Statistical Analysis of Random Fields

254 Pages·1989·6.825 MB·English

by  A. V. Ivanov, N. N. Leonenko (auth.)

Statistical Analysis of Random Fields
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIR.ILLOV, MGU, Moscow, U.S.s.R.
Yu.1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.s.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.s.s.R.
S. P. NOVIKOV, Landau Institute ofT heoretical Physics, Moscow, U.s.S.R.
M. C. POLYV ANOV, Steklov Institute ofM athematics, Moscow, U.s.s.R.
Yu. A. ROZANOV, Steldov Institute ofM athematics, Moscow, U.S.S.R.
VOLUME 28
Statistical Analysis of
Random Fields
by
A. V. Ivanov
Cybernetic Center, Kiev, U.s.S.R.
and
N. N. Leonenko
University of Kiev, U.s.s.R.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloaina in Publication Data
Ivanov. A. V.
Statistical analysis of random fields / A.V. Ivanov and N.N.
Leonenko ; translation edited by S. Kotz.
p .. cn. -- (Mathematics and its application. Soviet series)
Translated Russian.
fro~
Bibliography: p.
Includes index.
ISBN-13: 978-94-010-7027-0
1. Random fields. 2. Mathematical statistics. I. Leonenko. N.
N. II. Title. III. Series: Mathematics and its applications
(Kluwer Academic Publishers). Soviet series.
OA274.45.193 1989
519.2--dC19 88-8196
ISBN-13: 978-94-010-7027-0 e-ISBN-13: 978-94-009-1183-3
DOl: 10.1007/978-94-009-1183-3
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
This is an revised and updated edition of the original work
CTATHCTH'IECKHR AHAJlH3 CJlYIIARHblX nOJlER
Published by Vysa. Skola, Kiev, C 1986.
Translated from the Russian by A. I. Kochubinsky
Translation edited by S. Kotz.
printcd Oil acid.ti·cc paper
All Rights Reserved
This English edition C 1989 by Kluwer Academic Publishers.
Softcover reprint of the hardcover I st edition 1989
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, eleclronic or mechanical,
including photocopying, recording, or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Series Editor's Preface vii
Preface ix
1. Elements of the Theory of Random Fields 1
1.1. Basic concepts and notation 1
1.2. Homogeneous and isotropic random fields 9
1.3. Spectral properties of higher order moments
of random fields 20
1.4. Some properties of the uniform distribution 24
1.5. Variances of integrals of random fields 27
1.6. Weak dependence conditions for random fields 32
1.7. A central limit theorem 35
1.8. Moment inequalities 43
1.9. Invariance principle 49
2. Limit Theorems for Functionals of Gaussian Fields 54
2.1. Variances of integrals of local Gaussian functionals 54
2.2. Reduction conditions for strongly dependent random fields 61
2.3. Central limit theorem for non-linear transformations
of Gaussian fields 70
2.4. Approximation for distribution of geometric functionals
ro
~G~~an~~
2.5. Reduction conditions for weighted functionals 83
2.6. Reduction conditions for functionals
depending on a parameter 85
2.7. Reduction conditions for measures of excess
over a moving level 90
2.8. Reduction conditions for characteristics
of the excess over a radial surface 107
2.9. Multiple stochastic integrals 114
2.10. Conditions for attraction of functionals of homogeneous
isotropic Gaussian fields to semi-stable processes 117
v
vi CONTENTS
3. Estimation of Mathematical Expectation 129
3.l. Asymptotic properties of the least squares estimators
for linear regression coefficients 129
3.2. Consistency of the least squares estimate
under non-linear parametrization 138
3.3. Asymptotic expansion of least squares estimators 146
3.4. Asymptotic normality and convergence of moments
for least squares estimators 155
3.5. Consistency of the least moduli estimators 158
3.6. Asymptotic normality of the least moduli estimators 163
4. Estimation of the Correlation Function 174
4.l. Definition of estimators 174
4.2. Consistency 176
4.3. Asymptotic normality 184
4.4. Asymptotic normality. The case of a homogeneous
isotropic field 196
4.5. Estimation by means of several independent
sample functions 203
4.6. Confidence intervals 208
References 216
Comments 233
Index 240
SERIES EDITOR'S PREFACE
'Et moi, ...• si j'avait su comment en revcnir. One service mathematics has rendered the
je n'y scrais point aile.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shclf next
to the dusty canister labdlcd 'discarded non·
The series is divergent; therefore we may be sense'.
able to do something with it Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vii
viii SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no 1V; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no te1Iing where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Random fields, the 2-D analogues of stochastic processes are tough to work with as there are practi
cally no systematic books on the topic. This in spite of their obvious importance and relevance.
Here, apparently (as in analysis) the 'dimensional difficulty gap' is between I and 2 rather than
between 2 and 3, or 3 and 4, as is more customary in geometry, topology and mathematical physics.
There remains a great deal of fundamental work to be done as becomes already clear when one
starts thinking about what a 2D-Wiener process should be, an active area of current research.
It is well known how important limit theorems have been in the theory of stochastic processes
and how important they are and have been to built up intuition. Thus it is most fortunate and
valuable that the authors manage to say much about precisely limit theorems; in addition there is a
great deal of material on the estimation of mathematical expectations and correlation functions.
Precisely the stuff therefore which is needed for the many potential applications in image process
ing, statistical physics, meteorology, turbulence and many other fields.
This valuable book -in my opinion -is a substantially updated and expanded translation of its
RusSian original of some two years ago.
Perusing the present volume is not guaranteed to tum you into an instant expert, but it will
help, though perhaps only in the sense of the last quote on the right below.
The shonest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l'o ccasion de resoudre des problemes ... elle right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. Poinca:rC David Butler
Bussum, March 1989 Michiel Hazewinkel
Preface
This book is devoted to an investigation of the basic problems of the statistics
of random fields. It should be mentioned at the outset that up until now there
have been no books available in which these problems have been tackled. On
the whole, the theory of random fields is poorly represented in monographic
literature. I am aware of only two books: "Spectral Theory of Random Fields"
(1980) by M.1. Yadrenko and "The Geometry of Random Fields" (1981) by R.J.
Adler. This meagre output is in spite ofthe fact that the theory of random fields
continues to attract new applications such as in turbulence theory, meteorology,
statistical radiophysics, theory of surface roughness, pattern recognition and
identification of parameters of complex systems. Moreover, random functions
of many variables, namely, random fields, arise naturally in probability theory
proper.
The specific nature of random fields manifests itself when studying ran
dom functions whose properties are coordinated with the algebraic structure
of the space in the same manner as the specific nature of random processes (a
function of a single variable viewed as time) are revealed in coordination with
the ordering structure (this gives rise to the theory of martingales and Markov
processes) .
As is well known, limit theorems play an essential role in solving problems
in statistics. It is usually assumed in problems of statistics of random processes
that a process (as a rule a single realization of it) is observed on a time interval
that extends to infinity. Based on these observations:
(1) one constructs estimators of parameters and (2) one tests hypothe
ses about the distribution of the process or about the form of its basic charac
teristics and so on.
An analogous approach is utilized in this book for the study of homo
geneous isotropic random fields. It is assumed that a field is observed in an
expanding domain and based on these observations, problems (1) and (2) are
solved for random fields. Limit theorems for functionals (in particular, addi
tive ones) of sample functions of fields are required as a working tool. We note
that limit theorems for functionals of random fields are of interest in their own
right. The book devotes major attention to an investigation of limit distribu
tions for various specific functionals of a geometric nature for Gaussian random
ix
x PREFACE
fields possessing strongly and weakly decreasing correlations. For the first time
in monographic literature, functionals useful in applications are investigated,
such as measures of excess of a Gaussian field above a fixed or moving level and
above a spherical surface, and volumes constrained by realizations of Gaussian
fields over sets. The results obtained can be applied to the analysis of surface
roughness, for example.
Substantial results are obtained in estimation theory of the first two mo
ments of random fields with a continuous parameter, that is, the mathematical
expectation and correlation function. The least squares and least moduli esti
mators are studied for a multi-dimensional parameter of linear and non-linear
regressions. These estimators are not always optimal; however, they have an
important quality, being determined as the extremum point of simple integral
functionals of realization of an observed field. In addition to parametric es
timation problems, non-parametric estimation of the correlation function of
homogeneous (or homogeneous isotropic) fields having zero mean value is con
sidered. Correlation-type estimators are investigated. Conditions are provided
under which measures corresponding to these estimators converge weakly to
Gaussian measures. This allows us to solve the important problem of con
structing confidence intervals for an unknown correlation function. We note
that in practice it is often impossible to obtain a realization of a random field
or even measure an instantaneous value of such a field. However, measurements
of the correlograms of random fields can be obtained by means of physical de-
vIces.
Both specialists in the theory of random fields and scientists working in
related areas that utilize this theory will no doubt find a great deal of new and
useful information in this book.
A.V. Skorohod
Member of the Academy of Sciences of the Ukrainian SSR