Sufficient Dimension Reduction 1st edition by Bing Li – Ebook PDF Instant Download/Delivery: 1498704476 , 978-1498704472
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ISBN 10: 1498704476
ISBN 13: 978-1498704472
Author: Bing Li
Sufficient dimension reduction is a rapidly developing research field that has wide applications in regression diagnostics, data visualization, machine learning, genomics, image processing, pattern recognition, and medicine, because they are fields that produce large datasets with a large number of variables. Sufficient Dimension Reduction: Methods and Applications with R introduces the basic theories and the main methodologies, provides practical and easy-to-use algorithms and computer codes to implement these methodologies, and surveys the recent advances at the frontiers of this field.
Sufficient Dimension Reduction 1st Table of contents:
1 Preliminaries
1.1 Empirical Distribution and Sample Moments
1.2 Principal Component Analysis
1.3 Generalized Eigenvalue Problem
1.4 Multivariate Linear Regression
1.5 Generalized Linear Model
1.5.1 Exponential Family
1.5.2 Generalized Linear Models
1.6 Hilbert Space, Linear Manifold, Linear Subspace
1.7 Linear Operator and Projection
1.8 The Hilbert Space Rp(Σ)
1.9 Coordinate Representation
1.10 Generalized Linear Models under Link Violation
2 Dimension Reduction Subspaces
2.1 Conditional Independence
2.2 Sufficient Dimension Reduction Subspace
2.3 Transformation Laws of Central Subspace
2.4 Fisher Consistency, Unbiasedness, and Exhaustiveness
3 Sliced Inverse Regression
3.1 Sliced Inverse Regression: Population-Level Development
3.2 Limitation of SIR
3.3 Estimation, Algorithm, and R-codes
3.4 Application: The Big Mac Index
4 Parametric and Kernel Inverse Regression
4.1 Parametric Inverse Regression
4.2 Algorithm, R Codes, and Application
4.3 Relation of PIR with SIR
4.4 Relation of PIR with Ordinary Least Squares
4.5 Kernel Inverse Regression
5 Sliced Average Variance Estimate
5.1 Motivation
5.2 Constant Conditional Variance Assumption
5.3 Sliced Average Variance Estimate
5.4 Algorithm and R-code
5.5 Relation with SIR
5.6 The Issue of Exhaustiveness
5.7 SIR-II
5.8 Case Study: The Pen Digit Data
6 Contour Regression and Directional Regression
6.1 Contour Directions and Central Subspace
6.2 Contour Regression at the Population Level
6.3 Algorithm and R Codes for CR
6.4 Exhaustiveness of Contour Regression
6.5 Directional Regression
6.6 Representation of ΛDR Using Moments
6.7 Algorithm and R Codes for DR
6.8 Exhaustiveness Relation with SIR and SAVE
6.9 Pen Digit Case Study Continued
7 Elliptical Distribution and Predictor Transformation
7.1 Linear Conditional Mean and Elliptical Distribution
7.2 Box-Cox Transformation
7.3 Application to the Big Mac Data
7.4 Estimating Equations for Handling Non-Ellipticity
8 Sufficient Dimension Reduction for Conditional Mean
8.1 Central Mean Subspace
8.2 Ordinary Least Squares
8.3 Principal Hessian Direction
8.4 Iterative Hessian Transformation
9 Asymptotic Sequential Test for Order Determination
9.1 Stochastic Ordering and Von Mises Expansion
9.2 Von Mises Expansion and Influence Functions
9.3 Influence Functions of Some Statistical Functionals
9.4 Random Matrix with Affine Invariant Eigenvalues
9.5 Asymptotic Distribution of the Sum of Small Eigenvalues
9.6 General Form of the Sequential Tests
9.7 Sequential Test for SIR
9.8 Sequential Test for PHD
9.9 Sequential Test for SAVE
9.10 Sequential Test for DR
9.11 Applications
10 Other Methods for Order Determination
10.1 BIC Type Criteria for Order Determination
10.2 Bootstrapped Eigenvector Variation
10.3 Eigenvalue Magnitude and Eigenvector Variation
10.4 Ladle Estimator
10.5 Consistency of the Ladle Estimator
10.6 Application: Identification of Wine Cultivars
11 Forward Regressions for Dimension Reduction
11.1 Outer Product of Gradients
11.2 Fisher Consistency of Gradient Estimate
11.3 Minimum Average Variance Estimate
11.4 Refined MAVE and refined OPG
11.5 From Central Mean Subspace to Central Subspace
11.6 dOPG and Its Refinement
11.7 dMAVE and Its Refinement
11.8 Ensemble Estimators
11.9 Simulation Studies and Applications
11.10 Summary
12 Nonlinear Sufficient Dimension Reduction
12.1 Reproducing Kernel Hilbert Space
12.2 Covariance Operators in RKHS
12.3 Coordinate Mapping
12.4 Coordinate of Covariance Operators
12.5 Kernel Principal Component Analysis
12.6 Sufficient and Central σ-Field for Nonlinear SDR
12.7 Complete Sub σ-Field for Nonlinear SDR
12.8 Converting σ-Fields to Function Classes for Estimation
13 Generalized Sliced Inverse Regression
13.1 Regression Operator
13.2 Generalized Sliced Inverse Regression
13.3 Exhaustiveness and Completeness
13.4 Relative Universality
13.5 Implementation of GSIR
13.6 Precursors and Variations of GSIR
13.7 Generalized Cross Validation for Tuning ϵX and ϵY
13.8 k-Fold Cross Validation for Tuning ρX,ρY,ϵX,ϵ
13.9 Simulation Studies
13.10 Applications
13.10.1 Pen Digit Data
13.10.2 Face Sculpture Data
14 Generalized Sliced Average Variance Estimator
14.1 Generalized Sliced Average Variance Estimation
14.2 Relation with GSIR
14.3 Implementation of GSAVE
14.4 Simulation Studies and an Application
14.5 Relation between Linear and Nonlinear SDR
15 Broad Scope of Sufficient Dimension Reduction
15.1 Sufficient Dimension Reduction for Functional Data
15.2 Sufficient Dimension Folding for Tensorial Data
15.3 Sufficient Dimension Reduction for Grouped Data
15.4 Variable Selection via Sufficient Dimension Reduction
15.5 Efficient Dimension Reduction
15.6 Partial Dimension Reduction for Categorical Predictors
15.7 Measurement Error Problem
15.8 SDR via Support Vector Machine
15.9 SDR for Multivariate Responses
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