Geostatistical Analysis of Compositional Data

by Vera Pawlowsky-Glahn, Ricardo A. Olea

Overview

Pawlowsky-Glahn (Department of Informatics and Applied Mathematics, University of Girona) gives the general framework for regionalized compositions as well as for cokriging of a whole vector. She introduces basic concepts and definitions for regionalized compositions (r-compositions), then presents the definition of the spatial covariance structure for r-compositions and treats expressions of null cross-correlation for r-compositions. She describes the estimation technique known as cokriging, and illustrates procedures with a geological example. Annotation © 2004 Book News, Inc., Portland, OR

Product Details

ISBN-13:	9780198038313
Publisher:	Oxford University Press
Publication date:	06/03/2004
Series:	International Association for Mathematical Geology Studies in Mathematical Geology , #7
Sold by:	Barnes & Noble
Format:	eBook
File size:	5 MB

Table of Contents

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

Editorial Reviews

"Pawlowsky-Glahn (Department of Informatics and Applied Mathematics, University of Girona) gives the general framework for regionalized compositions as well as for cokriging of a whole vector. She introduces basic concepts and definitions for regionalized compositions (r-compositions), then presents the definition of the spatial covariance structure for r-compositions and treats expressions of null cross-correlation for r-compositions. She describes the estimation technique known as cokriging, and illustrates procedures with a geological example."—SciTech Book News

From the Publisher