Excursion Sets of Random Fields and its Applications

Table of Contents

1 Motivation

2 Random fields and dependence concepts

2.1 Random variables

2.2 Random fields and Kolmogorov’s existence theorem

2.3 Stationary and isotropic random fields

2.4 Measurability of random fields

2.5 Dependence concepts

2.5.1 Association, positive and negative association

2.5.2 Quasi-association

2.5.3 (BL, θ)-dependence

3 Excursion sets and the central limit theorem

3.1 Excursion sets of random fields

3.2 The CLT for (BL, θ)-dependent random fields

3.3 Covariance inequalities

3.4 The CLT for the volume of excursion sets of random fields

3.4.1 Quasi-associated random fields

3.4.2 Gaussian random fields

3.5 Statistical version of the CLT

3.5.1 The estimator involving local averaging

3.5.2 A covariance-based estimator

3.5.3 The subwindow estimator

3.6 Test of Gaussianity of random fields

4 Numerical Results

4.1 Application to simulated data

4.1.1 Comparison of subwindow size

4.1.2 Convergence rate

4.1.3 Theoretical value of Σ vs. numerical results

4.1.4 Computation time

4.2 Application to images of paper surface

4.2.1 Production process of paper

4.2.2 Estimation of Σ

4.2.3 Test of Gaussianity of paper data

4.2.4 What resolution should at least be taken?

5 Conclusion

A Proof of the entropy bound of Dudley

Objectives & Research Topics

This work investigates the geometric characteristics of random surfaces using the concept of excursion sets. The primary goal is to establish uni- and multivariate central limit theorems (CLT) for the volume of excursion sets for a general class of random fields with a specified dependence structure. This research bridges geometric probability and statistical applications, particularly in paper manufacturing.

• Mathematical foundations of random fields and dependence concepts.
• Establishment of central limit theorems for excursion sets.
• Development of consistent estimators for the asymptotic covariance matrix.
• Application of these models to real-world paper surface data.
• Statistical hypothesis testing for Gaussianity in random fields.

Excerpt from the Book

Summary of Chapters

1 Motivation: Introduces the practical background of the research by discussing the papermaking process and the need for analyzing random surfaces.

2 Random fields and dependence concepts: Establishes the mathematical foundation, covering definitions of random fields, measurability, stationarity, and various dependence concepts such as association and (BL, θ)-dependence.

3 Excursion sets and the central limit theorem: The main theoretical part, focusing on the definition of excursion sets and proving central limit theorems for their volume under different dependence assumptions.

4 Numerical Results: Evaluates the proposed estimators and central limit theorems using both simulated data and real-world paper surface images.

5 Conclusion: Summarizes the key findings, including the successful establishment of the CLT for general random fields and suggestions for future research.

Keywords

Random fields, Excursion sets, Central limit theorem, Dependence concepts, Quasi-association, Gaussian random fields, Covariance estimation, Statistical hypothesis testing, Papermaking, Geometric probability, Stochastic geometry, Asymptotic theory, Sobolev spaces, Borel sets.

Frequently Asked Questions

What is the core focus of this research?

The work focuses on investigating the geometric properties of random fields, specifically the volume of excursion sets, and proving central limit theorems for these volumes.

What are the primary mathematical fields involved?

The research integrates stochastic geometry, probability theory (specifically random fields), and statistics.

What is the main objective regarding the central limit theorem (CLT)?

The objective is to prove uni- and multivariate CLTs for the volume of excursion sets for a general class of stationary random fields that exhibit specific dependence structures.

Which estimation methods are discussed?

The author discusses and compares several estimators for the covariance matrix of excursion sets, including local averaging, covariance-based estimators, and subwindow estimators.

How is the paper industry relevant to this work?

The paper industry serves as a motivating application, where the surface structure of fiber arrangements is modeled as a random field to evaluate paper quality during the forming process.

Which statistical tests are implemented?

The work develops and implements a statistical hypothesis test based on the CLT to determine if a given random field can be considered Gaussian.

How does (BL, θ)-dependence characterize the random fields?

It provides a quantitative measure of dependence, allowing for the analysis of fields that are not necessarily independent, by considering how the correlation between points decays with distance.

Why are excursion sets used to model paper surfaces?

Excursion sets provide a method to quantify differences between real-world papermaking images and simulated Gaussian random fields by analyzing the behavior of the field at specific threshold levels.