Functional limit theorems for certain intrinsic volumes of excursion sets of random fields [Elektronische Ressource] / vorgelegt von Daniel Meschenmoser

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2011
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 2 Mo

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Functional Limit Theorems
for Certain Intrinsic Volumes
of Excursion Sets of Random Fields
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
vorgelegt von
Daniel Meschenmoser
aus Friedrichshafen
2011Amtierender Dekan: Prof. Dr. Werner Kratz
Erstgutachter: Prof. Dr. Evgeny Spodarev
Zweitgutachter: Prof. Dr. Ulrich Stadtmüller
Tag der Promotion: 28. 03. 2011
iiContents
Chapter 1. Introduction 1
1.1. Motivation 1
1.2. Overview of this Thesis 4
Chapter 2. Basics of the Geometry of Random Fields 7
2.1. Integral Geometry 8
2.2. Random Fields 13
Chapter 3. Computation of Intrinsic Volumes 21
3.1. Error Bound for a Classical Surface Area Estimator 23
3.2. Multigrid Convergent Computation of Intrinsic Volumes 46
3.3. Numerical Results 60
Chapter 4. Functional Limit Theorems for Dependent Random Fields 67
4.1. Functional Limit Theorem for the Volume of Excursion Sets 68
4.2. Limitem for the Surface Area of Excursion Sets 86
4.3. Large Deviations and Statistical Applications 121
Chapter 5. Conclusion 131
Bibliography 135
List of Figures 141
Zusammenfassung 145
iiiCHAPTER 1
Introduction
Coincidences, in general, are great stumbling-blocks in
the way of that class of thinkers who have been educated
to know nothing of the theory of probabilities – that
theory to which the most glorious objects of human
research are indebted for the most glorious of illustration.
Edgar Allan Poe: The Murders in the Rue Morgue
The title of this thesis contains three keywords: “Functional Limit Theorems”,
“Intrinsic Volumes”, and “Excursion Sets of Random Fields”. In the following,
we discuss these notions and the relations between them. Afterwards, we give
an overview of the structure of this thesis.
1.1. Motivation
In probability theory, the term “limit theorem” refers to a broad class of state-
ments about the distribution of a function of a sequence of random elements
as the length of this sequence grows to infinity. In its classical form the cen-
tral limit theorem states that the average of a large sequence of independent
and identically distributed random variables is approximately normally dis-
tributed. According to [Tijms, 2004, Section 5.4], its first version concerning
the distribution of the number of heads when repeatedly tossing a coin was
developed by de Moivre in 1733 in his work “Approximatio ad Summam Ter-
1nminorum Binomii a+ bn in Seriem expansi” . Hence, the term “Gaussian
distribution” for the limit distribution of the (normalized) sum is a misnomerpR x 2as it was de Moivre who used the expression exp t /2 dt/ 2p in his
¥
work more than 40 years before Gauss was born. Later, de Moivre’s result was
generalized by Laplace, Lyapunov, and others. The term “central limit theo-
rem” or, to be more precise, its German translation “Zentraler Grenzwertsatz”
n1 nIn modern mathematical notation, a+ bn is denoted by (a+ b) .
11 Introduction
was coined by Pólya in [Pólya, 1920] to emphasize the central role the theorem
plays in probability theory. A limit theorem is equipped with the prefix “func-
tional” if the limit is not a single Gaussian random variable but a Gaussian
process.
According to [Le Cam, 1986, p. 81], “Markov appears to be the first to try
to replace the independence condition” in the central limit theorem. Markov
considered a stochastic process which can be thought of as a random value
indexed by time, may the time pass by in discrete steps or continuously. If the
index of the random value is not considered as time but as a location in space
we usually speak of a random field.
Limit theorems are one of the central topics in the theory of random fields. In
that case the limit is understood in the sense that the domain where the field
is observed expands. Rosenblatt proved a central limit theorem under a con-
dition he called “strong mixing” [Rosenblatt, 1956]. Roughly speaking, this
means that two states of a random field are almost independent if the distance
between them is sufficiently large. Later, several variants of mixing conditions
were derived e.g. in [Volkonski and Rozanov, 1959] and [Ibragimov, 1962].
However, the “verification of strong mixing conditions for particular random
fields is a difficult task” [Ivanov and Leonenko, 1986, p. 234].
In the sixties of the last century, a complementary approach to model the de-
pendence structure of a random field was developed. The class of associated
random fields was introduced by Esary, Proschan, and Walkup in their fun-
damental article [Esary et al., 1967]. Slightly different conditions, namely pos-
itive association, negative association, and quasi-association appeared a cou-
ple of years later in [Burton et al., 1986], [Joag-Dev and Proschan, 1983], and
[Bulinski and Suquet, 2001], respectively. Association and related concepts
impose conditions on the covariance function of the random field which are
easier to verify compared to mixing conditions.
An important class of limit theorems for random fields deals with geomet-
ric functionals. It started with Rice, Cramér, and Belyaev who considered the
number of upcrossings of a stochastic process with a horizontal line. The ex-
tension to random fields was carried out only a couple of years later by Belyaev,
Malevich, and others; see [Adler, 1967] for a survey. In that case, the proper
generalization of the “number of upcrossings” is the Euler characteristic; a
functional related to the number of connected components of the excursion
set, i.e. the set of all points where the field exceeds a given value.
21.1 Motivation
The Euler characteristic belongs to a class of functionals called “intrinsic vol-
umes”. They are one of the main concepts in integral geometry and were origi-
nally defined on the class of compact convex sets by Blaschke in his pioneering
work [Blaschke, 1936]. Besides the Euler characteristic, they include the vol-
ume, the surface area (up to a factor), and other interesting functionals. The
importance of the intrinsic volumes is demonstrated in a fundamental result
by Hadwiger which states that any continuous, additive, and motion-invariant
functional on compact convex sets is a weighted sum of intrinsic volumes.
The intrinsic volumes of excursion sets of Gaussian and other classes of ran-
dom fields have been studied by several authors. The usual volume is con-
sidered e.g. in [Ivanov and Leonenko, 1986], [Adler and Taylor, 2007], and in
[Bulinski et al., 2010] where univariate and multivariate central limit theorems
are given. Adler and Taylor also show that the surface area of the excursion
set is closely related to the Hausdorff measure of the level set which con-
sists of all points where the field attains a given value. The measure of level
sets was considered e.g. by Azaïs, Kratz, Léon, and Wschebor. They derived
central limit theorems in [Kratz and Léon, 2001] and [Kratz and Léon, 2010]
and formulas for the moments in [Azaïs and Wschebor, 2009]. To our best
knowledge, so far no functional limit theorems for the intrinsic volumes of the
level or excursion sets are known in the literature. In this thesis, we estab-
lish two new functional limit theorems for the volume of excursion sets and
for the Hausdorff measure of level sets, respectively, in Chapter 4. These re-
sults are published in the articles [Meschenmoser and Shashkin, 2010a] and
[Meschenmoser and Shashkin, 2010b].
In order to compute the intrinsic volumes in practice, usually a digital im-
age of the set under consideration is the only information available. Consider
for example microscope images of cancer tissue where the knowledge of the
intrinsic volumes can help the pathologist classify the severity of the disease
[Böhm et al., 2008]. Further examples include the analysis of foams and other
porous media in materials science [Helfen et al., 2003] and the characterization
of galaxy distribution in cosmology [Kerscher et al., 1997]. However, concern-
ing the intrinsic volumes, it can make a big difference if you consider a set in
the real world or its digitization in the computer. It is obvious that two sets
which differ only slightly may result in the same digital image and hence the
intrinsic volumes computed in the digital world are the same. This implies that
it is not possible to compute the intrinsic volumes of a set given its digitization
for a wide class of sets without error.
31 Introduction
However, there are two ways to reduce the error. For fixed resolution of the dig-
ital image the error can be minimized by appropriate changes to existing meth-
ods. This is of importance for applications where the resolution is predefined
and cannot be changed. For increasing resolution, methods can be developed
to compute the intrinsic volumes without asymptotical error. This means that
the error tends to zero as the resolution increases. Although these problems
have been investigated for several years, there is no thorough solution known
in the literature. So far, only the volume and some special cases under rather
restrictive conditions have been considered; see e.g. [Coeurjolly et al., 2003]
and [Kiderlen, 2006]. We derive new weights to improve a classical method
to com