Geostatistics for Compositional Data with R (Use R!)

Description

This book provides a guided approach to the geostatistical modelling of compositional spatial data. These data are data in proportions, percentages or concentrations distributed in space which exhibit spatial correlation. The book can be divided into four blocks. The first block sets the framework and provides some background on compositional data analysis. Block two introduces compositional exploratory tools for both non-spatial and spatial aspects. Block three covers all necessary facets of multivariate spatial prediction for compositional data: variogram modelling, cokriging and validation. Finally, block four details strategies for simulation of compositional data, including transformations to multivariate normality, Gaussian cosimulation, multipoint simulation of compositional data, and common postprocessing techniques, valid for both Gaussian and multipoint methods.  All methods are illustrated via applications to two types of data sets: one a large-scale geochemical survey, comprised of a full suite of geochemical variables, and the other from a mining context, where only the elements of greatest importance are considered. R codes are included for all aspects of the methodology, encapsulated in the  R package "gmGeostats", available in CRAN. Preface Acknowledgements Contents List of Symbols List of Figures List of Tables 1 Introduction 1.1 What Is Compositional Geostatistics? 1.2 Why Use a Compositional Approach? 1.3 Data 1.3.1 Windarling Iron Ore Data 1.3.2 Tellus Horizon: A Soil Data 1.3.3 National Geochemical Survey of Australia 1.4 Relevant R Packages References 2 A Review of Compositional Data Analysis 2.1 Compositions 2.1.1 The Closure 2.1.2 The R-Package Compositions 2.1.3 Problems of Closed Data 2.1.4 Subcompositional Coherence and Scale Invariance 2.1.5 Alternative Frameworks of Compositions 2.2 Log-Ratio Transformations 2.3 Compositional Geometry of the Simplex 2.3.1 The Simplex 2.3.2 Compositional Distances 2.3.3 An Euclidean Vector Space Structure 2.3.4 Affine Equivariance and the ``Best'' Transformation 2.4 Zeroes, Missings and Values Below Detection Limit Problems References 3 Exploratory Data Analysis 3.1 Graphical Representations 3.2 First and Second Order Moments 3.3 PCA and Biplots 3.4 Additive Logistic Normality (ALN) and Mahalanobis Metric 3.5 Outliers and Robustness Problems References 4 Exploratory Spatial Analysis 4.1 The R-packages ``sp'', ``gstat'' and ``gmGeostats'' 4.2 Spatial Data Analysis 4.2.1 Maps and Plots 4.3 Variograms 4.3.1 Raw Variograms 4.3.2 Practical Aspects 4.3.3 Variograms for Compositional Data: Coordinate Variography 4.3.4 Variograms for Compositional Data: Variation-Variograms 4.3.5 Relationships Between Structural Functions 4.3.6 Anisotropy 4.4 MAF 4.4.1 Method 4.4.2 MAF Biplots 4.5 Checks of Spatial Structure 4.5.1 Spatial Decorrelation 4.5.2 Spatial Independence 4.6 Example: Tellus Problems References 5 Variogram Models 5.1 Linear Model of Regionalisation 5.2 Linear Model of Coregionalisation 5.2.1 Multivariate Random Function 5.2.2 Log-Ratio Invariance of the LMC 5.2.3 Practical Modelling Procedure 5.3 Model Functions and Model Fitting 5.3.1 Packages ``gmGeostats'', ``compositions'' and ``gstat'' 5.4 Factorial Representations 5.4.1 PCA 5.4.2 MAF 5.4.3 Rank-One Structures 5.5 Example: Variogram Models for the Windarling Data Problems References 6 Geostatistical Estimation 6.1 Cokriging 6.2 Cokriging of the Mean 6.3 Comments 6.3.1 Implementation Practicalities 6.3.2 Properties 6.3.3 Unbiased, in Which Scale? 6.3.4 Cokriging in R 6.4 Cokriging Estimation of Windarling Data 6.4.1 Ordinary Cokriging 6.4.2 Universal Cokriging 6.4.3 Estimation of the Local and Global Mean 6.4.4 Comparison of OCK and UCK Results 6.5 Estimation of Tellus Data Subcomposition Problems References 7 Cross-Validation 7.1 Introduction 7.2 Cross-Validation in the Univariate Case 7.3 Multivariate Cross-Validation 7.4 Accuracy and Precision 7.5 Some Caveats Problems References 8 Multivariate Normal Score Transformation 8.1 Introduction 8.2 Flow Anamorphosis 8.3 Properties 8.4 Application to Windarling Data Problems References 9 Simulation 9.1 Introduction 9.2 Simulation Algorithms 9.2.1 Sequential Gaussian Simulation 9.2.2 LU Decomposition Simulation 9.2.3 Turning Bands 9.2.4 Comments 9.3 Simulation of a Random Function via Univariate Simulation of PCA or MAF Factors 9.4 Accuracy and Precision Prior to Simulation 9.5 Example: Windarling Data 9.5.1 Cosimulation with an LMC 9.5.2 Simulation Through MAF Decomposition Problems References 10 Compositional Direct Sampling Simulation 10.1 Introduction 10.2 Compositional Direct Sampling Simulation 10.3 Comments 10.3.1 Implementation 10.3.2 Parameter Choices 10.3.3 Training Image Generation 10.4 Example: Direct Sampling Simulation of a Subcomposition of the Tellus Data 10.4.1 Training Image and Regridding 10.4.2 Conditional Spatial Model 10.4.3 Simulation 10.5 Example: Direct Sampling Simulation of Windarling East Data Based on Windarling West Problems References 11 Evaluation and Postprocessing of Results 11.1 Evaluation of Results 11.1.1 Validation of the Simulation Model 11.1.2 Individual Realisation Maps 11.1.3 Statistical Maps 11.1.4 Reproduction of Target Marginal and Two-Point Statistics 11.1.5 Selectivity Curves 11.2 Postprocessing 11.2.1 Block-COK Through Local Simulation 11.3 Case Study: Tellus Problems References A Matrix Decompositions A.1 LU Decomposition A.2 Spectral Decomposition A.3 Generalised Eigenvalue Problem A.4 Singular Value Decomposition References B Complete Data Analysis Workflows B.1 Exploratory Data Analysis B.1.1 Numerical EDA B.1.2 Spatial EDA B.2 Modelling the Spatial Continuity B.3 Estimation B.4 Simulation B.5 Evaluation and Postprocessing Index