多粒度覆盖粗糙模糊集模型不确定性研究

针对覆盖粗糙模糊集中存在的上下近似不一致问题.引入一种更为合理的覆盖粗糙模糊集模型,讨论了该模型的结构与相关性质,定义了基于此模型的粗糙度度量方法.基于覆盖粗糙模糊集中粗糙度相等的情形,提出模糊集中极大模糊集的概念,并利用模糊集与极大模糊集的距离问题定义了模糊集的优劣次序,从而有效解决了模糊集在覆盖粗糙模糊集中粗糙度的度量问题.通过引入粗糙熵等相关概念,证明了此模型中仍然存在随最简覆盖变细,两种度量单调减少的规律,并通过实例进行了验证.从而为进一步揭示粗糙集、粗糙模糊集及覆盖粗糙模糊集之间的不确定性度量规律提供了理论依据.     

粗糙集不确定性度量的矩阵运算

粗糙集的不确定性度量是粗糙集理论的重要研究内容之一。结合模糊理论和粒计算理论改进了粗糙集的不确定性度量方法。通过集合的相对知识粒度及边界熵给出了粗糙集的粗糙性度量函数与模糊性度量函数,随着近似空间知识粒的细分,粗糙集的粗糙度与模糊度均满足单调递减的性质。利用矩阵理论提出了易于实现的粗糙性度量与模糊性度量的矩阵算法。     

基于边界域的邻域知识距离度量模型

粗糙集的不确定性度量在知识获取中扮演着非常重要的角色。在邻域粗糙集理论中,当前不确定性度量方面的研究工作主要专注于度量单个知识空间的不确定性及其随粒度变化的单调性规律,其仍存在以下缺点:1)邻域粗糙集不确定性来自于邻域粒中属于目标概念的元素和不属于目标概念的元素,当前的方法没有同时考虑每个邻域信息粒的这两部分;2)不能反映不同知识空间对目标概念刻画能力的差异性;3)由于当前的知识距离包含了粒度划分的信息,已有方法在一些应用场合下不够准确,例如属性约简中的知识启发式搜索及其粒度选择。对此,文中首先构建了一种更加直观准确的邻域粗糙集的不确定性度量方法——邻域熵,并证明了不确定性度量随着粒度的细化具有单调性;为了反映不同邻域信息粒对目标概念刻画能力的差异性,提出了一种带近似描述能力的邻域粒距离,称为相对邻域粒距离,并介绍了它的相关性质;针对分层递阶的多粒度知识空间中的粒度选择问题,建立了基于边界域的邻域知识距离度量模型,该知识距离可以反映不同邻域知识空间对目标概念的刻画能力的差异性。     

不同知识粒度下粗糙集的不确定性研究

粗糙集的不确定性度量方法,目前主要包括粗糙集的粗糙度、粗糙熵、模糊度和模糊熵.在不同知识粒度下,从属性的角度,给出了分层递阶的知识空间链,发现在分层递阶的知识粒度下部分文献中定义的粗糙集的粗糙熵和模糊度随知识粒度的变化规律不一定符合人们的认识规律.从信息熵的角度提出了一种粗糙集不确定性的模糊度度量方法,证明了这种模糊度随知识粒度的减小而单调递减,弥补了现有粗糙熵和模糊度度量粗糙集不确定性的不足.最后,分析了在不同知识粒度下粗糙度和模糊度的变化关系.     

基于广义粗集覆盖约简的粗糙熵

在广义粗集覆盖约简理论中,由于集合的上下近似是由其覆盖约简来确定的,因此有必要寻求一种新的度量来刻画知识和粗集的粗糙性.通过引入信息熵以刻画广义粗集覆盖约简的知识粗糙性以及粗集粗糙性,提出了一种新的知识粗糙性和粗集粗糙性度量.得到知识粗糙熵和粗糙集的粗糙熵都随广义覆盖约简的变细而单调减少的结论,从信息论观点出发,对不完备信息系统粗集理论进行了探讨.     

覆盖粗糙直觉Fuzzy集模型的一点注记

通过对一类覆盖粗糙直觉模糊集模型中粗糙度定义的分析,对其所存在疏漏进行了改进;再将粗糙熵的概念引入到该模型,研究直觉模糊集的不确定度量;通过例子说明该度量的有效性。     

基于覆盖的粗糙模糊集的粗糙熵

覆盖约简是研究覆盖去冗余问题的一种有效方法。本文在基于最简覆盖的粗糙集模型的基础上,将粗糙度和粗糙熵的概念引入基于最简覆盖的粗糙模糊集,用来度量其不确定性程度;讨论了它们的一些性质,并通过实例说明粗糙熵比粗糙度更能精确地反映基于最简覆盖的粗糙模糊集的不确定性程度。     

信息系统不确定性的粗糙边界熵度量方法

不确定性度量是粗糙集理论研究的重要内容之一。分析了目前粗糙集不确定性度量主要方法的不足,给出了基于边界域的粗糙集粗糙边界熵的定义。证明了这种粗糙边界熵随着知识粒度的减小而单调减小,而且当负域的知识颗粒被细分时,粗糙边界熵不变。给出了粗糙边界熵的两条性质。     

模糊知识粒度计算与应用

根据自反模糊关系,将知识粒度的概念推广为模糊知识粒度.考虑传统模糊粗糙集的粗糙性度量和相似性度量,忽略了模糊集的粗糙近似处于不同知识粒度背景中这样一个重要因素,结合模糊知识粒度的计算,提出了模糊粗糙集的粗糙性度量和相似性度量的新方法.最后,在一个实际的模糊信息系统中,给出了基于模糊知识粒度的知识约简算法.     

覆盖粗糙模糊集的不确定性研究

不确定性度量是粗糙集理论中的基础问题之一。粗糙模糊集的不确定性一方面来自上、下近似集间差异产生的粗糙性,另一方面来自概念外延不清晰产生的模糊性。目前对于粗糙模糊集的不确定性研究仍不够透彻。针对覆盖近似空间下的粗糙模糊集不确定性,提出更加严格的度量修正准则,并借助上、下近似集隶属度与原模糊集隶属度之间的差异,给出修正粗糙度的概念。算例分析表明该方法能够更加准确地刻画实际问题。     

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Fuzzy集模型

考虑到经典粗糙集模型中等价关系过于严格的缺陷和直觉Fuzzy集在处理不确定信息时所具有的表达力,建立了覆盖粗糙直觉Fuzzy集模型,并给出了该模型下的一些性质;接着引入了覆盖粗糙直觉Fuzzy集模型的粗糙度和粗糙熵的概念,讨论其不确定性度量;最后给出了算例。     

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.     

Uncertainty measures for rough formulae in rough logic: An axiomatic approach

Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Various types of uncertainty measure such as accuracy measure, roughness measure, etc, which aim to quantify the imprecision of a rough set caused by its boundary region, have been extensively studied in the existing literatures. However, a few of these uncertainty measures are explored from the viewpoint of formal rough set theory, which, however, help to develop a kind of graded reasoning model in the framework of rough logic. To solve such a problem, a framework of uncertainty measure for formulae in rough logic is presented in this paper. Unlike the existing literatures, we adopt an axiomatic approach to study the uncertainty measure in rough logic, concretely, we define the notion of rough truth degree by some axioms, such a notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true. Then based on this fundamental notion, the notions of rough accuracy degree, roughness degree for any formula, and rough inclusion degree, rough similarity degree between any two formulae are also proposed. In addition, their properties are investigated in detail. These obtained results will be used to develop an approximate reasoning model in the framework of rough logic from the axiomatic viewpoint.     

基于知识含量的粗糙集不确定性度量

粗糙集的不确定性度量是粗糙集理论中一项重要的数值特征,而Z．Pawlak提出的粗糙集的不确定性度量,即传统的近似精度与粗糙度具有局限性。考虑导致粗集粗糙性的原因,将传统的粗糙度与知识的含量测度结合起来,提出了一种新的粗糙集不确定性的度量方法,讨论了这一度量的特性,通过实例说明这一新的度量方法的合理性及计算的简便性。     

Approximation operators based on vague relations and roughness measures of vague sets

Rough set theory and vague set theory are powerful tools for managing uncertain, incomplete and imprecise information. This paper extends the rough vague set model based on equivalence relations and the rough fuzzy set model based on fuzzy relations to vague sets. We mainly focus on the lower and upper approximation operators of vague sets based on vague relations, and investigate the basic properties of approximation operators on vague sets. Specially, we give some essential characterizations of the lower and upper approximation operators generated by reflexive, symmetric, and transitive vague relations. Finally, we structure a parameterized roughness measure of vague sets and similarity measure methods between two rough vague sets, and obtain some properties of the roughness measure and similarity measures. We also give some valuable counterexamples and point out some false properties of the roughness measure in the paper of Wang et al.     

Knowledge granulation, knowledge entropy and knowledge uncertainty measure in ordered information systems

In this paper, concepts of knowledge granulation, knowledge entropy and knowledge uncertainty measure are given in ordered information systems, and some important properties of them are investigated. From these properties, it can be shown that these measures provides important approaches to measuring the discernibility ability of different knowledge in ordered information systems. And relationship between knowledge granulation, knowledge entropy and knowledge uncertainty measure are considered. As an application of knowledge granulation, we introduce definition of rough entropy of rough sets in ordered information systems. By an example, it is shown that the rough entropy of rough sets is more accurate than classical rough degree to measure the roughness of rough sets in ordered information systems.     

Fuzzy probabilistic approximation spaces and their information measures

Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work.     

模糊粗糙集的相似度量和相似性方向

粗糙集理论是一种新的处理模糊和不确定性知识的软计算工具,在人工智能及认知科学等众多领域已经得到了广泛的应用。相似度量的研究是模糊集理论与粗糙集理论的热点问题之一。文章提出了一种更精确、更合理的相似度量方法,讨论了它的一些性质。然后,在此基础上提出了模糊粗糙集的相似性方向的概念,用于比较两个相似的模糊粗糙集所包含信息的精确性大小,并给出了一个关于相似性方向的判别函数。这在近似推理、模式识别和决策分析等领域有着广泛的应用。最后,通过一个实例,分析说明了这种相似度量方法和相似性方向的判别方法是更合理更有效的。