Excerpt
Ulm University
Faculty of Mathematics and Economics
Excursion Sets of Random Fields
and its Applications
Florian Timmermann
Foreword
The enormous interest, which my diploma thesis attracted on var-
ious conferences such as the International Conference on Stochas-
tic Analysis and Random Dynamical Systems (Lviv, Ukraine, 14th
20th June 2009), the DAAD-Summer Academy on Stochastic Geome-
try, Spatial Statistics and Random Fields (Hirschegg, Austria, 13th
26th September 2009), and the 9th German Open on Probability and
Statistics (Leipzig, Germany, 02nd 05th March 2010), was a great
motivation for me to provide it in an updated and extended version to
a larger group of people. Slight changes have been made in the proofs
of Chapter 3 and the layout has been re-worked. The main results
in a short form can also be found in [12], where a section about shot
noise elds is included.
This work combines two beautiful branches of mathematics: geome-
try and random elds. The mathematical basics needed to understand
the theory are developed carefully. Enriched with illustrative exam-
ples an easily implementable method for the analysis of a wide range
of surfaces, e.g. paper or metallic surfaces, is provided and thus suits
for direct application. For those interested only in application it is
recommended to skip Sections 2.2, 2.4 and the theorems of Section 2.5.
The diploma thesis in econo-mathematics was handed in on 13th
September 2009 and has been awarded at Ulm University on 7th
November 2009 by the Horbach Förderpreis.
I am grateful to the reviewers of my diploma thesis Professors A.
Bulinski and E. Spodarev for valuable remarks and suggestions per-
mitting me to improve the quality of this work.
Ulm,31st May 2011
Florian Timmermann
4
Contents
1 Motivation
7
2 Random elds and dependence concepts
11
2.1 Random variables . . . . . . . . . . . . . . . . . . . . 11
2.2 Random elds and Kolmogorov's existence theorem . . 14
2.3 Stationary and isotropic random elds . . . . . . . . . 18
2.4 Measurability of random elds . . . . . . . . . . . . . 21
2.5 Dependence concepts . . . . . . . . . . . . . . . . . . 26
2.5.1 Association, positive and negative association . 26
2.5.2 Quasi-association . . . . . . . . . . . . . . . . 32
2.5.3 (BL, )-dependence . . . . . . . . . . . . . . . 37
3 Excursion sets and the central limit theorem
41
3.1 Excursion sets of random elds . . . . . . . . . . . . . 41
3.2 The CLT for (BL,)-dependent random elds . . . . . 43
3.3 Covariance inequalities . . . . . . . . . . . . . . . . . 50
3.4 The CLT for the volume of excursion sets of random
elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Quasi-associated random elds . . . . . . . . . 53
3.4.2 Gaussian random elds . . . . . . . . . . . . . 60
3.5 Statistical version of the CLT . . . . . . . . . . . . . . 63
3.5.1 The estimator involving local averaging . . . . 65
3.5.2 A covariance-based estimator . . . . . . . . . . 66
3.5.3 The subwindow estimator . . . . . . . . . . . . 68
5
6
Contents
3.6 Test of Gaussianity of random elds . . . . . . . . . . 70
4 Numerical Results
71
4.1 Application to simulated data . . . . . . . . . . . . . 71
4.1.1 Comparison of subwindow size . . . . . . . . . 71
4.1.2Convergence rate . . . . . . . . . . . . . . . . 76
4.1.3 Theoretical value of vs. numerical results . . 76
4.1.4 Computation time . . . . . . . . . . . . . . . . 76
4.2Application to images of paper surface . . . . . . . . . 80
4.2.1 Production process of paper . . . . . . . . . . 80
4.2.2 Estimation of . . . . . . . . . . . . . . . . . 82
4.2.3 Test of Gaussianity of paper data . . . . . . . 85
4.2.4 What resolution should at least be taken? . . . 86
5 Conclusion
89
A Proof of the entropy bound of Dudley
91
List of Figures
97
List of Tables
99
Bibliography
101
Chapter 1
Motivation
Various applications in industry arise interest for the investigation
of geometric characteristics of random surfaces which now form an
important research area (see, e.g. [1] [3]). In this workwe mention
one motivating example.
The contemporary method of papermaking is said to have been
invented during the Han Dynasty (206 BC 220 AD) by the Chinese
court ocial Cai Lun (105 AD). Although paper already existed in
China he was responsible for the rst signicant improvement and
standardization of papermaking. Cai Lun created a sheet of paper
mainly using mulberry and other bast bres, and in contrast to earlier
methods his technique consisted of felting sheets of paper suspended
in water, draining o the water and then drying them. Nowadays, in
modern paper mills, the method is still the same, but machines with
a production speed of up to 33m/s operate much faster and bres of
soft- and hardwood are widely used. Still few years ago it has not been
possible to make statements about the surface structure during the
forming process, which to a great extent inuences the paper quality.
In a joint workwith Voith Paper students of the Cooperative State
University Heidenheim applied a test rig developed by scientists from
STFI-Packforsk, a research institute, to make this process visible.
They implemented a high-speed camera system into a forming rig and
7
8
Chapter 1. Motivation
Figure 1.1: Image section of paper surface (left) and a simulated Gaus-
sian random eld (right) based on estimated data. Source: Lochbrun-
ner [26].
gained large data sets of greyscale images. An example of a resulting
image is shown in Figure 1.1. It was now the main goal to derive
statements about the surface quality of paper already in the forming
process. To model paper surface, a reasonable rst choice can be
stationary random elds, such as shot noise (see [5]) or Gaussian ones.
Comparing by eye real paper image data and simulated realizations
of such elds it is dicult to distinguish between the images. But
how can we quantify the dierences between these two images? One
possibility to do this are excursion sets.
Belyaev (see, e.g. [3]) examined certain properties of these sets
during 1967 1975, such as the mean number of crossings for random
processes and elds. Later Adler and Taylor [2] gained interesting
results about the Minkowski functionals of excursion sets of random
elds. These results have been published from 1976 up to now.
Our idea lies in establishing central limit theorems for excursion
sets of a general class of elds X possessing a general dependence
structure. Examples of such stochastic models can be found in, e.g.
[10]. We will only assume that the covariance function of a random
eld decreases as the distance between two points grows, which is a
quite reasonable condition.
9
In this work we prove uni- and multivariate central limit theorems
(CLT) for volumes of excursion sets of a stationary random eld X =
{X(t), t T }, T R
d
, to characterize the surface generated by X.
After a detailled introduction to the mathematical basics which are
needed to establish our results we generalize a CLT in [20], page 80,
havingbeen obtained by other methods for volumes of excursion sets
of stationary and isotropic Gaussian random elds. We also discuss
consistent estimators for the asymptotic covariance matrix that arises
in the limitingdistribution and provide a statistical hypothesis test
to check if a considered random eld belongs to the desired class.
For a study of movinglevels for excursion sets, the estimate of the
convergence rate to the limit law and the analysis of functionals in
Gaussian random elds we refer to [20], [23] [25].
10
Chapter 1. Motivation
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