Table Of ContentGeostatistical Analysis of Compositional Data
International Association for Mathematical Geology
STUDIES IN MATHEMATICAL GEOLOGY
1. William B. Size, Editor
Use and Abuse of Statistical Methods in the Earth Sciences
2. Lawrence J. Drew
Oil and Gas Forecasting: Reflections of a Petroleum Geologist
3. Ricardo A. Olea, Editor
Geostatistical Glossary and Multilingual Dictionary
4. Regina L. Hunter and C. John Mann, Editors
Techniques for Determining Probabilities of Geologic Events and Processes
5. John C. Davis and Ute Christina Herzfeld, Editors
Computers in Geology—25 Years of Progress
6. George Christakos
Modern Spatiotemporal Geostatistics
1. Vera Pawlowsky-Glahn and Ricardo Olea
Geostatistical Analysis of Compositional Data
Geostatistical Analysis
of Compositional Data
Vera Pawlowsky-Glahn
RicardoA. Olea
OXFORD
UNIVERSITY PRESS
2004
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UNIVERSITY PRESS
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Library of Congress Cataloging-in-Publication Data
Pawlowsky-Glahn, Vera.
Geostatistical analysis of compositional data / Vera Pawlowsky-Glahn, Ricardo A. Olea.
p. cm. — (Studies in mathematical geology ; 7)
Includes bibliographical references and index.
ISBN 0-19-517166-7
1. Geology—Statistical methods. 2. Multivariate analysis. 3. Kriging.
I. Olea, R. A. (Ricardo A.) II. Title. III. Series.
QE33.2.M3P39 2003
551'.072—dc21 2003058014
9 8 7 6 5 4 3 21
Printed in the United States of America
on acid-free paper
Preface
In September 1982 when I decided to enroll at the Free University of Berlin for
my graduate studies, the interest expressed by Wolfdietrich Skala in composi-
tional data in a framework relevant to geology led me to devote my disserta-
tion to the study of this topic. I started to go through the literature—mainly
publications by Felix Chayes, but also by many other authors; the Journal
of the International Association for Mathematical Geology became my daily
companion.
I grew more and more interested in spatial statistics and began to wonder if
there were any problem in regionalized compositional data similar to spurious
correlation. I searched for months through papers and books on geostatistics,
and finally, not finding an answer to my question, I wrote a paper on spurious
spatial correlation and presented it at a meeting in Fontainebleau in June 1984.
I still remember how nervous I was during my presentation, not only because
it was the first time for me to speak in front of an international audience, not
only because it was in French, a language I knew only a little bit from school,
but mainly because I expected somebody to say: Haven't you seen that article
or that book? There you can find the answer. What you are talking about has
been known for a long time! When I finished speaking, there was just silence.
I know now that this is the case in most meetings, but at that moment I felt
like falling into a deep hole. Finally Georges Matheron said something like:
"There are certainly problems in the straight application of geostatistics to
compositional data." That was all I wanted to hear. This was the beginning of
my pursuit of a fascinating topic that has dominated my professional interest
ever since.
More or less at the same time, as I continued to read Mathematical Geology,
I came across one of the first papers by John Aitchison that treated composi-
tional data without regard to geographical location. I realized that here was
the answer to many of my questions, that integrating Aitchison's approach to
Matheron's theory was the way to proceed. I was lucky enough to get a grant
from the DAAD (Deutsche Akademische Austauschdienst) to visit John Aitchi-
son in Hong Kong in 1985, shortly before he finished his book on the statistical
analysis of compositional data. I went there for three months and was able to
vi Preface
work with a preprint of the book. The visit was a magnificent learning expe-
rience charmingly complemented by my familiarization with a fascinating new
culture.
I defended my dissertation successfully in 1986 and shortly thereafter re-
turned to Spain to begin teaching in my native Barcelona. The next transcen-
dental encounter of my professional career and my work in compositional data
took place in 1988 when John C. Davis of the Kansas Geological Survey visited
Wolfdietrich Skala and Heinz Burger in Berlin. Davis's interest in compositional
data in general and my research in particular resulted in the commissioning of
his research assistant, Jo Anne DeGraffenreid, to put her German to use by
translating my dissertation into English. After corresponding by post, John
Davis proposed that I give the translation a final review for publication as a
monograph and generously invited me to visit him in Lawrence, Kansas—a
visit that did not take place until 1991.
Reviewing Jo Anne's translation was an easy task, but the desire to revise,
update, and expand what originally had been my dissertation proved to be a
challenge utterly larger than expected, one that took several additional visits
to Lawrence and more time than I ever imagined.
So here is the publication of a much delayed work on compositional data
as it stood at the conclusion of the second millennium. Although the subject
remains open-ended, this seems coincidentally to be an opportune moment to
publish the findings before moving on to further developments.
Publication of a book is rarely the exclusive effort of a single individual. As
already mentioned, Wolfdietrich Skala, Heinz Burger, and particularly John
Aitchison have been extraordinarily supportive of the research summarized
here. I am grateful to John Davis and the Kansas Geological Survey for support
that was instrumental in advancing my research. Jo Anne DeGraffenreid, after
initiating this monograph with her translation, is now serendipitously bringing
it to a conclusion after becoming editor of Studies in Mathematical Geology
in 1998. Ricardo Olea of the Kansas Geological Survey, since shortly after
my first arrival in Kansas, became so supportive of my work that the only
way to properly acknowledge his contribution was to make him coauthor of
this monograph. Actually, without his constant pushing and particularly his
preparation of the case study that comprises the last chapter, there still would
be no end in sight. Sometimes I am tempted to add an "s" to Ricardo's surname
in order to extend my appreciation to all the Oleas in and from Lawrence who
gave me such a comfortable place in their family during my frequent visits.
Completion of this treatise has taken so much time that I am no longer
working at the same place where I started my task. Until September 2000, I
was a member of the Department of Applied Mathematics III at the Technical
Preface vii
University of Catalonia (UPC). Perhaps UPC's most significant contribution
to this monograph, and what has now become my crusade to advance and
promote the study of compositional data, was its support during the organi-
zation of an IAMG conference in 1997 in which the main topic of discussion
was compositional data. I also am indebted to Juan Jose Egozcue of UPC
for ongoing lengthy and productive discussions. At different stages of my ca-
reer I have also received the important support of Carles Barcelo Vidal and
Josep Antoni Martin Fernandez, both with the Department of Informatics and
Applied Mathematics—my new home at the University of Girona.
To my mother Helga and my daughter Tania, and to all the Pawlowsky
& Co. clan, thanks for your patience. You know you are the most important
people in my life, although sometimes it may not be apparent.
Vera Pawlowsky-Glahn
Girona, December 2000
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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
