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Preface | |

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List of Contributors | |

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Introduction | |

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A Short History of Compositional Data Analysis | |

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Introduction | |

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Spurious Correlation | |

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Log and Log-Ratio Transforms | |

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Subcompositional Dependence | |

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alr, clr, ilr: Which Transformation to Choose? | |

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Principles, Perturbations and Back to the Simplex | |

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Biplots and Singular Value Decompositions | |

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Mixtures | |

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Discrete Compositions | |

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Compositional Processes | |

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Structural, Counting and Rounded Zeros | |

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Conclusion 9 Acknowledgement | |

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References | |

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Basic Concepts and Procedures | |

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Introduction | |

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Election Data and Raw Analysis | |

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The Compositional Alternative | |

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

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

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

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Geometric Settings | |

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Centre and Variability | |

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Conclusion | |

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Acknowledgements | |

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References | |

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Theory - Statistical Modelling | |

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The Principle of Working on Coordinates | |

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Introduction | |

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The Role of Coordinates in Statistics | |

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The Simplex | |

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Basis of the Simplex | |

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Working on Orthonormal Coordinates | |

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Move or Stay in the Simplex | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Dealing with Zeros | |

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Introduction | |

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Rounded Zeros | |

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Non-Parametric Replacement of Rounded Zeros | |

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Parametric Modified EM Algorithm for Rounded Zeros | |

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Count Zeros | |

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Essential Zeros | |

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Difficulties, Troubles and Challenges | |

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Acknowledgements | |

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References | |

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Robust Statistical Analysis | |

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Introduction | |

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Elements of Robust Statistics from a Compositional Point of View | |

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Robust Methods for Compositional Data | |

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Multivariate Outlier Detection | |

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Principal Component Analysis | |

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Discriminant Analysis | |

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Case Studies | |

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Multivariate Outlier Detection | |

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Principal Component Analysis | |

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Discriminant Analysis | |

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Summary 70 Acknowledgement | |

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References | |

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Geostatistics for Compositions | |

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Introduction | |

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A Brief Summary of Geostatistics | |

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Cokriging of Regionalised Compositions | |

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Structural Analysis of Regionalised Composition | |

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Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging | |

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Application | |

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Delimiting the Body: Simplicial Indicator Kriging | |

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Interpolating the Oil-Brine-Solid Content | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Compositional VARIMA Time Series | |

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Introduction | |

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The Simplex SD as a Compositional Space | |

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Basic Concepts and Notation | |

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The Covariance Structure on the Simplex | |

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Compositional Time Series Models | |

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C-Stationary Processes | |

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C-VARIMA Processes | |

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CTS Modelling: An Example | |

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Expenditure Shares in the UK | |

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Model Selection | |

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Estimation of Parameters | |

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Interpretation and Comparison | |

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Discussion | |

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Acknowledgements | |

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References | |

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Compositional Data and Correspondence Analysis | |

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Introduction | |

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Comparative Technical Definitions | |

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Properties and Interpretation of LRA and CA | |

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Application to Fatty Acid Compositional Data | |

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Discussion and Conclusions | |

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Acknowledgements | |

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References | |

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Use of Survey Weights for the Analysis of Compositional Data | |

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Introduction | |

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Elements of Survey Design | |

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Randomization | |

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Design-Based Estimation | |

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Application to Compositional Data | |

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Weighted Arithmetic and Geometric Means | |

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Closed Arithmetic Mean of Amounts | |

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Centred Log-Ratio of the Geometric Mean Composition | |

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Closed Geometric Mean Composition | |

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Example: Swiss Earnings Structure Survey (SESS) | |

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Discussion | |

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References | |

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Notes on the Scaled Dirichlet Distribution | |

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Introduction | |

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Genesis of the Scaled Dirichlet Distribution | |

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Properties of the Scaled Dirichlet Distribution | |

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Graphical Comparison | |

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Membership in the Exponential Family | |

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Measures of Location and Variability | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Theory - Algebra and Calculus | |

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Elements of Simplicial Linear Algebra and Geometry | |

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Introduction | |

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Elements of Simplicial Geometry | |

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n-Part Simplex | |

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Vector Space | |

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Centred Log-Ratio Representation | |

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Metrics | |

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Orthonormal Basis and Coordinates | |

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Linear Functions | |

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Linear Functions Defined on the Simplex | |

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Simplicial Linear Function Defined on a Real Space | |

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Simplicial Linear Function Defined on the Simplex | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Calculus of Simplex-Valued Functions | |

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Introduction | |

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Limits, Continuity and Differentiability | |

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Limits and Continuity | |

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Differentiability | |

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Higher Order Derivatives | |

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Integration | |

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Antiderivatives. Indefinite Integral | |

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Integration of Continuous SV Functions | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Compositional Differential Calculus on the Simplex | |

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Introduction | |

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Vector-Valued Functions on the Simplex | |

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Scale-Invariant Vector-Valued Functions on Rn + | |

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Vector-Valued Functions on Sn | |

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C-Derivatives on the Simplex | |

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

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Directional C-Derivatives | |

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C-Derivative | |

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C-Gradient | |

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

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Example: Experiments with Mixtures | |

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Polynomial of Degree One | |

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Polynomial of Degree Two | |

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Polynomial of Degree One in Logarithms | |

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A numerical Example | |

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Discussion | |

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Acknowledgements | |

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References | |

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Applications | |

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Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? | |

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Introduction | |

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The Omics Imp and Two Bioscience Experiment Paradigms | |

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The Impact of Compositional Constraints in the Omics | |

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Univariate Impact of Compositional Constraints | |

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Impact of Compositional Constraints on Multivariate Distance Metrics | |

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Impact of Compositional Constraints on Correlation and Covariance | |

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Compositional Constraints, Covariance, Correlation and Log-Transformed Data | |

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A Simulation Approach to Understanding the Impact of Closure | |

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Implications | |

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Gathering Information to Infer Absolute Abundance | |

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Analysing Compositional Omics Data Appropriately | |

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Acknowledgements | |

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References | |

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Hardy-Weinberg Equilibrium: A Nonparametric Compositional Approach | |

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Introduction | |

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Genetic Data Sets | |

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Classical Tests for HWE | |

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A Compositional Approach | |

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Example | |

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Conclusion and Discussion | |

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Acknowledgements | |

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References | |

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Compositional Analysis in Behavioural and Evolutionary Ecology | |

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Introduction | |

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CODA in Population Genetics | |

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CODA in Habitat Choice | |

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Multiple Choice and Individual Variation in Preferences | |

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Ecological Specialization | |

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Time Budgets: More on Specialization | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates | |

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Introduction | |

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Flying Vertebrates - General Anatomical and Functional Characteristics | |

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Materials | |

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Methods | |

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Aitchison Distance Disparity Metrics | |

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Intragroup Aitchison Distance | |

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Intergroup Aitchison Distance | |

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Statistical Tests | |

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Biplots | |

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Chiroptera | |

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Pterosauria | |

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Balances | |

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Size Effect | |

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Final Remarks | |

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All Groups | |

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Aves | |

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Pterosauria | |

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Chiroptera | |

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Acknowledgements | |

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References | |

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Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach | |

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Introduction | |

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Geochemical Processes and Log-Ratio Approach | |

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Log-Ratio Approach and Water Chemistry | |

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Log-Ratio Approach and Volcanic Gas Chemistry | |

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Log-Ratio Approach and Subducting Sediment Composition | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury | |

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Introduction | |

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Mars | |

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Mercury | |

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Analysis of Surface Composition | |

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Compositional Analysis of Mars' Surface | |

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Compositional Analysis of Mercury's Surface | |

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Conclusion 278 Acknowledgement | |

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References | |

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Spectral Analysis of Compositional Data in Cyclostratigraphy | |

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Introduction | |

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The Method | |

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Case Study | |

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Discussion | |

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Conclusions 288 Acknowledgement | |

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References | |

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Multivariate Geochemical Data Analysis in Physical Geography | |

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Introduction | |

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Context | |

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Data | |

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Analysis | |

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Discussion | |

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Conclusion | |

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Acknowledgement | |

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References | |

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Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution | |

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Introduction | |

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Study Area | |

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Maresme | |

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Osona | |

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Llucï¿½an`es | |

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Empord`a | |

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Selva | |

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Analytical Methods | |

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Statistical Treatment | |

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Data Scaling | |

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Linear Discriminant Analysis | |

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Discriminant Biplots | |

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Results and Discussion | |

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Conclusions | |

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Acknowledgements | |

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References | |

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Applications in Economics | |

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Introduction | |

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Consumer Demand Systems | |

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Miscellaneous Applications | |

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Compositional Time Series | |

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New Directions | |

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Conclusion | |

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References | |

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Software | |

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Exploratory Analysis Using CoDaPack 3D | |

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CoDaPack 3D Description | |

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Data Set Description | |

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Exploratory Analysis | |

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Numerical Analysis | |

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Biplot | |

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The Ternary Diagram | |

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Principal Component Analysis | |

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Balance-Dendrogram | |

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By Groups Description | |

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Summary and Conclusions | |

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Acknowledgements | |

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References | |

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robCompositions: An R-package for Robust Statistical Analysis of Compositional Data | |

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General Information on the R-package robCompositions | |

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Data Sets Included in the Package | |

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Design Principles | |

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Expressing Compositional Data in Coordinates | |

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Multivariate Statistical Methods for Compositional Data Containing Outliers | |

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Multivariate Outlier Detection | |

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Principal Component Analysis and the Robust Compositional Biplot | |

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Discriminant Analysis | |

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Robust Imputation of Missing Values | |

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Summary | |

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References | |

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Linear Models with Compositions in R | |

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Introduction | |

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The Illustration Data Set | |

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The Data | |

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Descriptive Analysis of Compositional Characteristics | |

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Explanatory Binary Variable | |

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Explanatory Categorical Variable | |

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Explanatory Continuous Variable | |

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Explanatory Composition | |

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Conclusions | |

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Acknowledgement | |

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References | |

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Index | |