VERA PAWLOWSKY-GLAHN Department of Computer Science, Applied Mathematics, and Statistics, University of Girona, Spain JUAN JOSÉ EGOZCUE Department of Applied Mathematics III, Technical University of Catalonia, Barcelona, Spain RAIMON TOLOSANA-DELGADO Helmholtz Institute Freiberg for Resource Technology, Germany

• Preface xi About the Authors xvAcknowledgments xix1 Introduction 12 Compositional Data and Their Sample Space 82.1 Basic concepts 82.2 Principles of compositional analysis 122.2.1 Scale invariance 122.2.2 Permutation invariance 152.2.3 Subcompositional coherence 162.3 Zeros, missing values, and other irregular components 162.3.1 Kinds of irregular components 162.3.2 Strategies to analyze irregular data 192.4 Exercises 213 The Aitchison Geometry 233.1 General comments 233.2 Vector space structure 243.3 Inner product, norm and distance 263.4 Geometric figures 283.5 Exercises 304 Coordinate Representation 324.1 Introduction 324.2 Compositional observations in real space 334.3 Generating systems 334.4 Orthonormal coordinates 364.5 Balances 384.6 Working on coordinates 434.7 Additive logratio coordinates (alr) 464.8 Orthogonal projections 484.9 Matrix operations in the simplex 544.9.1 Perturbation-linear combination of compositions 544.9.2 Linear transformations of óKòù: endomorphisms 554.9.3 Other matrix transformations on óKòù: nonlinear transformations 574.10 Coordinates leading to alternative Euclidean structures 594.11 Exercises 615 Exploratory Data Analysis 655.1 General remarks 655.2 Sample center, total variance, and variation matrix 665.3 Centering and scaling 685.4 The biplot: a graphical display 705.4.1 Construction of a biplot 705.4.2 Interpretation of a 2D compositional biplot 725.5 Exploratory analysis of coordinates 765.6 A geological example 795.7 Linear trends along principal components 855.8 A nutrition example 895.9 A political example 965.10 Exercises 1006 Random Compositions 1036.1 Sample space 1036.1.1 Conventional approach to the sample space of compositions 1056.1.2 A compositional approach to the sample space of compositions 1066.1.3 Definitions related to random compositions 1076.2 Variability and center 1086.3 Probability distributions on the simplex 1126.3.1 The normal distribution on the simplex 1146.3.2 The Dirichlet distribution 1216.3.3 Other distributions 1276.4 Exercises 1287 Statistical Inference 1307.1 Point estimation of center and variability 1307.2 Testing hypotheses on compositional normality 1357.3 Testing hypotheses about two populations 1367.4 Probability and confidence regions for normal data 1427.5 Bayesian estimation with count data 1447.6 Exercises 1478 Linear Models 1498.1 Linear regression with compositional response 1508.2 Regression with compositional covariates 1568.3 Analysis of variance with compositional response 1608.4 Linear discrimination with compositional predictor 1638.5 Exercises 1659 Compositional Processes 1729.1 Linear processes 1739.2 Mixture processes 1769.3 Settling processes 1789.4 Simplicial derivative 1839.5 Elementary differential equations 1869.5.1 Constant derivative 1879.5.2 Forced derivative 1899.5.3 Complete first-order linear equation 1949.5.4 Harmonic oscillator 2009.6 Exercises 20410 Epilogue 206References 211Appendix A Practical Recipes 222A.1 Plotting a ternary diagram 222A.2 Parameterization of an elliptic region 224A.3 Matrix expressions of change of representation 226Appendix B Random Variables 228B.1 Probability spaces and random variables 228B.2 Description of probability 232List of Abbreviations and Symbols 234Author Index 237General Index 241

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VERA PAWLOWSKY-GLAHN Department of Computer Science, Applied Mathematics, and Statistics, University of Girona, Spain JUAN JOSÉ EGOZCUE Department of Applied Mathematics III, Technical University of Catalonia, Barcelona, Spain RAIMON TOLOSANA-DELGADO Helmholtz Institute Freiberg for Resource Technology, Germany

• Preface xi About the Authors xvAcknowledgments xix1 Introduction 12 Compositional Data and Their Sample Space 82.1 Basic concepts 82.2 Principles of compositional analysis 122.2.1 Scale invariance 122.2.2 Permutation invariance 152.2.3 Subcompositional coherence 162.3 Zeros, missing values, and other irregular components 162.3.1 Kinds of irregular components 162.3.2 Strategies to analyze irregular data 192.4 Exercises 213 The Aitchison Geometry 233.1 General comments 233.2 Vector space structure 243.3 Inner product, norm and distance 263.4 Geometric figures 283.5 Exercises 304 Coordinate Representation 324.1 Introduction 324.2 Compositional observations in real space 334.3 Generating systems 334.4 Orthonormal coordinates 364.5 Balances 384.6 Working on coordinates 434.7 Additive logratio coordinates (alr) 464.8 Orthogonal projections 484.9 Matrix operations in the simplex 544.9.1 Perturbation-linear combination of compositions 544.9.2 Linear transformations of óKòù: endomorphisms 554.9.3 Other matrix transformations on óKòù: nonlinear transformations 574.10 Coordinates leading to alternative Euclidean structures 594.11 Exercises 615 Exploratory Data Analysis 655.1 General remarks 655.2 Sample center, total variance, and variation matrix 665.3 Centering and scaling 685.4 The biplot: a graphical display 705.4.1 Construction of a biplot 705.4.2 Interpretation of a 2D compositional biplot 725.5 Exploratory analysis of coordinates 765.6 A geological example 795.7 Linear trends along principal components 855.8 A nutrition example 895.9 A political example 965.10 Exercises 1006 Random Compositions 1036.1 Sample space 1036.1.1 Conventional approach to the sample space of compositions 1056.1.2 A compositional approach to the sample space of compositions 1066.1.3 Definitions related to random compositions 1076.2 Variability and center 1086.3 Probability distributions on the simplex 1126.3.1 The normal distribution on the simplex 1146.3.2 The Dirichlet distribution 1216.3.3 Other distributions 1276.4 Exercises 1287 Statistical Inference 1307.1 Point estimation of center and variability 1307.2 Testing hypotheses on compositional normality 1357.3 Testing hypotheses about two populations 1367.4 Probability and confidence regions for normal data 1427.5 Bayesian estimation with count data 1447.6 Exercises 1478 Linear Models 1498.1 Linear regression with compositional response 1508.2 Regression with compositional covariates 1568.3 Analysis of variance with compositional response 1608.4 Linear discrimination with compositional predictor 1638.5 Exercises 1659 Compositional Processes 1729.1 Linear processes 1739.2 Mixture processes 1769.3 Settling processes 1789.4 Simplicial derivative 1839.5 Elementary differential equations 1869.5.1 Constant derivative 1879.5.2 Forced derivative 1899.5.3 Complete first-order linear equation 1949.5.4 Harmonic oscillator 2009.6 Exercises 20410 Epilogue 206References 211Appendix A Practical Recipes 222A.1 Plotting a ternary diagram 222A.2 Parameterization of an elliptic region 224A.3 Matrix expressions of change of representation 226Appendix B Random Variables 228B.1 Probability spaces and random variables 228B.2 Description of probability 232List of Abbreviations and Symbols 234Author Index 237General Index 241

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VERA PAWLOWSKY-GLAHN Department of Computer Science, Applied Mathematics, and Statistics, University of Girona, Spain JUAN JOSÉ EGOZCUE Department of Applied Mathematics III, Technical University of Catalonia, Barcelona, Spain RAIMON TOLOSANA-DELGADO Helmholtz Institute Freiberg for Resource Technology, Germany

• Preface xi About the Authors xvAcknowledgments xix1 Introduction 12 Compositional Data and Their Sample Space 82.1 Basic concepts 82.2 Principles of compositional analysis 122.2.1 Scale invariance 122.2.2 Permutation invariance 152.2.3 Subcompositional coherence 162.3 Zeros, missing values, and other irregular components 162.3.1 Kinds of irregular components 162.3.2 Strategies to analyze irregular data 192.4 Exercises 213 The Aitchison Geometry 233.1 General comments 233.2 Vector space structure 243.3 Inner product, norm and distance 263.4 Geometric figures 283.5 Exercises 304 Coordinate Representation 324.1 Introduction 324.2 Compositional observations in real space 334.3 Generating systems 334.4 Orthonormal coordinates 364.5 Balances 384.6 Working on coordinates 434.7 Additive logratio coordinates (alr) 464.8 Orthogonal projections 484.9 Matrix operations in the simplex 544.9.1 Perturbation-linear combination of compositions 544.9.2 Linear transformations of óKòù: endomorphisms 554.9.3 Other matrix transformations on óKòù: nonlinear transformations 574.10 Coordinates leading to alternative Euclidean structures 594.11 Exercises 615 Exploratory Data Analysis 655.1 General remarks 655.2 Sample center, total variance, and variation matrix 665.3 Centering and scaling 685.4 The biplot: a graphical display 705.4.1 Construction of a biplot 705.4.2 Interpretation of a 2D compositional biplot 725.5 Exploratory analysis of coordinates 765.6 A geological example 795.7 Linear trends along principal components 855.8 A nutrition example 895.9 A political example 965.10 Exercises 1006 Random Compositions 1036.1 Sample space 1036.1.1 Conventional approach to the sample space of compositions 1056.1.2 A compositional approach to the sample space of compositions 1066.1.3 Definitions related to random compositions 1076.2 Variability and center 1086.3 Probability distributions on the simplex 1126.3.1 The normal distribution on the simplex 1146.3.2 The Dirichlet distribution 1216.3.3 Other distributions 1276.4 Exercises 1287 Statistical Inference 1307.1 Point estimation of center and variability 1307.2 Testing hypotheses on compositional normality 1357.3 Testing hypotheses about two populations 1367.4 Probability and confidence regions for normal data 1427.5 Bayesian estimation with count data 1447.6 Exercises 1478 Linear Models 1498.1 Linear regression with compositional response 1508.2 Regression with compositional covariates 1568.3 Analysis of variance with compositional response 1608.4 Linear discrimination with compositional predictor 1638.5 Exercises 1659 Compositional Processes 1729.1 Linear processes 1739.2 Mixture processes 1769.3 Settling processes 1789.4 Simplicial derivative 1839.5 Elementary differential equations 1869.5.1 Constant derivative 1879.5.2 Forced derivative 1899.5.3 Complete first-order linear equation 1949.5.4 Harmonic oscillator 2009.6 Exercises 20410 Epilogue 206References 211Appendix A Practical Recipes 222A.1 Plotting a ternary diagram 222A.2 Parameterization of an elliptic region 224A.3 Matrix expressions of change of representation 226Appendix B Random Variables 228B.1 Probability spaces and random variables 228B.2 Description of probability 232List of Abbreviations and Symbols 234Author Index 237General Index 241

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