| Preface Part Ⅰ Geometry 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 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 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.I 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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