Compositional Data Analysis

It is difficult to imagine that the statistical analysis of
compositional data has been a major issue of concern for more than
100 years. It is even more difficult to realize that so many
statisticians and users of statistics are unaware of the particular
problems affecting compositional data, as well as their solutions.
The issue of ``spurious correlation'', as the situation was phrased
by Karl Pearson back in 1897, affects all data that measures parts
of some whole, such as percentages, proportions, ppm and ppb. Such
measurements are present in all fields of science, ranging from
geology, biology, environmental sciences, forensic sciences,
medicine and hydrology.

This book presents the history and development of compositional
data analysis along with Aitchison's log-ratio approach.
Compositional Data Analysis describes the state of the art
both in theoretical fields as well as applications in the different
fields of science.

Key Features:

• Reflects the state-of-the-art in compositional data
  analysis.

• Gives an overview of the historical development of
  compositional data analysis, as well as basic concepts and
  procedures.

• Looks at advances in algebra and calculus on the simplex.

• Presents applications in different fields of science,
  including, genomics, ecology, biology, geochemistry, planetology,
  chemistry and economics.

• Explores connections to correspondence analysis and the
  Dirichlet distribution.

• Presents a summary of three available software packages for
  compositional data analysis.

• Supported by an accompanying website featuring R code.

Applied scientists working on compositional data analysis in any
field of science, both in academia and professionals will benefit
from this book, along with graduate students in any field of
science working with compositional data.

Auteur

Vera Pawlowsky-Glahn, Department of Computer Science and Applied Mathematics, University of Girona, Spain. Antonella Buccianti, Department of Earth Sciences, University of Florence, Italy.

Résumé
It is difficult to imagine that the statistical analysis of compositional data has been a major issue of concern for more than 100 years. It is even more difficult to realize that so many statisticians and users of statistics are unaware of the particular problems affecting compositional data, as well as their solutions. The issue of ``spurious correlation'', as the situation was phrased by Karl Pearson back in 1897, affects all data that measures parts of some whole, such as percentages, proportions, ppm and ppb. Such measurements are present in all fields of science, ranging from geology, biology, environmental sciences, forensic sciences, medicine and hydrology. This book presents the history and development of compositional data analysis along with Aitchison's log-ratio approach. Compositional Data Analysis describes the state of the art both in theoretical fields as well as applications in the different fields of science.

Key Features:

• Reflects the state-of-the-art in compositional data analysis.
• Gives an overview of the historical development of compositional data analysis, as well as basic concepts and procedures.
• Looks at advances in algebra and calculus on the simplex.
• Presents applications in different fields of science, including, genomics, ecology, biology, geochemistry, planetology, chemistry and economics.
• Explores connections to correspondence analysis and the Dirichlet distribution.
• Presents a summary of three available software packages for compositional data analysis.
• Supported by an accompanying website featuring R code.
  Applied scientists working on compositional data analysis in any field of science, both in academia and professionals will benefit from this book, along with graduate students in any field of science working with compositional data.

Contenu
Preface xvii List of Contributors xix

Part I Introduction 1

1 A Short History of Compositional Data Analysis 3
John Bacon-Shone

1.1 Introduction 3

1.2 Spurious Correlation 3

1.3 Log and Log-Ratio Transforms 4

1.4 Subcompositional Dependence 5

1.5 alr, clr, ilr: Which Transformation to Choose? 5

1.6 Principles, Perturbations and Back to the Simplex 6

1.7 Biplots and Singular Value Decompositions 7

1.8 Mixtures 7

1.9 Discrete Compositions 8

1.10 Compositional Processes 8

1.11 Structural, Counting and Rounded Zeros 8

1.12 Conclusion 9

Acknowledgement 9

References 9

2 Basic Concepts and Procedures 12
Juan Jos´e Egozcue and Vera Pawlowsky-Glahn

2.1 Introduction 12

2.2 Election Data and Raw Analysis 13

2.3 The Compositional Alternative 15

2.3.1 Scale Invariance: Vectors with Proportional Positive Components Represent the Same Composition 15

2.3.2 Subcompositional Coherence: Analyses Concerning a Subset of Parts Must Not Depend on Other Non-Involved Parts 16

2.3.3 Permutation Invariance: The Conclusions of a Compositional Analysis Should Not Depend on the Order of the Parts 17

2.4 Geometric Settings 17

2.5 Centre and Variability 22

2.6 Conclusion 27

Acknowledgements 27

References 27

Part II Theory Statistical Modelling 29

3 The Principle of Working on Coordinates 31
Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue

3.1 Introduction 31

3.2 The Role of Coordinates in Statistics 32

3.3 The Simplex 33

3.3.1 Basis of the Simplex 34

3.3.2 Working on Orthonormal Coordinates 35

3.4 Move or Stay in the Simplex 38

3.5 Conclusions 40

Acknowledgements 41

References 41

4 Dealing with Zeros 43
Josep Antoni Martún-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea

4.1 Introduction 43

4.2 Rounded Zeros 44

4.2.1 Non-Parametric Replacement of Rounded Zeros 45

4.2.2 Parametric Modified EM Algorithm for Rounded Zeros 47

4.3 Count Zeros 50

4.4 Essential Zeros 53

4.5 Difficulties, Troubles and Challenges 55

Acknowledgements 57

References 57

5 Robust Statistical Analysis 59
Peter Filzmoser and Karel Hron

5.1 Introduction 59

5.2 Elements of Robust Statistics from a Compositional Point of View 60

5.3 Robust Methods for Compositional Data 63

5.3.1 Multivariate Outlier Detection 64

5.3.2 Principal Component Analysis 64

5.3.3 Discriminant Analysis 65

5.4 Case Studies 66

5.4.1 Multivariate Outlier Detection 66

5.4.2 Principal Component Analysis 68

5.4.3 Discriminant Analysis 68

5.5 Summary 70

Acknowledgement 71

References 71

6 Geostatistics for Compositions 73
Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn

6.1 Introduction 73

6.2 A Brief Summary of Geostatistics 74

6.3 Cokriging of Regionalised Compositions 76

6.4 Structural Analysis of Regionalised Composition 76

6.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 78

6.6 Application 79

6.6.1 Delimiting the Body: Simplicial Indicator Kriging 81

6.6.2 Interpolating the OilBrineSolid Content 82

6.7 Conclusions 84

Acknowledgements 84

References 84

7 Compositional VARIMA Time Series 87
Carles Barceló-Vidal, Lucúa Aguilar and Josep Antoni Martún-Fernández

7.1 Introduction 87 7.2 The Simplex SD as ...