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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