Statistical analysis of spatial point patterns [Elektronische Ressource] : applications to economical, biomedical and ecological data / vorgelegt von Stefanie Martina Eckel

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2008
• Nombre de lectures: 36
• Poids de l'ouvrage: 8 Mo

Extrait

Universit¨at Ulm
Institut fu¨r Stochastik
Statistical Analysis of Spatial Point Patterns
Applications to Economical, Biomedical and Ecological Data
Dissertation
zur Erlangung des Doktorgrades Dr.rer.nat. der Fakult¨at fu¨r
Mathematik und Wirtschaftswissenschaften der Universita¨t Ulm
vorgelegt von
Stefanie Martina Eckel
aus
Neu–Ulm
2008Amtierender Dekan: Prof. Dr. Frank Stehling
1. Gutachter: Prof. Dr. Volker Schmidt
2. Gutachter: Prof. Dr. Franz Schweiggert
Tag der Promotion: 07.11.2008Contents
1 Introduction 7
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Software Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Basic Concepts of Random (Marked) Point Processes 13
2.1 Random Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Palm Distributions for Random Point Processes . . . . . . . . . 18
2.1.4 Point Process Models . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.5 Characteristics for Random Point Processes . . . . . . . . . . . 27
2.2 Random Marked Point Processes . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Palm Distributions for Random Marked Point Processes . . . . 35
2.2.4 Marked Point Process Models . . . . . . . . . . . . . . . . . . . 38
2.2.5 Characteristics for Random Marked Point Processes . . . . . . . 38
34 Contents
3 Statistics for Random (Marked) Point Processes 44
3.1 Estimators for Random (Marked) Point Process Characteristics . . . . 44
3.1.1 Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Bivariate K–function and L–function . . . . . . . . . . . . . . . 45
3.1.3 Bivariate Pair Correlation Function . . . . . . . . . . . . . . . . 46
3.1.4 Bivariate Nearest Neighbor Distance Distribution Function . . . 46
3.1.5 Simpson Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.6 Mark Covariance Function and Mark Correlation Function . . . 47
3.1.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Fitting Methods for Point Process Models . . . . . . . . . . . . . . . . 50
3.2.1 Minimum Contrast Method . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Pseudolikelihood Method . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Simulation Algorithms for Random Point Processes . . . . . . . . . . . 53
3.3.1 Simulation of Poisson Point Processes . . . . . . . . . . . . . . . 53
3.3.2 Simulation of Gibbs Point Processes. . . . . . . . . . . . . . . . 56
3.4 Simulation–based Statistical Methods . . . . . . . . . . . . . . . . . . . 59
3.4.1 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Model Verification Tests . . . . . . . . . . . . . . . . . . . . . . 59
3.4.3 Computation of Acceptance Intervals . . . . . . . . . . . . . . . 65
4 Applications to Economical Data 66
4.1 Spatial Correlations of the Change of the Relative Purchasing Power in
Baden–Wu¨rttemberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.1 Data Description and Preprocessing . . . . . . . . . . . . . . . . 68
4.1.2 Analysis of Spatial Correlations in Baden–Wu¨rttemberg . . . . 69
4.1.3 Regional Analysis of Spatial Correlations . . . . . . . . . . . . . 72
4.1.4 Summary and Discussion of the Results . . . . . . . . . . . . . . 73
4.2 Spatial Correlations of S&P 500 Stock Returns . . . . . . . . . . . . . . 76Contents 5
4.2.1 Data Description and Preprocessing . . . . . . . . . . . . . . . . 76
4.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Estimated Spatial Correlations of Stock Returns . . . . . . . . . 83
4.2.4 Summary and Discussion of the Results. . . . . . . . . . . . . . 87
4.3 Reinsurance Risks – Tropical Cyclones in the North Atlantic . . . . . . 89
4.3.1 Stochastic Model for the Storm Tracks . . . . . . . . . . . . . . 89
4.3.2 Mark Correlation Function as a Tool for Model Validation . . . 92
4.3.3 Summary and Discussion of the Results. . . . . . . . . . . . . . 95
5 Applications to Biomedical Data 97
5.1 Spatial Distribution of Capillaries . . . . . . . . . . . . . . . . . . . . . 97
5.1.1 3–dimensional Fibre Processes . . . . . . . . . . . . . . . . . . . 98
5.1.2 Acquisition of Visual Fields . . . . . . . . . . . . . . . . . . . . 99
5.1.3 Descriptive Comparison of Normal Prostatic Tissue and Prostate
Cancer Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.4 Modelling the Geometry of the Capillaries . . . . . . . . . . . . 103
5.1.5 Summary and Discussion of the Results. . . . . . . . . . . . . . 109
5.2 Labelling of Mammary Carcinoma Cell Nuclei . . . . . . . . . . . . . . 111
5.2.1 Acquisition of Visual Fields . . . . . . . . . . . . . . . . . . . . 111
5.2.2 Distribution of Proliferating and Non–proliferating Cells . . . . 112
5.2.3 Summary and Discussion of the Results. . . . . . . . . . . . . . 118
5.3 Spatial Arrangement of Microglia and Astrocytes in the Mouse Hip-
pocampus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.1 Point Pattern Acquisition . . . . . . . . . . . . . . . . . . . . . 120
5.3.2 Spatial Interaction of Astrocytes and Microglia . . . . . . . . . 121
5.3.3 Summary and Discussion of the Results. . . . . . . . . . . . . . 1266 Contents
6 Applications to Ecological Data 128
6.1 Modelling Tree Roots in Pure Stands of Fagus sylvatica and Picea abies 128
6.1.1 Acquisition of Profile Walls . . . . . . . . . . . . . . . . . . . . 129
6.1.2 Vertical Homogenization of Root Data . . . . . . . . . . . . . . 130
6.1.3 Modelling by Inhomogeneous Matern Cluster Point Processes . 132
6.1.4 Summary and Discussion of the Results . . . . . . . . . . . . . . 135
6.2 Modelling Tree Roots in Mixed Stands of Fagus sylvatica and Picea abies 137
6.2.1 Acquisition of Profile Walls . . . . . . . . . . . . . . . . . . . . 139
6.2.2 Generalised Saturation Point Process . . . . . . . . . . . . . . . 141
6.2.3 Model Fitting using the Pseudolikelihood Method . . . . . . . . 146
6.2.4 Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2.5 Modelling by Inhomogeneous Bivariate Saturation Point Process 150
6.2.6 Summary and Discussion of the Results . . . . . . . . . . . . . . 152
7 Conclusions and Outlook 155
Bibliography 157
List of Figures 169
List of Tables 173
Nomenclature 175
Acknowledgement 179Chapter 1
Introduction
1.1 Motivation
The aim of this thesis is to investigate, implement and apply different techniques of
statistical point pattern analysis. In the last years, several projects at the Institute of
StochasticsatUlmUniversitydealtwithpointpatternanalysiscomingfromcompletely
different fields of research. The most important results which were obtained in these
projects are reported in the present thesis.
The first three projects are located in the field of economics. Spatial correlations of the
change of the relative purchasing power in Baden–Wu¨rttemberg are analyzed first. In
the second project we detect positive spatial correlations of monthly stock returns in
the S&P 500 index. A part of this joint project with the Institute of Finance at Ulm
University also consists of the application and comparison of two different methods for
analyzingspatialcorrelations. Thethirdapplicationshowsthevalidationofastochastic
storm track model by considering spatial correlations in the generated wind field.
In the next two projects which are performed jointly with the Institute of Pathol-
ogy at Ulm University the distribution of cell structures is investigated. We compare
the spatial behavior of blood capillaries from tumorous tissue and tumor–free tissue.
Therefore, we analyze point patterns consisting of the capillary centers on sections of
3–dimensional fibreprocesses. Furthermore, weanalyzethedistributionofproliferating
cell nuclei in comparison to cells which do not proliferate in tumor tissue. The next
project is a cooperation with the Department ofAnatomy and Neurobiology of Kyushu
University in Japan, where we investigate the spatial interaction of two neuronal cells
in the mouse hippocampus.
The third kind of projects is situated in ecology. They are performed jointly with the
Institute of Systematic Botany and Ecology at Ulm University, where we investigate
78 1 Introduction
and model the root distribution of two tree species in pure stands and in mixed stands,
respectively.
Although at first sight these projects do not seem to have much in common, it became
more and more obvious that the mathematical models, tools and techniques involved
are quite similar. The data sets that are analyzed represent point patterns in all cases:
• The point pattern is given on a macroscopic scale, e.g. locations of firms in the
USA, on amesoscopic scale, e.g. planar sections oftreeroots, or ona microscopic
scale, e.g. locations of cell nuclei in cancer cells.
• The locations can be 2–dimensional, e.g. townships