Approximations and reducts with covering generalized rough sets

The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. It is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attributes reduct. The existing covering generalized rough sets, however, mainly pay attention to constructing approximation operators. The ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable. Furthermore, less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets. In this paper we concentrate our discussion on the above two issues. We first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones. Then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets. With these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications.     

On minimization of axiom sets characterizing covering-based approximation operators

Rough set theory was proposed by Pawlak to deal with the vagueness and granularity in information systems. The classical relation-based Pawlak rough set theory has been extended to covering-based generalized rough set theory. The rough set axiom system is the foundation of the covering-based generalized rough set theory, because the axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators. In this paper, the equivalent characterizations for the independent axiom sets of four types of covering-based generalized rough sets are investigated, and more refined axiom sets are presented.     

Soft sets and soft rough sets

In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.     

新的模糊粗糙集的模糊性度量方法

研究了模糊粗糙集的模糊性度量方法。首先从模糊集支集的角度,给出了一般模糊关系下模糊集的粗糙隶属函数;在此基础上,设计了一种合理的模糊粗糙集的模糊性度量方法,并对其相关性质进行了详细的讨论。     

粗糙Vague集(值)相似性度量的一种新方法

对粗糙集和Vague集的概念、知识表示方法进行了讨论,把粗糙集和Vague集进行了融合,描述了粗糙Vague集概念,给出粗糙Vague值的相关概念及其相似性度量的一种新方法,研究了粗糙Vague集的相关性质及其相似性度量的方法。     

基于模糊加权相似度量的粗糙集数据补齐方法

目前基于粗糙集的数据补齐方法,大多都是通过计算决策信息系统中具有缺失值的对象与无缺失值的对象之间的相似性,选取相似性最大的对象的属性值来补齐缺失的数据。这类算法的问题在于：计算对象之间的相似性时所有条件属性对于决策属性的重要性是相同的,忽略了条件属性间的差异性。鉴于此,引入了模糊加权相似的概念,根据每个条件属性的重要性以及决策属性对条件属性的依赖度,计算对象间的相似性,提出基于模糊加权相似性度量的粗糙集数据补齐方法,并通过实例计算以及与现有算法的比较分析,说明了方法的有效性。     

改进的直觉模糊粗糙集相似性度量方法

针对现有的直觉模糊粗糙集相似性度量的问题,提出了一种改进的基于海明距离的直觉模糊粗糙集相似性度量方法。该方法考虑了犹豫度并引入加权参数,解决了相似性度量不精确的问题。首先给出了直觉模糊粗糙值间的相似性度量定义,并揭示其若干重要性质。在此基础上,提出了直觉模糊粗糙集间的相似性度量方法,并证明其具有同样性质。最后通过数值算例分析说明了该方法更合理、更有效。     

覆盖概率粗糙集的模糊性

在经典覆盖近似空间中定义了论域上任意元素x的最小子覆盖,基于任意元素的最小子覆盖给出了覆盖粗糙集上、下近似新的描述,进而给出了已有覆盖概率粗糙集模型在最小子覆盖意义下的描述。同时,以覆盖概率粗糙集的粗糙隶属函数为基础,应用经典模糊集熵的概念讨论了覆盖概率粗糙集模糊性的度量。     

变精度模糊粗糙集的一种定义

模糊粗糙集模型同经典粗糙集模型类似,容易受到噪音数据的影响.针对该问题,受变精度粗糙集模型的启发,提出了变精度模糊粗糙集的概念.针对现有变精度模糊粗糙集模型尚不能满足一些基本性质的缺陷,重新定义了模糊近似空间中某一模糊集的β-下近似和β-上近似,该定义方式能够满足上述的基本性质.     

覆盖粗糙模糊集的不确定性研究

不确定性度量是粗糙集理论中的基础问题之一。粗糙模糊集的不确定性一方面来自上、下近似集间差异产生的粗糙性,另一方面来自概念外延不清晰产生的模糊性。目前对于粗糙模糊集的不确定性研究仍不够透彻。针对覆盖近似空间下的粗糙模糊集不确定性,提出更加严格的度量修正准则,并借助上、下近似集隶属度与原模糊集隶属度之间的差异,给出修正粗糙度的概念。算例分析表明该方法能够更加准确地刻画实际问题。     

The algebraic structures of generalized rough set theory

Rough set theory is an important technique for knowledge discovery in databases, and its algebraic structure is part of the foundation of rough set theory. In this paper, we present the structures of the lower and upper approximations based on arbitrary binary relations. Some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended. Based on the concepts of definable sets in rough set theory, two important Boolean subalgebras in the generalized rough sets are investigated. An algorithm to compute atoms for these two Boolean algebras is presented.     

Textural approach to generalized rough sets based on relations

This paper presents a discussion on rough set theory from the textural point of view. A texturing is a family of subsets of a given universal set U satisfying certain conditions which are generally basic properties of the power set. The suitable morphisms between texture spaces are given by direlations defined as pairs (r,R) where r is a relation and R is a corelation. It is observed that the presections are natural generalizations for rough sets; more precisely, if (r,R) is a complemented direlation, then the inverse of the relation r (the corelation R) is actually a lower approximation operator (an upper approximation operator).     

Invertible approximation operators of generalized rough sets and fuzzy rough sets

This paper studies the classes of rough sets and fuzzy rough sets. We discuss the invertible lower and upper approximations and present the necessary and sufficient conditions for the lower approximation to coincide with the upper approximation in both rough sets and fuzzy rough sets. We also study the mathematical properties of a fuzzy rough set induced by a cyclic fuzzy relation.     

A comparison of two kinds of definitions of rough approximations based on a similarity relation

In the present paper, we investigate the three types of Yao’s lower and upper approximations of any set with respect to any similarity relation. These types based on a right neighborhood. Also, we define and investigate other three types of approximations for any similarity relations. These new types based on the intersection of the right neighborhoods. Moreover, we give a comparison between these types. Lastly, the relationship between these definitions is introduced.     

On the structure of generalized rough sets

In this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that

1.

Any reflexive binary relation determines a topology.

2.

If θ is a reflexive and symmetric relation on a set X, then O={A⊆X|θ_-(A)=A} is a topology such that A is open if and only if it is closed.

3.

Conversely, for every topological space (X,O) satisfying the condition that A is open if and only if it is closed, there exists a reflexive and symmetric relation R such that O={A⊆X|R_-(A)=A}.

4.

Let θ be an equivalence relation on X. For any pseudo ω-closed subset A of X, θ₋(A) is an ω-closed set if and only if ω(x, x, … , x) ∈ θ₋(A) for any x ∈ X.

Moreover we consider properties of generalized rough sets.     

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based (I,T)-intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of (I,T)-intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of (I,T)-intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.     

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.     

On axiomatic characterizations of three pairs of covering based approximation operators

Rough set theory is a useful tool for dealing with inexact, uncertain or vague knowledge in information systems. The classical rough set theory is based on equivalence relations and has been extended to covering based generalized rough set theory. This paper investigates three types of covering generalized rough sets within an axiomatic approach. Concepts and basic properties of each type of covering based approximation operators are first reviewed. Axiomatic systems of the covering based approximation operators are then established. The independence of axiom set for characterizing each type of covering based approximation operators is also examined. As a result, two open problems about axiomatic characterizations of covering based approximation operators proposed by Zhu and Wang in (IEEE Transactions on Knowledge and Data Engineering 19(8) (2007) 1131-1144, Proceedings of the Third IEEE International Conference on Intelligent Systems, 2006, pp. 444-449) are solved.     

粗糙Vague集（值）间k步相似度量新方法

对粗糙集、Vague集及粗糙Vague集的概念、知识表示方法进行了讨论,描述了粗糙Vague集的相关概念,在研究了粗糙Vague集相似性度量方法的基础上,提出了一种对踌躇度п_v（x）k步细化的k步相似度量新方法,并且研究了该方法的相关性质。进一步对k步相似度量在k趋向无穷时求极限,得出粗糙Vague集退化为模糊集的特例情况。     

A new extension of fuzzy sets using rough sets: R-fuzzy sets

This paper presents a new extension of fuzzy sets: R-fuzzy sets. The membership of an element of a R-fuzzy set is represented as a rough set. This new extension facilitates the representation of an uncertain fuzzy membership with a rough approximation. Based on our definition of R-fuzzy sets and their operations, the relationships between R-fuzzy sets and other fuzzy sets are discussed and some examples are provided.