微分几何在影响分析中的应用2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Part Ⅰ Geometry3

1 Preliminaries3

1.1 Linear algebra3

1.1.1 Vectors and matrices3

1.1.2 Symmetric bilinear forms5

1.1.3 Vector subspaces6

1.1.4 Linear maps from Rn to Rn7

1.1.5 A convention9

1.2 Vector calculus9

1.2.1 Vector-valued functions and differentials9

1.2.2 Taylor expansion and extrema11

1.2.3 Extrema and Lagrange multiplier theorem12

2 Euclidean Geometry15

2.1 Orthogonal transformations15

2.2 Rigid motions16

2.3 Translation of vector subspaces18

2.4 Conformal transformations20

2.5 Orthonormal basis20

2.6 Orthogonal projections23

2.7 Areas and volumes25

3 Geometry of Graphs29

3.1 Graphs in Euclidean spaces29

3.2 Normal sections31

3.3 Cross sections in high dimension33

3.4 First fundamental forms33

4 Curvatures35

4.1 Normal curvatures35

4.1.1 Definition35

4.1.2 Principal curvatures and principal directions37

4.2 Sectional curvatures40

5 Transformations and Invariance43

5.1 Change of coordinates43

5.2 Non-linear conformal transformations44

5.3 Invariant curvatures46

Part Ⅱ Statistics51

6 Discrete Random Variables and Related Concepts51

6.1 Preliminaries51

6.2 Discrete random variables52

6.2.1 Discrete random variables and probability function52

6.2.2 Relative frequency histogram55

6.2.3 Cumulative distribution function55

6.3 Population parameters and sample statistics56

6.3.1 Population mean and expected value56

6.3.2 Sample statistic57

6.3.3 Sample mean57

6.3.4 Sample and population variances58

6.4 Mathematical expectations60

6.5 Maximum likelihood estimation61

6.6 Maximum likelihood estimation of the probability of a Bernoulli experiment62

7 Continuous Random Variables and Related Concepts65

7.1 Continuous random variables65

7.2 Mathematical expectation for continuous random variables66

7.3 Mean and variance and their sample estimates66

7.4 Basic properties of expectations67

7.5 Normal distribution68

7.6 Maximum likelihood estimation for continuous variables72

7.7 Maximum likelihood estimation for the parameters of normal distribution73

7.8 Sampling distribution74

8 Bivariate and Multivariate Distribution77

8.1 Bivariate distribution for discrete random variables77

8.1.1 Joint probability function77

8.1.2 Marginal probability function78

8.1.3 Conditional probability function79

8.2 Bivariate distribution for continuous random variables80

8.3 Mathematical expectations80

8.3.1 Mathematical expectations for the functions of two random variables80

8.4 Covariance and correlation82

8.4.1 Sample covariance and correlation82

8.4.2 Population covariance and correlation84

8.4.3 Conditional expectations85

8.5 Bivariate normal distribution87

8.6 Independence88

8.7 Multivariate distribution89

9 Simple Linear Regression93

9.1 The model93

9.2 The least squares estimation95

9.3 The maximum likelihood estimation of regression parameters98

9.4 Residuals99

9.5 Coefficient of determination101

9.6 Weighted least squares estimates103

10 Topics on Linear Regression Analysis105

10.1 Multiple regression model105

10.2 Estimation and interpretation106

10.3 Influential observations and outliers110

10.4 Leverage111

10.5 Cook's distance113

10.6 Deletion influence,joint influence and masking effect114

107 Derivation of Cook's distances116

10.7.1 Weighted least squares and Cook's distance116

10.7.2 Cook's distance-deleting one data point118

Part Ⅲ Local Influence Analysis123

11 Basic Concepts123

11.1 Introduction123

11.2 Perturbation125

11.3 Likelihood displacement and infuence graph126

12 Measuring Local Influence129

12.1 Individual influence130

12.2 Derivation of normal curvature131

12.3 Case-weight perturbation—an example133

12.4 Roles of sectional curvature135

12.5 Joint influence138

13 Relations Among Various Measures141

13.1 A bound on influence measures141

13.2 Individual and overall joint influence143

13.3 Individual andjoint influence measures146

13.4 Competing eigenvalues147

13.5 Conclusions150

14 Conformal Modifications153

14.1 Modification and invariance153

14.2 Invariant measures154

14.3 Benchmarks155

14.4 Individual's contribution to joint influence—re-visited157

Appendix A Rank of Hat Matrix161

Appendix B Ricci Curvature163

Appendix C Cook's Distance—Deleting Two Data Points165

Bibliography167

Index171