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.
Features Provides comprehensive coverage of this emerging research field.
Synthesizes a wide variety of dimension reduction methods under a few unifying principles such as projection in Hilbert spaces, kernel mapping, and von Mises expansion.
Reflects most recent advances such as nonlinear sufficient dimension reduction, dimension folding for tensorial data, as well as sufficient dimension reduction for functional data.
Includes a set of computer codes written in R that are easily implemented by the readers.
Uses real data sets available online to illustrate the usage and power of the described methods.
Sufficient dimension reduction has undergone momentous development in recent years, partly due to the increased demands for techniques to process high-dimensional data, a hallmark of our age of Big Data. This book will serve as the perfect entry into the field for the beginning researchers or a handy reference for the advanced ones.
The author Bing Li obtained his Ph.
D. from the University of Chicago. He is currently a Professor of Statistics at the Pennsylvania State University. His research interests cover sufficient dimension reduction, statistical graphical models, functional data analysis, machine learning, estimating equations and quasilikelihood, and robust statistics. He is a fellow of the Institute of Mathematical Statistics and the American Statistical Association. He is an Associate Editor for The Annals of Statistics and the Journal of the American Statistical Association.
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
Bing Li obtained his Ph.D. from the University of Chicago. He is currently a Professor of Statistics at the Pennsylvania State University. His research interests cover sufficient dimension reduction, statistical graphical models, functional data analysis, machine learning, estimating equations and quasilikelihood, and robust statistics. He is a fellow of the Institute of Mathematical Statistics and the American Statistical Association. He is an Associate Editor for The Annals of Statistics and the Journal of the American Statistical Association.
Reviews for Sufficient Dimension Reduction: Methods and Applications with R
...Sufficient Dimension Reduction: Methods and Applications with R is a thorough overview of the key ideas and a detailed reference for advanced researchers...Professor Li gives careful discussions of the relevant details, rendering the text impressively self-contained. But as one would expect from a book based on graduate course notes, this manuscript is mainly accessible to those with advanced training in theoretical statistics...This book serves as an excellent introduction to the field of sufficient dimension reduction, and the depth of presentation and theoretical rigor are impressive. It would, of course, naturally serve as the basis for a deep graduate course, and provides a substantial foundation for anyone hoping to contribute in this thriving area. - Daniel J. McDonald, JASA 2020