The Statistical Analysis of Compositional Data

1 Compositional data: some challenging problems. - 1.1 Introduction. - 1.2 Geochemical compositions of rocks. - 1.3 Sediments at different depths. - 1.4 Ternary diagrams. - 1.5 Partial analyses and subcompositions. - 1.6 Supervisory behaviour. - 1.7 Household budget surveys. - 1.8 Steroid metabolite patterns in adults and children. - 1.9 Activity patterns of a statistician. - 1.10 Calibration of white-cell compositions. - 1.11 Fruit evaluation. - 1.12 Firework mixtures. - 1.13 Clam ecology. - 1.14 Bibliographic notes. - Problems. - 2 The simplex as sample space. - 2.1 Choice of sample space. - 2.2 Compositions and simplexes. - 2.3 Spaces, vectors, matrices. - 2.4 Bases and compositions. - 2.5 Subcompositions. - 2.6 Amalgamations. - 2.7 Partitions. - 2.8 Perturbations. - 2.9 Geometrical representations of compositional data. - 2.10 Bibliographic notes. - Problems. - 3 The special difficulties of compositional data analysis. - 3.1 Introduction. - 3.2 High dimensionality. - 3.3 Absence of an interpretable covariance structure. - 3.4 Difficulty of parametric modelling. - 3.5 The mixture variation difficulty. - 3.6 Bibliographic notes. - Problems. - 4 Covariance structure. - 4.1 Fundamentals. - 4.2 Specification of the covariance structure. - 4.3 The compositional variation array. - 4.4 Recovery of the compositional variation array from the crude mean vector and covariance matrix. - 4.5 Subcompositional analysis. - 4.6 Matrix specifications of covariance structures. - 4.7 Some important elementary matrices. - 4.8 Relationships between the matrix specifications. - 4.9 Estimated matrices for hongite compositions. - 4.10 Logratios and logcontrasts. - 4.11 Covariance structure of a basis. - 4.12 Commentary. - 4.13 Bibliographic notes. - Problems. - 5 Properties of matrix covariance specifications. - 5.1 Logratio notation. - 5.2 Logcontrast variances and covariances. - 5.3 Permutations. - 5.4 Properties of P and QP matrices. - 5.5 Permutation invariants involving ? - 5.6 Covariance matrix inverses. - 5.7 Subcompositions. - 5.8 Equivalence of characteristics of ? , ? , ? - 5.9 Logratio-uncorrelated compositions. - 5.10 Isotropic covariance structures. - 5.11 Bibliographic notes. - Problems. - 6 Logistic normal distributions on the simplex. - 6.1 Introduction. - 6.2 The additive logistic normal class. - 6.3 Density function. - 6.4 Moment properties. - 6.5 Composition of a lognormal basis. - 6.6 Class-preserving properties. - 6.7 Conditional subcompositional properties. - 6.8 Perturbation properties. - 6.9 A central limit theorem. - 6.10 A characterization by logcontrasts. - 6.11 Relationships with the Dirichlet class. - 6.12 Potential for statistical analysis. - 6.13 The multiplicative logistic normal class. - 6.14 Partitioned logistic normal classes. - 6.15 Some notation. - 6.16 Bibliographic notes. - Problems. - 7 Logratio analysis of compositions. - 7.1 Introduction. - 7.2 Estimation of ? and ? - 7.3 Validation: tests of logistic normality. - 7.4 Hypothesis testing strategy and techniques. - 7.5 Testing hypotheses about ? and ? - 7.6 Logratio linear modelling. - 7.7 Testing logratio linear hypotheses. - 7.8 Further aspects of logratio linear modelling. - 7.9 An application of logratio linear modelling. - 7.10 Predictive distributions, atypicality indices and outliers. - 7.11 Statistical discrimination. - 7.12 Conditional compositional modelling. - 7.13 Bibliographic notes. - Problems. - 8 Dimension-reducing techniques. - 8.1 Introduction. - 8.2 Crude principal component analysis. - 8.3 Logcontrast principal component analysis. - 8.4 Applications of logcontrast principal component analysis. - 8.5 Subcompositional analysis. - 8.6 Applications of subcompositional analysis. - 8.7 Canonical component analysis. - 8.8 Bibliographic notes. - Problems. - 9 Bases and compositions. - 9.1 Fundamentals. - 9.2 Covariance relationships. - 9.3 Principal and canonical component comparisons. - 9.4 Distributional relationships. - 9.5 Compositional invariance. - 9.6 An application to household budget analysis. - 9.7 An application to clinical biochemistry. - 9.8 Reappraisal of an early shape and size analysis. - 9.9 Bibliographic notes. - Problems. - 10 Subcompositions and partitions. - 10.1 Introduction. - 10.2 Complete subcompositional independence. - 10.3 Partitions of order 1. - 10.4 Ordered sequences of partitions. - 10.5 Caveat. - 10.6 Partitions of higher order. - 10.7 Bibliographic notes. - Problems. - 11 Irregular compositional data. - 11.1 Introduction. - 11.2 Modelling imprecision in compositions. - 11.3 Analysis of sources of imprecision. - 11.4 Imprecision and tests of independence. - 11.5 Rounded or trace zeros. - 11.6 Essential zeros. - 11.7 Missing components. - 11.8 Bibliographic notes. - Problems. - 12 Compositions in a covariate role. - 12.1 Introduction. - 12.2 Calibration. - 12.3 A before-and-after treatment problem. - 12.4 Experiments with mixtures. - 12.5 An application to firework mixtures. - 12.6 Classification from compositions. - 12.7 An application to geological classification. - 12.8 Bibliographic notes. - Problems. - 13 Further distributions on the simplex. - 13.1 Some generalizations of the Dirichlet class. - 13.2 Some generalizations of the logistic normal classes. - 13.3 Recapitulation. - 13.4 The Ad(? ,B) class. - 13.5 Maximum likelihood estimation. - 13.6 Neutrality and partition independence. - 13.7 Subcompositional independence. - 13.8 A generalized lognormal gamma distribution with compositional in variance. - 13.9 Discussion. - 13.10 Bibliographic notes. - Problems. - 14 Miscellaneous problems. - 14.1 Introduction. - 14.2 Multi-way compositions. - 14.3 Multi-stage compositions. - 14.4 Multiple compositions. - 14.5 Kernel density estimation for compositional data. - 14.6 Compositional stochastic processes. - 14.7 Relation to Bayesian statistical analysis. - 14.8 Compositional and directional data. - Problems. - Appendices. - A Algebraic properties of elementary matrices. - B Bibliography. - C Computer software for compositional data analysis. - D Data sets. - Author index.