粗糙集理论及其在数据挖掘中的应用

粗糙集理论是一种研究不完整、不确定知识处理的数学工具,近几年来已经在诸多领域得到应用,在机器学习、知识发现、算法研究、决策支持系统以及模式识别等应用中取得了较好的成果.本文在阐述粗糙集理论概念的基础上,介绍了粗糙集理论在数据挖掘中的两个应用.     

Rough Sets, Their Extensions and Applications

Rough set theory provides a useful mathematical foundation for developing automated computational systems that can help understand and make use of imperfect knowledge.Despite its recency,the theory and its extensions have been widely applied to many problems,including decision analysis,data mining,intelligent control and pattern recognition.This paper presents an outline of the basic concepts of rough sets and their major extensions,covering variable precision,tolerance and fuzzy rough sets.It also shows the diversity of successful applications these theories have entailed,ranging from financial and business,through biological and medicine,to physical,art,and meteorological.     

Rough approximations on a complete completely distributive lattice with applications to generalized rough sets

The generalizations of rough sets considered with respect to similarity relation, covers and fuzzy relations, are main research topics of rough set theory. However, these generalizations have shown less connection among each other and have not been brought into a unified framework, which has limited the in-depth research and application of rough set theory. In this paper the complete completely distributive (CCD) lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed. These definitions result from the concept of cover introduced on a CCD lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers. When T-similarity relation is considered, the existing fuzzy rough sets are the special cases of our proposed approximations on a CCD lattice. Thus these generalizations of rough sets are brought into a unified framework, and a wider mathematical foundation for rough set theory is established.     

Rough集之间的相似度量

Applications of rough set theory in incomplete information systems are a key of putting rough set into real applications. In this paper, after analyzing some basic concepts of classical rough set theory and extended rough set theory, the measure of similarity is developed between two rough sets in the classical rough set theory based on indiscernibility relation and between two rough sets in the extended rough set theory based on limited tolerance relation. Then,some properties of these two methods for measuring similarity are developed respectively. At last,these two measure methods of rough set theory are compared.     

基于两个集合上粗集模型的算法实现

为了处理人工智能中不精确和不确定的数据和知识,Pawlak 提出了粗集模型,之后粗集理论得到拓广,人们提出了许多新的粗集模型,拓广的方法主要有两种,一种是减弱对等价关系的依赖,另一种是把讨论问题的论域从一个拓展到两个,Y.Y.Yao 提出了一种基于两个论域的粗集模型,本文研究基于两个论域的粗集模型的具体算法实现,给出了上下近似的矩阵算法及其相应的焦点集的算法,并把相关结论及矩阵算法推广到模糊集上,还给出了相关模型的极为简洁的公理刻画,即仅用一条公理刻画该模型。     

一种基于并行遗传算法的粗糙集属性约简

指出现有粗糙集属性约简算法的不足,考虑并行遗传算法在处理大型数据库上的特有优势,将粗糙熵作为粗糙集不确定性的度量,给出一种求解信息系统约简集的三群体并行遗传算法.最后通过实例计算表明该算法能快速有效求解属性约简,而且对大规模数据样本的信息系统效果更为明显.     

Approximations in <Emphasis Type="Italic">n</Emphasis>-ary algebraic systems

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.     

广义近似空间粗模糊度研究

模糊粗集理论是粗集理论的重要推广,研究广义近似空间下模糊粗集的模糊性具有重要的意义。文章研究了由对象-属性值偶对构成的广义模糊近似空间下模糊集合的粗模糊度问题,给出了其模糊度表示,并讨论它的性质。     

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.     

Rough sets: past,present, and future

Introduction of rough sets by Professor Zdzis?aw Pawlak has completed 35 years. The theory has already attracted the attention of many researchers and practitioners, who have contributed essentially to its development, from all over the world. The methods, developed based on rough set theory alone or in combination with other approaches, found applications in many areas. In this article, we outline some selected past and present research directions of rough sets. In particular, we emphasize the importance of searching strategies for relevant approximation spaces as the basic tools in achieving computational building blocks (granules or patterns) required for approximation of complex vague concepts. We also discuss new challenges related to problem solving by intelligent systems (IS) or complex adaptive systems (CAS). The concern is to control problems using interactive granular computing, an extension of the rough set approach, for effective realization of computations realized in IS or CAS. These challenges are important for the development of natural computing too.     

粗糙集理论的新进展及其在智能信息处理中的应用

主要总结了近年来粗糙集理论的研究和进展,介绍了广义粗糙集模型研究的一些主要方面和最新成果,从逼近算子和粗糙隶属函数的角度,讨论了广义粗糙集模型的各种类型,并着重分析了粗集理论在智能信息处理中的应用情况。     

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.     

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.     

TEMPERATURE INFERENCE SYSTEM BY ROUGH-NEURO-FUZZY NETWORK

Rough set theory, suggested by Pawlak in 1982, has been useful in AI, machine learning, knowledge acquisition, knowledge discovery from databases, expert systems, inductive reasoning, etc. One of the main advantages of rough sets is that they do not need any preliminary or additional information about data and are capable to reduce superfluous information. However, their significant disadvantage in real applications is that inferences are not real values but disjoint intervals of real values. In order to overcome this difficulty, we propose a new approach in which rough set theory and neuro-fuzzy fusion are combined to obtain the optimal rule base from input/output data. These results are applied to the rule construction for inferring the temperature at specified points in a refrigerator.     

Roughness in modules

The theory of rough sets was proposed by Pawlak in 1982. The relations between rough sets and algebraic systems have been already considered by many mathematicians. Important algebraic structures are groups, rings and modules. Rough groups and rough rings have been investigated by Biswas and Nanda, Kuroki and Wang, and Davvaz. In this paper, we consider an R-module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an R-module, which is an extended notion of a submodule in an R-module. We also give some properties of the lower and the upper approximations in an R-module.     

Rough集理论与粒计算

近年来,粗糙集理论以其独特的优势在诸多科研领域取得了不俗的表现。在信息处理过程中,统计学方法需要知道数据的概率分布情况,模糊集的方法需要事先给定隶属度函数,而粗集理论不依赖于这些先验知识,利用上、下近似集这两个概念来描述不精确、不一致信息。本质上粗糙集理论是一种粒计算的模型框架。本文主要讨论粗糙集理论的基本概念、扩展模型以及未来的挑战。此外,对于与粒计算的相关概念以及它们在未来发展中的趋势、面临的主要问题本文也有所涉及。     

不完备信息系统中集对粗糙集模型

粗糙集理论在数据挖掘中的成功应用已成为近来人工智能领域研究的热点,人们将经典粗糙集中的等价关系放宽后使粗糙集理论的运用更加广泛,但在不完备信息系统中的运用仍受到限制。在已有的集对粗糙集模型的基础上,提出了针对不完备系统更加有效的集对粗糙集模型,通过实例说明了这种模型的可行性和有效性,使粗糙集模型在一定程度上得到了推广。     

Rough集理论与粒计算

近年来，粗糙集理论以其独特的优势在诸多科研领域取得了不俗的表现。在信息处理过程中，统计学方法需要知道数据的概率分布情况，模糊集的方法需要事先给定隶属度函数，而粗集理论不依赖于这些先验知识，利用上、下近似集这两个概念来描述不精确、不一致信息。本质上粗糙集理论是一种粒计算的模型框架。本文主要讨论粗糙集理论的基本概念、扩展模型以及未来的挑战。此外，对于与粒计算的相关概念以及它们在未来发展中的趋势、面临的主要问题本文也有所涉及。     

一种基于加权相似性的粗糙集数据补齐方法

近年来,对不完备数据的处理引起了人们的广泛关注。目前,在粗糙集理论中已经提出了多种不完备数据补齐方法,这些方法通常需要计算决策表中具有缺失值的对象与其他没有缺失值的对象之间的相似性,并以最相似对象的取值来代替缺失值。然而,这些方法普遍存在一个问题,即在计算决策表中对象之间的相似性时假设决策属性对所有条件属性的依赖性都是相等的,而且所有条件属性都是同等重要的,并没有考虑不同条件属性之间的差异性。针对这一问题,引入一个加权相似性的概念,以决策属性对条件属性的依赖性和条件属性的重要性作为权值来计算相似性。基于加权相似性,提出一种新的粗糙集数据补齐算法WSDCA。最后,在UCI数据集上,将WSDCA算法与现有的数据补齐算法进行了比较分析。实验结果表明,所提出的数据补齐方法是有效的。     

Three-way decision reduction in neighborhood systems

Rough set reduction has been used as an important preprocessing tool for pattern recognition, machine learning and data mining. As the classical Pawlak rough sets can just be used to evaluate categorical features, a neighborhood rough set model is introduced to deal with numerical data sets. Three-way decision theory proposed by Yao comes from Pawlak rough sets and probability rough sets for trading off different types of classification error in order to obtain a minimum cost ternary classifier. In this paper, we discuss reduction questions based on three-way decisions and neighborhood rough sets. First, the three-way decision reducts of positive region preservation, boundary region preservation and negative region preservation are introduced into the neighborhood rough set model. Second, three condition entropy measures are constructed based on three-way decision regions by considering variants of neighborhood classes. The monotonic principles of entropy measures are proved, from which we can obtain the heuristic reduction algorithms in neighborhood systems. Finally, the experimental results show that the three-way decision reduction approaches are effective feature selection techniques for addressing numerical data sets.