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.     

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.     

Generalized rough sets based on relations

Rough set theory has been proposed by Pawlak as a tool for dealing with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations. This paper studies arbitrary binary relation based generalized rough sets. In this setting, a binary relation can generate a lower approximation operation and an upper approximation operation, but some of common properties of classical lower and upper approximation operations are no longer satisfied. We investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations.This paper also explores the relationships between the lower or the upper approximation operation generated by the intersection of two binary relations and those generated by these two binary relations, respectively. Through these relationships, we prove that two different binary relations will certainly generate two different lower approximation operations and two different upper approximation operations.     

拟阵下的覆盖模糊粗糙集

拟阵是一种图和矩阵的同时推广的概念,而覆盖粗糙集是经典粗糙集的推广。利用拟阵理论研究覆盖模糊粗糙集,从而将两者进行了融合,提出了拟阵覆盖模糊粗糙集的概念,定义了拟阵覆盖近似空间的上下近似。分析了拟阵覆盖模糊粗糙集的相关性质,定义了拟阵覆盖粗糙集下的粗糙度,并通过它来衡量不确定程度,这也进一步推广了粗糙度。     

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。     

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.     

两类广义粗糙集的拟阵结构

基于邻域粗糙集模型和覆盖粗糙集模型,分别构造了两类拟阵结构,即邻域上近似数诱导的拟阵和覆盖上近似数诱导的拟阵。一方面,通过广义粗糙集定义了两类上近似数,并证明了它们满足拟阵理论中的秩公理,从而由秩函数的观点出发得到了两类拟阵;另一方面,利用粗糙集方法研究了这两类拟阵的独立集、极小圈、闭包、闭集等的表达形式,说明了粗糙集中的上近似算子与拟阵中的闭包算子的关系,进一步通过探讨覆盖和拟阵的关系,得到了覆盖中的元素及其任意并是由覆盖上近似数诱导的拟阵的闭集。     

覆盖Value集

覆盖粗糙集和Vague集都是处理不确定性问题的数学工具,它们分别是粗糙集和模糊集的扩展。已有的覆盖粗糙集模型在求上、下近似时,可能将一些实际上并非肯定属于给定集合的元素纳入到下近似中,而一些可能属于给定集合的元素却没有纳入到上近似中,这就会改变一些元素与给定集合的关系。通过深入分析论域中的元素与其相关覆盖元之间的关系,建立了覆盖Vague集。该覆盖Vague集能够从一种新的角度反映出论域中各元素与给定集合之间的从属程度。进一步研究了覆盖Vague集与覆盖粗糙集中一些重要概念之间的关系。最后讨论了当覆盖退化为划分时覆盖Vague集的特性。     

相容关系下的覆盖粗糙集

为了更有效地覆盖粗糙集理论应用到数据挖掘领域,所以对相容关系下的覆盖粗糙集进行了一系列的探究。首先介绍了基于相容关系的覆盖的定义以及基于相容关系的覆盖的特例--最大相容类的集合生成的覆盖的一些性质;其次对相容关系下由所有最大相容类的集合生成的覆盖中的可约元进行了讨论,并对这个条件下覆盖是否为单一的覆盖进行了探讨;接着借助于最小描述提出了k-最简覆盖这个概念,并对其做了简单的探究;最后探究了一些评价相容关系下的覆盖粗糙集的数值标准,并且分析了绝对覆盖率和相对覆盖率相等的情况。     

Connectedness of graphs and its application to connected matroids through covering-based rough sets

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of graph. Covering is a widely used form of data representation in data mining and covering-based rough sets provide a systematic approach to this type of representation. In this paper, we study the connectedness of graphs through covering-based rough sets and apply it to connected matroids. First, we present an approach to inducing a covering by a graph, and then study the connectedness of the graph from the viewpoint of covering approximation operators. Second, we construct a graph from a matroid, and find the matroid and the graph have the same connectedness, which makes us to use covering-based rough sets to study connected matroids. In summary, this paper provides a new approach to studying graph theory and matroid theory.     

类对称关系下的广义粗糙集模型

针对等价关系下的经典粗糙集,定义弱对称与局部强对称二元关系,构造相应的广义粗糙集模型。给出这2种模型的公理化特征,并将两者结合,得到强对称二元关系下的广义粗糙集模型。理论分析证明,论域上任何集合均为广义精确集的充要条件是其二元关系为强对称关系,即可以利用该模型刻画经典粗糙集中的广义精确集。     

Topological approaches to covering rough sets

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the topological properties of this type of rough sets, study the interdependency between the lower and the upper approximation operations, and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. Lastly, axiomatic systems for the lower approximation operation and the upper approximation operation are constructed.     

双论域粗糙集的矩阵表示

在双论域粗糙集中,不论是理论上还是应用上,矩阵方法都是一种简单且高效的计算方法。利用矩阵的方法来研究双论域粗糙集。提出了双论域上的关系矩阵,通过关系矩阵以及关系矩阵的转置,构造了两个布尔方阵。利用这两个布尔方阵的特征研究了双论域覆盖粗糙集的一些性质。定义了一种布尔矩阵之间新的运算,并且利用这个运算,简洁地表示了双论域覆盖粗糙集的上近似算子和关系粗糙集的下近似算子。     

类传递关系下的广义粗糙集及其公理化

提出了正向类传递与反向类传递二元关系,分别考虑了基于这两种二元关系的广义粗糙集,探讨了它们各自的性质,给出了相应粗糙近似算子的公理化特征。分析了这两类广义粗糙集与其它相关二元关系下广义粗糙集之间的联系,得到了一些重要的结果。     

基于粗隶属函数的粗糙集模型的推广

通过粗隶属函数,将粗糙集理论与模糊集理论联系起来,建立一种粗糙集理论与模糊集理论间的关系。把粗隶属函数视为论域上的一个特殊模糊集,用它的!-截集和强"-截集的概念,将经典粗糙集模型进行推广,提出基于等价关系的隶属度粗糙集模型,验证一些有用的性质,并证明该模型比Pawlak粗糙集模型具有更好的精度。最后将基于等价关系的隶属度粗糙集模型拓展到基于一般二元关系的广义隶属度粗糙集模型,并给出其相应的性质。     

复合覆盖粗糙集模型及其应用

针对覆盖粗糙集仅适用于单一数据类型的论域覆盖的问题,提出复合覆盖粗糙集模型。在研究邻域覆盖粗糙集、集值覆盖粗糙集、区间值覆盖粗糙集的基础上,在复合数据模型下,通过建立多种覆盖关系（邻域覆盖、集值覆盖、区间值覆盖等）,提出复合覆盖粗糙集模型,并给出复合覆盖粗糙集相关概念及性质。该模型适用于多种数据类型（符号数据、区间数据、集合数据、数值数据等）的论域覆盖问题,通过实例说明了该模型在复合信息系统中的应用,进一步加深对复合覆盖粗糙集相关概念的理解。     

On definable concepts of rough set models

In various generalized rough set models, definable concepts are used to approximate other concepts. Based on a unified abstract rough set theory, this paper investigates the mathematical structure of the set of definable concepts in several generalized rough set models such as relation based models, covering based models, and fuzzy based models. It is shown that there exist two kinds of interesting structures for the set of definable concepts, which are often used in rough set models: the complementary structure and the equational structure. In addition, relations between different rough set models are discussed.     

A novel approach to fuzzy rough sets based on a fuzzy covering

This paper proposes an approach to fuzzy rough sets in the framework of lattice theory. The new model for fuzzy rough sets is based on the concepts of both fuzzy covering and binary fuzzy logical operators (fuzzy conjunction and fuzzy implication). The conjunction and implication are connected by using the complete lattice-based adjunction theory. With this theory, fuzzy rough approximation operators are generalized and fundamental properties of these operators are investigated. Particularly, comparative studies of the generalized fuzzy rough sets to the classical fuzzy rough sets and Pawlak rough set are carried out. It is shown that the generalized fuzzy rough sets are an extension of the classical fuzzy rough sets as well as a fuzzification of the Pawlak rough set within the framework of complete lattices. A link between the generalized fuzzy rough approximation operators and fundamental morphological operators is presented in a translation-invariant additive group.     

条件概率描述下的多粒度覆盖粗糙集模型研究

在覆盖空间中,利用元素的最小描述并结合条件概率的概念,将经典多粒度粗糙集进行拓展,提出了3种条件概率描述下的多粒度覆盖粗糙集模型;研究了模型的一些基本性质,指出它们是一些已有多粒度覆盖粗糙集模型的泛化形式;最后探讨了几种模型之间的关系.