粗糙集理论在属性约简和数据压缩中的应用
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摘要
粗糙集理论是一种有效的处理不确定和不精确问题的数学工具，其已经与模糊集、概率论等不确定性理论相结合并广泛应用于数据挖掘、人工智能和模式识别等很多领域。粗糙集的研究主要包括理论研究和应用研究。其中，属性约简是粗糙集理论研究最核心的问题。属性约简即在不影响分类精度的前提下，用最少的属性表示原有的信息。针对属性约简领域的研究，虽然很多研究者提出了许多有效的属性重要度函数，但是有些属性重要度函数并不适用于不完备信息系统。还有研究者基于信息系统之间的同态提出了一种压缩信息系统进行属性约简的方法。然而很多信息系统的属性约简并未采取基于信息系统同态的方法，并且目前对动态信息系统属性约简和概念近似方面的研究比较少。
在计算属性重要度方面，Yamaguchi为完备信息系统提出了一种新的属性重要度函数，但这个模型在用于不完备信息系统时存在一些问题。为了在不完备信息系统中更精确的计算属性的重要度，我们首先为不完备信息系统定义了三种属性重要度函数，并用12个数据集验证其有效性。然后，利用定义的函数简化不完备信息系统的区分矩阵。数据实验表明我们给出的重要度函数在不完备信息系统中能有效地计算条件属性的重要度。
在概念近似方面，王石平等基于特征矩阵和近似算子之间的关系把计算集合的近似转换成特征矩阵的计算，但他们没有研究如何有效的计算特征矩阵。为了计算概念的近似，我们提出了两种有效计算特征矩阵的方法。然后，基于递增式算法研究了如何计算动态覆盖的特征矩阵。主要包括三种情形：(1)覆盖中等价类的变化；(2)对象集合的变化；(3)属性值的变化。数据实验表明我们提出的方法能够明显地减少计算动态覆盖特征矩阵的时间复杂度。
在覆盖近似空间的压缩方面，为了研究覆盖近似空间之间的关系，我们提出了上、下同态和同态的概念。然后，我们引入覆盖近似子空间和乘积空间的概念，并研究了它们的基本性质。基于覆盖近似空间之间的同态，我们研究了覆盖近似空间和覆盖信息系统的压缩，进而研究了覆盖近似空间的粒约简和覆盖信息系统的属性约简。针对覆盖近似空间和覆盖信息系统的动态性，我们基于原始空间和信息系统的压缩研究了动态覆盖近似空间和信息系统的压缩。数据实验表明我们引入的同态为覆盖近似空间和覆盖信息系统的压缩提供了一种有效的方法。
在集值信息系统的压缩方面，我们提出了三种关系并研究了它们的基本性质。针对集值信息系统的属性约简，我们研究了集值信息系统的数据压缩。然后，基于原始集值信息系统的压缩，研究了动态集值信息系统的压缩。数据实验表明我们提出的方法能够简化集值信息系统属性约简的计算过程。
总之，本文为信息系统的属性约简、动态信息系统的属性约简和计算动态概念的近似提出了一些有效的方法，完善了信息系统属性约简的理论体系，进一步丰富了粗糙集理论。
Rough set theory as an efective mathematical tool to deal with uncertain andimprecise problems has been applied to various areas such as data mining, artifcialintelligence and pattern recognition. Current research regarding rough sets main-ly focuses on two aspects: theory and its applications. Among them, attributereductions is the most important problem for rough set theory. For attribute re-ductions, on one hand, researchers have proposed efective attribute dependencydegree functions. But some functions are not ft for incomplete information sys-tems. On the other hand, on the basis of homomorphisms between informationsystems, researchers have presented an approach to attribute reductions by com-pressing information systems. But there is not much research on other types ofinformation systems. Furthermore, researchers paid a little attention to attributereductions and approximations of concepts with respect to dynamic informationsystems.
In terms of computing the attribute dependency degree, Yamaguchi present-ed a new attribute dependency degree function for complete information systems.But there are some issues on applying it to incomplete information systems. Forcomputing the attribute dependency degree, we introduce three attribute depen-dency degree functions for incomplete information systems, and apply them to12data sets. Then we conduct the simplifcation for discernibility matrixes of incom-plete information systems with the proposed function. The experimental resultsshow that our proposed functions are more fexible to calculate the degree of eachconditional attribute related to the decision attribute for incomplete informationsystems.
With respect to approximations of concepts, Wang Shiping et al. transformedcomputing approximations of concepts into the computing of characteristic matrix-es, but there are few studies on the approach to computing characteristic matrixes.To compute approximation of concepts, we propose two approaches to computingcharacteristic matrixes. Then, by using an incremental approach, we computecharacteristic matrixes of the dynamic covering without running the matrix acqui-sition algorithm repeatedly. We mainly address the characteristic matrix updatingfrom three aspects: the variations of blocks in the covering, the immigration andemigration of objects and the changes of attribute values. Several illustrative exam-ples are employed to show that the time complexity of constructing characteristic matrixes of the dynamic covering can be reduced signifcantly with the proposedapproach.
For the compression of covering approximation spaces, we introduce the con-cepts of upper and lower homomorphisms as well as homomorphisms in order tostudy the relationship between covering approximation spaces. Then the notionsof covering approximation subspaces and product spaces are presented and theirfundamental properties are examined. Afterwards, we investigate the compres-sion of covering approximation spaces and covering information systems with theaim of reductions. Finally, we discuss the compression of dynamic covering ap-proximation spaces and dynamic covering information systems by utilizing thecompressions of the original spaces and systems, respectively. Several illustrativeexamples are employed to demonstrate that the homomorphisms provide an efec-tive approach to the compression of covering approximation spaces and coveringinformation systems.
For the compression of set-valued information systems, we put forward threerelations for set-valued information systems and explore their basic properties indetail. Then the compression is investigated for attribute reductions of set-valuedinformation systems. Afterwards, we discuss the compression of dynamic set-valued information systems by utilizing the precious compression of the originalsystems. Several illustrative examples are employed to show that attribute re-ductions of set-valued information systems can be simplifed signifcantly by ourproposed approach.
This dissertation has developed efective approaches to approximations of con-cepts and attribute reductions of information systems especially for dynamic in-formation systems. It enriches the rough set theory and attribute reductions ofinformation systems.
在计算属性重要度方面，Yamaguchi为完备信息系统提出了一种新的属性重要度函数，但这个模型在用于不完备信息系统时存在一些问题。为了在不完备信息系统中更精确的计算属性的重要度，我们首先为不完备信息系统定义了三种属性重要度函数，并用12个数据集验证其有效性。然后，利用定义的函数简化不完备信息系统的区分矩阵。数据实验表明我们给出的重要度函数在不完备信息系统中能有效地计算条件属性的重要度。
在概念近似方面，王石平等基于特征矩阵和近似算子之间的关系把计算集合的近似转换成特征矩阵的计算，但他们没有研究如何有效的计算特征矩阵。为了计算概念的近似，我们提出了两种有效计算特征矩阵的方法。然后，基于递增式算法研究了如何计算动态覆盖的特征矩阵。主要包括三种情形：(1)覆盖中等价类的变化；(2)对象集合的变化；(3)属性值的变化。数据实验表明我们提出的方法能够明显地减少计算动态覆盖特征矩阵的时间复杂度。
在覆盖近似空间的压缩方面，为了研究覆盖近似空间之间的关系，我们提出了上、下同态和同态的概念。然后，我们引入覆盖近似子空间和乘积空间的概念，并研究了它们的基本性质。基于覆盖近似空间之间的同态，我们研究了覆盖近似空间和覆盖信息系统的压缩，进而研究了覆盖近似空间的粒约简和覆盖信息系统的属性约简。针对覆盖近似空间和覆盖信息系统的动态性，我们基于原始空间和信息系统的压缩研究了动态覆盖近似空间和信息系统的压缩。数据实验表明我们引入的同态为覆盖近似空间和覆盖信息系统的压缩提供了一种有效的方法。
在集值信息系统的压缩方面，我们提出了三种关系并研究了它们的基本性质。针对集值信息系统的属性约简，我们研究了集值信息系统的数据压缩。然后，基于原始集值信息系统的压缩，研究了动态集值信息系统的压缩。数据实验表明我们提出的方法能够简化集值信息系统属性约简的计算过程。
总之，本文为信息系统的属性约简、动态信息系统的属性约简和计算动态概念的近似提出了一些有效的方法，完善了信息系统属性约简的理论体系，进一步丰富了粗糙集理论。
Rough set theory as an efective mathematical tool to deal with uncertain andimprecise problems has been applied to various areas such as data mining, artifcialintelligence and pattern recognition. Current research regarding rough sets main-ly focuses on two aspects: theory and its applications. Among them, attributereductions is the most important problem for rough set theory. For attribute re-ductions, on one hand, researchers have proposed efective attribute dependencydegree functions. But some functions are not ft for incomplete information sys-tems. On the other hand, on the basis of homomorphisms between informationsystems, researchers have presented an approach to attribute reductions by com-pressing information systems. But there is not much research on other types ofinformation systems. Furthermore, researchers paid a little attention to attributereductions and approximations of concepts with respect to dynamic informationsystems.
In terms of computing the attribute dependency degree, Yamaguchi present-ed a new attribute dependency degree function for complete information systems.But there are some issues on applying it to incomplete information systems. Forcomputing the attribute dependency degree, we introduce three attribute depen-dency degree functions for incomplete information systems, and apply them to12data sets. Then we conduct the simplifcation for discernibility matrixes of incom-plete information systems with the proposed function. The experimental resultsshow that our proposed functions are more fexible to calculate the degree of eachconditional attribute related to the decision attribute for incomplete informationsystems.
With respect to approximations of concepts, Wang Shiping et al. transformedcomputing approximations of concepts into the computing of characteristic matrix-es, but there are few studies on the approach to computing characteristic matrixes.To compute approximation of concepts, we propose two approaches to computingcharacteristic matrixes. Then, by using an incremental approach, we computecharacteristic matrixes of the dynamic covering without running the matrix acqui-sition algorithm repeatedly. We mainly address the characteristic matrix updatingfrom three aspects: the variations of blocks in the covering, the immigration andemigration of objects and the changes of attribute values. Several illustrative exam-ples are employed to show that the time complexity of constructing characteristic matrixes of the dynamic covering can be reduced signifcantly with the proposedapproach.
For the compression of covering approximation spaces, we introduce the con-cepts of upper and lower homomorphisms as well as homomorphisms in order tostudy the relationship between covering approximation spaces. Then the notionsof covering approximation subspaces and product spaces are presented and theirfundamental properties are examined. Afterwards, we investigate the compres-sion of covering approximation spaces and covering information systems with theaim of reductions. Finally, we discuss the compression of dynamic covering ap-proximation spaces and dynamic covering information systems by utilizing thecompressions of the original spaces and systems, respectively. Several illustrativeexamples are employed to demonstrate that the homomorphisms provide an efec-tive approach to the compression of covering approximation spaces and coveringinformation systems.
For the compression of set-valued information systems, we put forward threerelations for set-valued information systems and explore their basic properties indetail. Then the compression is investigated for attribute reductions of set-valuedinformation systems. Afterwards, we discuss the compression of dynamic set-valued information systems by utilizing the precious compression of the originalsystems. Several illustrative examples are employed to show that attribute re-ductions of set-valued information systems can be simplifed signifcantly by ourproposed approach.
This dissertation has developed efective approaches to approximations of con-cepts and attribute reductions of information systems especially for dynamic in-formation systems. It enriches the rough set theory and attribute reductions ofinformation systems.
引文
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