摘要
形式概念分析是从数据中进行概念发现的一种新型的数学工具,以概念格为其核心数据结构体现概念之间的层次关系。近年来,形式概念分析日益受到各领域学者的广泛关注,并己在信息检索、数字图书馆、软件工程和知识发现等方面得到应用。粗糙集理论是处理不确定知识的一种重要的数学工具,已在人工智能与知识发现,模式识别与分类,故障检测等方面得到了较好应用。公理模糊集,简称AFS(Axiomatic Fuzzy Sets)理论,是一种处理模糊信息的新语义方法,其本质是如何把蕴涵在训练样本、原始数据或数据库中的内在规律和模式转化到模糊集及其逻辑运算中的一种新方法,现已经被应用于聚类分析、模糊分类器、知识表示等方面。本文以粗糙集和AFS理论为工具主要针对形式概念分析的扩展形式及应用展开研究,主要研究工作包括:
1)从回顾概念格与其它理论结合的发展情况入手,分析了概念格的扩展模型—单调概念的不足之处,在此基础上把AFS理论与概念格结合并提出AFS形式概念,并证明了AFS形式概念的外延和内涵相互唯一确定及所有AFS形式概念在引入的序关系下构成一个完备格。此外,本文还给出了一种利用粗糙集技术进行概念逼近的方法,克服了利用单调概念进行概念逼近导致的不唯一性。进一步地,将AFS结构与形式背景结合引入新的形式背景,并提出了基于AFS结构的模糊形式概念,它可用来描述对象和属性之间的不确定关系,而且从原始数据直接可以确定对象对属性的隶属度。
2)基于形式背景建立一种新的AFS代数—E~CⅡ代数,并在E~CⅡ代数系统下讨论了经典概念格的代数性质,其中部分性质可以用来发现形式背景中的形式概念。E~CⅡ代数进一步地揭示了AFS理论与概念格是互补的。
3)利用概念格的不可约属性(对象)引入一个新的评概念相似度量模型,并讨论了它的可行性。此模型不但利用属性信息而且充分利用了格的结构信息,且模型的实现算法简便,避免了Souza和Davis的模型中利用Hasse图判断交不可约元方法的繁琐,是Souza和Davis的模型的一个改进。本文还讨论了它的两种扩展形式,即利用粗糙集给出了度量不可定义对象与属性集对之间的相似程度模型;提出了在模糊形式背景下的概念相似度量模型。
4)提出一种基于AFS理论的邻近集。在邻近集中,每一个对象都具有一个确定语义的模糊描述,并最大程度地区分于其它对象。它不依据某种距离度量评价对象间的邻近程度而是仅依赖它们的模糊描述相近程度,可以用来研究空间不邻近而具有相似的模糊描述的对象集。此外,通过结合逼近空间和AFS理论提出了两种新的逼近空间,它们可以看作是逼近空间及基于邻近关系的逼近空间的多粒度形式。
Formal concept analysis (FCA) is an effective tool for concept discovery from data, in which the relationship of concepts is embodied by concept lattice. Formal concept analysis has been widely used in information retrieval, digital library, software engineering and knowledge discovery, etc. Rough set theory is a new mathematical tool dealing with vagueness and uncertainty, has been successfully used in many areas such as knowledge discovery, pattern recognition and classification and fault diagnostication. Axiomatic fuzzy sets (AFS) theory is another method to deal with fuzzy information, which provides an effective tool to concert the information in the training examples and databases into the membership functions and their fuzzy logic operations. Recently, AFS theory has been developed further and applied to many fields such as fuzzy clustering analysis, fuzzy decision trees and concept representations et al. In this paper, some new theories and applications about FCA are discussed based on rough set and AFS theory. Main topics include:
Firstly, in order to overcome shortcoming of monotone concept, AFS formal concept is proposed by combining AFS theory and concept lattice, in which the intent and extent can be determined by each other. AFS formal concept can be viewed as the generalization and development of monotone concept. Moreover, we show that the set of all AFS formal concepts forms a complete lattice under the order relation. Furthermore, we give an approach to find some AFS formal concepts whose intents (extents) approximate any element of AFS algebras by virtue of rough set theory, which overcomes the shortcoming of concept approximating by using monotone concept. Moreover, we introduce fuzzy formal concept based AFS structure by combining AFS structure and formal context, which can be used to express the uncertainty relations between the objects and the attributes, and the relation is directly determined by the origin data and facts.
Secondly, with the aim of deriving the mathematical properties of formal concept and the relationships between concept lattice and AFS theory, we propose a new AFS algebra system on formal context, called E~CII algebra, by which the algebra properties of FCA can be explored and show that AFS theory is closely related to FCA.
Thirdly, a new similarity model is proposed, which using irreducible attributes and objects according to structure elements to evaluate the similarity degree of the two concepts of concept lattice. We give a new method to find irreducible elements of concept lattice by using attributes classes and objects classes, rather than constructing Hasse Diagram. The proposed method combines featural and structural information into decision and has a higher correlation with human judgement, which can be viewed as the generalization of Souza and Davis's similarity model. In order further to extent the ability of the above model in real world, two extension models are discussed. A measure evaluating non-definable pairs of objects and attributes is proposed based on the proposed model and rough set. Moreover, the proposed model is also extended to fuzzy formal context.
Fourthly, a new near set is established based on AFS theory, in which every object has an AFS fuzzy description with definitely semantics and distinguished among other objects at maximum extent. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using any distance metric, but depend on their fuzzy descriptions. Near set based on AFS logic can be used to discover the "nearness" of objects that are possibly disjoint and, yet, qualitatively near each other. Furthermore, by combining approximation space and AFS theory, two new approximation spaces are established, which can be viewed as multi-granulations forms of approximation spaces and approximation spaces based on nearness relation.
引文
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