Compositional data analysis: theory and applications/ edited by Vera Pawlowsky-Glahn and Antonella Buccianti

Preface xvii 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 Gloria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan Jose 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 Martun-Fernandez, 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 Barcelo-Vidal, Lucua Aguilar and Josep Antoni Martun-Fernandez 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, Gloria 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 Jose Egozcue, Carles Barcelo-Vidal, Josep Antoni Martun-Fernandez, Eusebi Jarauta-Bragulat, Jose Luis Duaz-Barrero and Gloria 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 Jose Egozcue, Eusebi Jarauta-Bragulat and Jose Luis Diaz-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 Barcelo-Vidal, Josep Antoni Martun-Fernandez and Gloria 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 Muller, 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 Jose 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 Martun-Fernandez 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, Gloria 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 Martun-Fernandez 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-Iguzquiza 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 Thio-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