Computational Statistics: An Introduction to R Book

Introduction v

1 Basic Data Analysis 1

1.1 R Programming Conventions 1

1.2 Generation of Random Numbers and Patterns 4

1.2.1 Random Numbers 4

1.2.2 Patterns 9

1.3 Case Study: Distribution Diagnostics 10

1.3.1 Distribution Functions 13

1.3.2 Histograms 17

Barcharts 21

1.3.3 Statistics of Distribution Functions: Kolmogorov-Smirnov Tests 22

Monte Carlo Confidence Bands 23

1.3.4 Statistics of Histograms and Related Plots; X2-Tests 29

1.4 Moments and Quantiles 34

1.5 R Complements 39

1.5.1 Random Numbers 39

1.5.2 Graphical Comparisons 40

1.5.3 Functions 46

1.5.4 Enhancing Graphical Displays 50

1.5.5 R Internals 53

parse 53

eval 53

print 54

Executing Files 54

1.5.6 Packages 54

1.6 Statistical Summary 56

1.7 Literature and Additional References 57

2 Regression 59

2.1 General Regression Model 59

2.2 Linear Model 60

2.2.1 Factors 63

2.2.2 Least Squares Estimation 64

2.2.3 Regression Diagnostics 69

2.2.4 More Examples for Linear Models 75

2.2.5 Model Formulae 76

2.2.6 Gauss-Markov Estimator and Residuals 77

2.3 Variance Decomposition and Analysis of Variance 79

2.4 Simultaneous Inference 85

2.4.1 Scheffé's Confidence Bands 85

2.4.2 Tukey's Confidence Intervals 87

Case Study: Titre Plates 88

2.5 Beyond Linear Regression 96

Transformations 96

2.5.1 Generalised Linear Models 96

2.5.2 Local Regression 97

2.6 R Complements 101

2.6.1 Discretisation 101

2.6.2 External Data 101

2.6.3 Testing Software 101

2.6.4 R Data Types 102

2.6.5 Classes and Polymorphic Functions 103

2.6.6 Extractor Functions 104

2.7 Statistical Summary 105

2.8 Literature and Additional References 105

3Comparisons 107

3.1 Shift/Scale Families, and Stochastic Order 109

3.2 QQ Plot, PP Plot, and Comparison of Distributions 111

3.2.1 Kolmogorov-Smirnov Tests 116

3.3 Tests for Shift Alternatives 117

3.4 A Road Map 125

3.5 Power and Confidence 126

3.5.1 Theoretical Power and Confidence 126

3.5.2 Simulated Power and Confidence 130

3.5.3 Quantile Estimation 133

3.6 Qualitative Features of Distributions 135

3.7 Statistical Summary 136

3.8 Literature and Additional References 137

4 Dimensions 1, 2, 3, ..., α 139

4.1 R Complements 140

4.2 Dimensions 143

4.3 Selections 145

4.4 Projections 145

4.4.1 Marginal Distributions and Scatter Plot Matrices 145

4.4.2 Projection Pursuit 150

4.4.3 Projections for Dimensions 1, 2, 3, ...7 153

4.4.4 Parallel Coordinates 154

4.5 Sections, Conditional Distributions and Coplots 156

4.6 Transformation and Dimension Reduction 162

4.7 Higher Dimensions 167

4.7.1 Linear Case 167

Partial Residuals and Added Variable Plots 168

4.7.2 Non-Linear Case 169

Example: Cusp Non-Linearity 169

4.7.3 Case Study: Melbourne Temperature Data 173

4.7.4 Curse of Dimensionality 174

4.7.5 Case Study: Body Fat 175

4.8 High Dimensions 189

4.9 Statistical Summary 190

R as a Programming Language and Environment 193

A.1 Help and Information 193

A.2 Names and Search Paths 195

A.3 Administration and Customisation 196

A.4 Basic Data Types 197

A.5 Output for Objects 199

A.6 Object Inspection 200

A.7 System Inspection 201

A.8 Complex Data Types 202

A.9 Accessing Components 204

A.10 Data Manipulation 206

A.11 Operators 208

A.12 Functions 209

A.13 Debugging and Profiling 211

A.14 Control Structures 213

A.15 Input and Output to Data Streams; External Data 215

A.16 Libraries, Packages 218

A.17 Mathematical Operators and Functions; Linear Algebra 220

A.18 Model Descriptions 221

A.19 Graphic Functions 223

A.19.1 High-Level Graphics 223

A.19.2 Low-Level Graphics 224

A.19.3 Annotations and Legends 225

A.19.4 Graphic Parameters and Layout 226

A.20 Elementary Statistical Functions 227

A.21 Distributions, Random Numbers, Densities... 228

A.22 Computing on the Language 231

References 233

Functions and Variables by Topic 237

Function and Variable Index 245

Subject Index 249