R. Dennis Cook

An Introduction to Envelopes

Dimension Reduction for Efficient Estimation in Multivariate Statistics

• Gebundenes Buch

Produktbeschreibung

Written by the leading expert in the field, this text reviews the major new developments in envelope models and methods

An Introduction to Envelopes provides an overview of the theory and methods of envelopes, a class of procedures for increasing efficiency in multivariate analyses without altering traditional objectives. The author offers a balance between foundations and methodology by integrating illustrative examples that show how envelopes can be used in practice. He discusses how to use envelopes to target selected coefficients and explores predictor envelopes and their connection with partial least squares regression. The book reveals the potential for envelope methodology to improve estimation of a multivariate mean.

The text also includes information on how envelopes can be used in generalized linear models, regressions with a matrix-valued response, and reviews work on sparse and Bayesian response envelopes. In addition, the text explores relationships between envelopes and other dimension reduction methods, including canonical correlations, reduced-rank regression, supervised singular value decomposition, sufficient dimension reduction, principal components, and principal fitted components. This important resource:

_ Offers a text written by the leading expert in this field

_ Describes groundbreaking work that puts the focus on this burgeoning area of study

_ Covers the important new developments in the field and highlights the most important directions

_ Discusses the underlying mathematics and linear algebra

_ Includes an online companion site with both R and Matlab support

Written for researchers and graduate students in multivariate analysis and dimension reduction, as well as practitioners interested in statistical methodology, An Introduction to Envelopes offers the first book on the theory and methods of envelopes.

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Produktdetails

• Wiley Series in Probability and Statistics
• Verlag: Wiley / Wiley & Sons
• Artikelnr. des Verlages: 1W119422930
• 1. Auflage
• Seitenzahl: 320
• Erscheinungstermin: 9. Oktober 2018
• Englisch
• Abmessung: 231mm x 157mm x 20mm
• Gewicht: 635g
• ISBN-13: 9781119422938
• ISBN-10: 1119422930
• Artikelnr.: 51483364

Autorenporträt

R. DENNIS COOK, PHD, is Full Professor, School of Statistics, University of Minnesota. He served as Director of the School of Statistics, Chair of the Department of Applied Statistics, and as Director of the Statistical Center, all at the University of Minnesota. He is Fellow of the American Statistical Association and the Institute of Mathematical Statistics. His research areas include dimension reduction, linear and nonlinear regression, experimental design, statistical diagnostics, statistical graphics, and population genetics.

Inhaltsangabe

Preface xv

Notation and Definitions xix

1 Response Envelopes 1

1.1 The Multivariate Linear Model 2

1.1.1 Partitioned Models and Added Variable Plots 5

1.1.2 Alternative Model Forms 6

1.2 Envelope Model for Response Reduction 6

1.3 Illustrations 10

1.3.1 A Schematic Example 10

1.3.2 Compound Symmetry 13

1.3.3 Wheat Protein: Introductory Illustration 13

1.3.4 Cattle Weights: Initial Fit 14

1.4 More on the Envelope Model 19

1.4.1 Relationship with Sufficiency 19

1.4.2 Parameter Count 19

1.4.3 Potential Gains 20

1.5 Maximum Likelihood Estimation 21

1.5.1 Derivation 21

1.5.2 Cattle Weights: Variation of the X-Variant Parts of Y 23

1.5.3 Insights into ÊSigma (B)24

1.5.4 Scaling the Responses 25

1.6 Asymptotic Distributions 25

1.7 Fitted Values and Predictions 28

1.8 Testing the Responses 29

1.8.1 Test Development 29

1.8.2 Testing Individual Responses 32

1.8.3 Testing Containment Only 34

1.9 Nonnormal Errors 34

1.10 Selecting the Envelope Dimension, u 36

1.10.1 Selection Methods 36

1.10.1.1 Likelihood Ratio Testing 36

1.10.1.2 Information Criteria 37

1.10.1.3 Cross-validation 37

1.10.2 Inferring About rank (beta) 38

1.10.3 Asymptotic Considerations 38

1.10.4 Overestimation Versus Underestimation of u 41

1.10.5 Cattle Weights: Influence of u 43

1.11 Bootstrap and Uncertainty in the Envelope Dimension 45

1.11.1 Bootstrap for Envelope Models 45

1.11.2 Wheat Protein: Bootstrap and Asymptotic Standard Errors, u Fixed 46

1.11.3 Cattle Weights: Bootstrapping u 47

1.11.4 Bootstrap Smoothing 48

1.11.5 Cattle Data: Bootstrap Smoothing 49

2 Illustrative Analyses Using Response Envelopes 51

2.1 Wheat Protein: Full Data 51

2.2 Berkeley Guidance Study 51

2.3 Banknotes 54

2.4 Egyptian Skulls 55

2.5 Australian Institute of Sport: Response Envelopes 58

2.6 Air Pollution 59

2.7 Multivariate Bioassay 63

2.8 Brain Volumes 65

2.9 Reducing Lead Levels in Children 67

3 Partial Response Envelopes 69

3.1 Partial Envelope Model 69

3.2 Estimation 71

3.2.1 Asymptotic Distribution of 72

3.2.2 Selecting u1 73

3.3 Illustrations 74

3.3.1 Cattle Weight: Incorporating Basal Weight 74

3.3.2 Mens' Urine 74

3.4 Partial Envelopes for Prediction 77

3.4.1 Rationale 77

3.4.2 Pulp Fibers: Partial Envelopes and Prediction 78

3.5 Reducing Part of the Response 79

4 Predictor Envelopes 81

4.1 Model Formulations 81

4.1.1 Linear Predictor Reduction 81

4.1.1.1 Predictor Envelope Model 83

4.1.1.2 Expository Example 83

4.1.2 Latent Variable Formulation of Partial Least Squares Regression 84

4.1.3 Potential Advantages 86

4.2 SIMPLS 88

4.2.1 SIMPLS Algorithm 88

4.2.2 SIMPLS When n

4.2.2.1 Behavior of the SIMPLS Algorithm 90

4.2.2.2 Asymptotic Properties of SIMPLS 91

4.3 Likelihood-Based Predictor Envelopes 94

4.3.1 Estimation 95

4.3.2 Comparisions with SIMPLS and Principal Component Regression 97

4.3.2.1 Principal Component Regression 98

4.3.2.2 SIMPLS 98

4.3.3 Asymptotic Properties 98

4.3.4 Fitted Values and Prediction 100