Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

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.     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

基于覆盖的直觉模糊粗糙集

通过直觉模糊覆盖概念将覆盖粗糙集模型进行推广,提出一种基于直觉模糊覆盖的直觉模糊粗糙集模型.首先,介绍了直觉模糊集、直觉模糊覆盖和直觉模糊逻辑算子等概念;然后,利用直觉模糊三角模和直觉模糊蕴涵,构建两对基于直觉模糊覆盖的下直觉模糊粗糙近似算子和上直觉模糊粗糙近似算子;最后,给出了这些算子的基本性质并研究了它们之间的对偶性.     

On intuitionistic fuzzy rough sets and their topological structures

In this paper, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of intuitionistic fuzzy approximation operators are examined. Relationships between intuitionistic fuzzy rough set approximations and intuitionistic fuzzy topologies are then discussed. It is proved that the set of all lower approximation sets based on an intuitionistic fuzzy reflexive and transitive approximation space forms an intuitionistic fuzzy topology; and conversely, for an intuitionistic fuzzy rough topological space, there exists an intuitionistic fuzzy reflexive and transitive approximation space such that the topology in the intuitionistic fuzzy rough topological space is just the set of all lower approximation sets in the intuitionistic fuzzy reflexive and transitive approximation space. That is to say, there exists an one-to-one correspondence between the set of all intuitionistic fuzzy reflexive and transitive approximation spaces and the set of all intuitionistic fuzzy rough topological spaces. Finally, intuitionistic fuzzy pseudo-closure operators in the framework of intuitionistic fuzzy rough approximations are investigated.     

粗糙模糊集的构造与公理化方法

用构造性方法和公理化研究了粗糙模糊集．由一个一般的二元经典关系出发构造性地定义了一对对偶的粗糙模糊近似算子,讨论了粗糙模糊近似算子的性质,并且由各种类型的二元关系通过构造得到了各种类型的粗糙模糊集代数．在公理化方法中,用公理形式定义了粗糙模糊近似算子,各种类型的粗糙模糊集代数可以被各种不同的公理集所刻画．阐明了近似算子的公理集可以保证找到相应的二元经典关系,使得由关系通过构造性方法定义的粗糙模糊近似算子恰好就是用公理化定义的近似算子。     

粗糙support-intuitionistic模糊集及其聚类分析应用

粗糙集和模糊集理论已经被用于各种类型的不确定性建模中。Dubois和Prade研究了将模糊集和粗糙集结合的问题。提出了粗糙support-intuitionistic模糊集。介绍了粗糙集、粗糙直觉模糊集和support-intuitionistic模糊集等的概念;定义了在Pawlak近似空间中的support-intuitionistic模糊集的上下近似,讨论了一些粗糙support-intuitionistic模糊集近似算子的性质,给出了其相似度表达式;将其应用到聚类分析问题中,并通过一个实例验证其合理性。     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

一般等价关系下直觉模糊粗糙集模型及其性质的研究

系统研究了一般等价关系下直觉模糊粗糙集模型,给出该模型多种定义形式,详细阐述了上下近似的性质,定义了直觉模糊粗糙集截集的概念,最后给出了该直觉模糊粗糙集模型的分解定理和表现定理,并证明定理的正确性。     

区间值Fuzzy集分解定理上的近似算子研究

将广义粗糙模糊下、上近似算子拓展到区间上,并利用区间值模糊集分解定理给出一组新的广义区间值粗糙模糊下、上近似算子,证明二者在由任意二元经典关系构成的广义近似空间中是等价的,最后讨论了在一般二元关系下,两组近似算子的性质。     

模糊近似空间上的粗糙模糊集的公理系统

粗糙集理论是近年来发展起来的一种有效的处理不精确、不确定、含糊信息的理论,在机器学习及数据挖掘等领域获得了成功的应用．粗糙集的公理系统是粗糙集理论与应用的基础．粗糙模糊集是粗糙集理论的自然的有意义的推广．作者研究了模糊近似空间上的粗糙模糊集的公理系统,用三条简洁的相互独立的公理完全刻划了模糊近似空间上的粗糙模糊集,同时还把作者给出的公理系统与粗糙集的公理系统做了对比,指出了两者的区别．     

改进的粗糙直觉模糊集

在Pawlak近似空间中,针对直觉模糊目标集合,假设在信息粒度不变的情况下,试图寻求目标集合更好的近似集。在现有的粗糙直觉模糊集的基础之上,利用直觉模糊粗糙隶属函数,采用分段函数的形式建立直觉模糊集新的下近似与上近似算子,并讨论新模型的一些基本性质。与现有的粗糙直觉模糊集相比,改进后的模型无论在近似精度方面,还是与目标集合的相似度方面,都有了较大的改善和提高。最后通过数值算例验证了所给结论的正确性。     

二型模糊粗糙集

基于二型模糊关系,研究二型模糊粗糙集.首先,在二型模糊近似空间中定义了二型模糊集的上近似和下近似;然后,研究二型模糊粗糙上下近似算子的基本性质,讨论二型模糊关系与二型模糊粗糙近似算子的特征联系;最后,给出二型模糊粗糙近似算子的公理化描述.     

基于直觉模糊三角模的直觉模糊粗糙集

提出一种基于直觉模糊三角模的直觉模糊粗糙集.首先,定义了直觉模糊集上的T模及其剩余蕴涵,研究了直觉模糊T模的剩余蕴涵的性质,并推导了通用计算表达式;然后,将模糊T粗糙集扩展成直觉模糊粗糙集,证明了模糊T粗糙集、粗糙模糊集和Pawlak粗糙集都是直觉模糊粗糙集的特殊情形;最后,证明了直觉模糊粗糙集的一些性质.     

覆盖粗糙直觉模糊集模型的研究

粗糙集和直觉模糊集的融合是一个研究热点。在粗糙集、直觉模糊集和覆盖理论基础上,给出了模糊覆盖粗糙隶属度和非隶属度的定义。考虑到元素自身与最小描述元素的隶属度和非隶属度之间的关系,构建了两种新的模型——覆盖粗糙直觉模糊集和覆盖粗糙区间值直觉模糊集,证明了这两种模型的一些重要性质,与此同时定义了一种新的直觉模糊集的相似性度量公式,并用实例说明其应用。     

基于最小/最大描述的多粒度覆盖粗糙直觉模糊集模型

覆盖粗糙集和直觉模糊集都是处理不确定性问题的基础理论,它们有着很强的互补性,且覆盖粗糙集和直觉模糊集的融合研究是一个新的热点。对多粒度覆盖粗糙集和直觉模糊集的融合进行深入研究。首先将最小描述、最大描述从单一粒度推广到多个粒度,提出了多粒度的最小描述和最大描述,讨论了多粒度的融合;其次,分别给出了基于最小描述和最大描述的模糊覆盖粗糙隶属度、非隶属度的概念,构建了两种新的模型即基于最小描述的多粒度覆盖粗糙直觉模糊集和基于最大描述的多粒度覆盖粗糙直觉模糊集,并讨论了它们的性质,同时举例说明;最后,分析和研究了两种模型的关系。该研究为多粒度覆盖粗糙集和直觉模糊集的融合提供了一种方法。     

广义区间值模糊粗糙近似算子的构造研究

构造了一组新的广义模糊粗糙近似算子,将其拓展到区间上.在由任意的二元区间值模糊关系构成的广义近似空间中,证明了该组近似算子与区间化的广义Dubois模糊粗糙近似算子是等价的,最后在一般二元区间值模糊关系下对该组近似算子的性质进行了讨论.     

On the generalization of fuzzy rough sets

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, T-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.     

Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.     

Algorithm and axiomatization of rough fuzzy sets based finite dimensional fuzzy vectors

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.