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· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 확률과 통계 > 일반
· ISBN : 9781498704472
· 쪽수 : 284쪽
· 출판일 : 2018-05-01
목차
List of Figures
List of Tables
Foreword
Preface
Author Bios
Contributors
Preliminaries
Empirical Distribution and Sample Moments
Principal Component Analysis
Generalized Eigenvalue Problem
Multivariate Linear Regression
Generalized Linear Model
Exponential family
Generalized Linear Models
Hilbert Space, Linear Manifold, Linear Subspace
Linear Operator and Projection
The Hilbert space Rp(S)
Coordinate Representation
Behavior of Generalized Linear Models under Link Violation
Dimension Reduction Subspaces
Conditional Independence
Sufficient Dimension Reduction Subspace
Behavior of the central subspace under transformations
Fisher Consistency, Unbiasedness, and Exhaustiveness
Sliced Inverse Regression
Sliced Inverse Regression: Population-Level Development
Limitation of SIR
Estimation, Algorithm, and R-codes
Application: the Big Mac index
Parametric and Kernel Inverse Regression
Parametric Inverse Regression
Algorithm, R Codes, and Application
Relation of PIR with SIR
Relation of PIR with Ordinary Least Squares
Kernel Inverse Regression
Sliced Average Variance Estimate
Motivation
Constant Conditional Variance Assumption
Sliced Average Variance Estimate
Algorithm and R-code
Relation with SIR
The Issue of Exhaustiveness
SIR-II
Case Study: The Pen Digit Data
Contour Regression and Directional Regression
Contour Directions and Central Subspace
Contour Regression at the Population Level
Algorithm and R Codes
Exhaustiveness of Contour Regression
Directional Regression
Representation of LDR using moments
Algorithm and R Codes
Exhaustiveness relation with SIR and SAVE
Pen-Digit Case Study Continued
Elliptical Distribution and Transformation of Predictors
Linear Conditional Mean and Elliptical Distribution
Box-Cox Transformation
Application to the Big Mac data
Sufficient Dimension Reduction for Conditional Mean
Central Mean Subspace
Ordinary Least Squares
Principal Hessian Direction
Iterative Hessian Transformation
Asymptotic Sequential Test for Order Determination
Stochastic ordering and von Mises Expansion
von Mises expansion and Influence functions
Influence functions of some useful statistical functionals
Random matrix with Affine invariant eigenvalues
Asymptotic distribution of the sum of small eigenvalues
General form of the sequential tests
Sequential test for SIR
Sequential test for PHD
Sequential test for SAVE
Sequential test for DR
Applications
Other Methods for Order Determination
BIC type criteria for order determination
Order determination by bootstrapped eigenvector variation
Eigenvalue magnitude and eigenvector variation
Ladle estimator
Consistency of the ladle estimator
Application: identification of wine cultivars
Forward Regressions for Dimension Reduction
Local linear regression and outer product of gradients
Fisher consistency of gradient estimate
Minimum Average Variance Estimate
Refined OPG and MAVE
From central mean subspace to central subspace
dOPG and its refinement
dMAVE and its refinement
Ensemble Estimators
Simulation studies and applications
Summary
Nonlinear Sufficient Dimension Reduction
Reproducing Kernel Hilbert Space
Mean element and covariance operator in RKHS
Coordinate representations
Coordinate of covariance operators
Kernel principal component analysis
Sufficient and central s-field for nonlinear SDR
Complete sub s-field for nonlinear SDR
Converting s-fields to function classes for estimation
Generalized Sliced Inverse Regression
Regression operator
Generalized Sliced Inverse Regression
Exhaustiveness and Completeness
Relative universality
Implementation of GSIR
Precursors and variations of GSIR
Generalized Cross Validation for tuning eX and eY
k-fold Cross Validation for tuning rX ;rY ; eX ; eY
Simulation studies
Applications
Pen Digit data
Face Sculpture data
Generalized Sliced Average Variance Estimator
Generalized Sliced Average Variance Estimation
Relation with GSIR
Implementation of GSAVE
Simulation studies and an application
Relation between linear and nonlinear SDR
Bibliography
