the CP is lost in many existing variable precision (fuzzy) rough sets. Therefore, a novel variable precision fuzzy rough set model with CP is proposed and an associated three-way decision model is developed. Firstly, a pair of variable precision fuzzy upper and lower approximation operators are defined and studied through fuzzy implication and co-implication operators. We show that this pair of approximation operators satisfy various properties (including CP) that the existing variable precision fuzzy approximation operators do not possess. Secondly, new methods for constructing loss functions and conditional probabilities are given by using CP, and then a new three-way decision model with three strategies (optimistic, pessimistic, and compromise strategies) is established. Finally, an explanatory example is provided to examine validity, stability and sensitivity of the proposed three-way decision model.

Rough Pythagorean fuzzy approximations with neighborhood systems and information granulation

A rough set approximates a subset of a universal set on the basis of some binary relation and is significant for the reduction of attributes of an information system. On the other hand, a Pythagorean fuzzy set provides information about the extent of truthness and falsity of a statement. Both these theories deal with different forms of uncertainty and can be united to get their combined benefits. This paper contributes a new blend of rough sets and Pythagorean fuzzy sets namely, rough Pythagorean fuzzy sets. This model can encapsulate two distinct types of uncertainties that appear in imprecise available data through the approximation of Pythagorean fuzzy sets in crisp approximation space. We define rough Pythagorean fuzzy sets on the basis of equivalence relation and generalize it for arbitrary binary relations. The manuscript also provides a general framework to study rough Pythagorean fuzzy approximations of different k-step neighborhood systems. The identities and properties of upper and lower rough Pythagorean fuzzy approximation operators are discussed for the neighborhood systems induced from different types of binary relations. Further, we develop algorithms that compute reduct family, core and rough Pythagorean fuzzy approximations of single-valued and set-valued information systems using indiscernibility relation and similarity relation, respectively. These algorithms are subjected to simple yet interesting applications.

Enriched Fuzzy  -Algebra and Fuzzy RoughApproximation Operators

In this paper, we introduce the concept of enriched fuzzy -algebra and establish its relation to fuzzy rough approximation operators. Firstly, the concept of enriched fuzzy  -algebra is proposed as an extension of existing fuzzy -algebras, and several concrete examples are provided. Particularly, it has been proven that each - algebra can naturally generate an enriched fuzzy  -algebra. Secondly, the methods for constructing enriched fuzzy  -algebra using reflexive fuzzy rough approximation operators and serial rough fuzzy approximation operators are given, respectively. Finally, it is shown that enriched fuzzy  -algebras generated by -algebras must be induced by some rough fuzzy approximation operator.

A Spectral Feature Selection Approach With Kernelized Fuzzy Rough Sets

Feature evaluation is an important issue in constructing a feature selection algorithm in kernelized fuzzy rough sets, which has been proven to be an effective approach to deal with nonlinear classification tasks and uncertainty in learning problems. However, the feature evaluation function developed with kernelized fuzzy rough sets cannot better reflect the affinity relationship of samples and is time-consuming. To overcome these drawbacks, in this article, the problem of feature selection with kernelized fuzzy rough sets is studied based on the spectral graph theory. First, the within-class and between-class sample similarity matrices by using kernelized fuzzy approximation operators are constructed. Two operators, which can capture the affinity relationship of samples, are then introduced based on the sample similarity matrices. The proposed operator can be regarded as the sum of the weighted kernelized fuzzy approximation operators. Second, based on the ratio criterion, a feature evaluation function and its corresponding feature selection algorithm FRKF are presented, which can effectively evaluate the importance of features. Third, to illustrate the performance of the proposed algorithm, extensive experiments have been carried out to compare FRKF and other well-known feature selection methods, including the feature ranking methods and feature subset selection methods on various classification tasks. The experimental results on real-world datasets demonstrate that FRKF achieves the high performances in terms of the robustness, efficiency, and effectiveness.

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.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

Interval-valued pythagorean fuzzy rough approximation operators and its application

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model.

<i>L</i>-fuzzy upper approximation operators associated with <i>L</i>-generalized fuzzy remote neighborhood systems of <i>L</i>-fuzzy points

It is well known that the approximation operators are always primitive concepts in kinds of general rough set theories. In this paper, considering L to be a completely distributive lattice, we introduce a notion of L-fuzzy upper approximation operator based on L-generalized fuzzy remote neighborhood systems of L-fuzzy points. It is shown that the new approximation operator is a fuzzification of the upper approximation operator in the rough set theory based on general remote neighborhood systems of classical points. Then the basic properties, axiomatic characterizations and the reduction theory on the L-fuzzy upper approximation operator are presented. Furthermore, the L-fuzzy upper approximation operators corresponding to the serial, reflexive, unary and transitive Lgeneralized fuzzy remote neighborhood systems, are discussed and characterized respectively.

Pythagorean fuzzy soft rough sets and their applications in decision-making

ABSTRACT This paper aims to originate two new notions that are soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set, and investigate some important properties of soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set in detail. Furthermore, classical representations of Pythagorean fuzzy soft rough approximation operators are presented. Then the proposed operators are applied on decision-making problem in which the experts provide their preferences in Pythagorean fuzzy soft rough environment. Finally through an illustrative example, it is shown that how the proposed operators work in decision-making problems.

Rough approximation operators based on quantale-valued fuzzy generalized neighborhood systems

Let $L$ be an integral and commutative quantale. In this paper, by fuzzifying the notion of generalized neighborhood systems, the notion of $L$-fuzzy generalized neighborhoodsystem is introduced and then a pair of lower and upperapproximation operators based on it are defined and discussed. It is proved that these approximation operators include generalized neighborhood system-based approximation operators, $L$-fuzzyrelation-based approximation operators and $L$-fuzzycovering-based approximation operators as their specialcircumstances. Therefore, the research on $L$-fuzzy generalizedneighborhood system-based approximation operators has more generalsignificance. In addition, when the $L$-fuzzy generalizedneighborhood system is serial, reflexive, unary and transitive,then the corresponding approximation operators are discussed and characterized, respectively.

On novel hesitant fuzzy rough sets

This paper presents a novel framework for the study of hesitant fuzzy rough sets by integrating rough sets with hesitant fuzzy sets. Lower and upper approximations of hesitant fuzzy sets with respect to a hesitant fuzzy approximation space are first defined. Properties of hesitant fuzzy approximation operators are examined. Relationships between hesitant fuzzy approximation spaces and hesitant fuzzy topological spaces are then established. It is proved that the set of all lower approximation sets based on a hesitant fuzzy reflexive and transitive approximation space forms a hesitant fuzzy topology. And conversely, for a hesitant fuzzy rough topological space, there exists a hesitant fuzzy reflexive and transitive approximation space such that the topology in the hesitant fuzzy rough topological space is exactly the set of all lower approximation sets in the hesitant fuzzy reflexive and transitive approximation space. That is to say, there exists a one-to-one correspondence between the set of all hesitant fuzzy reflexive and transitive approximation spaces and the set of all hesitant fuzzy rough topological spaces.

New decision-making hybrid model: intuitionistic fuzzy N-soft rough sets

In this paper, we introduce three novel hybrid models, namely, intuitionistic fuzzy N-soft sets (IFNSSs), N-soft rough intuitionistic fuzzy sets and intuitionistic fuzzy N-soft rough sets (IFNSRSs). We discuss some of their respective properties. We also present the lower and upper intuitionistic fuzzy N-soft rough approximation operators and investigate their properties. Furthermore, we present respective new approaches to decision-making problems based on the IFNSS and IFNSRS models in an algorithmic format. Finally, we present illustrative examples in order to show that these decision-making methods can be fruitfully employed to solve real life problems.

A characterization of novel rough fuzzy sets of information systems and their application in decision making

AbstracVVt A novel concept of α-rough fuzzy sets is proposed as a generalization of rough fuzzy sets in information systems. A pair of rough fuzzy approximation operators is presented based on one type of correlation between attributes, namely, α-correlative relation. In addition, the properties of α-rough fuzzy sets are investigated. Finally, we develop an approach for decision-making problems based on lower and upper α-approximations and provide an actual example.

Multi‐adjoint intuitionistic fuzzy rough sets

The combination of fuzzy information systems (ISs) and multi-adjoint theory has become a hot issue in the study and applications of artificial intelligence. An intuitionistic fuzzy set has more flexible and practical ability to represent information and is better in dealing with ambiguity and uncertainty when compared with the fuzzy set. Multi- adjoint intuitionistic fuzzy rough sets are constructed by using adjoint triples under intuitionistic fuzzy IS. For this purpose, the authors propose intuitionistic fuzzy indiscernibility relation and multi-adjoint approximation operators. The basic results in the multi-adjoint fuzzy rough set model are generalised to multi-adjoint intuitionistic fuzzy rough set model. The analogous results are also verified. After that, a novel approach of attribute reduction is proposed. First, a kind of approximate reduction to keep the dependence of the positive region to a degree α is formulated. Second, they propose a heuristic algorithm to compute the attribute reduction. At last, they employ an example to describe the processing of the algorithm.

A fast analytical approximation type-reduction method for a class of spiked concave type-2 fuzzy sets

In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design.

Feature selection based on generalized variable-precision $$(\vartheta ,\sigma )$$ ( ϑ , σ ) -fuzzy granular rough set model over two universes

Fuzzy rough set theory provides us an important theoretical tool for feature selection in machine learning and pattern recognition. In this paper, based on an arbitrary fuzzy binary relation and fuzzy granules, we construct a novel fuzzy granular rough set model for feature selection of real-valued data. Firstly, we propose variable-precision $$(\vartheta ,\sigma )$$ -fuzzy granular rough set model based on fuzzy granules derived from an arbitrary fuzzy binary relation. Then the properties of the newly proposed variable-precision fuzzy approximation operators and the feature selection based on this model are studied in detail. The discernibility matrix is presented and the related reduction algorithm is constructed to find the minimal fuzzy feature subsets. Thirdly, generalized fuzzy rough sets over two universes are presented and their properties are discussed. In addition, the generalized fuzzy rough sets over two universes are used to illness diagnosis. Two examples are given to show the validity of the two new models.

The axiomatic characterizations on L-generalized fuzzy neighborhood system-based approximation operators

The rough sets based on L-fuzzy relations and L-fuzzy coverings are the two most well-known L-fuzzy rough sets. Quite recently, we prove that some of these rough sets can be unified into one framework—rough sets based on L-generalized fuzzy neighborhood systems. So, the study on the rough sets based on L-generalized fuzzy neighborhood system has more general significance. Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations, L-fuzzy coverings that reproduce the approximation operators. In this paper, we shall give an axiomatic study on L-generalized fuzzy neighborhood system-based approximation operators. In particular, we will seek the axiomatic sets to characterize the approximation operators generated by serial, reflexive, unary and transitive L-generalized fuzzy neighborhood systems, respectively.

A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets

We introduce notions of soft rough m-polar fuzzy sets and m-polar fuzzy soft rough sets as novel hybrid models for soft computing, and investigate some of their fundamental properties. We discuss the relationship between m-polar fuzzy soft rough approximation operators and crisp soft rough approximation operators. We also present applications of m-polar fuzzy soft rough sets to decision-making.