| Description |
1 online resource (307 pages) |
| Series |
Monographs on Statistics and Applied Probability ; 161 |
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Monographs on statistics and applied probability (Series) ; 161.
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| Contents |
Preliminaries -- Dimension reduction subspaces -- Sliced inverse regression -- Parametric and kernel inverse regression -- Sliced average variance estimate -- Contour regression and directional regression -- Elliptical distribution and predictor transformation -- Sufficient dimension reduction for conditional mean -- Asymptotic sequential test for order determination -- Other methods for order determination -- Forward regressions for dimension reduction -- Nonlinear sufficient dimension reduction -- Generalized sliced inverse regression -- Generalized sliced average variance estimator -- The broad scope of sufficient dimension reduction |
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Cover; Half Title; Monographs; Title; Copyright; Dedication; Contents; List of Figures; List of Tables; Preface; Author; Chapter 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(S; 1.9 Coordinate Representation; 1.10 Generalized Linear Models under Link Violation |
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Chapter 2 Dimension Reduction Subspaces2.1 Conditional Independence; 2.2 Sufficient Dimension Reduction Subspace; 2.3 Transformation Laws of Central Subspace; 2.4 Fisher Consistency, Unbiasedness, and Exhaustiveness; Chapter 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; Chapter 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 |
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4.4 Relation of PIR with Ordinary Least Squares4.5 Kernel Inverse Regression; Chapter 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; Chapter 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 |
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6.5 Directional Regression6.6 Representation of LDR 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; Chapter 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; Chapter 8 Sufficient Dimension Reduction for Conditional Mean; 8.1 Central Mean Subspace; 8.2 Ordinary Least Squares; 8.3 Principal Hessian Direction |
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8.4 Iterative Hessian TransformationChapter 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; Chapter 10 Other Methods for Order Determination |
| Summary |
Sufficient dimension reduction 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. This book 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 |
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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.--Provided by publisher |
| Notes |
10.1 BIC Type Criteria for Order Determination |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Dimension reduction (Statistics) -- Data processing
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Regression analysis -- Data processing.
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R (Computer program language)
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MATHEMATICS -- Applied.
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MATHEMATICS -- Probability & Statistics -- General.
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R (Computer program language)
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Regression analysis -- Data processing
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| Form |
Electronic book
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| ISBN |
9781351645737 |
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1351645730 |
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9781315119427 |
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1315119420 |
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1498704476 |
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9781498704472 |
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9781498704489 |
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1498704484 |
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