Application of Elementary Differential Geometry to Influence Analysis (微分几何在影响分析

样章

作者：

潘日新潘伟贤著

定价：

59.00 元

版面字数：

250千字

开本：

16开

装帧形式：

精装

版次：

1

最新版次
印刷时间：

2012年

ISBN：

978-7-04-035700-4

物料号：

35700-00

出版时间：

2012-08-24

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

数学与统计其他

《微分几何在影响分析中的应用（英文版）》讨论微分几何在统计学影响分析中的应用，适合数学及统计学本科生或研究生阅读。对于研习数学的学生，本书描述微分几何在数学范畴以外的具体应用；对于研习统计的学生，本书则能帮助他们理解统计领域中的微分几何概念。

《微分几何在影响分析中的应用（英文版）》要求读者具备线性代数及向量微积分的基础知识。书的第一部分围绕法曲率、截面曲率和高斯曲率概念介绍了图的几何学知识；第二部分回顾了统计学的一些基本概念及模型，为理解影响分析提供必要的基础知识；第三部分则集中讨论上述几何概念在局部影响分析中的应用，并探讨如何有效地应用几何概念以提高局部影响分析估计的效力。

《微分几何在影响分析中的应用（英文版）》为研习统计学或数学的学生架起了知识理解的桥梁，为数学与统计学的跨学科研究合作及相互推进发挥创新性的作用。

目录
- 前辅文
- Part I Geometry
  - 1 Preliminaries
    - 1.1 Linear algebra
      - 1.1.1 Vectors and matrices
      - 1.1.2 Symmetric bilinear forms
      - 1.1.3 Vector subspaces
      - 1.1.4 Linear maps from Rn to Rn
      - 1.1.5 A convention
    - 1.2 Vector calculus
      - 1.2.1 Vector-valued functions and differentials
      - 1.2.2 Taylor expansion and extrema
      - 1.2.3 Extrema and Lagrange multiplier theorem
  - 2 Euclidean Geometry
    - 2.1 Orthogonal transformations
    - 2.2 Rigid motions
    - 2.3 Translation of vector subspaces
    - 2.4 Conformal transformations
    - 2.5 Orthonormal basis
    - 2.6 Orthogonal projections
    - 2.7 Areas and volumes
  - 3 Geometry of Graphs
    - 3.1 Graphs in Euclidean spaces
    - 3.2 Normal sections
    - 3.3 Cross sections in high dimension
    - 3.4 First fundamental forms
  - 4 Curvatures
    - 4.1 Normal curvatures
      - 4.1.1 Definition
      - 4.1.2 Principal curvatures and principal directions
    - 4.2 Sectional curvatures
  - 5 Transformations and Invariance
    - 5.1 Change of coordinates
    - 5.2 Non-linear conformal transformations
    - 5.3 Invariant curvatures
- Part II Statistics
  - 6 Discrete Random Variables and Related Concepts
    - 6.1 Preliminaries
    - 6.2 Discrete random variables
      - 6.2.1 Discrete random variables and probability function
      - 6.2.2 Relative frequency histogram
      - 6.2.3 Cumulative distribution function
    - 6.3 Population parameters and sample statistics
      - 6.3.1 Population mean and expected value
      - 6.3.2 Sample statistic
      - 6.3.3 Sample mean
      - 6.3.4 Sample and population variances
    - 6.4 Mathematical expectations
    - 6.5 Maximum likelihood estimation
    - 6.6 Maximum likelihood estimation of the probability of a Bernoulli experiment
  - 7 Continuous Random Variables and Related Concepts
    - 7.1 Continuous random variables
    - 7.2 Mathematical expectation for continuous random variables
    - 7.3 Mean and variance and their sample estimates
    - 7.4 Basic properties of expectations
    - 7.5 Normal distribution
    - 7.6 Maximum likelihood estimation for continuous variables
    - 7.7 Maximum likelihood estimation for the parameters of normal distribution
    - 7.8 Sampling distribution
  - 8 Bivariate and Multivariate Distribution
    - 8.1 Bivariate distribution for discrete random variables
      - 8.1.1 Joint probability function
      - 8.1.2 Marginal probability function
      - 8.1.3 Conditional probability function
    - 8.2 Bivariate distribution for continuous random variables
    - 8.3 Mathematical expectations
      - 8.3.1 Mathematical expectations for the functions of two random variables
    - 8.4 Covariance and correlation
      - 8.4.1 Sample covariance and correlation
      - 8.4.2 Population covariance and correlation
      - 8.4.3 Conditional expectations
    - 8.5 Bivariate normal distribution
    - 8.6 Independence
    - 8.7 Multivariate distribution
  - 9 Simple Linear Regression
    - 9.1 The model
    - 9.2 The least squares estimation
    - 9.3 The maximum likelihood estimation of regression parameters
    - 9.4 Residuals
    - 9.5 Coefficient of determination
    - 9.6 Weighted least squares estimates
  - 10 Topics on Linear Regression Analysis
    - 10.1 Multiple regression model
    - 10.2 Estimation and interpretation
    - 10.3 Influential observations and outliers
    - 10.4 Leverage
    - 10.5 Cook's distance
    - 10.6 Deletion influence, joint influence and masking effect
    - 10.7 Derivation of Cook's distances
      - 10.7.1 Weighted least squares and Cook's distance
      - 10.7.2 Cook's distance-deleting one data point
- Part III Local Influence Analysis
  - 11 Basic Concepts
    - 11.1 Introduction
    - 11.2 Perturbation
    - 11.3 Likelihood displacement and influence graph
  - 12 Measuring Local Influence
    - 12.1 Individual influence
    - 12.2 Derivation of normal curvature
    - 12.3 Case-weight perturbation—an example
    - 12.4 Roles of sectional curvature
    - 12.5 Joint influence
  - 13 Relations Among Various Measures
    - 13.1 A bound on influence measures
    - 13.2 Individual and overall joint influence
    - 13.3 Individual and joint influence measures
    - 13.4 Competing eigenvalues
    - 13.5 Conclusions
  - 14 Conformal Modifications
    - 14.1 Modification and invariance
    - 14.2 Invariant measures
    - 14.3 Benchmarks
    - 14.4 Individual's contribution to joint influence—re-visited
- Appendix A Rank of Hat Matrix
- Appendix B Ricci Curvature
- Appendix C Cook's Distance—Deleting Two Data Points
- Bibliography
- Index

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Application of Elementary Differential Geometry to Influence Analysis (微分几何在影响分析

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