高斯过程协方差函数的正定性与Karhunen-Loeve展开

国内图书分类号：O211.61 国际图书分类号：519.21
பைடு நூலகம்
学校代码：10213 密级： 公开
理学硕士学位论文
高斯过程协方差函数的正定性与 Karhunen-Loeve 展开
硕 士 研 究 生 ：何昀锋 导 师：刘国庆副教授
申 请 学 位：理学硕士 学 科：概率论与数理统计
所 在 单 位：理学院数学系 答 辩 日 期：2011 年 6 月 授予学位单位：哈尔滨工业大学
Degree-Conferring-Institution： Harbin Institute of Technology
哈尔滨工业大学理学硕士学位论文
摘
要
随机过程是研究偶然现象的重要手段。高斯过程是最重要的随机过程。它是 信号处理的重要工具，与核函数、紧算子、小偏差概率联系紧密，且有十分良好 的性质：线性组合正定性、协方差函数的正定性、独立与不相关等价性等等。 本文“高斯过程协方差函数的正定性与 Karhunen-Loeve 展开”以高斯过程为 核心，讨论了几类协方差函数的正定性，及其 Karhunen-Loeve 展开，最后用它的 协方差函数构造了相应的再生核希尔伯特空间。 本文主要研究内容如下： （1）讨论了复数域上的正定函数的定义、性质、判别法。研究了利用内积的 分解判别法和利用半正定矩阵的判别法，并且根据矩阵构造了正定函数。 （2）研究了高斯过程协方差函数的正定性，得到了几类高斯过程的 Karhunen-Loeve 展开。 （3） 根据高斯过程 Karhunen-Loeve 展开与再生核希尔伯特空间的联系构造了 高斯过程对应的再生核希尔伯特空间。 本文结构如下：第一章绪论介绍了本课题的背景、现状等相关情况；第二章 系统研究了二元函数的正定性；第三章给出了几类高斯过程的 Karhunen-Loeve 展 开；第四章用第三章的高斯过程构造对应的再生核希尔伯特空间。 关键词：正定函数；正定矩阵；Karhunen-Loeve 展开；高斯过程；再生核希尔伯 特空间
Abstract ...........................................................................................................................II 第1章 绪 1.1 课题来源背景及研究的目的和意义 ................................................................. 1 1.1.1 课题的来源及背景 ...................................................................................... 1 1.1.2 本文的研究目的 .......................................................................................... 1 1.2 国内外现状 ......................................................................................................... 2 1.3 本文的主要研究内容 ......................................................................................... 3 第 2 章 正定函数 ............................................................................................................ 4 2.1 正定函数的定义与性质 ..................................................................................... 4 2.2 正定函数的判别 ................................................................................................. 7 2.3 本章小结 ........................................................................................................... 15 第 3 章 几类高斯过程的Karhunen-Loeve展开 ........................................................... 16 3.1 基本定义 ........................................................................................................... 16 3.2 高斯过程协方差函数的正定性及Karhunen-Loeve展开 ................................ 17 3.3 几类高斯过程的Karhunen-Loeve展开 ............................................................ 19 3.4 本章小结 ........................................................................................................... 27 第 4 章 再生核希尔伯特空间 ...................................................................................... 28 4.1 再生核函数 ....................................................................................................... 28 4.1.1 再生核函数的定义 .................................................................................... 28 4.2 再生核函数的性质 ........................................................................................... 29 4.3 高斯过程对应的再生核希尔伯特空间 ........................................................... 31 4.4 本章小节 ........................................................................................................... 37 结 论 ............................................................................................................................ 38 参考文献 ........................................................................................................................ 39 哈尔滨工业大学学位论文原创性声明及使用授权说明 ............................................ 43 学位论文出版授权书 .................................................................................................... 44 致 谢 ............................................................................................................................ 45
Classified Index:O211.61 U.D.C.:519.21
Dissertation for the Master Degree in science
KARHUNEN-LOEVE EXPANSIONS AND POSITIVE DEFINITE OF COVARIANCE FUNCTION OF GAUSSIAN PROCESS
II
哈尔滨工业大学理学硕士学位论文
目
摘
录
要 ............................................................................................................................... I 论 ................................................................................................................ 1
硕士学位论文
高斯过程协方差函数的正定性与 Karhunen-Loeve 展开
KARHUNEN-LOEVE EXPANSIONS AND POSITIVE DEFINITE OF COVARIANCE FUNCTION OF GAUSSIAN PROCESS 何昀锋
哈尔滨工业大学 2011 年 6 月
Candidate： Supervisor：
He Yunfeng Assoc. Prof. Liu Guoqing
Academic Degree Applied for： Master of Science Speciality： Probability Theory and Mathematical Statistics Affiliation： Date of Defence： Department of Mathematics June, 2011
I
哈尔滨工业大学理学硕士学位论文
Abstract
A random process is an important means of researching an accidental phenomenon. Gaussian process is the most important random process. It is the important tool of signal processing, closely link to kernel function, compact operators and small deviation probability. And there are very good characteres: the linear combination and covariance function is positive definite, independent is equivalent to unrelated, etc. In this dissertation, the gaussian process are the core, we discussed several types of the covariance function positive definite, and Karhunen-Loeve expansions, finally constructed corresponding reproducing kernel Hilbert spaces with its covariance function. This paper mainly researches the following problems: (1) researches the positive definite function's definition, character, discriminant method on complex field. By using the inner product of the decomposition and half a positive definite matrices, we researched positive definite function's discriminant, and according to the positive definite matrix constructed the positive definite function. (2) researches the positive definite of gaussian process covariance function, and obtained some kinds of gaussian process's Karhunen-Loeve expansion. (3) according to the gaussian process Karhunen-Loeve expansion contact with reproducing kernel Hilbert spaces, constructed the reproducing kernel Hilbert spaces correspond with the gaussian process. This paper's structure is as follows: In the first chapter, we introduced this topic's background, present situation and other relevant circumstances; in the second chapter, we systematically researched the positive definite of binary function; in the third chapter, we given several types of Karhunen-Loeve expansion of gaussian process; in the fourth chapter, we structured corresponding reproducing kernel Hilbert space with gaussian process which is yielded in third chapter. Keywords ： positive definite function ， positive definite matrix ， Karhunen-Loeve expansion，gaussian process，reproducing kernel Hilbert spaces