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Compositional Data Analysis Theory and Applications

by: Pawlowsky-Glahn, Vera; Buccianti, Antonella

Compositional Data Analysis Theory and Applications

by: Pawlowsky-Glahn, Vera; Buccianti, Antonella

ISBN 13:
9780470711354
ISBN 10:
0470711353
Edition: 1st
Format: Hardcover
Copyright: 09/19/2011
Publisher: Wiley

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Summary

This book presents the state-of-the-art in compositional data analysis and will feature a collection of papers covering theory, applications to various fields of science and software. Areas covered will range from geology, biology, environmental sciences, forensic sciences, medicine and hydrology. Key features: Provides the state-of-the-art text in compositional data analysis Covers a variety of subject areas, from geology to medicine Written by leading researchers in the field Is supported by a website featuring R code

Preface	p. xvii
List of Contributors	p. xix
Introduction	p. 1
A Short History of Compositional Data Analysis	p. 3
Introduction	p. 3
Spurious Correlation	p. 3
Log and Log-Ratio Transforms	p. 4
Subcompositional Dependence	p. 5
alr, clr, ilr: Which Transformation to Choose?	p. 5
Principles, Perturbations and Back to the Simplex	p. 6
Biplots and Singular Value Decompositions	p. 7
Mixtures	p. 7
Discrete Compositions	p. 8
Compositional Processes	p. 8
Structural, Counting and Rounded Zeros	p. 8
Conclusion 9 Acknowledgement	p. 9
References	p. 9
Basic Concepts and Procedures	p. 12
Introduction	p. 12
Election Data and Raw Analysis	p. 13
The Compositional Alternative	p. 15
Scale Invariance: Vectors with Proportional Positive Components Represent the Same Composition	p. 15
Subcompositional Coherence: Analyses Concerning a Subset of Parts Must Not Depend on Other Non-Involved Parts	p. 16
Permutation Invariance: The Conclusions of a Compositional Analysis Should Not Depend on the Order of the Parts	p. 17
Geometric Settings	p. 17
Centre and Variability	p. 22
Conclusion	p. 27
Acknowledgements	p. 27
References	p. 27
Theory - Statistical Modelling	p. 29
The Principle of Working on Coordinates	p. 31
Introduction	p. 31
The Role of Coordinates in Statistics	p. 32
The Simplex	p. 33
Basis of the Simplex	p. 34
Working on Orthonormal Coordinates	p. 35
Move or Stay in the Simplex	p. 38
Conclusions	p. 40
Acknowledgements	p. 41
References	p. 41
Dealing with Zeros	p. 43
Introduction	p. 43
Rounded Zeros	p. 44
Non-Parametric Replacement of Rounded Zeros	p. 45
Parametric Modified EM Algorithm for Rounded Zeros	p. 47
Count Zeros	p. 50
Essential Zeros	p. 53
Difficulties, Troubles and Challenges	p. 55
Acknowledgements	p. 57
References	p. 57
Robust Statistical Analysis	p. 59
Introduction	p. 59
Elements of Robust Statistics from a Compositional Point of View	p. 60
Robust Methods for Compositional Data	p. 63
Multivariate Outlier Detection	p. 64
Principal Component Analysis	p. 64
Discriminant Analysis	p. 65
Case Studies	p. 66
Multivariate Outlier Detection	p. 66
Principal Component Analysis	p. 68
Discriminant Analysis	p. 68
Summary 70 Acknowledgement	p. 71
References	p. 71
Geostatistics for Compositions	p. 73
Introduction	p. 73
A Brief Summary of Geostatistics	p. 74
Cokriging of Regionalised Compositions	p. 76
Structural Analysis of Regionalised Composition	p. 76
Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging	p. 78
Application	p. 79
Delimiting the Body: Simplicial Indicator Kriging	p. 81
Interpolating the Oil-Brine-Solid Content	p. 82
Conclusions	p. 84
Acknowledgements	p. 84
References	p. 84
Compositional VARIMA Time Series	p. 87
Introduction	p. 87
The Simplex SD as a Compositional Space	p. 89
Basic Concepts and Notation	p. 89
The Covariance Structure on the Simplex	p. 90
Compositional Time Series Models	p. 91
C-Stationary Processes	p. 92
C-VARIMA Processes	p. 93
CTS Modelling: An Example	p. 94
Expenditure Shares in the UK	p. 94
Model Selection	p. 95
Estimation of Parameters	p. 96
Interpretation and Comparison	p. 96
Discussion	p. 99
Acknowledgements	p. 99
References	p. 100
	p. 102
Compositional Data and Correspondence Analysis	p. 104
Introduction	p. 104
Comparative Technical Definitions	p. 105
Properties and Interpretation of LRA and CA	p. 107
Application to Fatty Acid Compositional Data	p. 107
Discussion and Conclusions	p. 111
Acknowledgements	p. 112
References	p. 112
Use of Survey Weights for the Analysis of Compositional Data	p. 114
Introduction	p. 114
Elements of Survey Design	p. 115
Randomization	p. 115
Design-Based Estimation	p. 118
Application to Compositional Data	p. 122
Weighted Arithmetic and Geometric Means	p. 123
Closed Arithmetic Mean of Amounts	p. 123
Centred Log-Ratio of the Geometric Mean Composition	p. 124
Closed Geometric Mean Composition	p. 124
Example: Swiss Earnings Structure Survey (SESS)	p. 125
Discussion	p. 126
References	p. 126
Notes on the Scaled Dirichlet Distribution	p. 128
Introduction	p. 128
Genesis of the Scaled Dirichlet Distribution	p. 129
Properties of the Scaled Dirichlet Distribution	p. 131
Graphical Comparison	p. 131
Membership in the Exponential Family	p. 133
Measures of Location and Variability	p. 134
Conclusions	p. 136
Acknowledgements	p. 137
References	p. 137
Theory - Algebra and Calculus	p. 139
Elements of Simplicial Linear Algebra and Geometry	p. 141
Introduction	p. 141
Elements of Simplicial Geometry	p. 142
n-Part Simplex	p. 142
Vector Space	p. 143
Centred Log-Ratio Representation	p. 146
Metrics	p. 147
Orthonormal Basis and Coordinates	p. 149
Linear Functions	p. 151
Linear Functions Defined on the Simplex	p. 152
Simplicial Linear Function Defined on a Real Space	p. 153
Simplicial Linear Function Defined on the Simplex	p. 154
Conclusions	p. 156
Acknowledgements	p. 156
References	p. 156
Calculus of Simplex-Valued Functions	p. 158
Introduction	p. 158
Limits, Continuity and Differentiability	p. 161
Limits and Continuity	p. 161
Differentiability	p. 163
Higher Order Derivatives	p. 169
Integration	p. 171
Antiderivatives. Indefinite Integral	p. 171
Integration of Continuous SV Functions	p. 172
Conclusions	p. 174
Acknowledgements	p. 175
References	p. 175
Compositional Differential Calculus on the Simplex	p. 176
Introduction	p. 176
Vector-Valued Functions on the Simplex	p. 177
Scale-Invariant Vector-Valued Functions on Rn +	p. 177
Vector-Valued Functions on Sn	p. 178
C-Derivatives on the Simplex	p. 178
Derivative of a Scale-Invariant Vector-Valued Function on Rn +	p. 178
Directional C-Derivatives	p. 180
C-Derivative	p. 182
C-Gradient	p. 184
Critical Points of a C-Differentiable Real-Valued Function on Sn	p. 184
Example: Experiments with Mixtures	p. 185
Polynomial of Degree One	p. 185
Polynomial of Degree Two	p. 186
Polynomial of Degree One in Logarithms	p. 187
A numerical Example	p. 188
Discussion	p. 189
Acknowledgements	p. 190
References	p. 190
Applications	p. 191
Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right?	p. 193
Introduction	p. 193
The Omics Imp and Two Bioscience Experiment Paradigms	p. 194
The Impact of Compositional Constraints in the Omics	p. 197
Univariate Impact of Compositional Constraints	p. 197
Impact of Compositional Constraints on Multivariate Distance Metrics	p. 199
Impact of Compositional Constraints on Correlation and Covariance	p. 201
Compositional Constraints, Covariance, Correlation and Log-Transformed Data	p. 202
A Simulation Approach to Understanding the Impact of Closure	p. 202
Implications	p. 204
Gathering Information to Infer Absolute Abundance	p. 204
Analysing Compositional Omics Data Appropriately	p. 205
Acknowledgements	p. 206
References	p. 206
Hardy-Weinberg Equilibrium: A Nonparametric Compositional Approach	p. 208
Introduction	p. 208
Genetic Data Sets	p. 209
Classical Tests for HWE	p. 210
A Compositional Approach	p. 210
Example	p. 214
Conclusion and Discussion	p. 215
Acknowledgements	p. 215
References	p. 215
Compositional Analysis in Behavioural and Evolutionary Ecology	p. 218
Introduction	p. 218
CODA in Population Genetics	p. 219
CODA in Habitat Choice	p. 222
Multiple Choice and Individual Variation in Preferences	p. 224
Ecological Specialization	p. 228
Time Budgets: More on Specialization	p. 229
Conclusions	p. 231
Acknowledgements	p. 231
References	p. 231
Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates	p. 235
Introduction	p. 235
Flying Vertebrates - General Anatomical and Functional Characteristics	p. 236
Materials	p. 236
Methods	p. 238
Aitchison Distance Disparity Metrics	p. 239
Intragroup Aitchison Distance	p. 239
Intergroup Aitchison Distance	p. 240
Statistical Tests	p. 243
Biplots	p. 244
Chiroptera	p. 244
Pterosauria	p. 245
Balances	p. 246
Size Effect	p. 249
Final Remarks	p. 249
All Groups	p. 250
Aves	p. 250
Pterosauria	p. 250
Chiroptera	p. 251
Acknowledgements	p. 252
References	p. 252
Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach	p. 255
Introduction	p. 255
Geochemical Processes and Log-Ratio Approach	p. 256
Log-Ratio Approach and Water Chemistry	p. 258
Log-Ratio Approach and Volcanic Gas Chemistry	p. 261
Log-Ratio Approach and Subducting Sediment Composition	p. 263
Conclusions	p. 265
Acknowledgements	p. 265
References	p. 265
Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury	p. 267
Introduction	p. 267
Mars	p. 267
Mercury	p. 269
Analysis of Surface Composition	p. 270
Compositional Analysis of Mars' Surface	p. 270
Compositional Analysis of Mercury's Surface	p. 274
Conclusion 278 Acknowledgement	p. 278
References	p. 278
Spectral Analysis of Compositional Data in Cyclostratigraphy	p. 282
Introduction	p. 282
The Method	p. 283
Case Study	p. 285
Discussion	p. 287
Conclusions 288 Acknowledgement	p. 288
References	p. 288
Multivariate Geochemical Data Analysis in Physical Geography	p. 290
Introduction	p. 290
Context	p. 291
Data	p. 293
Analysis	p. 295
Discussion	p. 299
Conclusion	p. 300
Acknowledgement	p. 300
References	p. 300
Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution	p. 302
Introduction	p. 302
Study Area	p. 303
Maresme	p. 304
Osona	p. 305
Lluc¸an`es	p. 305
Empord`a	p. 306
Selva	p. 306
Analytical Methods	p. 306
Statistical Treatment	p. 307
Data Scaling	p. 307
Linear Discriminant Analysis	p. 309
Discriminant Biplots	p. 310
Results and Discussion	p. 311
Conclusions	p. 314
Acknowledgements	p. 315
References	p. 315
Applications in Economics	p. 318
Introduction	p. 318
Consumer Demand Systems	p. 319
Miscellaneous Applications	p. 322
Compositional Time Series	p. 323
New Directions	p. 323
Conclusion	p. 325
References	p. 325
Software	p. 327
Exploratory Analysis Using CoDaPack 3D	p. 329
CoDaPack 3D Description	p. 329
Data Set Description	p. 331
Exploratory Analysis	p. 333
Numerical Analysis	p. 333
Biplot	p. 334
The Ternary Diagram	p. 335
Principal Component Analysis	p. 336
Balance-Dendrogram	p. 336
By Groups Description	p. 338
Summary and Conclusions	p. 339
Acknowledgements	p. 340
References	p. 340
robCompositions: An R-package for Robust Statistical Analysis of Compositional Data	p. 341
General Information on the R-package robCompositions	p. 341
Data Sets Included in the Package	p. 342
Design Principles	p. 343
Expressing Compositional Data in Coordinates	p. 343
Multivariate Statistical Methods for Compositional Data Containing Outliers	p. 345
Multivariate Outlier Detection	p. 345
Principal Component Analysis and the Robust Compositional Biplot	p. 347
Discriminant Analysis	p. 350
Robust Imputation of Missing Values	p. 351
Summary	p. 354
References	p. 354
Linear Models with Compositions in R	p. 356
Introduction	p. 356
The Illustration Data Set	p. 357
The Data	p. 357
Descriptive Analysis of Compositional Characteristics	p. 358
Explanatory Binary Variable	p. 360
Explanatory Categorical Variable	p. 363
Explanatory Continuous Variable	p. 365
Explanatory Composition	p. 367
Conclusions	p. 370
Acknowledgement	p. 371
References	p. 371
Index	p. 373
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Amazon no longer offers textbook rentals. We do!

Compositional Data Analysis Theory and Applications

Compositional Data Analysis Theory and Applications

9780470711354

0470711353

Summary

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Table of Contents

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