Compositional Data Analysis -- Contents -- Preface -- List of Contributors -- Part I Introduction -- 1 A Short History of Compositional Data Analysis -- 1.1 Introduction -- 1.2 Spurious Correlation -- 1.3 Log and Log-Ratio Transforms -- 1.4 Subcompositional Dependence -- 1.5 alr, clr, ilr: Which Transformation to Choose? -- 1.6 Principles, Perturbations and Back to the Simplex -- 1.7 Biplots and Singular Value Decompositions -- 1.8 Mixtures -- 1.9 Discrete Compositions -- 1.10 Compositional Processes -- 1.11 Structural, Counting and Rounded Zeros -- 1.12 Conclusion -- Acknowledgement -- References -- 2 Basic Concepts and Procedures -- 2.1 Introduction -- 2.2 Election Data and Raw Analysis -- 2.3 The Compositional Alternative -- 2.3.1 Scale Invariance: Vectors with Proportional Positive Components Represent the Same Composition -- 2.3.2 Subcompositional Coherence: Analyses Concerning a Subset of Parts Must Not Depend on Other Non-Involved Parts -- 2.3.3 Permutation Invariance: The Conclusions of a Compositional Analysis Should Not Depend on the Order of the Parts -- 2.4 Geometric Settings -- 2.5 Centre and Variability -- 2.6 Conclusion -- Acknowledgements -- References -- Part II Theory - Statistical Modelling -- 3 The Principle of Working on Coordinates -- 3.1 Introduction -- 3.2 The Role of Coordinates in Statistics -- 3.3 The Simplex -- 3.3.1 Basis of the Simplex -- 3.3.2 Working on Orthonormal Coordinates -- 3.4 Move or Stay in the Simplex -- 3.5 Conclusions -- Acknowledgements -- References -- 4 Dealing with Zeros -- 4.1 Introduction -- 4.2 Rounded Zeros -- 4.2.1 Non-Parametric Replacement of Rounded Zeros -- 4.2.2 Parametric Modified EM Algorithm for Rounded Zeros -- 4.3 Count Zeros -- 4.4 Essential Zeros -- 4.5 Difficulties, Troubles and Challenges -- Acknowledgements -- References -- 5 Robust Statistical Analysis -- 5.1 Introduction.

5.2 Elements of Robust Statistics from a Compositional Point of View -- 5.3 Robust Methods for Compositional Data -- 5.3.1 Multivariate Outlier Detection -- 5.3.2 Principal Component Analysis -- 5.3.3 Discriminant Analysis -- 5.4 Case Studies -- 5.4.1 Multivariate Outlier Detection -- 5.4.2 Principal Component Analysis -- 5.4.3 Discriminant Analysis -- 5.5 Summary -- Acknowledgement -- References -- 6 Geostatistics for Compositions -- 6.1 Introduction -- 6.2 A Brief Summary of Geostatistics -- 6.3 Cokriging of Regionalised Compositions -- 6.4 Structural Analysis of Regionalised Composition -- 6.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging -- 6.6 Application -- 6.6.1 Delimiting the Body: Simplicial Indicator Kriging -- 6.6.2 Interpolating the Oil-Brine-Solid Content -- 6.7 Conclusions -- Acknowledgements -- References -- 7 Compositional VARIMA Time Series -- 7.1 Introduction -- 7.2 The Simplex SD as a Compositional Space -- 7.2.1 Basic Concepts and Notation -- 7.2.2 The Covariance Structure on the Simplex -- 7.3 Compositional Time Series Models -- 7.3.1 C-Stationary Processes -- 7.3.2 C-VARIMA Processes -- 7.4 CTS Modelling: An Example -- 7.4.1 Expenditure Shares in the UK -- 7.4.2 Model Selection -- 7.4.3 Estimation of Parameters -- 7.4.4 Interpretation and Comparison -- 7.5 Discussion -- Acknowledgements -- References -- Appendix -- 8 Compositional Data and Correspondence Analysis -- 8.1 Introduction -- 8.2 Comparative Technical Definitions -- 8.3 Properties and Interpretation of LRA and CA -- 8.4 Application to Fatty Acid Compositional Data -- 8.5 Discussion and Conclusions -- Acknowledgements -- References -- 9 Use of Survey Weights for the Analysis of Compositional Data -- 9.1 Introduction -- 9.2 Elements of Survey Design -- 9.2.1 Randomization -- 9.2.2 Design-Based Estimation.

9.3 Application to Compositional Data -- 9.3.1 Weighted Arithmetic and Geometric Means -- 9.3.2 Closed Arithmetic Mean of Amounts -- 9.3.3 Centred Log-Ratio of the Geometric Mean Composition -- 9.3.4 Closed Geometric Mean Composition -- 9.3.5 Example: Swiss Earnings Structure Survey (SESS) -- 9.4 Discussion -- References -- 10 Notes on the Scaled Dirichlet Distribution -- 10.1 Introduction -- 10.2 Genesis of the Scaled Dirichlet Distribution -- 10.3 Properties of the Scaled Dirichlet Distribution -- 10.3.1 Graphical Comparison -- 10.3.2 Membership in the Exponential Family -- 10.3.3 Measures of Location and Variability -- 10.4 Conclusions -- Acknowledgements -- References -- Part III Theory - Algebra and Calculus -- 11 Elements of Simplicial Linear Algebra and Geometry -- 11.1 Introduction -- 11.2 Elements of Simplicial Geometry -- 11.2.1 n-Part Simplex -- 11.2.2 Vector Space -- 11.2.3 Centred Log-Ratio Representation -- 11.2.4 Metrics -- 11.2.5 Orthonormal Basis and Coordinates -- 11.3 Linear Functions -- 11.3.1 Linear Functions Defined on the Simplex -- 11.3.2 Simplicial Linear Function Defined on a Real Space -- 11.3.3 Simplicial Linear Function Defined on the Simplex -- 11.4 Conclusions -- Acknowledgements -- References -- 12 Calculus of Simplex-Valued Functions -- 12.1 Introduction -- 12.2 Limits, Continuity and Differentiability -- 12.2.1 Limits and Continuity -- 12.2.2 Differentiability -- 12.2.3 Higher Order Derivatives -- 12.3 Integration -- 12.3.1 Antiderivatives. Indefinite Integral -- 12.3.2 Integration of Continuous SV Functions -- 12.4 Conclusions -- Acknowledgements -- References -- 13 Compositional Differential Calculus on the Simplex -- 13.1 Introduction -- 13.2 Vector-Valued Functions on the Simplex -- 13.2.1 Scale-Invariant Vector-Valued Functions on Rn -- 13.2.2 Vector-Valued Functions on Sn -- 13.3 C-Derivatives on the Simplex.

13.3.1 Derivative of a Scale-Invariant Vector-Valued Function on Rn -- 13.3.2 Directional C-Derivatives -- 13.3.3 C-Derivative -- 13.3.4 C-Gradient -- 13.3.5 Critical Points of a C-Differentiable Real-Valued Function on Sn -- 13.4 Example: Experiments with Mixtures -- 13.4.1 Polynomial of Degree One -- 13.4.2 Polynomial of Degree Two -- 13.4.3 Polynomial of Degree One in Logarithms -- 13.4.4 A numerical Example -- 13.5 Discussion -- Acknowledgements -- References -- Part IV Applications -- 14 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? -- 14.1 Introduction -- 14.2 The Omics Imp and Two Bioscience Experiment Paradigms -- 14.3 The Impact of Compositional Constraints in the Omics -- 14.3.1 Univariate Impact of Compositional Constraints -- 14.3.2 Impact of Compositional Constraints on Multivariate Distance Metrics -- 14.4 Impact of Compositional Constraints on Correlation and Covariance -- 14.4.1 Compositional Constraints, Covariance, Correlation and Log-Transformed Data -- 14.4.2 A Simulation Approach to Understanding the Impact Closure -- 14.5 Implications -- 14.5.1 Gathering Information to Infer Absolute Abundance -- 14.5.2 Analysing Compositional Omics Data Appropriately -- Acknowledgements -- References -- 15 Hardy-Weinberg Equilibrium: A Nonparametric Compositional Approach -- 15.1 Introduction -- 15.2 Genetic Data Sets -- 15.3 Classical Tests for HWE -- 15.4 A Compositional Approach -- 15.5 Example -- 15.6 Conclusion and Discussion -- Acknowledgements -- References -- 16 Compositional Analysis in Behavioural and Evolutionary Ecology -- 16.1 Introduction -- 16.2 CODA in Population Genetics -- 16.3 CODA in Habitat Choice -- 16.4 Multiple Choice and Individual Variation in Preferences -- 16.5 Ecological Specialization -- 16.6 Time Budgets: More on Specialization -- 16.7 Conclusions -- Acknowledgements.

References -- 17 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates -- 17.1 Introduction -- 17.2 Flying Vertebrates - General Anatomical and Functional Characteristics -- 17.3 Materials -- 17.4 Methods -- 17.5 Aitchison Distance Disparity Metrics -- 17.5.1 Intragroup Aitchison Distance -- 17.5.2 Intergroup Aitchison Distance -- 17.6 Statistical Tests -- 17.7 Biplots -- 17.7.1 Chiroptera -- 17.7.2 Pterosauria -- 17.8 Balances -- 17.9 Size Effect -- 17.10 Final Remarks -- 17.10.1 All Groups -- 17.10.2 Aves -- 17.10.3 Pterosauria -- 17.10.4 Chiroptera -- Acknowledgements -- References -- 18 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach -- 18.1 Introduction -- 18.2 Geochemical Processes and Log-Ratio Approach -- 18.3 Log-Ratio Approach and Water Chemistry -- 18.4 Log-Ratio Approach and Volcanic Gas Chemistry -- 18.5 Log-Ratio Approach and Subducting Sediment Composition -- 18.6 Conclusions -- Acknowledgements -- References -- 19 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury -- 19.1 Introduction -- 19.1.1 Mars -- 19.1.2 Mercury -- 19.1.3 Analysis of Surface Composition -- 19.2 Compositional Analysis of Mars' Surface -- 19.3 Compositional Analysis of Mercury's Surface -- 19.4 Conclusion -- Acknowledgement -- References -- 20 Spectral Analysis of Compositional Data in Cyclostratigraphy -- 20.1 Introduction -- 20.2 The Method -- 20.3 Case Study -- 20.4 Discussion -- 20.5 Conclusions -- Acknowledgement -- References -- 21 Multivariate Geochemical Data Analysis in Physical Geography -- 21.1 Introduction -- 21.2 Context -- 21.3 Data -- 21.4 Analysis -- 21.5 Discussion -- 21.6 Conclusion -- Acknowledgement -- References -- 22 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution.

22.1 Introduction.