1 | Introduction | 1 |
1.1 | Statement of the problem | 1 |
1.2 | Compositions | 3 |
1.3 | Coregionalization | 5 |
1.4 | Organization of the book | 8 |
2 | Regionalized compositions | 11 |
2.1 | First concepts of regionalized compositions | 12 |
2.2 | Basis of a regionalized composition | 13 |
2.3 | Regionalized subcompositions | 15 |
2.4 | Regionalized amalgamations and partitions | 16 |
2.5 | alr and clr transformations | 18 |
2.6 | Hypothesis of stationarity | 20 |
2.7 | The additive logistic normal distribution | 22 |
3 | Spatial covariance structure | 25 |
3.1 | Second-order stationary case | 28 |
3.1.1 | Spurious spatial correlation | 28 |
3.1.2 | Defining spatial covariance structure | 29 |
3.1.3 | lr autocovariance | 33 |
3.1.4 | alr cross-covariance | 35 |
3.1.5 | clr cross-covariance | 36 |
3.1.6 | Relationships between specifications | 39 |
3.1.7 | Symmetry of the spatial covariance structure | 43 |
3.2 | Spatial covariance structure under intrinsic hypothesis | 44 |
3.2.1 | Intrinsic spatial covariance structure | 44 |
3.2.2 | lr semivariogram | 45 |
3.2.3 | alr cross-semivariogram | 46 |
3.2.4 | clr cross-semivariogram | 47 |
3.2.5 | Further relationships between specifications | 48 |
3.3 | Spatial covariance structure of an r-basis | 50 |
4 | Concepts of null correlation | 53 |
4.1 | Null lr cross-correlation | 54 |
4.2 | Null lr autocorrelation | 57 |
4.3 | Null alr cross-correlation | 59 |
4.4 | Null clr cross-correlation | 60 |
4.5 | Composition invariance | 61 |
4.6 | Relationship between concepts of null cross-correlation | 62 |
4.7 | Subcompositional invariance and partition independence | 63 |
5 | Cokriging | 67 |
5.1 | The general case of cokriging | 69 |
5.1.1 | Cokriging with known mean | 72 |
5.1.2 | Cokriging with unknown mean | 75 |
5.2 | Normal cokriging | 76 |
5.3 | Lognormal cokriging | 79 |
5.3.1 | Lognormal cokriging with known mean | 82 |
5.3.2 | Lognormal cokriging with unknown mean | 83 |
5.3.3 | Comments on lognormal cokriging | 86 |
5.4 | alr cokriging | 87 |
5.4.1 | alr cokriging with known and unknown mean | 87 |
5.4.2 | alr cokriging of a subvector | 88 |
5.4.3 | alr autocorrelation and alr cokriging | 89 |
5.4.4 | Permutation invariance of alr cokriging estimators | 90 |
5.5 | Intrinsic vector random functions | 91 |
6 | Practical aspects of compositional data analysis | 95 |
6.1 | Dealing with zeros in compositional data | 95 |
6.2 | Modeling alr cross-covariance matrices | 97 |
6.3 | Exploratory analysis of compositional data | 101 |
6.4 | Back transforming alr means and variances | 102 |
6.5 | Confidence intervals and confidence regions | 107 |
6.5.1 | General concepts | 107 |
6.5.2 | Confidence intervals and confidence regions in the non-regionalized case | 109 |
6.5.3 | Confidence intervals and confidence regions in the regionalized case | 114 |
6.6 | H-data files | 117 |
6.7 | Criteria for comparing results | 118 |
6.7.1 | Distance between observed samples and estimates | 118 |
6.7.2 | STRESS between observed and estimated data set | 121 |
7 | Application to real data | 123 |
7.1 | The Lyons West oil field of Kansas | 123 |
7.1.1 | Description of the field | 124 |
7.1.2 | Preparation of the oil-field data | 125 |
7.2 | Direct estimation | 127 |
7.3 | The alr method | 129 |
7.3.1 | The particular case of Lyons West | 129 |
7.3.2 | Spatial correlation | 130 |
7.3.3 | Comparison of estimation methods | 145 |
7.4 | The basis method | 151 |
7.4.1 | The basis for Lyons West | 152 |
7.4.2 | Modeling of covariances | 152 |
7.4.3 | The kriging and cokriging of the basis | 152 |
7.4.4 | Explanation of the reversal of the optimal method | 160 |
7.5 | Last exercise | 162 |
7.6 | Concluding comparisons | 163 |
| Summary and prospects | 165 |
| References | 167 |
| Index | 177 |