Title Details

Vera Pawlowsky-Glahn, Department of Computer Science and Applied Mathematics, University of Girona, Spain.

Antonella Buccianti, Department of Earth Sciences, University of Florence, Italy.

List of Contributors xix

Part I Introduction 1

1 A Short History of Compositional Data Analysis 3
John Bacon-Shone

1.1 Introduction 3

1.2 Spurious Correlation 3

1.3 Log and Log-Ratio Transforms 4

1.4 Subcompositional Dependence 5

1.5 alr, clr, ilr: Which Transformation to Choose? 5

1.6 Principles, Perturbations and Back to the Simplex 6

1.7 Biplots and Singular Value Decompositions 7

1.8 Mixtures 7

1.9 Discrete Compositions 8

1.10 Compositional Processes 8

1.11 Structural, Counting and Rounded Zeros 8

1.12 Conclusion 9

Acknowledgement 9

References 9

2 Basic Concepts and Procedures 12
Juan Jos´e Egozcue and Vera Pawlowsky-Glahn

2.1 Introduction 12

2.2 Election Data and Raw Analysis 13

2.3 The Compositional Alternative 15

2.3.1 Scale Invariance: Vectors with Proportional Positive Components Represent the Same Composition 15

2.3.2 Subcompositional Coherence: Analyses Concerning a Subset of Parts Must Not Depend on Other Non-Involved Parts 16

2.3.3 Permutation Invariance: The Conclusions of a Compositional Analysis Should Not Depend on the Order of the Parts 17

2.4 Geometric Settings 17

2.5 Centre and Variability 22

2.6 Conclusion 27

Acknowledgements 27

References 27

Part II Theory – Statistical Modelling 29

3 The Principle of Working on Coordinates 31
Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue

3.1 Introduction 31

3.2 The Role of Coordinates in Statistics 32

3.3 The Simplex 33

3.3.1 Basis of the Simplex 34

3.3.2 Working on Orthonormal Coordinates 35

3.4 Move or Stay in the Simplex 38

3.5 Conclusions 40

Acknowledgements 41

References 41

4 Dealing with Zeros 43
Josep Antoni Martún-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea

4.1 Introduction 43

4.2 Rounded Zeros 44

4.2.1 Non-Parametric Replacement of Rounded Zeros 45

4.2.2 Parametric Modified EM Algorithm for Rounded Zeros 47

4.3 Count Zeros 50

4.4 Essential Zeros 53

4.5 Difficulties, Troubles and Challenges 55

Acknowledgements 57

References 57

5 Robust Statistical Analysis 59
Peter Filzmoser and Karel Hron

5.1 Introduction 59

5.2 Elements of Robust Statistics from a Compositional Point of View 60

5.3 Robust Methods for Compositional Data 63

5.3.1 Multivariate Outlier Detection 64

5.3.2 Principal Component Analysis 64

5.3.3 Discriminant Analysis 65

5.4 Case Studies 66

5.4.1 Multivariate Outlier Detection 66

5.4.2 Principal Component Analysis 68

5.4.3 Discriminant Analysis 68

5.5 Summary 70

Acknowledgement 71

References 71

6 Geostatistics for Compositions 73
Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn

6.1 Introduction 73

6.2 A Brief Summary of Geostatistics 74

6.3 Cokriging of Regionalised Compositions 76

6.4 Structural Analysis of Regionalised Composition 76

6.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 78

6.6 Application 79

6.6.1 Delimiting the Body: Simplicial Indicator Kriging 81

6.6.2 Interpolating the Oil–Brine–Solid Content 82

6.7 Conclusions 84

Acknowledgements 84

References 84

7 Compositional VARIMA Time Series 87
Carles Barceló-Vidal, Lucúa Aguilar and Josep Antoni Martún-Fernández

7.1 Introduction 87

7.2 The Simplex SD as a Compositional Space 89

7.2.1 Basic Concepts and Notation 89

7.2.2 The Covariance Structure on the Simplex 90

7.3 Compositional Time Series Models 91

7.3.1 C-Stationary Processes 92

7.3.2 C-VARIMA Processes 93

7.4 CTS Modelling: An Example 94

7.4.1 Expenditure Shares in the UK 94

7.4.2 Model Selection 95

7.4.3 Estimation of Parameters 96

7.4.4 Interpretation and Comparison 96

7.5 Discussion 99

Acknowledgements 99

References 100

Appendix 102

8 Compositional Data and Correspondence Analysis 104
Michael Greenacre

8.1 Introduction 104

8.2 Comparative Technical Definitions 105

8.3 Properties and Interpretation of LRA and CA 107

8.4 Application to Fatty Acid Compositional Data 107

8.5 Discussion and Conclusions 111

Acknowledgements 112

References 112

9 Use of Survey Weights for the Analysis of Compositional Data 114
Monique Graf

9.1 Introduction 114

9.2 Elements of Survey Design 115

9.2.1 Randomization 115

9.2.2 Design-Based Estimation 118

9.3 Application to Compositional Data 122

9.3.1 Weighted Arithmetic and Geometric Means 123

9.3.2 Closed Arithmetic Mean of Amounts 123

9.3.3 Centred Log-Ratio of the Geometric Mean Composition 124

9.3.4 Closed Geometric Mean Composition 124

9.3.5 Example: Swiss Earnings Structure Survey (SESS) 125

9.4 Discussion 126

References 126

10 Notes on the Scaled Dirichlet Distribution 128
Gianna Serafina Monti, Glòria Mateu-Figueras and Vera Pawlowsky-Glahn

10.1 Introduction 128

10.2 Genesis of the Scaled Dirichlet Distribution 129

10.3 Properties of the Scaled Dirichlet Distribution 131

10.3.1 Graphical Comparison 131

10.3.2 Membership in the Exponential Family 133

10.3.3 Measures of Location and Variability 134

10.4 Conclusions 136

Acknowledgements 137

References 137

Part III Theory – Algebra and Calculus 139

11 Elements of Simplicial Linear Algebra and Geometry 141
Juan José Egozcue, Carles Barceló-Vidal, Josep Antoni Martún-Fernández, Eusebi Jarauta-Bragulat, José Luis Dúaz-Barrero and Glòria Mateu-Figueras

11.1 Introduction 141

11.2 Elements of Simplicial Geometry 142

11.2.1 n-Part Simplex 142

11.2.2 Vector Space 143

11.2.3 Centred Log-Ratio Representation 146

11.2.4 Metrics 147

11.2.5 Orthonormal Basis and Coordinates 149

11.3 Linear Functions 151

11.3.1 Linear Functions Defined on the Simplex 152

11.3.2 Simplicial Linear Function Defined on a Real Space 153

11.3.3 Simplicial Linear Function Defined on the Simplex 154

11.4 Conclusions 156

Acknowledgements 156

References 156

12 Calculus of Simplex-Valued Functions 158
Juan José Egozcue, Eusebi Jarauta-Bragulat and José Luis Díaz-Barrero

12.1 Introduction 158

12.2 Limits, Continuity and Differentiability 161

12.2.1 Limits and Continuity 161

12.2.2 Differentiability 163

12.2.3 Higher Order Derivatives 169

12.3 Integration 171

12.3.1 Antiderivatives. Indefinite Integral 171

12.3.2 Integration of Continuous SV Functions 172

12.4 Conclusions 174

Acknowledgements 175

References 175

13 Compositional Differential Calculus on the Simplex 176
Carles Barceló-Vidal, Josep Antoni Martún-Fernández and Glòria Mateu-Figueras

13.1 Introduction 176

13.2 Vector-Valued Functions on the Simplex 177

13.2.1 Scale-Invariant Vector-Valued Functions on Rn + 177

13.2.2 Vector-Valued Functions on Sn 178

13.3 C-Derivatives on the Simplex 178

13.3.1 Derivative of a Scale-Invariant Vector-Valued Function on Rn + 178

13.3.2 Directional C-Derivatives 180

13.3.3 C-Derivative 182

13.3.4 C-Gradient 184

13.3.5 Critical Points of a C-Differentiable Real-Valued Function on Sn 184

13.4 Example: Experiments with Mixtures 185

13.4.1 Polynomial of Degree One 185

13.4.2 Polynomial of Degree Two 186

13.4.3 Polynomial of Degree One in Logarithms 187

13.4.4 A numerical Example 188

13.5 Discussion 189

Acknowledgements 190

References 190

Part IV Applications 191

14 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? 193
David Lovell, Warren Müller, Jen Taylor, Alec Zwart and Chris Helliwell

14.1 Introduction 193

14.2 The Omics Imp and Two Bioscience Experiment Paradigms 194

14.3 The Impact of Compositional Constraints in the Omics 197

14.3.1 Univariate Impact of Compositional Constraints 197

14.3.2 Impact of Compositional Constraints on Multivariate

Distance Metrics 199

14.4 Impact of Compositional Constraints on Correlation and Covariance 201

14.4.1 Compositional Constraints, Covariance, Correlation and Log-Transformed Data 202

14.4.2 A Simulation Approach to Understanding the Impact of Closure 202

14.5 Implications 204

14.5.1 Gathering Information to Infer Absolute Abundance 204

14.5.2 Analysing Compositional Omics Data Appropriately 205

Acknowledgements 206

References 206

15 Hardy–Weinberg Equilibrium: A Nonparametric Compositional Approach 208
Jan Graffelman and Juan José Egozcue

15.1 Introduction 208

15.2 Genetic Data Sets 209

15.3 Classical Tests for HWE 210

15.4 A Compositional Approach 210

15.5 Example 214

15.6 Conclusion and Discussion 215

Acknowledgements 215

References 215

16 Compositional Analysis in Behavioural and Evolutionary Ecology 218
Michele Edoardo Raffaele Pierotti and Josep Antoni Martún-Fernández

16.1 Introduction 218

16.2 CODA in Population Genetics 219

16.3 CODA in Habitat Choice 222

16.4 Multiple Choice and Individual Variation in Preferences 224

16.5 Ecological Specialization 228

16.6 Time Budgets: More on Specialization 229

16.7 Conclusions 231

Acknowledgements 231

References 231

17 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates 235
Luis Azevedo Rodrigues, Josep Daunis-i-Estadella, Glòria Mateu-Figueras and Santiago Thi´o-Henestrosa

17.1 Introduction 235

17.2 Flying Vertebrates – General Anatomical and Functional Characteristics 236

17.3 Materials 236

17.4 Methods 238

17.5 Aitchison Distance Disparity Metrics 239

17.5.1 Intragroup Aitchison Distance 239

17.5.2 Intergroup Aitchison Distance 240

17.6 Statistical Tests 243

17.7 Biplots 244

17.7.1 Chiroptera 244

17.7.2 Pterosauria 245

17.8 Balances 246

17.9 Size Effect 249

17.10 Final Remarks 249

17.10.1 All Groups 250

17.10.2 Aves 250

17.10.3 Pterosauria 250

17.10.4 Chiroptera 251

Acknowledgements 252

References 252

18 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach 255
Antonella Buccianti

18.1 Introduction 255

18.2 Geochemical Processes and Log-Ratio Approach 256

18.3 Log-Ratio Approach and Water Chemistry 258

18.4 Log-Ratio Approach and Volcanic Gas Chemistry 261

18.5 Log-Ratio Approach and Subducting Sediment Composition 263

18.6 Conclusions 265

Acknowledgements 265

References 265

19 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury 267
Helmut Lammer, Peter Wurz, Josep Antoni Martún-Fernández and Herbert Iwo Maria Lichtenegger

19.1 Introduction 267

19.1.1 Mars 267

19.1.2 Mercury 269

19.1.3 Analysis of Surface Composition 270

19.2 Compositional Analysis of Mars’ Surface 270

19.3 Compositional Analysis of Mercury’s Surface 274

19.4 Conclusion 278

Acknowledgement 278

References 278

20 Spectral Analysis of Compositional Data in Cyclostratigraphy 282
Eulogio Pardo-Igúzquiza and Javier Heredia

20.1 Introduction 282

20.2 The Method 283

20.3 Case Study 285

20.4 Discussion 287

20.5 Conclusions 288

Acknowledgement 288

References 288

21 Multivariate Geochemical Data Analysis in Physical Geography 290
Jennifer McKinley and Christopher David Lloyd

21.1 Introduction 290

21.2 Context 291

21.3 Data 293

21.4 Analysis 295

21.5 Discussion 299

21.6 Conclusion 300

Acknowledgement 300

References 300

22 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution 302
Roger Puig, Raimon Tolosana-Delgado, Neus Otero and Albert Folch

22.1 Introduction 302

22.2 Study Area 303

22.2.1 Maresme 304

22.2.2 Osona 305

22.2.3 Lluc¸an`es 305

22.2.4 Empord`a 306

22.2.5 Selva 306

22.3 Analytical Methods 306

22.4 Statistical Treatment 307

22.4.1 Data Scaling 307

22.4.2 Linear Discriminant Analysis 309

22.4.3 Discriminant Biplots 310

22.5 Results and Discussion 311

22.6 Conclusions 314

Acknowledgements 315

References 315

23 Applications in Economics 318
Tim Fry

23.1 Introduction 318

23.2 Consumer Demand Systems 319

23.3 Miscellaneous Applications 322

23.4 Compositional Time Series 323

23.5 New Directions 323

23.6 Conclusion 325

References 325

Part V Software 327

24 Exploratory Analysis Using CoDaPack 3D 329
Santiago Thió-Henestrosa and Josep Daunis-i-Estadella

24.1 CoDaPack 3D Description 329

24.2 Data Set Description 331

24.3 Exploratory Analysis 333

24.3.1 Numerical Analysis 333

24.3.2 Biplot 334

24.3.3 The Ternary Diagram 335

24.3.4 Principal Component Analysis 336

24.3.5 Balance-Dendrogram 336

24.3.6 By Groups Description 338

24.4 Summary and Conclusions 339

Acknowledgements 340

References 340

25 robCompositions: An R-package for Robust Statistical Analysis of Compositional Data 341
Matthias Templ, Karel Hron and Peter Filzmoser

25.1 General Information on the R-package robCompositions 341

25.1.1 Data Sets Included in the Package 342

25.1.2 Design Principles 343

25.2 Expressing Compositional Data in Coordinates 343

25.3 Multivariate Statistical Methods for Compositional Data Containing Outliers 345

25.3.1 Multivariate Outlier Detection 345

25.3.2 Principal Component Analysis and the Robust Compositional Biplot 347

25.3.3 Discriminant Analysis 350

25.4 Robust Imputation of Missing Values 351

25.5 Summary 354

References 354

26 Linear Models with Compositions in R 356
Raimon Tolosana-Delgado and Karl Gerald van den Boogaart

26.1 Introduction 356

26.2 The Illustration Data Set 357

26.2.1 The Data 357

26.2.2 Descriptive Analysis of Compositional Characteristics 358

26.3 Explanatory Binary Variable 360

26.4 Explanatory Categorical Variable 363

26.5 Explanatory Continuous Variable 365

26.6 Explanatory Composition 367

26.7 Conclusions 370

Acknowledgement 371

References 371

Index 373