粒度计算的模型研究
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摘要
粒度计算的思想起源于上世纪70年代末，它是模仿人类思考问题的方式，正如张钹院士和张铃教授所说：“人类智能的一个公认的特点，就是人们能从极不相同的粒度上观察和分析同一问题。人们不仅能在不同粒度(granularity)的世界上进行问题的求解，而且能够很快地从一个粒度世界跳到另一个粒度世界，往返自如，毫无困难。这种处理不同粒度世界的能力，正是人类问题求解的强有力的表现。”它象一把大伞，覆盖了所有有关粒度的理论、方法论、技术和工具的研究，是用来处理不完全、不可靠、不精确、不一致和不确定的知识。它是人工智能研究的最重要基础，现已成为人工智能领域的热点之一，主要包括商空间理论、粗糙集理论和模糊集理论等。本文的主要内容如下：
1．介绍了粒度计算的产生与发展背景，粒度计算的研究现状和主要理论，并对商空间理论、粗糙集理论和模糊集理论作了祥细的比较，说明了本文的研究背景及意义。
2．对粒度计算的基础进行了研究，主要包括粒度的描述，给出了粒度的宏观结构层次图和微观结构层次图。给出了最一般的粒的定义，并在商空间理论、粗糙集理论以及模糊集理论的基础上提出了粒度空间的动态模型，讨论了粒的基本性质。重点研究了粒的度量，从研究问题的需要出发，给出了普通等价关系下的粒度、细度、粒度熵、条件粒度、条件细度以及条件粒度熵的定义，指出了定义的合理性，并得到了相应的性质，揭示出了它们之间的关系。还讨论了无限集的度量，给出了粒度的一般性定义思想。
3．介绍了模糊关系矩阵的主要性质，通过关系矩阵给出任意关系下的粒度、细度、粒度熵、条件粒度、条件细度和条件粒度熵的一般性定义。通过关系矩阵定义了贴近度和差异度，并讨论了它们的主要性质。从而可以将商空间理论、粗糙集理论和模糊集理论统一起来进行研究。
4．定义并讨论了粒的基本运算：商交、商并、商非和商差等，并给出了这些运算的主要性质。还研究了粒的知识空间，根据知识的宏观性或微观性可以得到相应的宏观或微观的知识空间。给出知识基的定义，可以通过知识基来定义或描述知识空间，形象地给出了粗糙集的一个属性约简的知识空间算法(求最优约简的算法)。
5．介绍了粒度空间的构成，商粒的构成方法主要有属性划分法、结构划分法、约束划分法和综合划分法等；当论域是拓扑结构和半序结构时，商结构的构成方法；当论域有结构和无结构时，商属性的构成方法。随后讨论了粒度空间的主要运算和主要原理。
6．研究了有结构的知识空间，在研究有结构的知识空间时，不但要考虑粒的商交和商并，还要考虑结构的“商交”和“商并”，在研究其知识基时，不但要考虑到粒的知识基，还要考虑到结构的知识基。一般当结构只有一种时，问题可以简化为无结构的知识空间，当结构有多种时，对应的知识空间要复杂得多。并以拓扑结构为例研究了有结构的知识空间，这样的知识空间是由粒度基和拓扑基共同生成的。
7．着重从与以往不同的角度，即着重从结构方面去探索粒度计算的应用问题。从空间坐标上取粒度对时间序列进行分析，得出：当系统是一个马尔科夫链时，对此马尔科夫链进行粗粒度观察，将得到一个对应的隐马尔科夫模型；反之，任一HMM必存在一个对应的马尔科夫链M及一个适当的粒度T，HMM是M在粒度T下的观察。
8．讨论构造性学习方法在AVE局部模型划分上的应用，初步研究了基于结构的粒度问题，包括结构关系矩阵的构造；基于结构的粒度的划分；商结构关系矩阵的构成；运用构造性学习方法进行粒度划分等。
The idea of granular computing (GrC) emerged in the late 1970's. It imitates the manner of human thinking, just as Zhang and Zhang said: A well known feature of human intelligence is that human can not only observe and analyze a problem at different grain-sizes but also translate from one granule world to the others with no difficulty. As a new tool dealing with incomplete, uncertain, imprecise and inconsistent knowledge, GrC is a big umbrella which covers all the research of the theories, methodologies, technologies, and tools about granules. It is an important base of artificial intelligence and now becomes a hot research topic domestically and abroad which includes quotient space theory, rough set theory and fuzzy set theory. The thesis covers the following aspects.
1. The background, status and main theories of GrC are introduced. The relations and differences among quotient space theory, rough set theory and fuzzy set theory are discussed in detail, thus explaining the research background and significance of the thesis.
2. We research into the basics of GrC, including the description and measurement of granules. We present the micro and macro hierarchical figure and the general definition of granules. Then we introduce the dynamic model of granular space on the basis of quotient space theory, rough set theory and fuzzy set theory and discuss its main properties. We place emphasis on the measurement of granules and define the granularity, fineness, granularity entropy, conditional granularity, conditional fineness and conditional granularity entropy under a normal equivalence relation. Then we analyze these definitions and get their basic properties. Besides, we also discuss the measurement of infinite set and put forward a general idea concerning the definition of granules.
3. The main characteristics of fuzzy relation matrix are introduced. By the relation matrix we present the general definitions of (conditional) granularity. (conditional) fineness, and (conditional) granularity entropy tinder any relation. We also define closeness and difference between two granules and discuss their properties. In this way, the quotient space theory, rough set theory and fuzzy set theory are unified.
4. The basic operations such as quotient intersection, quotient union, quotient not, quotient subtraction and so on are defined and analyzed.. The knowledge space of granules is researched. We can obtain its corresponding macro and micro knowledge space according to macro and micro characteristic of knowledge. We introduce the definition of knowledge base, picture the knowledge space by the base, and thus visually get a knowledge space algorithm of feature reduction in rough set theory (the optimal feature reduction algorithm).
5. Constructing methods of granular space are introduced: constructing methods of quotient granular including feature partition, structure partition, constraint partition and so on; constructing methods of quotient structure when the domain is a topological or semi-order structure; constructing method of quotient feature when the domain has or has no structure. Then the main operations and principle of granular space are discussed.
6. The knowledge space with structure is discussed. We not only think about the quotient intersection and quotient union of granules, but also the quotient intersection and quotient union of structures. When we research on the knowledge base, we should think about both the base of granules and that of structures. In general when there is only one kind of structure, a problem is simplified as a knowledge space with no structure; when there are several kinds of structures, the corresponding knowledge space is much more complicated. Then taking the topological structure as an example we research on the knowledge space with structure, which is produced by granular base and topological base.
7. We investigate the applications of granular computing focusing on structure. We carry out granularity analysis of time sequence based on space, and get the following result: when a system is a Markov chain and is observed in a coarser-grain space, we can get a corresponding HMM; on the contrary, for any HMM, there must be a corresponding Markov chain M and a relevant granularity T, that is, HMM is the observation of M based on T.
8. We discuss the application of constructing learning method in AVE local model partition and study granular problems based on structure including constructing of structure relation matrix, granular partition based on structure, constructing of relation matrix of quotient structure, carrying out granular partition by constructing learning methods, and so on.
1．介绍了粒度计算的产生与发展背景，粒度计算的研究现状和主要理论，并对商空间理论、粗糙集理论和模糊集理论作了祥细的比较，说明了本文的研究背景及意义。
2．对粒度计算的基础进行了研究，主要包括粒度的描述，给出了粒度的宏观结构层次图和微观结构层次图。给出了最一般的粒的定义，并在商空间理论、粗糙集理论以及模糊集理论的基础上提出了粒度空间的动态模型，讨论了粒的基本性质。重点研究了粒的度量，从研究问题的需要出发，给出了普通等价关系下的粒度、细度、粒度熵、条件粒度、条件细度以及条件粒度熵的定义，指出了定义的合理性，并得到了相应的性质，揭示出了它们之间的关系。还讨论了无限集的度量，给出了粒度的一般性定义思想。
3．介绍了模糊关系矩阵的主要性质，通过关系矩阵给出任意关系下的粒度、细度、粒度熵、条件粒度、条件细度和条件粒度熵的一般性定义。通过关系矩阵定义了贴近度和差异度，并讨论了它们的主要性质。从而可以将商空间理论、粗糙集理论和模糊集理论统一起来进行研究。
4．定义并讨论了粒的基本运算：商交、商并、商非和商差等，并给出了这些运算的主要性质。还研究了粒的知识空间，根据知识的宏观性或微观性可以得到相应的宏观或微观的知识空间。给出知识基的定义，可以通过知识基来定义或描述知识空间，形象地给出了粗糙集的一个属性约简的知识空间算法(求最优约简的算法)。
5．介绍了粒度空间的构成，商粒的构成方法主要有属性划分法、结构划分法、约束划分法和综合划分法等；当论域是拓扑结构和半序结构时，商结构的构成方法；当论域有结构和无结构时，商属性的构成方法。随后讨论了粒度空间的主要运算和主要原理。
6．研究了有结构的知识空间，在研究有结构的知识空间时，不但要考虑粒的商交和商并，还要考虑结构的“商交”和“商并”，在研究其知识基时，不但要考虑到粒的知识基，还要考虑到结构的知识基。一般当结构只有一种时，问题可以简化为无结构的知识空间，当结构有多种时，对应的知识空间要复杂得多。并以拓扑结构为例研究了有结构的知识空间，这样的知识空间是由粒度基和拓扑基共同生成的。
7．着重从与以往不同的角度，即着重从结构方面去探索粒度计算的应用问题。从空间坐标上取粒度对时间序列进行分析，得出：当系统是一个马尔科夫链时，对此马尔科夫链进行粗粒度观察，将得到一个对应的隐马尔科夫模型；反之，任一HMM必存在一个对应的马尔科夫链M及一个适当的粒度T，HMM是M在粒度T下的观察。
8．讨论构造性学习方法在AVE局部模型划分上的应用，初步研究了基于结构的粒度问题，包括结构关系矩阵的构造；基于结构的粒度的划分；商结构关系矩阵的构成；运用构造性学习方法进行粒度划分等。
The idea of granular computing (GrC) emerged in the late 1970's. It imitates the manner of human thinking, just as Zhang and Zhang said: A well known feature of human intelligence is that human can not only observe and analyze a problem at different grain-sizes but also translate from one granule world to the others with no difficulty. As a new tool dealing with incomplete, uncertain, imprecise and inconsistent knowledge, GrC is a big umbrella which covers all the research of the theories, methodologies, technologies, and tools about granules. It is an important base of artificial intelligence and now becomes a hot research topic domestically and abroad which includes quotient space theory, rough set theory and fuzzy set theory. The thesis covers the following aspects.
1. The background, status and main theories of GrC are introduced. The relations and differences among quotient space theory, rough set theory and fuzzy set theory are discussed in detail, thus explaining the research background and significance of the thesis.
2. We research into the basics of GrC, including the description and measurement of granules. We present the micro and macro hierarchical figure and the general definition of granules. Then we introduce the dynamic model of granular space on the basis of quotient space theory, rough set theory and fuzzy set theory and discuss its main properties. We place emphasis on the measurement of granules and define the granularity, fineness, granularity entropy, conditional granularity, conditional fineness and conditional granularity entropy under a normal equivalence relation. Then we analyze these definitions and get their basic properties. Besides, we also discuss the measurement of infinite set and put forward a general idea concerning the definition of granules.
3. The main characteristics of fuzzy relation matrix are introduced. By the relation matrix we present the general definitions of (conditional) granularity. (conditional) fineness, and (conditional) granularity entropy tinder any relation. We also define closeness and difference between two granules and discuss their properties. In this way, the quotient space theory, rough set theory and fuzzy set theory are unified.
4. The basic operations such as quotient intersection, quotient union, quotient not, quotient subtraction and so on are defined and analyzed.. The knowledge space of granules is researched. We can obtain its corresponding macro and micro knowledge space according to macro and micro characteristic of knowledge. We introduce the definition of knowledge base, picture the knowledge space by the base, and thus visually get a knowledge space algorithm of feature reduction in rough set theory (the optimal feature reduction algorithm).
5. Constructing methods of granular space are introduced: constructing methods of quotient granular including feature partition, structure partition, constraint partition and so on; constructing methods of quotient structure when the domain is a topological or semi-order structure; constructing method of quotient feature when the domain has or has no structure. Then the main operations and principle of granular space are discussed.
6. The knowledge space with structure is discussed. We not only think about the quotient intersection and quotient union of granules, but also the quotient intersection and quotient union of structures. When we research on the knowledge base, we should think about both the base of granules and that of structures. In general when there is only one kind of structure, a problem is simplified as a knowledge space with no structure; when there are several kinds of structures, the corresponding knowledge space is much more complicated. Then taking the topological structure as an example we research on the knowledge space with structure, which is produced by granular base and topological base.
7. We investigate the applications of granular computing focusing on structure. We carry out granularity analysis of time sequence based on space, and get the following result: when a system is a Markov chain and is observed in a coarser-grain space, we can get a corresponding HMM; on the contrary, for any HMM, there must be a corresponding Markov chain M and a relevant granularity T, that is, HMM is the observation of M based on T.
8. We discuss the application of constructing learning method in AVE local model partition and study granular problems based on structure including constructing of structure relation matrix, granular partition based on structure, constructing of relation matrix of quotient structure, carrying out granular partition by constructing learning methods, and so on.
引文
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