Applied Compositional Data Analysis: With Worked Examples in R (Springer Series in Statistics)

This book presents the statistical analysis of compositional data using the log-ratio approach. It includes a wide range of classical and robust statistical methods adapted for compositional data analysis, such as supervised and unsupervised methods like PCA, correlation analysis, classification and regression. In addition, it considers special data structures like high-dimensional compositions and compositional tables. The methodology introduced is also frequently compared to methods which ignore the specific nature of compositional data. It focuses on practical aspects of compositional data analysis rather than on detailed theoretical derivations, thus issues like graphical visualization and preprocessing (treatment of missing values, zeros, outliers and similar artifacts) form an important part of the book. Since it is primarily intended for researchers and students from applied fields like geochemistry, chemometrics, biology and natural sciences, economics, and social sciences, all the proposed methods are accompanied by worked-out examples in R using the package robCompositions.-- Provided by publisher Preface......Page 3 Contents......Page 5 Acronyms......Page 10 1.1 What Are Compositional Data?......Page 11 1.2.1 PhD Students Example......Page 15 1.2.2 Beer Data Example......Page 18 1.2.3 Geochemical Data Example......Page 20 1.3 Principles of Compositional Data Analysis......Page 21 1.4 Steps to a Concise Methodology......Page 24 References......Page 25 2.1 Brief Overview on Packages Related to Compositional Data Analysis......Page 27 2.1.2 robCompositions......Page 28 2.1.6 CoDaPack......Page 31 2.3 Basics in R......Page 32 2.3.2 Install robCompositions......Page 34 2.3.3 Help......Page 35 2.3.4 The R Workspace and the Working Directory......Page 36 2.3.5 Data Types......Page 37 2.3.6 Generic Functions, Methods and Classes......Page 42 References......Page 43 3.1 Motivation......Page 45 3.2 Aitchison Geometry on the Simplex......Page 50 3.3 Coordinate Representations of Compositions......Page 53 3.3.1 Additive Logratio (alr) Coordinates......Page 54 3.3.2 Centered Logratio (clr) Coefficients......Page 55 3.3.3 Isometric Logratio (ilr) and Pivot Coordinates......Page 58 3.3.4 Special Coordinate Systems: Generalization of Pivot Coordinates......Page 62 3.3.5 Special Coordinate Systems: Symmetric Pivot Coordinates......Page 64 3.3.6 Special Coordinate Systems: Balances......Page 66 3.4 Examples......Page 69 References......Page 77 4.1 Descriptive Statistics of Compositional Data......Page 79 4.2 Univariate Graphics......Page 83 4.3 Bivariate Plotting......Page 87 4.4 Multivariate Visualization......Page 89 References......Page 92 5.1 Distributions and Statistical Inference......Page 94 5.1.1 Normality Testing......Page 96 5.1.2 Statistical Inference in Coordinates......Page 97 5.2 Classical and Robust Statistical Analysis......Page 99 5.2.2 Univariate Scale......Page 100 5.2.3 Multivariate Location and Covariance......Page 101 5.2.4 Center and Variation Matrix......Page 102 5.3 Outlier Detection......Page 103 5.3.1 Univariate Outliers......Page 104 5.3.2 Multivariate Outliers......Page 107 5.3.3 Interpretation of Multivariate Outliers......Page 110 5.4 Example......Page 112 References......Page 115 6.1 Distance Measures and Dissimilarities......Page 116 6.2.1 Agglomerative Clustering Algorithms......Page 119 6.2.1.1 Single Linkage......Page 120 6.2.1.2 Complete Linkage......Page 121 6.2.2 Tree Cutting......Page 122 6.3 Partitioning Methods......Page 123 6.4 Model-Based Clustering......Page 126 6.6 Clustering Parts: Q-Mode Clustering......Page 128 6.7 Evaluation......Page 131 6.8 Examples......Page 133 References......Page 139 7.1 Introductory Remarks......Page 140 7.2.1 Estimation by SVD......Page 141 7.2.2 Estimation by Decomposing the Covariance Matrix......Page 144 7.3 Compositional Biplot......Page 146 7.4.2 Example: Household Expenditures at EU Level......Page 149 7.4.3 Example: Beer Data......Page 152 7.4.5 Example for PCA Including External Non-compositional Variables......Page 153 References......Page 157 8.1 Correlation Measures......Page 158 8.2 Relating Two Compositional Parts......Page 160 8.3 Multiple Correlation......Page 161 8.4 Correlation Between Groups of Compositional Parts......Page 162 8.5.1 Example for Correlation Between Single Compositional Parts......Page 163 8.5.2 Example for Multiple Correlation......Page 166 8.5.3 Example for Correlation Between Groups of Compositional Parts......Page 167 References......Page 171 9.1 Introductory Remarks......Page 172 9.2 Bayes Discriminant Rule......Page 174 9.3 Fisher Discriminant Rule......Page 176 9.4.1 Example for LDA and QDA......Page 178 9.4.3 Example with Appropriate Evaluationof the Error Rate......Page 183 References......Page 188 10.1 Introductory Remarks......Page 189 10.2 Regression with Compositional Response......Page 190 10.3.1 Real Response......Page 194 10.3.2 Compositional Response......Page 196 10.4 Regression Within a Composition......Page 197 10.5 Variable Selection......Page 200 10.6 Robustness Issues......Page 202 10.7.1 Example for Regression with CompositionalResponse......Page 203 10.7.2 Example for Regression with Compositional Covariates and Real Response......Page 205 10.7.3 Example for Regression with Compositional Covariates and Compositional Response......Page 208 10.7.4 Example for Regression Within a Composition......Page 209 References......Page 212 11.1 Specific Problems of High-Dimensional Compositions......Page 214 11.2 Partial Least Squares for Regression and Classification......Page 216 11.3 Marker Identification Using Pairwise Logratios......Page 219 11.4 Principal Balances......Page 222 11.5.1 Example for PLS for Two-Group Classification......Page 223 11.5.2 Example for Marker Identification......Page 227 References......Page 232 12.1 Motivation and Geometry......Page 233 12.2 Independent and Interaction Parts of Compositional Tables......Page 235 12.2.1 Decomposition of 22 Compositional Tables......Page 237 12.2.2 Coordinate Representation of Compositional Tables......Page 239 12.3 Extension to the General Case......Page 240 12.4 Examples......Page 242 12.4.1 Gender Based Cancer Data......Page 243 12.4.2 Social Expenditures According to Funding Sources......Page 245 References......Page 248 13.1 Specific Problems with Data Preprocessing of Compositions......Page 250 13.2 Missing Values......Page 253 13.2.1 k-Nearest Neighbor (knn) Imputation......Page 254 13.2.2 Iterative Model-Based Imputation......Page 257 13.3 Rounded and Count Zeros......Page 259 13.3.1 Rounded Zeros......Page 260 13.3.2 Count Zeros......Page 264 13.4 Rounded Zeros in High-Dimensional Data......Page 265 13.5 Structural Zeros......Page 268 References......Page 275 Software Versions used in the Book......Page 278 Biblio......Page 279 Index......Page 280