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El. knyga: Matheron's Theory of Regionalised Variables

Edited by (Centre of Mathematical Morphology), Edited by (Professor of Statistics), (Professor, Paris School of Mines)
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In the summer of 1970, Georges Matheron, the father of geostatistics, presented a series of lectures at the Centre de Morphologie Mathmatique in France. These lectures would go on to become Matheron's Theory of Regionalized Variables, a seminal work that would inspire hundreds of papers and become the bedrock of numerous theses and books on the topic; however, despite their importance, the notes were never formally published.

In this volume, Matheron's influential work is presented as a published book for the first time. Originally translated into English by Charles Huijbregts, and carefully curated here, this book stays faithful to Matheron's original notes. The text has been ordered with a common structure, and equations and figures have been redrawn and numbered sequentially for ease of reference.

While not containing any mathematical technicalities or case studies, the reader is invited to wonder about the physical meaning of the notions Matheron deals with. When Matheron wrote them, he considered the theory of linear geostatistics complete and the book his final one on the subject; however, this end for Matheron has been the starting point for most geostatisticians.
1 Introduction
1(8)
1.1 Notation
1(1)
1.2 Convolution (Moving Average)
2(2)
1.2.1 Regularisation of a Function ƒ (weighted moving average of this function)
3(1)
1.3 Geostatistics and the Theory of Regionalised Variables
4(1)
1.3.1 Field and Support of a Regionalised Variable
4(1)
1.4 Transitive Methods and Intrinsic Theory
5(2)
1.5 Bibliography
7(2)
2 Transitive Methods
9(36)
2.1 Introductory Example
9(2)
2.1.1 Properties of the Geometrical Covariogram K(h)
10(1)
2.2 The Transitive Covariogram
11(3)
2.2.1 Properties of the Transitive Covariogram g(h)
11(1)
2.2.2 Isotropic Case
12(2)
2.3 Regularisation and Grading
14(5)
2.3.1 Regularisation of a Regionalised Variable
14(1)
2.3.2 Grading
14(1)
2.3.3 Grading in the Isotropic Case
15(4)
2.4 Estimation of a Regionalised Variable
19(18)
2.4.1 Exact Expression of the Estimation Variance (Regular Grid)
19(4)
2.4.2 Case of a Stratified Random Grid
23(1)
2.4.3 Approximation Formulae in the One-Dimensional Space
24(3)
2.4.4 Approximation Formulae in Rn
27(5)
2.4.5 Application to the Estimation of a Surface
32(2)
2.4.6 Passage to the Probabilistic Version of the Theory
34(3)
2.5 Exercises on Transitive Methods
37(8)
2.5.1 Exercises on Sections 2.1 and 2.2
37(3)
2.5.2 Exercises on Grading
40(2)
2.5.3 Exercises on Estimation
42(1)
2.5.4 Exercises on Transitive Theory for Measures
43(2)
3 Theory of Intrinsic Random Functions
45(56)
3.1 General Definitions
45(3)
3.1.1 Notion of a Random Function
45(1)
3.1.2 Stationary Random Functions
46(1)
3.1.3 Expectation
46(1)
3.1.4 The Covariance K(h)
46(1)
3.1.5 Second Order Stationary Hypothesis
47(1)
3.1.6 Infinite A Priori Variance
47(1)
3.1.7 Intrinsic Hypothesis
47(1)
3.2 Properties of the Covariance and of the Semi-Variogram
48(5)
3.2.1 Authorised Linear Combinations
49(2)
3.2.2 Continuity in Quadratic Mean and Other Properties
51(2)
3.3 Regularisation of an Intrinsic Random Function
53(4)
3.3.1 Stochastic Integral
53(1)
3.3.2 Stochastic Convolution
54(2)
3.3.3 Grading
56(1)
3.4 Extension and Estimation Variances
57(6)
3.4.1 Extension Variance
57(1)
3.4.2 Estimation Variance
58(1)
3.4.3 Variance of v within V
59(3)
3.4.4 Application: Random or Stratified Random Networks
62(1)
3.5 Approximation Methods in One Dimension
63(3)
3.5.1 The Correspondence Principle
63(1)
3.5.2 Principle of Composition of Elementary Extension Variances
64(2)
3.6 Approximation Method in Rn
66(5)
3.6.1 Case of Three Dimensions
70(1)
3.7 The Nugget Effect
71(3)
3.7.1 Genesis of the Nugget Effect
71(1)
3.7.2 Macroscopic Influence of the Nugget Effect
72(2)
3.8 The de Wijsian Scheme
74(7)
3.8.1 Linear Equivalents
74(2)
3.8.2 The Two-Dimensional de Wijsian Scheme
76(2)
3.8.3 The de Wijsian Scheme in Three-Dimensional Space
78(3)
3.9 The Spherical Scheme
81(2)
3.10 Statistical Inference and Quasi-Stationarity
83(5)
3.10.1 Quasi-Stationary Random Function
83(2)
3.10.2 Computation of the Estimation Variance
85(2)
3.10.3 Possibility of Statistical Inference
87(1)
3.11 Exercises on Intrinsic Random Functions
88(13)
3.11.1 Exercises on the Construction of Intrinsic Random Functions
88(2)
3.11.2 Exercises on Estimation Variances
90(1)
3.11.3 Exercises on the Nugget Effect, de Wijsian Schemes, and Spherical Schemes
91(1)
3.11.4 Exercises on Large Grids
92(5)
3.11.5 Exercises on Statistical Inference for Random Functions
97(4)
4 Kriging
101(22)
4.1 The Object of Kriging
101(3)
4.2 Notation
104(2)
4.3 Stationary Random Functions with Zero or A Priori Known Expectation
106(2)
4.4 Stationary Random Functions with Unknown Expectation
108(4)
4.4.1 Kriging Equations
108(1)
4.4.2 Optimal Estimation of the Expectation
109(1)
4.4.3 The Additivity Theorem
110(2)
4.5 Case of an Intrinsic Random Function without Covariance
112(1)
4.6 Exercises on Kriging
113(10)
5 Universal Kriging
123(56)
5.1 Introduction
123(4)
5.1.1 Review of Least Squares Methods
123(1)
5.1.2 Statement of the Problem and General Hypotheses
124(3)
5.2 Optimal Estimation of the Drift
127(17)
5.2.1 Estimation of the Drift at a Point
127(3)
5.2.2 Estimation of the Coefficients of a Drift
130(2)
5.2.3 Tensorial Invariance
132(2)
5.2.4 The Variogram of the Residuals
134(2)
5.2.5 Comparison with the Method of Maximum Likelihood
136(1)
5.2.6 Case of a Drift Given in an Implicit Form
137(2)
5.2.7 Comparison with Least Squares Methods
139(5)
5.3 Kriging
144(8)
5.3.1 Equations of Universal Kriging
144(2)
5.3.2 Additivity Theorem
146(2)
5.3.3 Kriging Considered as an Interpolator
148(4)
5.4 Universal Kriging for a Random Drift
152(7)
5.4.1 Hypotheses
152(2)
5.4.2 Estimation of a Drift
154(1)
5.4.3 An Equivocal Example
155(3)
5.4.4 The Problem of the Constant Term A0
158(1)
5.4.5 Kriging
159(1)
5.5 Cokriging
159(4)
5.5.1 Notation
160(1)
5.5.2 Optimal Estimation of the Drift
161(1)
5.5.3 Punctual Cokriging
162(1)
5.6 The Indeterminability of the Underlying Variogram
163(6)
5.6.1 Statement of the Problem
163(2)
5.6.2 Universal Quadratic Estimators
165(2)
5.6.3 General Form of the Admissible Covariance Matrices
167(1)
5.6.4 Consequences for the Optimal Estimators
168(1)
5.7 Exercises on
Chapter 5
169(10)
5.7.1 Exercises on Drifts
169(4)
5.7.2 Exercises on Kriging and Cokriging
173(4)
5.7.3 Exercises on the Indeterminability of the Underlying Covariance
177(2)
References 179(2)
Epilogue 181(8)
Index 189
Dr. Georges Matheron (December 2, 1930 - August 7, 2000) was a French mining engineer and mathematician. Starting from the works of Krige and de Wijs, from South Africa, he created a theory for estimating mining resources that he named geostatistics. During the Sixties, he extended geostatistics by developing a regression method for the cartography of natural phenomena, known as universal kriging. From 1965 to 1968, Dr Matheron co-founded the new discipline of mathematical morphology for describing set shapes and textures. He then developed a theory of the random structures involved in mathematical morphology. During the 1980s he generalized the theory of mathematical morphology to complete lattices, thus providing a common approach to sets, functions and partitions, and introduced morphological filtering. He left the Paris School of Mines in 1995, when he retired.



Dr. Vera Pawlowsky-Glahn's main research topic since 1982 has been the statistical analysis of compositional data. She was the leader of a research group on compositional data analysis involving professors from different Spanish universities from 1986 until 2008. She was also the first President of the Association for Compositional Data, founded in L'Escala (Spain) in June 2015. For the period 2017-2021 she is Past-President of the same association.

Dr. Jean Serra, emeritus professor at University of Paris-Est, established the field of mathematical morphology in 1965, before later founding the Centre de Morphologie Mathématique. His three major contributions to mathematics and physics are morphological filtering, the formulation of mathematical morphology in the convenient framework of complete lattices, and a new concept for connectivity that is the core for set segmentation theory. He founded the International Society for Mathematical Morphology, and was its first president.
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