覆盖粗糙集研究
摘要
现在我们正处于信息革命浪潮中,信息化的浪潮已经引领交通信息工程进入了智能交通时代。构建智能交通系统所需的信息具有多源、异构等典型特征,对具有复杂性、模糊性和不确定性交通信息的分析处理,已成为智能交通系统发展和运用的关键。粗糙集理论是一种处理不确定性问题的数学工具,在交通拥堵与交通事故的原因分析和提前预警等方面已有大量研究成果。覆盖粗糙集模型是Pawlak粗糙集模型的重要推广形式。本文研究覆盖粗糙集模型,研究内容主要包括覆盖粗糙集理论以及基于覆盖粗糙集理论的决策表约简理论与方法。
在覆盖粗糙集理论研究方面,本文主要研究基于覆盖以及基于覆盖产生的邻域系统的覆盖粗糙集模型,并且讨论论域为完全分配格(简称为CD格)的覆盖粗糙集。
(1)对于基于一般覆盖的粗糙集模型,本文首先回顾了Bonkowski关于覆盖粗糙集的相关研究工作,提出了基于拟单层覆盖的覆盖粗糙集模型并研究了相关近似算子的基本性质；给出了扩展相等意义下的上、下近似算子的特征刻画,并给出了一个覆盖是拟单层覆盖的若干必要条件以及基于改进近似算子的近似空间中存在拟单层覆盖的判定定理。
(2)对于基于邻域的覆盖粗糙集模型,本文侧重研究基于最小邻域的覆盖粗糙集以及基于极小描述的覆盖粗糙集,作为这两种粗糙近似算子的通用形式,我们定义了5对上、下近似算子(Ⅰ)-(Ⅴ),讨论了这五种近似算子的基本性质及其之间的关系,给出了基于这些近似算子的近似集构成的拓扑空间,并研究了这些拓扑空间中内部算子、闭包算子与近似算子之间的关系。
(3)基于CD格的近似空间与近似算子是粗糙集模型的一种重要推广形式,旨在CD格的框架下为各种近似算子提供统一的描述方法。本文在已有文献研究工作的基础上给出了基于CD格的近似算子的若干性质；提出了一种改进的上、下近似算子,使之具备了上近似保并,下近似保交这一性质；基于拓扑分子格理论,讨论了CD格上近似算子的拓扑性质,构造了由相关近似集合构成的拓扑空间；讨论了CD格上覆盖的约简问题以及约简对近似算子的影响。
在覆盖粗糙集应用方面,本文讨论了基于覆盖粗糙集模型的覆盖信息系统的约简理论以及约简方法。针对覆盖信息表与协调覆盖决策表分别提出了覆盖约简与d覆盖约简的概念,借助区分矩阵与区分函数给出了约简方法；针对不协调覆盖决策表提出了正域约简与分配约简的概念,给出了正域约简与分配约简的判定定理并借助区分矩阵与区分函数给出了约简方法；对于以上提出的约简方法设计了约简算法,并通过对比试验说明了基于覆盖粗糙集模型的约简理论的可行性。
最后是结论与展望。
Today, we are in the tide of information revoluation. The tide of informationization has guided traffic information engineering into intelligent traffic era. The desired information constructing of intelligent transport systems have some typical characteristics, such as multi-source and heterogeneous. Analysis and treatment for the traffic information which has complexity, fuzziness and uncertainty, plays a key role in the development and application of intelligent transport systems. Rough set theory is a mathematical tool of dealing with uncertainties. The theory has applied for the reason analysis, warning of traffic congestion and traffic accident, and obtains many research achievements. Cover ing rough set model is an important form of generalization for Pawlak rough set model. This paper mainly researches the covering rough set model, including the theory of covering rough set and the approaches of decision table reduction that bases on covering rough set theory.
In the aspects of covering rough set theory, this study mainly researches the covering rough set models based on general covering and neighborhood system which is generated from covering, and discusses complete completely distributive lattices (referred to as the CD lattice) based covering rough sets.
(1) For the rough set model based on general covering, this paper first reviews the related work of Bonkowski about covering rough sets and proposes the covering rough set model based on the single coverage and studies the relevant basic properties of the approximation operators; and this paper gives characteristics of the upper and lower approximation operators in the sense of expansion equal, and gives a number of necessary conditions on which the covering is the single covering and the judging theorem of single covering in approximate space based on improved approximate operators.
(2) For the rough set model based on the neighborhood system, this paper focuses on the research of the rough set models based on smallest neighborhood and the one based on minimal description. As the common forms of these two rough approximation operators, we define five kinds of upper and lower approximation operators (I)-(V), and discuss their properties and their relationship, and construct the topological spaces based on these approximation operators. The paper also researches the relationship among the approximate operators and the inner operator, closure operator in the topological spaces.
(3) The approximation spaces and approximation operators based on the CD lattice are important models for the generalizations of the rough set model. Under the framework of CD lattice, it aims to provide a unified description method for a variety of approximation operators. Based on the existing researches, this work gives some properties of approximation operators rooting in the CD-based grid. It also shows improved upper and lower approximation operators that have the properties that the lower approximation operator is closed under intersection and the upper approximation operator is closed under union. Based on the topological molecular lattice theory, it also discusses the topological properties of the approximate operators in the CD-based grid. And it also constructs the topological space of the relevant approximate sets, and discusses the reduction problem of covering and its affection for approximate operators.
In the aspect of the covering rough set applications, this paper discusses the reduction theory and method of covering information systems based on the covering rough set model. This paper proposes the concepts of covering reduction and d-covering reduction of covering information system and consistent covering decision table respectively, and also presents the reduction method with the assistance of discernibility matrix and discernibility function. This paper also provides the concepts of positive domain reduction and distribution reduction for the inconsistent covering decision tables, and their judging theorems and the reduction methods with the help of discernibility matrix and discernibility function. The work design reduction algorithm for all the reduction methods above, and also shows the feasibility of the covering rough set model based reduction theory.
At last, it's the conclusion and expectation.
在覆盖粗糙集理论研究方面,本文主要研究基于覆盖以及基于覆盖产生的邻域系统的覆盖粗糙集模型,并且讨论论域为完全分配格(简称为CD格)的覆盖粗糙集。
(1)对于基于一般覆盖的粗糙集模型,本文首先回顾了Bonkowski关于覆盖粗糙集的相关研究工作,提出了基于拟单层覆盖的覆盖粗糙集模型并研究了相关近似算子的基本性质；给出了扩展相等意义下的上、下近似算子的特征刻画,并给出了一个覆盖是拟单层覆盖的若干必要条件以及基于改进近似算子的近似空间中存在拟单层覆盖的判定定理。
(2)对于基于邻域的覆盖粗糙集模型,本文侧重研究基于最小邻域的覆盖粗糙集以及基于极小描述的覆盖粗糙集,作为这两种粗糙近似算子的通用形式,我们定义了5对上、下近似算子(Ⅰ)-(Ⅴ),讨论了这五种近似算子的基本性质及其之间的关系,给出了基于这些近似算子的近似集构成的拓扑空间,并研究了这些拓扑空间中内部算子、闭包算子与近似算子之间的关系。
(3)基于CD格的近似空间与近似算子是粗糙集模型的一种重要推广形式,旨在CD格的框架下为各种近似算子提供统一的描述方法。本文在已有文献研究工作的基础上给出了基于CD格的近似算子的若干性质；提出了一种改进的上、下近似算子,使之具备了上近似保并,下近似保交这一性质；基于拓扑分子格理论,讨论了CD格上近似算子的拓扑性质,构造了由相关近似集合构成的拓扑空间；讨论了CD格上覆盖的约简问题以及约简对近似算子的影响。
在覆盖粗糙集应用方面,本文讨论了基于覆盖粗糙集模型的覆盖信息系统的约简理论以及约简方法。针对覆盖信息表与协调覆盖决策表分别提出了覆盖约简与d覆盖约简的概念,借助区分矩阵与区分函数给出了约简方法；针对不协调覆盖决策表提出了正域约简与分配约简的概念,给出了正域约简与分配约简的判定定理并借助区分矩阵与区分函数给出了约简方法；对于以上提出的约简方法设计了约简算法,并通过对比试验说明了基于覆盖粗糙集模型的约简理论的可行性。
最后是结论与展望。
Today, we are in the tide of information revoluation. The tide of informationization has guided traffic information engineering into intelligent traffic era. The desired information constructing of intelligent transport systems have some typical characteristics, such as multi-source and heterogeneous. Analysis and treatment for the traffic information which has complexity, fuzziness and uncertainty, plays a key role in the development and application of intelligent transport systems. Rough set theory is a mathematical tool of dealing with uncertainties. The theory has applied for the reason analysis, warning of traffic congestion and traffic accident, and obtains many research achievements. Cover ing rough set model is an important form of generalization for Pawlak rough set model. This paper mainly researches the covering rough set model, including the theory of covering rough set and the approaches of decision table reduction that bases on covering rough set theory.
In the aspects of covering rough set theory, this study mainly researches the covering rough set models based on general covering and neighborhood system which is generated from covering, and discusses complete completely distributive lattices (referred to as the CD lattice) based covering rough sets.
(1) For the rough set model based on general covering, this paper first reviews the related work of Bonkowski about covering rough sets and proposes the covering rough set model based on the single coverage and studies the relevant basic properties of the approximation operators; and this paper gives characteristics of the upper and lower approximation operators in the sense of expansion equal, and gives a number of necessary conditions on which the covering is the single covering and the judging theorem of single covering in approximate space based on improved approximate operators.
(2) For the rough set model based on the neighborhood system, this paper focuses on the research of the rough set models based on smallest neighborhood and the one based on minimal description. As the common forms of these two rough approximation operators, we define five kinds of upper and lower approximation operators (I)-(V), and discuss their properties and their relationship, and construct the topological spaces based on these approximation operators. The paper also researches the relationship among the approximate operators and the inner operator, closure operator in the topological spaces.
(3) The approximation spaces and approximation operators based on the CD lattice are important models for the generalizations of the rough set model. Under the framework of CD lattice, it aims to provide a unified description method for a variety of approximation operators. Based on the existing researches, this work gives some properties of approximation operators rooting in the CD-based grid. It also shows improved upper and lower approximation operators that have the properties that the lower approximation operator is closed under intersection and the upper approximation operator is closed under union. Based on the topological molecular lattice theory, it also discusses the topological properties of the approximate operators in the CD-based grid. And it also constructs the topological space of the relevant approximate sets, and discusses the reduction problem of covering and its affection for approximate operators.
In the aspect of the covering rough set applications, this paper discusses the reduction theory and method of covering information systems based on the covering rough set model. This paper proposes the concepts of covering reduction and d-covering reduction of covering information system and consistent covering decision table respectively, and also presents the reduction method with the assistance of discernibility matrix and discernibility function. This paper also provides the concepts of positive domain reduction and distribution reduction for the inconsistent covering decision tables, and their judging theorems and the reduction methods with the help of discernibility matrix and discernibility function. The work design reduction algorithm for all the reduction methods above, and also shows the feasibility of the covering rough set model based reduction theory.
At last, it's the conclusion and expectation.
引文
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