發(fā)布時間：2018-07-25 07:14

【摘要】：本文研究了統(tǒng)計流形上的纖維叢和李群結(jié)構(gòu).首先回顧了微分幾何的發(fā)展歷史、信息幾何理論的一些基本概念、基本結(jié)果.然后回顧了纖維叢理論的基本概念以及纖維叢上的微分幾何理論,并將纖維叢結(jié)構(gòu)引進統(tǒng)計流形.統(tǒng)計流形上的幾何結(jié)構(gòu)的計算可以轉(zhuǎn)移到其標架叢上,而后者的計算更為簡潔,因為它是線性的.以正態(tài)分布流形為例,進行了具體計算,驗證了前面的結(jié)果.本文還研究了具有李群結(jié)構(gòu)的統(tǒng)計流形,給出了統(tǒng)計李群的定義,并證明了幾個主要性質(zhì).作為統(tǒng)計李群的例子,給出了一元正態(tài)分布、指數(shù)分布、多元獨立正態(tài)分布流形上的統(tǒng)計李群結(jié)構(gòu).此外,提出并證明了兩種從已知統(tǒng)計李群構(gòu)造新的統(tǒng)計李群的方法.作為例子,多個一元正態(tài)分布統(tǒng)計李群可以用來構(gòu)造多元獨立正態(tài)分布統(tǒng)計李群.最后對全文進行總結(jié).
[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.
【學(xué)位授予單位】：北京理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O186.12
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本文編號：2143043