Vera Pawlowsky-Glahn, Department of Computer Science and Applied Mathematics, University of Girona, Spain. Antonella Buccianti, Department of Earth Sciences, University of Florence, Italy.

• Preface xviiList of Contributors xixPart I Introduction 11 A Short History of Compositional Data Analysis 3John Bacon-Shone1.1 Introduction 31.2 Spurious Correlation 31.3 Log and Log-Ratio Transforms 41.4 Subcompositional Dependence 51.5 alr, clr, ilr: Which Transformation to Choose? 51.6 Principles, Perturbations and Back to the Simplex 61.7 Biplots and Singular Value Decompositions 71.8 Mixtures 71.9 Discrete Compositions 81.10 Compositional Processes 81.11 Structural, Counting and Rounded Zeros 81.12 Conclusion 9Acknowledgement 9References 92 Basic Concepts and Procedures 12Juan José Egozcue and Vera Pawlowsky-Glahn2.1 Introduction 122.2 Election Data and Raw Analysis 132.3 The Compositional Alternative 152.4 Geometric Settings 172.5 Centre and Variability 222.6 Conclusion 27Acknowledgements 27References 27Part II Theory – Statistical Modelling 293 The Principle of Working on Coordinates 31Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue3.1 Introduction 313.2 The Role of Coordinates in Statistics 323.3 The Simplex 333.4 Move or Stay in the Simplex 383.5 Conclusions 40Acknowledgements 41References 414 Dealing with Zeros 43Josep Antoni Martín-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea4.1 Introduction 434.2 Rounded Zeros 444.3 Count Zeros 504.4 Essential Zeros 534.5 Difficulties, Troubles and Challenges 55Acknowledgements 57References 575 Robust Statistical Analysis 59Peter Filzmoser and Karel Hron5.1 Introduction 595.2 Elements of Robust Statistics from a Compositional Point of View 605.3 Robust Methods for Compositional Data 635.4 Case Studies 665.5 Summary 70Acknowledgement 71References 716 Geostatistics for Compositions 73Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn6.1 Introduction 736.2 A Brief Summary of Geostatistics 746.3 Cokriging of Regionalised Compositions 766.4 Structural Analysis of Regionalised Composition 766.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 786.6 Application 796.7 Conclusions 84Acknowledgements 84References 847 Compositional VARIMA Time Series 87Carles Barceló-Vidal, Lucía Aguilar and Josep Antoni Martín-Fernández7.1 Introduction 877.2 The Simplex S D as a Compositional Space 897.3 Compositional Time Series Models 917.4 CTS Modelling: An Example 947.5 Discussion 99Acknowledgements 99References 100Appendix 1028 Compositional Data and Correspondence Analysis 104Michael Greenacre8.1 Introduction 1048.2 Comparative Technical Definitions 1058.3 Properties and Interpretation of LRA and CA 1078.4 Application to Fatty Acid Compositional Data 1078.5 Discussion and Conclusions 111Acknowledgements 112References 1129 Use of Survey Weights for the Analysis of Compositional Data 114Monique Graf9.1 Introduction 1149.2 Elements of Survey Design 1159.3 Application to Compositional Data 1229.4 Discussion 126References 12610 Notes on the Scaled Dirichlet Distribution 128Gianna Serafina Monti, Glòria Mateu-Figueras and Vera Pawlowsky-Glahn10.1 Introduction 12810.2 Genesis of the Scaled Dirichlet Distribution 12910.3 Properties of the Scaled Dirichlet Distribution 13110.4 Conclusions 136Acknowledgements 137References 137Part III Theory – Algebra and Calculus 13911 Elements of Simplicial Linear Algebra and Geometry 141Juan José Egozcue, Carles Barceló-Vidal, Josep Antoni Martín-Fernández, Eusebi Jarauta-Bragulat, José LuisDíaz-Barrero and Glòria Mateu-Figueras11.1 Introduction 14111.2 Elements of Simplicial Geometry 14211.3 Linear Functions 15111.4 Conclusions 156Acknowledgements 156References 15612 Calculus of Simplex-Valued Functions 158Juan José Egozcue, Eusebi Jarauta-Bragulat and José LuisDíaz-Barrero12.1 Introduction 15812.3 Integration 17112.4 Conclusions 174Acknowledgements 175References 17513 Compositional Differential Calculus on the Simplex 176Carles Barceló-Vidal, Josep Antoni Martín-Fernández and Glòria Mateu-Figueras13.1 Introduction 17613.2 Vector-Valued Functions on the Simplex 17713.3 C-Derivatives on the Simplex 17813.4 Example: Experiments with Mixtures 18513.5 Discussion 189Acknowledgements 190References 190Part IV Applications 19114 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? 193David Lovell, Warren Müller, Jen Taylor, Alec Zwart and Chris Helliwell14.1 Introduction 19314.2 The Omics Imp and Two Bioscience Experiment Paradigms 19414.3 The Impact of Compositional Constraints in the Omics 19714.4 Impact of Compositional Constraints on Correlation and Covariance 20114.5 Implications 204Acknowledgements 206References 20615 Hardy–Weinberg Equilibrium: A Nonparametric Compositional Approach 208Jan Graffelman and Juan José Egozcue15.1 Introduction 20815.2 Genetic Data Sets 20915.3 Classical Tests for HWE 21015.4 A Compositional Approach 21015.5 Example 21415.6 Conclusion and Discussion 215Acknowledgements 215References 21516 Compositional Analysis in Behavioural and Evolutionary Ecology 218Michele Edoardo Raffaele Pierotti and Josep Antoni Martín-Fernández16.1 Introduction 21816.2 CODA in Population Genetics 21916.3 CODA in Habitat Choice 22216.4 Multiple Choice and Individual Variation in Preferences 22416.5 Ecological Specialization 22816.6 Time Budgets: More on Specialization 22916.7 Conclusions 231Acknowledgements 231References 23117 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates 235Luis Azevedo Rodrigues, Josep Daunis-i-Estadella, Glòria Mateu-Figueras and Santiago Thió-Henestrosa17.1 Introduction 23517.2 Flying Vertebrates – General Anatomical and Functional Characteristics 23617.3 Materials 23617.4 Methods 23817.5 Aitchison Distance Disparity Metrics 23917.6 Statistical Tests 24317.7 Biplots 24417.8 Balances 24617.9 Size Effect 24917.10 Final Remarks 249Acknowledgements 252References 25218 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach 255Antonella Buccianti18.1 Introduction 25518.2 Geochemical Processes and Log-Ratio Approach 25618.3 Log-Ratio Approach and Water Chemistry 25818.4 Log-Ratio Approach and Volcanic Gas Chemistry 26118.5 Log-Ratio Approach and Subducting Sediment Composition 26318.6 Conclusions 265Acknowledgements 265References 26519 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury 267Helmut Lammer, Peter Wurz, Josep Antoni Martín-Fernández and Herbert Iwo Maria Lichtenegger19.1 Introduction 26719.2 Compositional Analysis of Mars’ Surface 27019.3 Compositional Analysis of Mercury’s Surface 27419.4 Conclusion 278Acknowledgement 278References 27820 Spectral Analysis of Compositional Data in Cyclostratigraphy 282Eulogio Pardo-Igúzquiza and Javier Heredia20.1 Introduction 28220.2 The Method 28320.3 Case Study 28520.4 Discussion 28720.5 Conclusions 288Acknowledgement 288References 28821 Multivariate Geochemical Data Analysis in Physical Geography 290Jennifer McKinley and Christopher David Lloyd21.1 Introduction 29021.2 Context 29121.3 Data 29321.4 Analysis 29521.5 Discussion 29921.6 Conclusion 300Acknowledgement 300References 30022 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution 302Roger Puig, Raimon Tolosana-Delgado, Neus Otero and Albert Folch22.1 Introduction 30222.2 Study Area 30322.3 Analytical Methods 30622.4 Statistical Treatment 30722.5 Results and Discussion 31122.6 Conclusions 314Acknowledgements 315References 31523 Applications in Economics 318Tim Fry23.1 Introduction 31823.2 Consumer Demand Systems 31923.3 Miscellaneous Applications 32223.4 Compositional Time Series 32323.5 New Directions 32323.6 Conclusion 325References 325Part V Software 32724 Exploratory Analysis Using CoDaPack 3D 329Santiago Thió-Henestrosa and Josep Daunis-i-Estadella24.1 CoDaPack 3D Description 32924.2 Data Set Description 33124.3 Exploratory Analysis 33324.4 Summary and Conclusions 339Acknowledgements 340References 34025 robCompositions: An R-package for Robust Statistical Analysis of Compositional Data 341Matthias Templ, Karel Hron and Peter Filzmoser25.1 General Information on the R-package robCompositions 34125.2 Expressing Compositional Data in Coordinates 34325.3 Multivariate Statistical Methods for Compositional Data Containing Outliers 34525.4 Robust Imputation of Missing Values 35125.5 Summary 354References 35426 Linear Models with Compositions in R 356Raimon Tolosana-Delgado and Karl Gerald van den Boogaart26.1 Introduction 35626.2 The Illustration Data Set 35726.3 Explanatory Binary Variable 36026.4 Explanatory Categorical Variable 36326.5 Explanatory Continuous Variable 36526.6 Explanatory Composition 36726.7 Conclusions 370Acknowledgement 371References 371Index 373

Vera Pawlowsky-Glahn, Department of Computer Science and Applied Mathematics, University of Girona, Spain. Antonella Buccianti, Department of Earth Sciences, University of Florence, Italy.

• Preface xviiList of Contributors xixPart I Introduction 11 A Short History of Compositional Data Analysis 3John Bacon-Shone1.1 Introduction 31.2 Spurious Correlation 31.3 Log and Log-Ratio Transforms 41.4 Subcompositional Dependence 51.5 alr, clr, ilr: Which Transformation to Choose? 51.6 Principles, Perturbations and Back to the Simplex 61.7 Biplots and Singular Value Decompositions 71.8 Mixtures 71.9 Discrete Compositions 81.10 Compositional Processes 81.11 Structural, Counting and Rounded Zeros 81.12 Conclusion 9Acknowledgement 9References 92 Basic Concepts and Procedures 12Juan José Egozcue and Vera Pawlowsky-Glahn2.1 Introduction 122.2 Election Data and Raw Analysis 132.3 The Compositional Alternative 152.4 Geometric Settings 172.5 Centre and Variability 222.6 Conclusion 27Acknowledgements 27References 27Part II Theory – Statistical Modelling 293 The Principle of Working on Coordinates 31Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue3.1 Introduction 313.2 The Role of Coordinates in Statistics 323.3 The Simplex 333.4 Move or Stay in the Simplex 383.5 Conclusions 40Acknowledgements 41References 414 Dealing with Zeros 43Josep Antoni Martín-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea4.1 Introduction 434.2 Rounded Zeros 444.3 Count Zeros 504.4 Essential Zeros 534.5 Difficulties, Troubles and Challenges 55Acknowledgements 57References 575 Robust Statistical Analysis 59Peter Filzmoser and Karel Hron5.1 Introduction 595.2 Elements of Robust Statistics from a Compositional Point of View 605.3 Robust Methods for Compositional Data 635.4 Case Studies 665.5 Summary 70Acknowledgement 71References 716 Geostatistics for Compositions 73Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn6.1 Introduction 736.2 A Brief Summary of Geostatistics 746.3 Cokriging of Regionalised Compositions 766.4 Structural Analysis of Regionalised Composition 766.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 786.6 Application 796.7 Conclusions 84Acknowledgements 84References 847 Compositional VARIMA Time Series 87Carles Barceló-Vidal, Lucía Aguilar and Josep Antoni Martín-Fernández7.1 Introduction 877.2 The Simplex S D as a Compositional Space 897.3 Compositional Time Series Models 917.4 CTS Modelling: An Example 947.5 Discussion 99Acknowledgements 99References 100Appendix 1028 Compositional Data and Correspondence Analysis 104Michael Greenacre8.1 Introduction 1048.2 Comparative Technical Definitions 1058.3 Properties and Interpretation of LRA and CA 1078.4 Application to Fatty Acid Compositional Data 1078.5 Discussion and Conclusions 111Acknowledgements 112References 1129 Use of Survey Weights for the Analysis of Compositional Data 114Monique Graf9.1 Introduction 1149.2 Elements of Survey Design 1159.3 Application to Compositional Data 1229.4 Discussion 126References 12610 Notes on the Scaled Dirichlet Distribution 128Gianna Serafina Monti, Glòria Mateu-Figueras and Vera Pawlowsky-Glahn10.1 Introduction 12810.2 Genesis of the Scaled Dirichlet Distribution 12910.3 Properties of the Scaled Dirichlet Distribution 13110.4 Conclusions 136Acknowledgements 137References 137Part III Theory – Algebra and Calculus 13911 Elements of Simplicial Linear Algebra and Geometry 141Juan José Egozcue, Carles Barceló-Vidal, Josep Antoni Martín-Fernández, Eusebi Jarauta-Bragulat, José LuisDíaz-Barrero and Glòria Mateu-Figueras11.1 Introduction 14111.2 Elements of Simplicial Geometry 14211.3 Linear Functions 15111.4 Conclusions 156Acknowledgements 156References 15612 Calculus of Simplex-Valued Functions 158Juan José Egozcue, Eusebi Jarauta-Bragulat and José LuisDíaz-Barrero12.1 Introduction 15812.3 Integration 17112.4 Conclusions 174Acknowledgements 175References 17513 Compositional Differential Calculus on the Simplex 176Carles Barceló-Vidal, Josep Antoni Martín-Fernández and Glòria Mateu-Figueras13.1 Introduction 17613.2 Vector-Valued Functions on the Simplex 17713.3 C-Derivatives on the Simplex 17813.4 Example: Experiments with Mixtures 18513.5 Discussion 189Acknowledgements 190References 190Part IV Applications 19114 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? 193David Lovell, Warren Müller, Jen Taylor, Alec Zwart and Chris Helliwell14.1 Introduction 19314.2 The Omics Imp and Two Bioscience Experiment Paradigms 19414.3 The Impact of Compositional Constraints in the Omics 19714.4 Impact of Compositional Constraints on Correlation and Covariance 20114.5 Implications 204Acknowledgements 206References 20615 Hardy–Weinberg Equilibrium: A Nonparametric Compositional Approach 208Jan Graffelman and Juan José Egozcue15.1 Introduction 20815.2 Genetic Data Sets 20915.3 Classical Tests for HWE 21015.4 A Compositional Approach 21015.5 Example 21415.6 Conclusion and Discussion 215Acknowledgements 215References 21516 Compositional Analysis in Behavioural and Evolutionary Ecology 218Michele Edoardo Raffaele Pierotti and Josep Antoni Martín-Fernández16.1 Introduction 21816.2 CODA in Population Genetics 21916.3 CODA in Habitat Choice 22216.4 Multiple Choice and Individual Variation in Preferences 22416.5 Ecological Specialization 22816.6 Time Budgets: More on Specialization 22916.7 Conclusions 231Acknowledgements 231References 23117 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates 235Luis Azevedo Rodrigues, Josep Daunis-i-Estadella, Glòria Mateu-Figueras and Santiago Thió-Henestrosa17.1 Introduction 23517.2 Flying Vertebrates – General Anatomical and Functional Characteristics 23617.3 Materials 23617.4 Methods 23817.5 Aitchison Distance Disparity Metrics 23917.6 Statistical Tests 24317.7 Biplots 24417.8 Balances 24617.9 Size Effect 24917.10 Final Remarks 249Acknowledgements 252References 25218 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach 255Antonella Buccianti18.1 Introduction 25518.2 Geochemical Processes and Log-Ratio Approach 25618.3 Log-Ratio Approach and Water Chemistry 25818.4 Log-Ratio Approach and Volcanic Gas Chemistry 26118.5 Log-Ratio Approach and Subducting Sediment Composition 26318.6 Conclusions 265Acknowledgements 265References 26519 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury 267Helmut Lammer, Peter Wurz, Josep Antoni Martín-Fernández and Herbert Iwo Maria Lichtenegger19.1 Introduction 26719.2 Compositional Analysis of Mars’ Surface 27019.3 Compositional Analysis of Mercury’s Surface 27419.4 Conclusion 278Acknowledgement 278References 27820 Spectral Analysis of Compositional Data in Cyclostratigraphy 282Eulogio Pardo-Igúzquiza and Javier Heredia20.1 Introduction 28220.2 The Method 28320.3 Case Study 28520.4 Discussion 28720.5 Conclusions 288Acknowledgement 288References 28821 Multivariate Geochemical Data Analysis in Physical Geography 290Jennifer McKinley and Christopher David Lloyd21.1 Introduction 29021.2 Context 29121.3 Data 29321.4 Analysis 29521.5 Discussion 29921.6 Conclusion 300Acknowledgement 300References 30022 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution 302Roger Puig, Raimon Tolosana-Delgado, Neus Otero and Albert Folch22.1 Introduction 30222.2 Study Area 30322.3 Analytical Methods 30622.4 Statistical Treatment 30722.5 Results and Discussion 31122.6 Conclusions 314Acknowledgements 315References 31523 Applications in Economics 318Tim Fry23.1 Introduction 31823.2 Consumer Demand Systems 31923.3 Miscellaneous Applications 32223.4 Compositional Time Series 32323.5 New Directions 32323.6 Conclusion 325References 325Part V Software 32724 Exploratory Analysis Using CoDaPack 3D 329Santiago Thió-Henestrosa and Josep Daunis-i-Estadella24.1 CoDaPack 3D Description 32924.2 Data Set Description 33124.3 Exploratory Analysis 33324.4 Summary and Conclusions 339Acknowledgements 340References 34025 robCompositions: An R-package for Robust Statistical Analysis of Compositional Data 341Matthias Templ, Karel Hron and Peter Filzmoser25.1 General Information on the R-package robCompositions 34125.2 Expressing Compositional Data in Coordinates 34325.3 Multivariate Statistical Methods for Compositional Data Containing Outliers 34525.4 Robust Imputation of Missing Values 35125.5 Summary 354References 35426 Linear Models with Compositions in R 356Raimon Tolosana-Delgado and Karl Gerald van den Boogaart26.1 Introduction 35626.2 The Illustration Data Set 35726.3 Explanatory Binary Variable 36026.4 Explanatory Categorical Variable 36326.5 Explanatory Continuous Variable 36526.6 Explanatory Composition 36726.7 Conclusions 370Acknowledgement 371References 371Index 373

Vera Pawlowsky-Glahn, Department of Computer Science and Applied Mathematics, University of Girona, Spain. Antonella Buccianti, Department of Earth Sciences, University of Florence, Italy.

• Preface xviiList of Contributors xixPart I Introduction 11 A Short History of Compositional Data Analysis 3John Bacon-Shone1.1 Introduction 31.2 Spurious Correlation 31.3 Log and Log-Ratio Transforms 41.4 Subcompositional Dependence 51.5 alr, clr, ilr: Which Transformation to Choose? 51.6 Principles, Perturbations and Back to the Simplex 61.7 Biplots and Singular Value Decompositions 71.8 Mixtures 71.9 Discrete Compositions 81.10 Compositional Processes 81.11 Structural, Counting and Rounded Zeros 81.12 Conclusion 9Acknowledgement 9References 92 Basic Concepts and Procedures 12Juan José Egozcue and Vera Pawlowsky-Glahn2.1 Introduction 122.2 Election Data and Raw Analysis 132.3 The Compositional Alternative 152.4 Geometric Settings 172.5 Centre and Variability 222.6 Conclusion 27Acknowledgements 27References 27Part II Theory – Statistical Modelling 293 The Principle of Working on Coordinates 31Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue3.1 Introduction 313.2 The Role of Coordinates in Statistics 323.3 The Simplex 333.4 Move or Stay in the Simplex 383.5 Conclusions 40Acknowledgements 41References 414 Dealing with Zeros 43Josep Antoni Martín-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea4.1 Introduction 434.2 Rounded Zeros 444.3 Count Zeros 504.4 Essential Zeros 534.5 Difficulties, Troubles and Challenges 55Acknowledgements 57References 575 Robust Statistical Analysis 59Peter Filzmoser and Karel Hron5.1 Introduction 595.2 Elements of Robust Statistics from a Compositional Point of View 605.3 Robust Methods for Compositional Data 635.4 Case Studies 665.5 Summary 70Acknowledgement 71References 716 Geostatistics for Compositions 73Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn6.1 Introduction 736.2 A Brief Summary of Geostatistics 746.3 Cokriging of Regionalised Compositions 766.4 Structural Analysis of Regionalised Composition 766.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 786.6 Application 796.7 Conclusions 84Acknowledgements 84References 847 Compositional VARIMA Time Series 87Carles Barceló-Vidal, Lucía Aguilar and Josep Antoni Martín-Fernández7.1 Introduction 877.2 The Simplex S D as a Compositional Space 897.3 Compositional Time Series Models 917.4 CTS Modelling: An Example 947.5 Discussion 99Acknowledgements 99References 100Appendix 1028 Compositional Data and Correspondence Analysis 104Michael Greenacre8.1 Introduction 1048.2 Comparative Technical Definitions 1058.3 Properties and Interpretation of LRA and CA 1078.4 Application to Fatty Acid Compositional Data 1078.5 Discussion and Conclusions 111Acknowledgements 112References 1129 Use of Survey Weights for the Analysis of Compositional Data 114Monique Graf9.1 Introduction 1149.2 Elements of Survey Design 1159.3 Application to Compositional Data 1229.4 Discussion 126References 12610 Notes on the Scaled Dirichlet Distribution 128Gianna Serafina Monti, Glòria Mateu-Figueras and Vera Pawlowsky-Glahn10.1 Introduction 12810.2 Genesis of the Scaled Dirichlet Distribution 12910.3 Properties of the Scaled Dirichlet Distribution 13110.4 Conclusions 136Acknowledgements 137References 137Part III Theory – Algebra and Calculus 13911 Elements of Simplicial Linear Algebra and Geometry 141Juan José Egozcue, Carles Barceló-Vidal, Josep Antoni Martín-Fernández, Eusebi Jarauta-Bragulat, José LuisDíaz-Barrero and Glòria Mateu-Figueras11.1 Introduction 14111.2 Elements of Simplicial Geometry 14211.3 Linear Functions 15111.4 Conclusions 156Acknowledgements 156References 15612 Calculus of Simplex-Valued Functions 158Juan José Egozcue, Eusebi Jarauta-Bragulat and José LuisDíaz-Barrero12.1 Introduction 15812.3 Integration 17112.4 Conclusions 174Acknowledgements 175References 17513 Compositional Differential Calculus on the Simplex 176Carles Barceló-Vidal, Josep Antoni Martín-Fernández and Glòria Mateu-Figueras13.1 Introduction 17613.2 Vector-Valued Functions on the Simplex 17713.3 C-Derivatives on the Simplex 17813.4 Example: Experiments with Mixtures 18513.5 Discussion 189Acknowledgements 190References 190Part IV Applications 19114 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? 193David Lovell, Warren Müller, Jen Taylor, Alec Zwart and Chris Helliwell14.1 Introduction 19314.2 The Omics Imp and Two Bioscience Experiment Paradigms 19414.3 The Impact of Compositional Constraints in the Omics 19714.4 Impact of Compositional Constraints on Correlation and Covariance 20114.5 Implications 204Acknowledgements 206References 20615 Hardy–Weinberg Equilibrium: A Nonparametric Compositional Approach 208Jan Graffelman and Juan José Egozcue15.1 Introduction 20815.2 Genetic Data Sets 20915.3 Classical Tests for HWE 21015.4 A Compositional Approach 21015.5 Example 21415.6 Conclusion and Discussion 215Acknowledgements 215References 21516 Compositional Analysis in Behavioural and Evolutionary Ecology 218Michele Edoardo Raffaele Pierotti and Josep Antoni Martín-Fernández16.1 Introduction 21816.2 CODA in Population Genetics 21916.3 CODA in Habitat Choice 22216.4 Multiple Choice and Individual Variation in Preferences 22416.5 Ecological Specialization 22816.6 Time Budgets: More on Specialization 22916.7 Conclusions 231Acknowledgements 231References 23117 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates 235Luis Azevedo Rodrigues, Josep Daunis-i-Estadella, Glòria Mateu-Figueras and Santiago Thió-Henestrosa17.1 Introduction 23517.2 Flying Vertebrates – General Anatomical and Functional Characteristics 23617.3 Materials 23617.4 Methods 23817.5 Aitchison Distance Disparity Metrics 23917.6 Statistical Tests 24317.7 Biplots 24417.8 Balances 24617.9 Size Effect 24917.10 Final Remarks 249Acknowledgements 252References 25218 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach 255Antonella Buccianti18.1 Introduction 25518.2 Geochemical Processes and Log-Ratio Approach 25618.3 Log-Ratio Approach and Water Chemistry 25818.4 Log-Ratio Approach and Volcanic Gas Chemistry 26118.5 Log-Ratio Approach and Subducting Sediment Composition 26318.6 Conclusions 265Acknowledgements 265References 26519 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury 267Helmut Lammer, Peter Wurz, Josep Antoni Martín-Fernández and Herbert Iwo Maria Lichtenegger19.1 Introduction 26719.2 Compositional Analysis of Mars’ Surface 27019.3 Compositional Analysis of Mercury’s Surface 27419.4 Conclusion 278Acknowledgement 278References 27820 Spectral Analysis of Compositional Data in Cyclostratigraphy 282Eulogio Pardo-Igúzquiza and Javier Heredia20.1 Introduction 28220.2 The Method 28320.3 Case Study 28520.4 Discussion 28720.5 Conclusions 288Acknowledgement 288References 28821 Multivariate Geochemical Data Analysis in Physical Geography 290Jennifer McKinley and Christopher David Lloyd21.1 Introduction 29021.2 Context 29121.3 Data 29321.4 Analysis 29521.5 Discussion 29921.6 Conclusion 300Acknowledgement 300References 30022 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution 302Roger Puig, Raimon Tolosana-Delgado, Neus Otero and Albert Folch22.1 Introduction 30222.2 Study Area 30322.3 Analytical Methods 30622.4 Statistical Treatment 30722.5 Results and Discussion 31122.6 Conclusions 314Acknowledgements 315References 31523 Applications in Economics 318Tim Fry23.1 Introduction 31823.2 Consumer Demand Systems 31923.3 Miscellaneous Applications 32223.4 Compositional Time Series 32323.5 New Directions 32323.6 Conclusion 325References 325Part V Software 32724 Exploratory Analysis Using CoDaPack 3D 329Santiago Thió-Henestrosa and Josep Daunis-i-Estadella24.1 CoDaPack 3D Description 32924.2 Data Set Description 33124.3 Exploratory Analysis 33324.4 Summary and Conclusions 339Acknowledgements 340References 34025 robCompositions: An R-package for Robust Statistical Analysis of Compositional Data 341Matthias Templ, Karel Hron and Peter Filzmoser25.1 General Information on the R-package robCompositions 34125.2 Expressing Compositional Data in Coordinates 34325.3 Multivariate Statistical Methods for Compositional Data Containing Outliers 34525.4 Robust Imputation of Missing Values 35125.5 Summary 354References 35426 Linear Models with Compositions in R 356Raimon Tolosana-Delgado and Karl Gerald van den Boogaart26.1 Introduction 35626.2 The Illustration Data Set 35726.3 Explanatory Binary Variable 36026.4 Explanatory Categorical Variable 36326.5 Explanatory Continuous Variable 36526.6 Explanatory Composition 36726.7 Conclusions 370Acknowledgement 371References 371Index 373