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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.

5