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

157 pages
Functional Limit Theoremsfor Certain Intrinsic Volumesof Excursion Sets of Random FieldsDissertationzur Erlangung des Doktorgrades Dr. rer. nat.der Fakultät für Mathematik und Wirtschaftswissenschaftender Universität Ulmvorgelegt vonDaniel Meschenmoseraus Friedrichshafen2011Amtierender Dekan: Prof. Dr. Werner KratzErstgutachter: Prof. Dr. Evgeny SpodarevZweitgutachter: Prof. Dr. Ulrich StadtmüllerTag der Promotion: 28. 03. 2011iiContentsChapter 1. Introduction 11.1. Motivation 11.2. Overview of this Thesis 4Chapter 2. Basics of the Geometry of Random Fields 72.1. Integral Geometry 82.2. Random Fields 13Chapter 3. Computation of Intrinsic Volumes 213.1. Error Bound for a Classical Surface Area Estimator 233.2. Multigrid Convergent Computation of Intrinsic Volumes 463.3. Numerical Results 60Chapter 4. Functional Limit Theorems for Dependent Random Fields 674.1. Functional Limit Theorem for the Volume of Excursion Sets 684.2. Limitem for the Surface Area of Excursion Sets 864.3. Large Deviations and Statistical Applications 121Chapter 5. Conclusion 131Bibliography 135List of Figures 141Zusammenfassung 145iiiCHAPTER 1IntroductionCoincidences, in general, are great stumbling-blocks inthe way of that class of thinkers who have been educatedto know nothing of the theory of probabilities – thattheory to which the most glorious objects of humanresearch are indebted for the most glorious of illustration.
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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 compute the intrinsic volumes and show that the error is close to the min-
imum. Further, we propose two methods to compute the intrinsic volumes
without asymptotical error for a large family of sets and give numerical re-
sults illustrating the convergence of the error to zero in Chapter 3. These
results are published in the articles [Kiderlen and Meschenmoser, 2009] and
[Meschenmoser and Spodarev, 2010].
The main results of this thesis are:
The computation of all intrinsic volumes without asymptotical error for
a wide class of sets and for various digitization models. This was possi-
ble only in restrictive special cases but not in this generality before.
Two functional limit theorems for the volume of excursion sets and for
the Hausdorff measure of level sets of random fields, respectively. These
are new results and no similar results are known in the literature.
In the following section, we give details on how these topics are organized.
1.2. Overview of this Thesis
Chapter 2 contains basic mathematical definitions and concepts from integral
geometry and from the theory of random fields. In Section 2.1, the intrinsic
volumes for compact convex sets are defined via Steiner’s formula and some
of their key properties are given. We recall three of the most important results
for intrinsic volumes, namely Hadwiger’s characterization theorem, Crofton’s
formula, and the principal kinematic formula. The definition of the intrinsic
volumes is extended to wider classes of sets in two steps. First, the
v are generalized to polyconvex sets via the inclusion-exclusion formula
and then toU -sets which are sets that may have smooth concave inlets andPR
hence overcome the restrictions of polyconvex sets. Section 2.2 contains the def-
inition of random fields and some of their essential properties like stationarity,
isotropy, continuity, and differentiability. Further, we give a short overview of
association and related concepts and state some relationships between them.
41.2 Overview of this Thesis
Chapter 3 deals with the computation of intrinsic volumes of a set given its
digitization. We illustrate two methods to digitize a set on a rectangular grid
and give a mathematical definition of what is meant by “without asymptotical
error” in the previous section. In Section 3.1, we analyze a classical approach
to compute the intrinsic volumes given in [Ohser and Mücklich, 2000] and
derive bounds for the worst case error. In the next step, we apply Bonnesen’s
improved isoperimetric inequality which leads to error bounds considerably
tighter than the previous ones. The new error bounds are close to the minimal
error which is obtained numerically. In Section 3.2, we present two methods to
compute the intrinsic volumes without asymptotical error for a wide class of
sets and for various digitization models. For that we approximate the bound-
ary of the set by line segments of flexible length. We give an explicit description
of the approximation procedure adapted to different classes of sets. Numer-
ical results for various test sets in the plane complete this chapter in Section 3.3.
Chapter 4 is devoted to two functional limit theorems for the volume of ex-
cursion sets and for the Hausdorff measure of level sets of random fields,
respectively. We recall some basic concepts, namely weak convergence, rel-
ative compactness, and tightness. Prokhorov’s theorem gives conditions under
which the latter two concepts are equivalent. In Section 4.1, we establish a
functional limit theorem for the volume of excursion sets of associated random
fields in the Skorokhod space of càdlàg functions. For the proof we use a result
by Billingsley stating that tightness and convergence of the finite-dimensional
distributions imply weak convergence. Section 4.2 contains a functional limit
theorem for the Hausdorff measure of level sets of Gaussian random fields in
the space of square integrable functions with respect to the standard Gauss-
ian measure. In that case, due to a result by Prokhorov, relative compactness
and convergence of the characteristic functions have to be proven to achieve
weak convergence. The functional limit theorem for the volume of excursion
sets enables us to prove two large deviation principles in Section 4.3. The first
one gives an upper bound for the probability of high excursions. Under the
additional condition of Gaussianity the second one gives the exact asymptotics
which allows to define an asymptotical test.
The thesis concludes with Chapter 5 in which the main results are summarized
and open problems as well as implications for further research are stated.
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