Modeling and Analysis of Compositional Data

Preface xi About the Authors xv Acknowledgments xix 1 Introduction 1 2 Compositional Data and Their Sample Space 8 2.1 Basic concepts 8 2.2 Principles of compositional analysis 12 2.2.1 Scale invariance 12 2.2.2 Permutation invariance 15 2.2.3 Subcompositional coherence 16 2.3 Zeros, missing values, and other irregular components 16 2.3.1 Kinds of irregular components 16 2.3.2 Strategies to analyze irregular data 19 2.4 Exercises 21 3 The Aitchison Geometry 23 3.1 General comments 23 3.2 Vector space structure 24 3.3 Inner product, norm and distance 26 3.4 Geometric figures 28 3.5 Exercises 30 4 Coordinate Representation 32 4.1 Introduction 32 4.2 Compositional observations in real space 33 4.3 Generating systems 33 4.4 Orthonormal coordinates 36 4.5 Balances 38 4.6 Working on coordinates 43 4.7 Additive logratio coordinates (alr) 46 4.8 Orthogonal projections 48 4.9 Matrix operations in the simplex 54 4.9.1 Perturbation-linear combination of compositions 54 4.9.2 Linear transformations of óKòù: endomorphisms 55 4.9.3 Other matrix transformations on óKòù: nonlinear transformations 57 4.10 Coordinates leading to alternative Euclidean structures 59 4.11 Exercises 61 5 Exploratory Data Analysis 65 5.1 General remarks 65 5.2 Sample center, total variance, and variation matrix 66 5.3 Centering and scaling 68 5.4 The biplot: a graphical display 70 5.4.1 Construction of a biplot 70 5.4.2 Interpretation of a 2D compositional biplot 72 5.5 Exploratory analysis of coordinates 76 5.6 A geological example 79 5.7 Linear trends along principal components 85 5.8 A nutrition example 89 5.9 A political example 96 5.10 Exercises 100 6 Random Compositions 103 6.1 Sample space 103 6.1.1 Conventional approach to the sample space of compositions 105 6.1.2 A compositional approach to the sample space of compositions 106 6.1.3 Definitions related to random compositions 107 6.2 Variability and center 108 6.3 Probability distributions on the simplex 112 6.3.1 The normal distribution on the simplex 114 6.3.2 The Dirichlet distribution 121 6.3.3 Other distributions 127 6.4 Exercises 128 7 Statistical Inference 130 7.1 Point estimation of center and variability 130 7.2 Testing hypotheses on compositional normality 135 7.3 Testing hypotheses about two populations 136 7.4 Probability and confidence regions for normal data 142 7.5 Bayesian estimation with count data 144 7.6 Exercises 147 8 Linear Models 149 8.1 Linear regression with compositional response 150 8.2 Regression with compositional covariates 156 8.3 Analysis of variance with compositional response 160 8.4 Linear discrimination with compositional predictor 163 8.5 Exercises 165 9 Compositional Processes 172 9.1 Linear processes 173 9.2 Mixture processes 176 9.3 Settling processes 178 9.4 Simplicial derivative 183 9.5 Elementary differential equations 186 9.5.1 Constant derivative 187 9.5.2 Forced derivative 189 9.5.3 Complete first-order linear equation 194 9.5.4 Harmonic oscillator 200 9.6 Exercises 204 10 Epilogue 206 References 211 Appendix A Practical Recipes 222 A.1 Plotting a ternary diagram 222 A.2 Parameterization of an elliptic region 224 A.3 Matrix expressions of change of representation 226 Appendix B Random Variables 228 B.1 Probability spaces and random variables 228 B.2 Description of probability 232 List of Abbreviations and Symbols 234 Author Index 237 General Index 241