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。     

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。     

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.     

程度多粒度软粗糙集模型

多粒度粗糙集的研究是近几年来研究的热门课题之一。提出了一种介于乐观和悲观多粒度软粗糙集的新模型——程度多粒度软粗糙集。首先,通过计数函数建立了程度多粒度软粗糙集模型;其次,讨论了程度多粒度软粗糙近似算子的性质;再次,定义并研究了程度多粒度软粗糙集的不确定性度量及性质;最后,通过医院对病人诊断的案例验证了模型的实用性。     

变精度下近似算子与程度上近似算子的逻辑与运算模型

基于精度与程度的逻辑与需求,提出了变精度下近似算子与程度上近似算子的逻辑与运算模型。在该模型中,得到了变精度下近似算子与程度上近似算子的逻辑与运算的精确描述与基本性质,提出了宏观算法与微观算法,进行了算法分析与比较,得到了微观算法更具空间优势的结论。最后用医疗实例对模型与算法进行了说明。变精度下近似算子与程度上近似算子的逻辑与运算模型,部分拓展了变精度粗糙集模型、程度粗糙集模型和经典粗糙集模型,并在这些模型中得到了近似算子的相应性质。     

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。     

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.     

广义变精度粗糙集模型中近似算子研究

定义了多数包含关系；借助引入的误差参数β（0≤β〈0．5），提出了基于后继邻域的广义变精度粗糙集模型的β上近似aprβX、β下近似aprβX、β边界bnrβX和β负域negrβX的定义；详细讨论了β上、下近似算子aprβX与aprβX的性质；从对偶性角度出发推广了β上近似、β下近似算子aprβX与aprβX，得到了两对对偶的上、下近似算子aprβX与aprβX和aprβX与aprβX；最后全面讨论了推广后的两对上、下近似算子APRβX与aprβX和aprβX与aprβX的性质，详细分析了它们同广义变精度粗糙集模型中上、下近似算子aprβX与aprβX和一般关系下的变精度粗糙集模型中上、下近似算子RβX与RβX的关系。     

变精度相容粗糙集模型

将变精度粗糙集的思想引入相容粗糙集,提出了两种变精度相容粗糙集模型,在模型I中,目标概念的下近似和边界域的交集非空;在模型II中,目标概念的下近似和边界域的交集为空。研究了两种模型中上、下近似算子的基本性质、两种模型之间的关系,以及与其他粗糙集模型之间的关系。     

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.

     

变精度粗糙模糊集模型研究

介绍了Ziarko’s变精度粗糙集模型和粗糙模糊集模型,找出了它们的不足。基于支集相对错误分类率及误差参数β（0≤β<0.5）,提出了变精度粗糙模糊集模型,讨论了模型中β上、下近似算子的性质;分析了该模型与Ziarko’s变精度粗糙集模型和粗糙模糊集模型的关系;最后给出了该模型中近似约简的定义和方法,并通过实例分析说明了约简算法的有效性。     

Generalized interval-valued fuzzy variable precision rough sets determined by fuzzy logical operators

The fuzzy rough set model and interval-valued fuzzy rough set model have been introduced to handle databases with real values and interval values, respectively. Variable precision rough set was advanced by Ziarko to overcome the shortcomings of misclassification and/or perturbation in Pawlak rough sets. By combining fuzzy rough set and variable precision rough set, a variety of fuzzy variable precision rough sets were studied, which cannot only handle numerical data, but are also less sensitive to misclassification. However, fuzzy variable precision rough sets cannot effectively handle interval-valued data-sets. Research into interval-valued fuzzy rough sets for interval-valued fuzzy data-sets has commenced; however, variable precision problems have not been considered in interval-valued fuzzy rough sets and generalized interval-valued fuzzy rough sets based on fuzzy logical operators nor have interval-valued fuzzy sets been considered in variable precision rough sets and fuzzy variable precision rough sets. These current models are incapable of wide application, especially on misclassification and/or perturbation and on interval-valued fuzzy data-sets. In this paper, these models are generalized to a more integrative approach that not only considers interval-valued fuzzy sets, but also variable precision. First, we review generalized interval-valued fuzzy rough sets based on two fuzzy logical operators: interval-valued fuzzy triangular norms and interval-valued fuzzy residual implicators. Second, we propose generalized interval-valued fuzzy variable precision rough sets based on the above two fuzzy logical operators. Finally, we confirm that some existing models, including rough sets, fuzzy variable precision rough sets, interval-valued fuzzy rough sets, generalized fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy logical operators, are special cases of the proposed models.     

Variable precision rough set model over two universes and its properties

The extension of rough set model is an important research direction in the rough set theory. In this paper, based on the rough set model over two universes, we firstly propose the variable precision rough set model (VPRS-model) over two universes using the inclsion degree. Meantime, the concepts of the reverse lower and upper approximation operators are presented. Afterwards, the properties of the approximation operators are studied. Finally, the approximation operators with two parameters are introduced as a generalization of the VPRS-model over two universes, and the related conclusions are discussed.     

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.     

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。     

变精度与程度粗糙集的一种推广

粗糙集理论是一种处理模糊和不确定性知识的数学工具,在人工智能及数据挖掘等众多领域已经得到了广泛的应用。在程度粗糙集和变精度粗糙集的基础上,通过引入误差参数,给出了一种新的程度变精度粗糙集模型并得出了所给模型上、下近似的一些性质。最后,通过一个具体的例子,说明了这种模型在信息系统中处理模糊和不确定性知识的可行性和有效性。     

The further investigation of covering-based rough sets: Uncertainty characterization, similarity measure and generalized models

The notion of rough sets was originally proposed by Pawlak. In Pawlak’s rough set theory, the equivalence relation or partition plays an important role. However, the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems. This limits the theory’s application to a certain extent. Therefore covering-based rough sets are derived by replacing the partitions of a universe with its coverings. This paper focuses on the further investigation of covering-based rough sets. Firstly, we discuss the uncertainty of covering in the covering approximation space, and show that it can be characterized by rough entropy and the granulation of covering. Secondly, since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition, image processing and fuzzy reasoning, we present an approach which measures these similarities using a triangular norm. We show that in a covering approximation space, a triangular norm can induce an inclusion degree, and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree. Thirdly, two generalized covering-based rough set models are proposed, and we employ practical examples to illustrate their applications. Finally, relationships between the proposed covering-based rough set models and the existing rough set models are also made.     

基于分子格及其类闭包子系统的粗近似

粗糙集的代数研究方法一直吸引着众多的研究人员,其中一个重要的研究方法是用算子的观点来看到粗糙集中的近似,并基于一般抽象代数结构来定义相应的粗糙近似算子。论文将分子格引入到粗糙集理论中作为基本代数系统,在分子格中构造了一个类似于闭包的子系统,并基于它们定义了更为一般和抽象的近似算子。文中还研究了相关粗近似结构的性质。