覆盖粒度空间的多层次粒度结构研究

针对覆盖粒度空间中的知识表示、基本运算、层次结构及粒度结构度量问题进行分析与研究.首先,定义覆盖近似空间中对象的相容类,构造覆盖粗糙集模型的相容关系,定义相容类中对象之间的相容度,由此相容关系诱导出覆盖粒度空间的概念.其次,给出覆盖粒度空间下对象的矩阵表示,定义覆盖粒度空间中基本运算,并诱导出覆盖信息粒的概念,从而对覆盖粒度空间中粒度的大小进行了度量.接着,定义覆盖粒度空间的三种偏序关系,以此揭示覆盖粒度空间的层次关系.最后,定义覆盖粒度空间的信息粒度、粗糙度和粗糙熵,研究在覆盖粒度空间中多层次粒度结构度量的各种关系.研究结果统一了覆盖粒度空间下信息粒度的相关度量,从而为粒计算的多层次粒结构理论进一步的完善提供依据.     

多粒度粗糙集和覆盖粗糙集间的近似与约简关系*

多粒度粗糙集和覆盖粗糙集是2种重要的数据处理机制.文中从近似集和属性约简2个角度探讨完备信息系统与不完备信息系统中多粒度粗糙集和覆盖粗糙集的关系.通过构造信息系统的粒空间,证明乐观多粒度粗糙集近似等价于松覆盖粗糙集近似,悲观多粒度粗糙集近似等价于紧覆盖粗糙集近似,即乐观多粒度粗糙集和悲观多粒度粗糙集可分别表示为松覆盖粗糙集和紧覆盖粗糙集.进一步指出信息系统的2类多粒度粗糙集的协调集可转化为2类覆盖粗糙集的协调集,并刻画多粒度粗糙集约简与覆盖粗糙集约简间的密切联系.     

粗糙集的近似集

粗糙集是1982年由Pawlak教授提出的解决集合边界不确定的重要方法,它通过两个精确的上、下近似集作为边界线来刻画目标集合(概念)X的不确定性,但它没有给出如何用已知的知识基(知识粒)来精确或近似地描述边界不确定的目标集合(概念)X的方法.首先给出了集合之间的相似度概念,然后分析了分别用上近似集R(X)和下近似集R(X)作为目标集合(概念)X近似描述的不足,提出了在已有知识基(粒)空间下寻找目标集合(概念)X的近似集的方法,并分析了用R0.5(X)作为X(概念)的近似集的优越性.最后讨论了不同知识粒度空间下R0.5(X)与X的相似度随知识粒度的变化关系.从新的角度提出了目标集合(概念)X近似集的构造方法,促进了粗糙集模型的发展.     

Approximation Sets of Rough Sets

粗糙集是1982年由Pawlak教授提出的解决集合边界不确定的重要方法,它通过两个精确的上、下近似集作为边界线来刻画目标集合(概念)X的不确定性,但它没有给出如何用已知的知识基(知识粒)来精确或近似地描述边界不确定的目标集合(概念)X的方法.首先给出了集合之间的相似度概念,然后分析了分别用上近似集(-R)(X)和下近似集(R-)(X)作为目标集合(概念)X近似描述的不足,提出了在已有知识基(粒)空间下寻找目标集合(概念)X的近似集的方法,并分析了用R0.5(X)作为X(概念)的近似集的优越性.最后讨论了不同知识粒度空间下R0.5(X)与X的相似度随知识粒度的变化关系.从新的角度提出了目标集合(概念)X近似集的构造方法,促进了粗糙集模型的发展.     

一种元素最大描述下的多粒度覆盖粗糙集模型

利用元素的最大描述,将传统多粒度粗糙集拓展到覆盖空间,首先提出了两种新的多粒度粗糙集模型,然后对模型的一些基本性质进行了研究,给出了不同多粒度覆盖粗糙集产生相同上、下近似的条件,最后研究了两种模型之间的关系。     

粒度计算中的商结构

基于商空间的粒度计算理论是目前三个主要的粒度计算理论之一.主要讨论商空间理论中的结构问题,并与粗糙集方法进行比较,指出结构在粒度计算理论中的重要性.讨论如何从结构着手来建立商空间模型.文中给出了从结构上取不同粒度来构造商空间的新方法,最后通过相关例子说明所提出的方法的合理性、可行性.     

最小描述的多粒度覆盖粗糙集模型

在覆盖广义粗糙集理论中,对最小描述的定义是建立在单一粒度基础上。将最小描述从单一粒度推广到多个粒度,建立了多粒度覆盖粗糙集模型。在此基础上,用最小描述建立了两类不同的上下近似算子,研究其性质,给出了一种基于最小描述下求属性约简的新算法。     

覆盖多粒度粗糙集的数值特征

通过极大描述和极小描述获取的覆盖多粒度粗糙集,可以更好地应用于实际.首先通过极小描述和极大描述的交并运算定义4个悲观覆盖多粒度粗糙集模型,并讨论其基本性;在此基础上进一步分析其证据结构,并得出覆盖多粒度粗糙集具有信任结构的充分条件,即上、下近似满足对偶性、可加性和可乘性.通过上述研究,进一步丰富了多粒度粗糙集的研究.     

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。     

多粒度决策粗糙集中的粒度约简方法

多粒度决策粗糙集模型是一种泛化的多粒度粗糙集模型,该模型结合决策粗糙集数据分析理论和多粒度思想,实现了在多个粒空间进行决策粗糙集理论的建模。在此基础上,利用贝叶斯决策理论具体分析了在多粒度粗糙集模型中乐观和悲观的融合策略下多个粒空间中的概率融合关系,推导出基于最大条件概率和最小条件概率的粗糙集近似表示,进而构建了乐观多粒度决策粗糙集模型和悲观多粒度决策粗糙集模型。在该模型中引入近似分布约简的概念,分析了多个粒空间中的粒度选择问题。基于多粒度近似分布质量定义了多粒度决策粗糙集的粒度重要度,并且基于此给出了悲观和乐观融合策略α-下近似分布约简的粒度约简算法。通过实例验证了该算法的有效性。     

The relationship among soft sets,soft rough sets and topologies

Molodtsov’s soft set theory is a newly emerging tool to deal with uncertain problems. Based on the novel granulation structures called soft approximation spaces, Feng et al. initiated soft rough approximations and soft rough sets. Feng’s soft rough sets can be seen as a generalized rough set model based on soft sets, which could provide better approximations than Pawlak’s rough sets in some cases. This paper is devoted to establishing the relationship among soft sets, soft rough sets and topologies. We introduce the concept of topological soft sets by combining soft sets with topologies and give their properties. New types of soft sets such as keeping intersection soft sets and keeping union soft sets are defined and supported by some illustrative examples. We describe the relationship between rough sets and soft rough sets. We obtain the structure of soft rough sets and the topological structure of soft sets, and reveal that every topological space on the initial universe is a soft approximating space.     

Granular computing based on fuzzy similarity relations

Rough sets and fuzzy rough sets serve as important approaches to granular computing, but the granular structure of fuzzy rough sets is not as clear as that of classical rough sets since lower and upper approximations in fuzzy rough sets are defined in terms of membership functions, while lower and upper approximations in classical rough sets are defined in terms of union of some basic granules. This limits further investigation of the existing fuzzy rough sets. To bring to light the innate granular structure of fuzzy rough sets, we develop a theory of granular computing based on fuzzy relations in this paper. We propose the concept of granular fuzzy sets based on fuzzy similarity relations, investigate the properties of the proposed granular fuzzy sets using constructive and axiomatic approaches, and study the relationship between granular fuzzy sets and fuzzy relations. We then use the granular fuzzy sets to describe the granular structures of lower and upper approximations of a fuzzy set within the framework of granular computing. Finally, we characterize the structure of attribute reduction in terms of granular fuzzy sets, and two examples are also employed to illustrate our idea in this paper.     

粗糙集理论与应用研究综述

在阐释粗糙集理论基本体系结构的基础上,从多个角度探讨粗糙集模型的研究思路,分析粗糙集理论与模糊集、证据理论、粒计算、形式概念分析、知识空间等其它理论之间的联系,介绍国内外关于粗糙集理论研究的主要方向和发展状况,讨论当前粗糙集理论研究的热点研究领域以及将来需要重点研究的主要问题.     

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.     

粒计算的一种覆盖模型

为了在一种更为广泛的背景之下研究粒计算的基本问题(诸如粒化、粒的计算及粒空间之间信息粒转化等),在放弃等价关系的3个条件的基础上提出了一种基于覆盖的粒计算模型,进一步推广了已有的工作。在该模型下,重新定义了Zoom-in算子与Zoom-out算子。对于论域及粒化了的论域而言,Zoom-in算子与Zoom-out算子的不同复合会产生不同的近似算子。研究了这些近似算子的性质并建立起它们与拓扑空间及Galois联络之间的联系。     

Theory and applications of granular labelled partitions in multi-scale decision tables

Granular computing and acquisition of if-then rules are two basic issues in knowledge representation and data mining. A formal approach to granular computing with multi-scale data measured at different levels of granulations is proposed in this paper. The concept of labelled blocks determined by a surjective function is first introduced. Lower and upper label-block approximations of sets are then defined. Multi-scale granular labelled partitions and multi-scale decision granular labelled partitions as well as their derived rough set approximations are further formulated to analyze hierarchically structured data. Finally, the concept of multi-scale information tables in the context of rough set is proposed and the unravelling of decision rules at different scales in multi-scale decision tables is discussed.     

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.     

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.     

Application of covering rough granular computing model in collaborative filtering recommendation algorithm optimization

Data sparseness will reduce the accuracy and diversity of collaborative filtering recommendation algorithms. In response to this problem, using granular computing model to realize the nearest neighbor clustering, and a covering rough granular computing model for collaborative filtering recommendation algorithm optimization is proposed. First of all, our method is built on the historical record of the user's rating of the item, the user’s predilection threshold is set under the item type layer to find the user's local rough granular set to avoid data sparsity. Then it combines the similarity between users. Configuring the covering coefficient for target user layer, it obtained the global covering rough granular set of the target user. So it solved the local optimal problem caused by data sparsity. Completed the coarse–fine-grained conversion in the covering rough granular space, obtain a rough granular computing model with multiple granular covering of target users, it improved the diversity of the recommendation system. All in all, predict the target users’ score and have the recommendation. Compared experiments with six classic algorithms on the public MovieLens data set, the results showed that the optimized algorithm not only has enhanced robustness under the premise of equivalent time complexity, but also has significantly higher recommendation diversity as well as accuracy.