Understanding Biplots

Biplots are a graphical method for simultaneously displaying two
kinds of information; typically, the variables and sample units
described by a multivariate data matrix or the items labelling the
rows and columns of a two-way table. This book aims to popularize
what is now seen to be a useful and reliable method for the
visualization of multidimensional data associated with, for
example, principal component analysis, canonical variate analysis,
multidimensional scaling, multiplicative interaction and various
types of correspondence analysis.

Understanding Biplots:

• Introduces theory and techniques which can be applied to
  problems from a variety of areas, including ecology, biostatistics,
  finance, demography and other social sciences.

• Provides novel techniques for the visualization of
  multidimensional data and includes data mining techniques.

• Uses applications from many fields including finance,
  biostatistics, ecology, demography.

• Looks at dealing with large data sets as well as smaller
  ones.

• Includes colour images, illustrating the graphical
  capabilities of the methods.

• Is supported by a Website featuring R code and
  datasets.

Researchers, practitioners and postgraduate students of
statistics and the applied sciences will find this book a useful
introduction to the possibilities of presenting data in informative
ways.

Autorentext

John C. Gower, Department of Mathematics, The Open University, Milton Keynes, UK.
Over 100 papers. Books include Gower & Hand (1996) Biplots, in which the authors developed a unified theory of biplots. Sugnet Gardner, British American Tobacco, Stellenbosch, South Africa. Niel J. le Roux, Department of Statistics and Actuarial Science, University of Stellenbosch , South Africa.

Zusammenfassung
Biplots are a graphical method for simultaneously displaying two kinds of information; typically, the variables and sample units described by a multivariate data matrix or the items labelling the rows and columns of a two-way table. This book aims to popularize what is now seen to be a useful and reliable method for the visualization of multidimensional data associated with, for example, principal component analysis, canonical variate analysis, multidimensional scaling, multiplicative interaction and various types of correspondence analysis. Understanding Biplots:

• Introduces theory and techniques which can be applied to problems from a variety of areas, including ecology, biostatistics, finance, demography and other social sciences.

• Provides novel techniques for the visualization of multidimensional data and includes data mining techniques.

• Uses applications from many fields including finance, biostatistics, ecology, demography.

• Looks at dealing with large data sets as well as smaller ones.

• Includes colour images, illustrating the graphical capabilities of the methods.

• Is supported by a Website featuring R code and datasets.

Researchers, practitioners and postgraduate students of statistics and the applied sciences will find this book a useful introduction to the possibilities of presenting data in informative ways.

Inhalt

Preface xi

1 Introduction 1

1.1 Types of biplots 2

1.2 Overview of the book 5

1.3 Software 7

1.4 Notation 7

1.4.1 Acronyms 9

2 Biplot basics 11

2.1 A simple example revisited 11

2.2 The biplot as a multidimensional scatterplot 14

2.3 Calibrated biplot axes 20

2.3.1 Lambda scaling 24

2.4 Refining the biplot display 32

2.5 Scaling the data 36

2.6 A closer look at biplot axes 37

2.7 Adding new variables: the regression method 44

2.8 Biplots and large data sets 47

2.9 Enclosing a configuration of sample points 50

2.9.1 Spanning ellipse 53

2.9.2 Concentration ellipse 54

2.9.3 Convex hull 57

2.9.4 Bagplot 58

2.9.5 Bivariate density plots 62

2.10 Buying by mail order catalogue data set revisited 64

2.11 Summary 66

3 Principal component analysis biplots 67

3.1 An example: risk management 67

3.2 Understanding PCA and constructing its biplot 71

3.2.1 Representation of sample points 72

3.2.2 Interpolation biplot axes 74

3.2.3 Prediction biplot axes 77

3.3 Measures of fit for PCA biplots 80

3.4 Predictivities of newly interpolated samples 94

3.5 Adding new axes to a PCA biplot and defining their predictivities 98

3.6 Scaling the data in a PCA biplot 103

3.7 Functions for constructing a PCA biplot 107

3.7.1 Function PCAbipl 107

3.7.2 Function PCAbipl.zoom 115

3.7.3 Function PCAbipl.density 115

3.7.4 Function PCAbipl.density.zoom 116

3.7.5 Function PCA.predictivities 117

3.7.6 Function PCA.predictions.mat 117

3.7.7 Function vector.sum.interp 117

3.7.8 Function circle.projection.interactive 118

3.7.9 Utility functions 118

3.8 Some novel applications and enhancements of PCA biplots 119

3.8.1 Risk management example revisited 119

3.8.2 Quality as a multidimensional process 123

3.8.3 Using axis predictivities in biplots 128

3.8.4 One-dimensional PCA biplots 128

3.8.5 Three-dimensional PCA biplots 135

3.8.6 Changing the scaffolding axes in conventional two-dimensional PCA biplots 138

3.8.7 Alpha-bags, kappa-ellipses, density surfaces and zooming 139

3.8.8 Predictions by circle projection 139

3.9 Conclusion 144

4 Canonical variate analysis biplots 145

4.1 An example: revisiting the Ocotea data 145

4.2 Understanding CVA and constructing its biplot 153

4.3 Geometric interpretation of the transformation to the canonical space 157

4.4 CVA biplot axes 160

4.4.1 Biplot axes for interpolation 160

4.4.2 Biplot axes for prediction 160

4.5 Adding new points and variables to a CVA biplot 162

4.5.1 Adding new sample points 162

4.5.2 Adding new variables 162

4.6 Measures of fit for CVA biplots 163

4.6.1 Predictivities of new samples and variables 168

4.7 Functions for constructing a CVA biplot 169

4.7.1 Function CVAbipl 169

4.7.2 Function CVAbipl.zoom 170

4.7.3 Function CVAbipl.density 170

4.7.4 Function CVAbipl.density.zoom 170

4.7.5 Function CVAbipl.pred.regions 170

4.7.6 Function CVA.predictivities 171

4.7.7 Function CVA.predictions.mat 172

4.8 Continuing the Ocotea example 172

4.9 CVA biplots for two classes 178

4.9.1 An example of two-class CVA biplots 178

4.10 A five-class CVA biplot example 185

4.11 Overlap in two-dimensional biplots 189

4.11.1 Describing the structure of overlap 189

4.11.2 Quantifying overlap 191

5 Multidimensional scaling and nonlinear biplots 205

5.1 Introduction 205

5.2 The regression method 206 5.3 Nonlinear biplots 208&lt...