多重概率粗糙集模型

基于多重集合,对Z.Pawlak粗集意义下的概率粗糙集模型的论域进行了扩展,提出了基于多重集的概率粗糙集模型,即多重概率粗糙集模型,给出了该模型的完整定义、相关定理和重要性质,其中包括多重论域定义、多重概率粗糙近似集的定义及其各种性质的证明、多重概率粗糙集的近似精度定义、可定义集与属性约简的定义、多重集意义下的粗糙近似算子之间的关系及其与Z.Pawlak意义下的粗糙近似算子之间的关系等。多重概率粗糙集可充分反映知识颗粒间的重叠性,对象的重要度差别及其多态性,这样有利于用粗糙集理论从保存在关系数据库中的具有一对多、多对多依赖性的且具有不完全性或存在统计性的数据中挖掘知识。     

基于概念格的粗糙集近似

粗糙集理论的一个重要研究方面是用已定义的概念来近似未定义的概念,而如何构建可定义概念以及如何确定近似运算是这一工作的基础.利用粗糙集这一工具,从概念格的角度来确定可定义概念,并在此基础上研究了概念的粗糙近似.根据粗糙集上下近似的包含关系,得到概念的一种新的上下近似的运算的定义.粗糙集近似理论利用两种不同的近似运算,产生两种不同的近似来描述概念格背景下的对象集合.     

基于近似集与粒子群的粗糙熵图像分割方法

基于经典粗糙集理论的图像分割方法缺少对目标图像不确定性边界域的精确划分,其根据先验粒度构建的图像粗糙集信息系统,并没有客观准确地反映出不同粒度之间的粗糙性信息。基于粗糙集近似集理论模型,首先采用自适应粒化方法得到图像的最优粒度,接着基于该粒度划分构建图像的目标和背景的上下近似集,再根据近似集思想对目标集合的边界域进行精确刻画,同时结合粒子群算法提高求解粗糙集近似集最大粗糙熵的效率,最终得到图像分割的最优分割阈值,并通过仿真实验表明该方法具有可行性和有效性。     

变精度粗糙集

本文定义了变精度粗糙集,从精度的取值情况分类讨论了其基本结构.对比一般粗糙集性质,研究了变精度粗糙集三个方面的性质:集合与其变精度近似集的关系、变精度近似算子的幂作用、变精度边界算子对粗糙集性质的修正.得出了若干具有理论和应用价值的结果,并从算子论和集合论的角度丰富了粗糙集理论.     

基于对象的广义粗糙近似算子的拓扑性质

粗糙集理论是一种处理不确定性问题的数学工具。近似算子是粗糙集理论中的核心概念，基于等价关系的Pawlak近似算子可以推广为基于一般二元关系的广义粗糙近似算子。近似算子的拓扑结构是粗糙集理论的重点研究方向。文中主要研究基于对象的广义粗糙近似算子诱导拓扑的性质，证明了广义近似空间中所有可定义集形成拓扑的充分条件也是其必要条件，研究了该拓扑的正则、正规性等拓扑性质；给出了串行二元关系与其传递闭包可以生成相同拓扑的等价条件；讨论了该拓扑与任意二元关系下基于对象的广义粗糙近似算子所诱导拓扑之间的相互关系。     

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.     

变精度覆盖粗糙集

介绍了Ziarko变精度粗糙集模型和覆盖粗糙集模型;定义了多数包含关系;借助引入的误差参数β（0≤β<0.5）,给出了基于对象邻域的变精度覆盖粗糙集模型中β上近似、β下近似、β边界和β负域的定义以及β近似质量和β粗糙性测度定义;详细讨论了β上、下近似算子的性质、集合的相对可辨别性、该模型与Ziarko变精度粗糙集模型和覆盖粗糙集模型的关系;最后探讨了变精度覆盖粗糙集模型中的约简问题并在所给模型的基础上举例说明了它们在信息处理中的应用。     

粗糙近似算子的公理化刻画:综述

从近似空间导出的一对下近似算子与上近似算子是粗糙集理论研究与应用发展的核心基础,近似算子的公理化刻画是粗糙集的理论研究的主要方向.文中回顾基于二元关系的各种经典粗糙近似算子、粗糙模糊近似算子和模糊粗糙近似算子的构造性定义,总结与分析这些近似算子的公理化刻画研究的进展.最后,展望粗糙近似算子的公理化刻画的进一步研究和与其它数学结构之间关系的研究.     

覆盖粗糙集上下近似的对偶性分析

覆盖粗糙集是经典粗糙集的推广。然而,覆盖粗糙集的上下近似定义的方法有很多,上下近似是否对偶一直是争论的焦点。本文分析覆盖粗糙集上下近似的对偶性质,讨论对偶下的正域、负域及边界的可定义性。通过对偶性质的分析,对不同问题使用不同上下近似的方法。进一步研究约简与对偶运算的关系,分析覆盖粗糙集中满足对偶的两对重要的上下近似。     

基于相似度的粗糙集近似算子快速求解

由于经典粗糙集只能处理精确分类问题,基于相似度的粗糙集模型被提出并用于解决不完备信息系统的相关问题.粗糙集通过近似算子对某一给定的概念进行近似表示,科学的求解这些算子对粗糙集理论的发展具有重要意义.本文提出一种新的近似算子快速求解方法,分析证明了所提快速方法比经典方法具有更高的求解效率.文章定义了元素覆盖度、集合覆盖度等概念,使用覆盖度等价关系可以将覆盖粗糙集转化为经典粗糙集,从而简化覆盖粗糙集的相关问题的解决.     

An axiomatic characterization of a fuzzy generalization of rough sets

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find the assumptions which permit a given fuzzy set-theoretic operator to represent a upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.     

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.     

变精度覆盖粗糙集模型的比较

介绍覆盖粗糙集和Ziarko变精度粗糙集模型,将Ziarko变精度粗糙近似算子应用于覆盖近似空间,借助引入的误差参数β (0 ≤β<0.5),给出2种变精度覆盖粗糙集模型的β上近似、β下近似、β边界和β负域的定义。讨论2种模型中β上、下近似算子的基本性质、2种模型之间的关系以及变精度覆盖粗糙集模型与其他粗糙集模型的关系。     

Generalized rough sets over fuzzy lattices

This paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches. From the viewpoint of the constructive approach, the basic properties of generalized rough sets over fuzzy lattices are obtained. The matrix representation of the lower and upper approximations is given. According to this matrix view, a simple algorithm is obtained for computing the lower and upper approximations. As for the axiomatic approach, a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices.     

模糊信息系统中基于OWA算子的模糊粗糙集模型

在模糊信息系统中,属性值并不是一个确定的值,而是一个隶属度函数。因此,通过利用有序加权平均(OWA)算子聚合对象间在每个属性上的差异,刻画出对象之间的相似性,定义对象的相似度并讨论其相关性质。借助对象相似度,通过逻辑关系和相应的函数运算,分别给出了对象隶属于上、下近似集合的隶属度。最后,通过实例分析说明在模糊信息系统中,该相似度能较准确地刻画出对象的相似性,同时,对象对于上、下近似的隶属度能更直观、合理地反应对象隶属于某一集合的上、下近似的情况,且能更合理地描述这一粗糙集合。     

关于覆盖粗糙集模型性质的一个注记

对覆盖粗糙集模型的部分性质进行了推广,即通过引进一对新的算子,把并与交的上下近似集之包含关系推广到了相等关系,从而得到了更好的结果。     

On generalized intuitionistic fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak’s rough approximations have been made in the literature over the years. This paper proposes a general framework for the study of relation-based intuitionistic fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation are defined. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established. Finally, an operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators.     

变精度粗糙集模型及其应用研究

介绍了广义粗糙集模型和Ziarko变精度粗糙集模型,找出了它们的不足;借助引入的误差参数β(0≤β<0.5),给出了基于后继邻域的一般二元关系下变精度粗糙集模型的β上近似、β下近似、3边界和β负域的定义以及β近似质量和β粗糙性测度定义;详细讨论了β上、下近似算子的性质、该模型与其他粗糙集模型的关系以及一般二元关系下两种变精度粗糙集模型的关系;最后,举例说明了该模型在信息处理中的应用。     

On Three Types of Covering-Based Rough Sets

Rough set theory is a useful tool for data mining. It is based on equivalence relations and has been extended to covering-based generalized rough set. This paper studies three kinds of covering generalized rough sets for dealing with the vagueness and granularity in information systems. First, we examine the properties of approximation operations generated by a covering in comparison with those of the Pawlak's rough sets. Then, we propose concepts and conditions for two coverings to generate an identical lower approximation operation and an identical upper approximation operation. After the discussion on the interdependency of covering lower and upper approximation operations, we address the axiomization issue of covering lower and upper approximation operations. In addition, we study the relationships between the covering lower approximation and the interior operator and also the relationships between the covering upper approximation and the closure operator. Finally, this paper explores the relationships among these three types of covering rough sets.