Collecting spatial data: Optimum design of experiments for random fields

250 Pages·2007·2.051 MB·English

by  Werner G. Müller

Collecting Spatial Data
Werner G. Müller
Collecting
Spatial Data
Optimum Design of Experiments
for Random Fields
ThirdRevisedandExtendedEdition
With37Figuresand8Tables
123
Univ.Prof.Dr.WernerG.Müller
DepartmentofAppliedStatistics
JohannesKeplerUniversityLinz
Altenbergerstraße69
4040Linz
Austria
[email protected]
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Preface to the Third Edition
In the past years the gap that this monograph had intended to close
seems to have narrowed considerably, looking at the ever-growing sci-
entific literature on theory and practice of spatial sampling. However,
it also seems of convenience to have a concise collection of methods
and references at hand, as this book aims to provide. This need may
explain that a third edition of it was required.
Of course recent research made it not only necessary to simply
reprint the book but to add a sizable amount of text, especially con-
cerningdesignsforthesimultaneousestimationoftrendandvariogram
parameters. The new developments in this area were taken into ac-
count by a rearrangement of the previous Chapter 6, which is now
split in two. Most of the new material stems from joint work with
E. Glatzer, D. Gumprecht, J. Lop´ez-Fidalgo, J. Rodr´ıguez-D´ıaz, and
M. Stehl´ık, whom I am indebted for their collaboration. However, I
take sole responsibility for any mistakes that may have found their
way into this edition. I also wish to thank J. Laca for providing an
additional example and M. Hainy and the participants of the statGIS
2006 summer school in Klagenfurt for their valuable feedback.
I am grateful to Mrs. Wetzel-Vandai from Springer Verlag for the
support of the new edition project and especially about her patience
in the phase of realizing it.
In the period between the editions my professional life has substan-
tially changed by my move to the Johannes-Kepler University in Linz.
I appreciate tremendously the warm welcome and the acceptance I
have been offered by my new colleagues in the department. With-
out their support the work on this third edition would not have been
possible.
Gramatneusiedl, June 28, 2007 Werner G. Mu¨ller
Preface to the Second Edition
I was very pleased when Dr. Bihn proposed me to publish a second
edition of this monograph. Not only did this allow me to correct quite
a number of misprints (a few disturbing amongst them), but also to
take account of the rapid current development in the area of spatial
statistics (and within it the one of spatial design) by including and
addressing many new relevant references (and updating older ones).
New to this addition is also the inclusion of exercises at the end of
each chapter (with suggested solutions to be published on a web site).
This allows a deeper practical involvement of the readers with the
material and also a flexible way of extending it as new results become
available. IamgratefultoDr. C.MilotafromtheNieder¨osterreichischen
Landesregierung for making a corresponding data set available to me,
and to G. Grafeneder for providing his initial geostatistical analysis of
these data.
I would like to offer my thanks to E. Glatzer, M. Holtmann, I.
Molchanov, and A. Pa´zman for pointing out inaccuracies that would
otherwise have been left undetected. My special gratitude goes to D.
Ucin`ski for his very thorough reading of the first edition and his long
list of suggested improvements.
Gramatneusiedl, August 2, 2000 Werner G. Mu¨ller
Preface to the First Edition
The aim of this monograph is to provide an overview over classical
as well as recently developed methods for efficient collection of spatial
data. In the past 10 years, starting with my involvement with the
International Institute for Applied Systems Analysis (IIASA), I have
devoted much of my research to this subject. This monograph is a
compilation of the results of this work with additional material on
kriging, variogram estimation and design techniques. Some parts of
the book are adapted from papers coauthored with V.V. Fedorov, A.
Pa´zman and D.L. Zimmerman. I am indebted to them for the fruitful
cooperation and many helpful advices on the preparation of this text.
The theoretical elaborations are accompanied by an applied exam-
ple, the redesign of the Upper-Austrian SO2 monitoring network. I
am most grateful to Dr. E. Danninger from the Obero¨sterreichischen
Landesregierung for making the corresponding data available to me.
Additionally I would like to acknowledge the positive impact of my
pastandpresentworkingenvironmentsattheDepartmentofStatistics
oftheUniversityofVienna,theInstituteforAdvancedStudiesVienna,
the Department of Statistics and Actuarial Sciences of the University
of Iowa and currentlythe Department of Statistics of the University of
Economics Vienna. Manythankstoallthecolleagues fortheirsupport
and friendship.
MydeepgratitudegoestoE.Glatzer,W.Katzenbeisser,J.Ledolter,
and K. Po¨tzelberger for attentive readings of and for detecting a num-
ber of inaccuracies in previous versions of this text. I am especially
grateful to P. Hackl for his committed efforts in this respect that led
to a considerable improvement of the manuscript.
Gramatneusiedl, June 3, 1998 Werner G. Mu¨ller
Contents
1 Introduction 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Fundamentals of Spatial Statistics 11
2.1 Estimation of Spatial Trend . . . . . . . . . . . . . . . . 13
2.2 Universal Kriging . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Local Regression . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Variogram Fitting . . . . . . . . . . . . . . . . . . . . . 23
2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Fundamentals of Experimental Design 43
3.1 Information Matrices . . . . . . . . . . . . . . . . . . . . 45
3.2 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Numerical Algorithms . . . . . . . . . . . . . . . . . . . 57
3.4 Further Design Topics Useful in the Spatial Setting . . . 60
3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Exploratory Designs 77
4.1 Deterministic and Random Sampling . . . . . . . . . . . 79
4.2 Space Filling Designs . . . . . . . . . . . . . . . . . . . . 82
4.3 Designs for Local Regression . . . . . . . . . . . . . . . 85
4.4 Model Discriminating Designs . . . . . . . . . . . . . . . 88
4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Designs for Spatial Trend Estimation 101
5.1 Approximate Information Matrices . . . . . . . . . . . . 102
5.2 Replication-Free Designs . . . . . . . . . . . . . . . . . . 105
x Contents
5.3 Designs for Correlated Fields . . . . . . . . . . . . . . . 110
5.4 Designs for Spatial Prediction . . . . . . . . . . . . . . . 125
5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6 Design and Dependence 141
6.1 Designs for Detecting Spatial Dependence . . . . . . . . 142
6.2 Designs for Variogram Estimation . . . . . . . . . . . . 151
6.3 Methods Based on Likelihood Approaches . . . . . . . . 161
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 167
References . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7 Multipurpose Designs 173
7.1 Combining Different Purpose Designs . . . . . . . . . . 175
7.2 Likelihood-based Approaches . . . . . . . . . . . . . . . 177
7.3 Alternative ‘Direct’ Approaches . . . . . . . . . . . . . . 180
7.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 186
References . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Appendix 193
A.1 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.2 Proofs for Chapter 2 . . . . . . . . . . . . . . . . . . . . 195
A.3 Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . 196
A.4 Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . 198
A.5 Proofs for Chapter 5 . . . . . . . . . . . . . . . . . . . . 201
A.6 Proofs for Chapter 6 . . . . . . . . . . . . . . . . . . . . 216
A.7 Proofs for Chapter 7 . . . . . . . . . . . . . . . . . . . . 217
A.8 D2PT Description . . . . . . . . . . . . . . . . . . . . . 218
References . . . . . . . . . . . . . . . . . . . . . . . . . . 225
List of Figures 227
Author Index 229
Subject Index 237
List of Symbols
A support of replication-free L lag space
design
m dimension of parameter
c covariance function vector β
C error covariance matrix M information matrix
d prediction variance M set of information matrices
D covariancematrixofparameters n number of support points
D discrepancy measure N number of observations
E entropy N numberofcandidatesupport
points
f arbitrary functions
p order of polynomial or
F cumulative distribution
design weights
function
F standard normal cdf p(·) probability density
F induced design space q(·),r(·) general functions
r remainder term
h distancebetweentwospatial
locations
s indexinsequentialprocedures
H set of distances
S supporting set
H distance matrix
S generalized sum of squares
i,j,k general indices
t temporal index
I indicator function
T number of time points
I Moran’s statistic
U c.d.f. of a uniform
J measure of information distribution
K observationcovariancematrix U mapped design space
l general vector v,w weights
L general matrix V,W weighting matrices