统计流形上的纤维丛与李群结构
发布时间：2018-07-25 07:14
【摘要】：本文研究了统计流形上的纤维丛和李群结构.首先回顾了微分几何的发展历史、信息几何理论的一些基本概念、基本结果.然后回顾了纤维丛理论的基本概念以及纤维丛上的微分几何理论,并将纤维丛结构引进统计流形.统计流形上的几何结构的计算可以转移到其标架丛上,而后者的计算更为简洁,因为它是线性的.以正态分布流形为例,进行了具体计算,验证了前面的结果.本文还研究了具有李群结构的统计流形,给出了统计李群的定义,并证明了几个主要性质.作为统计李群的例子,给出了一元正态分布、指数分布、多元独立正态分布流形上的统计李群结构.此外,提出并证明了两种从已知统计李群构造新的统计李群的方法.作为例子,多个一元正态分布统计李群可以用来构造多元独立正态分布统计李群.最后对全文进行总结.
[Abstract]:In this paper, the fiber bundles and Li Qun structures on the statistical manifold are studied. First, the development history of differential geometry, the basic concepts and the basic results of the information geometry theory are reviewed. Then the basic concepts of the fiber bundle theory and the differential geometry theory on the fiber bundle are reviewed, and the fibrous plexus structure is introduced to the statistical manifold. The calculation of the geometric structure can be transferred to the frame cluster. The calculation of the latter is more concise because it is linear. Taking the normal distribution manifold as an example, the previous results are calculated. The statistical manifold with the Li Qun structure is studied. The definition of the unified plan Li Qun is given, and several main properties are proved. For the example of Li Qun, a statistical Li Qun structure on a manifold of normal distribution, exponential distribution, and multivariate independent normal distribution is given. In addition, two new statistical Li Qun methods from known statistical Li Qun constructs are proposed and proved. As an example, multiple single normal distribution statistics Li Qun can be used to construct a multivariate independent normal distribution. Statistics Li Qun. Finally, a summary of the full text.
【学位授予单位】：北京理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O186.12
，
本文编号：2143043
[Abstract]:In this paper, the fiber bundles and Li Qun structures on the statistical manifold are studied. First, the development history of differential geometry, the basic concepts and the basic results of the information geometry theory are reviewed. Then the basic concepts of the fiber bundle theory and the differential geometry theory on the fiber bundle are reviewed, and the fibrous plexus structure is introduced to the statistical manifold. The calculation of the geometric structure can be transferred to the frame cluster. The calculation of the latter is more concise because it is linear. Taking the normal distribution manifold as an example, the previous results are calculated. The statistical manifold with the Li Qun structure is studied. The definition of the unified plan Li Qun is given, and several main properties are proved. For the example of Li Qun, a statistical Li Qun structure on a manifold of normal distribution, exponential distribution, and multivariate independent normal distribution is given. In addition, two new statistical Li Qun methods from known statistical Li Qun constructs are proposed and proved. As an example, multiple single normal distribution statistics Li Qun can be used to construct a multivariate independent normal distribution. Statistics Li Qun. Finally, a summary of the full text.
【学位授予单位】：北京理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O186.12
，
本文编号：2143043
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