矩阵型灰色关联分析建模技术研究
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摘要
灰色系统理论是研究贫信息不确定性系统的理论,灰色关联分析是灰色系统理论的重要组成部分,在系统辨识、优势分析、聚类评估、模式识别、数据挖掘等方面得到了广泛的应用,展现出了重要的学术意义和实际应用价值。尽管灰色关联分析理论已经取得一些成果,但是在理论与方法研究方面还有不完善之处。
本文采用定量研究与实例分析相结合的方法,讨论多维对象数据描述方法,建立了较为完整的多维对象行为矩阵理论基础,深入研究矩阵灰色关联分析公理和性质体系,构造较为完整的矩阵关联度模型工具群,并进行相关的应用研究。本文的主要研究成果有:
(1)推广关联分析适用范围,探讨多维对象行为矩阵理论。将灰色关联分析的对象从系统行为序列推广到行为矩阵和矩阵序列,系统地分析行为矩阵的数据结构、数据规模、物理意义;构造行为矩阵的初值化、初始零化、均值化、紧邻生成等算子,探讨多元序列数据预处理和残缺数据修补的方法;提出行为矩阵的曲面描述方法和矩阵序列的规则正六面体示意方法,形成较为完善的行为矩阵理论体系,为模型构造奠定了理论基础。
(2)继承关联分析基本思想,拓展经典序列关联度模型。对传统序列关联度中应用广泛、性质良好、性能优越的邓氏关联度、绝对关联度、几何相似关联度等模型进行拓展。在继承关联分析基本思想的基础上,将上述模型推广至三维、四维数据空间中,并证明序列邓氏、绝对、几何相似等关联度模型是其对应三维、四维模型的特殊形式,拓展模型则是上述序列模型在多维空间中的自然延伸。在模型拓展的同时,采用粒子群算法对矩阵一般关联度模型进行参数优化,提高关联分析的区分度,并构造适用于灰矩阵的关联度模型。
(3)适应多维对象数据特征,构造新的矩阵关联分析工具。根据系统因素行为矩阵的数据结构,针对序列拓展模型的缺陷和不足,将行为矩阵的局部体积特征、行为曲面的法向量、对应点增量向量作为构造灰色关联系数的物理量,分别构造适应行为矩阵数据结构的体积关联度、法向量关联度和增量关联度模型以及矩阵序列的增量模型。从相近性、相似性、综合度量三个方面系统的研究关联度模型,形成较为完备的多维数据关联分析模型体系。
(4)完善关联分析公理体系,挖掘矩阵关联分析模型性质。对序列关联分析四公理进行推广,并将序列关联分析中的平行性、一致性、仿射变换保序性推广到多维空间中,提出完整的矩阵关联度模型公理体系和性质框架,为关联度建模提供理论指导和检验标准。在模型构造中注重模型的性质研究,研究表明:本文构造的一般关联度、绝对关联度、体积关联度等相近性关联度满足平行性要求,梯度、比值等相似性关联度模型满足一致性要求。特别地,梯度关联度模型同时满足平行性、一致性和仿射变换不变性,是一种性质较完备的关联度模型。
(5)借助多元序列测试数据,对比模型分析性能。为检验文中模型性能,客观比较模型优劣,借用多元时间序列分析中常用的小规模测试数据集REF和大规模测试数据集EEG,对本文提出的三类相近性关联分析模型进行测试,记录性能指标,并与多元时间序列挖掘中的PCA方法、Euclid方法进行比较。结果表明:体积关联度、矩阵一般关联度等模型在小规模行为矩阵相似性分析中具有较好的性能表现,是处理小规模多维数据的新方法;矩阵关联度模型对于大规模数据矩阵的处理能力较弱,不能直接应用于大规模行为矩阵的相关性分析。
(6)注重理论与实践相结合,解决管理决策中的难题。以解决广泛存在于经济管理中的多元相关性问题为研究背景,将构造的关联分析模型应用于面板数据聚类分析、金融多元时间序列分析、动态多属性决策中,取得了良好的应用效果,解决了传统数据分析和决策方法不能反映数据动态化过程的问题。最后,根据研究的模型和方法,开发出矩阵型灰色关联分析软件,为进一步的研究和应用提供交互性能良好的工具。
The grey relational analysis was an important part of grey system theory which study the uncertainty and poor information systems. Grey relational analysis(GRA) has been widely applied in system analysis, advantage analysis, data clustering, pattern identification and data mining. The application proved important academic and practical value of GRA. Although GRA theory has achieved many success, but there were still some imperfections.
In the thesis, Multidimensional objects data description methods was discussed, more comprehensive multidimensional object behavior matrix theory was established. On the basis, matrix analysis axiom, property, modeling and application were studied deeply, with quantitative research and case study method.The main contents of this thesis were:
First,application of grey relational analysis was extended,and the behavior matrix theory of multi-dimensional object was explored. The object of GRA was extended from behavior sequence to the behavior of matrix and matrix sequence. Then, The behavior matrix data structure, data size and the physical meaning were analyzed. Meanwhile behavior matrix initial operator, the starting zero operator, neighbor operator, data preprocessing and repair methods for behavior matrix also were disscussed. Last, behavior matrix and matrix sequence description methods were proposed, a comparatively perfect behavior matrix data model system was formed to lay the theoretical foundation for matrix GRA construction.
Second, application scope of GRA models was extended by inheriting the basic thought of GAR. Some superior traditional model, for example: Deng-si, absolute, incidence and so on, were developed. On the basic idea of GRA, these models were extended to three-dimensional and four-dimensional data space, and these exeteded models were improved the general forms of the original ones for sequences. At the same time, parameters of general matrix incidence degree was optimized for improving the discrimination of analysis by the particle swarm algorithm, and GRA model for grey matrix aslo was constructed.
Thired, adapting multidimensional objects data characteristics, some new matrix GRA models were constructed. Accroding to shortcomings of extended models, volume, normal vector and incremental incidence degree model were constructed depending on behavior of the local volume matrix, the behavior of the surface normal vector, and incremental vector feature. Then, proximity, similarity and comprehensive GRA models were study,,to form a more complete multidimensional data GRA model system.
Forth, relational analysis axiom system was perfected, new GRA models’properties were studied deeply. The sequence relational analysis axioms, parallel, consistent and affine transform rank properties were extended to multidimensional space, then a complete matrix model axiom system and property framework were formed to guide and test models. In modeling, more attention was paid to the property research. These studies show that: Deng-si, absolute and volume matrix GRA models meet the requirements in parallel, gradient, ratios and similar models satisfied the consistency preperty. In particular, the gradient model,which meet parallel, consistency and invariance of affine transform perprety, was an excellent matrix GRA model.
Fifth, With multi-sequence test data, model performance was compared. In order to test the performance of the models, and to compare merits of models objectively, three proposed proximity model was tested, their performance was recorded, and the results were compared with PCA and Euclid methods, in virtue of small-scale test data set REF multiple sequence and large-scale multi-sequence test data sets EEG which were commonly used in multivariate time series analysis. The results showed that: the volume and general matrix GRA models perform excellently in similarity analysis for small-scale matrix, so they were new method dealing with small-scale multidimensional data. But matrix GRA model performed weakly in large-scale data, and can not be directly used for large-scale behavior matrix.
Sixth, more attention was paied to practice, and some challenges of management were solved. To solve multiple issues which were widespread in the economic management, the proposed matrix GRA model was applied to panel data analysis, financial multivariate time series analysis, dynamic multi-attribute decision making. The applications achieved better results, and solve the problem that the traditional methods of data analysis and decision-making method did not reflect the dynamicprocess.
本文采用定量研究与实例分析相结合的方法,讨论多维对象数据描述方法,建立了较为完整的多维对象行为矩阵理论基础,深入研究矩阵灰色关联分析公理和性质体系,构造较为完整的矩阵关联度模型工具群,并进行相关的应用研究。本文的主要研究成果有:
(1)推广关联分析适用范围,探讨多维对象行为矩阵理论。将灰色关联分析的对象从系统行为序列推广到行为矩阵和矩阵序列,系统地分析行为矩阵的数据结构、数据规模、物理意义;构造行为矩阵的初值化、初始零化、均值化、紧邻生成等算子,探讨多元序列数据预处理和残缺数据修补的方法;提出行为矩阵的曲面描述方法和矩阵序列的规则正六面体示意方法,形成较为完善的行为矩阵理论体系,为模型构造奠定了理论基础。
(2)继承关联分析基本思想,拓展经典序列关联度模型。对传统序列关联度中应用广泛、性质良好、性能优越的邓氏关联度、绝对关联度、几何相似关联度等模型进行拓展。在继承关联分析基本思想的基础上,将上述模型推广至三维、四维数据空间中,并证明序列邓氏、绝对、几何相似等关联度模型是其对应三维、四维模型的特殊形式,拓展模型则是上述序列模型在多维空间中的自然延伸。在模型拓展的同时,采用粒子群算法对矩阵一般关联度模型进行参数优化,提高关联分析的区分度,并构造适用于灰矩阵的关联度模型。
(3)适应多维对象数据特征,构造新的矩阵关联分析工具。根据系统因素行为矩阵的数据结构,针对序列拓展模型的缺陷和不足,将行为矩阵的局部体积特征、行为曲面的法向量、对应点增量向量作为构造灰色关联系数的物理量,分别构造适应行为矩阵数据结构的体积关联度、法向量关联度和增量关联度模型以及矩阵序列的增量模型。从相近性、相似性、综合度量三个方面系统的研究关联度模型,形成较为完备的多维数据关联分析模型体系。
(4)完善关联分析公理体系,挖掘矩阵关联分析模型性质。对序列关联分析四公理进行推广,并将序列关联分析中的平行性、一致性、仿射变换保序性推广到多维空间中,提出完整的矩阵关联度模型公理体系和性质框架,为关联度建模提供理论指导和检验标准。在模型构造中注重模型的性质研究,研究表明:本文构造的一般关联度、绝对关联度、体积关联度等相近性关联度满足平行性要求,梯度、比值等相似性关联度模型满足一致性要求。特别地,梯度关联度模型同时满足平行性、一致性和仿射变换不变性,是一种性质较完备的关联度模型。
(5)借助多元序列测试数据,对比模型分析性能。为检验文中模型性能,客观比较模型优劣,借用多元时间序列分析中常用的小规模测试数据集REF和大规模测试数据集EEG,对本文提出的三类相近性关联分析模型进行测试,记录性能指标,并与多元时间序列挖掘中的PCA方法、Euclid方法进行比较。结果表明:体积关联度、矩阵一般关联度等模型在小规模行为矩阵相似性分析中具有较好的性能表现,是处理小规模多维数据的新方法;矩阵关联度模型对于大规模数据矩阵的处理能力较弱,不能直接应用于大规模行为矩阵的相关性分析。
(6)注重理论与实践相结合,解决管理决策中的难题。以解决广泛存在于经济管理中的多元相关性问题为研究背景,将构造的关联分析模型应用于面板数据聚类分析、金融多元时间序列分析、动态多属性决策中,取得了良好的应用效果,解决了传统数据分析和决策方法不能反映数据动态化过程的问题。最后,根据研究的模型和方法,开发出矩阵型灰色关联分析软件,为进一步的研究和应用提供交互性能良好的工具。
The grey relational analysis was an important part of grey system theory which study the uncertainty and poor information systems. Grey relational analysis(GRA) has been widely applied in system analysis, advantage analysis, data clustering, pattern identification and data mining. The application proved important academic and practical value of GRA. Although GRA theory has achieved many success, but there were still some imperfections.
In the thesis, Multidimensional objects data description methods was discussed, more comprehensive multidimensional object behavior matrix theory was established. On the basis, matrix analysis axiom, property, modeling and application were studied deeply, with quantitative research and case study method.The main contents of this thesis were:
First,application of grey relational analysis was extended,and the behavior matrix theory of multi-dimensional object was explored. The object of GRA was extended from behavior sequence to the behavior of matrix and matrix sequence. Then, The behavior matrix data structure, data size and the physical meaning were analyzed. Meanwhile behavior matrix initial operator, the starting zero operator, neighbor operator, data preprocessing and repair methods for behavior matrix also were disscussed. Last, behavior matrix and matrix sequence description methods were proposed, a comparatively perfect behavior matrix data model system was formed to lay the theoretical foundation for matrix GRA construction.
Second, application scope of GRA models was extended by inheriting the basic thought of GAR. Some superior traditional model, for example: Deng-si, absolute, incidence and so on, were developed. On the basic idea of GRA, these models were extended to three-dimensional and four-dimensional data space, and these exeteded models were improved the general forms of the original ones for sequences. At the same time, parameters of general matrix incidence degree was optimized for improving the discrimination of analysis by the particle swarm algorithm, and GRA model for grey matrix aslo was constructed.
Thired, adapting multidimensional objects data characteristics, some new matrix GRA models were constructed. Accroding to shortcomings of extended models, volume, normal vector and incremental incidence degree model were constructed depending on behavior of the local volume matrix, the behavior of the surface normal vector, and incremental vector feature. Then, proximity, similarity and comprehensive GRA models were study,,to form a more complete multidimensional data GRA model system.
Forth, relational analysis axiom system was perfected, new GRA models’properties were studied deeply. The sequence relational analysis axioms, parallel, consistent and affine transform rank properties were extended to multidimensional space, then a complete matrix model axiom system and property framework were formed to guide and test models. In modeling, more attention was paid to the property research. These studies show that: Deng-si, absolute and volume matrix GRA models meet the requirements in parallel, gradient, ratios and similar models satisfied the consistency preperty. In particular, the gradient model,which meet parallel, consistency and invariance of affine transform perprety, was an excellent matrix GRA model.
Fifth, With multi-sequence test data, model performance was compared. In order to test the performance of the models, and to compare merits of models objectively, three proposed proximity model was tested, their performance was recorded, and the results were compared with PCA and Euclid methods, in virtue of small-scale test data set REF multiple sequence and large-scale multi-sequence test data sets EEG which were commonly used in multivariate time series analysis. The results showed that: the volume and general matrix GRA models perform excellently in similarity analysis for small-scale matrix, so they were new method dealing with small-scale multidimensional data. But matrix GRA model performed weakly in large-scale data, and can not be directly used for large-scale behavior matrix.
Sixth, more attention was paied to practice, and some challenges of management were solved. To solve multiple issues which were widespread in the economic management, the proposed matrix GRA model was applied to panel data analysis, financial multivariate time series analysis, dynamic multi-attribute decision making. The applications achieved better results, and solve the problem that the traditional methods of data analysis and decision-making method did not reflect the dynamicprocess.
引文
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