微分幾何在影響分析中的應(yīng)用

內(nèi)容概要

　　《微分幾何在影響分析中的應(yīng)用（英文版）》討論微分幾何在統(tǒng)計學(xué)影響分析中的應(yīng)用，適合數(shù)學(xué)及統(tǒng)計學(xué)本科生或研究生閱讀。對于研習(xí)數(shù)學(xué)的學(xué)生，本書描述微分幾何在數(shù)學(xué)范疇以外的具體應(yīng)用；對于研習(xí)統(tǒng)計的學(xué)生，本書則能幫助他們理解統(tǒng)計領(lǐng)域中的微分幾何概念。
　　《微分幾何在影響分析中的應(yīng)用（英文版）》要求讀者具備線性代數(shù)及向量微積分的基礎(chǔ)知識。書的第一部分圍繞法曲率、截面曲率和高斯曲率概念介紹了圖的幾何學(xué)知識；第二部分回顧了統(tǒng)計學(xué)的一些基本概念及模型，為理解影響分析提供必要的基礎(chǔ)知識；第三部分則集中討論上述幾何概念在局部影響分析中的應(yīng)用，并探討如何有效地應(yīng)用幾何概念以提高局部影響分析估計的效力。
　　《微分幾何在影響分析中的應(yīng)用（英文版）》為研習(xí)統(tǒng)計學(xué)或數(shù)學(xué)的學(xué)生架起了知識理解的橋梁，為數(shù)學(xué)與統(tǒng)計學(xué)的跨學(xué)科研究合作及相互推進(jìn)發(fā)揮創(chuàng)新性的作用。

書籍目錄

Preface
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
6.6Maximum 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
9 Simple Linear Regression
10 Topics on Linear Regression Analysis
11 Basic Concepts
12 Measuring Local Influence
13 Relations Among Various Measures
14 Conformal Modifications
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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《微分幾何在影響分析中的應(yīng)用》為研習(xí)統(tǒng)計學(xué)或數(shù)學(xué)的學(xué)生架起了知識理解的橋梁，為數(shù)學(xué)與統(tǒng)計學(xué)的跨學(xué)科研究合作及相互推進(jìn)發(fā)揮創(chuàng)新性的作用。

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