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

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Universit¨at UlmInstitut fu¨r StochastikStatistical Analysis of Spatial Point PatternsApplications to Economical, Biomedical and Ecological DataDissertationzur Erlangung des Doktorgrades Dr.rer.nat. der Fakult¨at fu¨rMathematik und Wirtschaftswissenschaften der Universita¨t Ulmvorgelegt vonStefanie Martina EckelausNeu–Ulm2008Amtierender Dekan: Prof. Dr. Frank Stehling1. Gutachter: Prof. Dr. Volker Schmidt2. Gutachter: Prof. Dr. Franz SchweiggertTag der Promotion: 07.11.2008Contents1 Introduction 71.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Software Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Basic Concepts of Random (Marked) Point Processes 132.1 Random Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.3 Palm Distributions for Random Point Processes . . . . . . . . . 182.1.4 Point Process Models . . . . . . . . . . . . . . . . . . . . . . . . 202.1.5 Characteristics for Random Point Processes . . . . . . . . . . . 272.2 Random Marked Point Processes . . . . . . . . . . . . . . . . . . . . . 322.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.2 Basic Properties . . . . . . . . . . . .
Publié le : mardi 1 janvier 2008
Lecture(s) : 35
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Source : VTS.UNI-ULM.DE/DOCS/2008/6602/VTS_6602_9020.PDF
Nombre de pages : 187
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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 in Baden–Wu¨rttemberg, 3–
dimensional, e.g. neuronal cells, or even spherical, e.g. geographical coordinates
of firm locations in the USA.
• The points can have marks, i.e. additionally information to each location, e.g.
tree rootsin mixed stands, or can be unmarked, e.g. tree rootsin pure stands. In
case of a marked point pattern, the mark space is not limited to a discrete space,
but the marks can also take values in IR, e.g. the change in the purchasing power
in Baden–Wu¨rttemberg.
Further applications of statistical point pattern analysis in various scientific research
fields, in which the Institute of Stochastics at Ulm University is involved, can be found
in [12], [13], [24], [122] and [144].
In the following we distinguish between a point pattern and a point process, where
“point pattern” describes a realization of a point process model or observed real data
and “point process” means the mathematical model.
This thesis presents some of the results that were obtained in the above mentioned
research projects and thereby tries to convince the reader of the universality of the
applied techniques of stochastic geometry w.r.t. to their applicability in various fields
ofscientific andindustrial research. Basicallythismeans thatmethodsthatareapplied
in order to solve some specific problem for one scientific field can often be used to solve
a problem in a different field of application after a slight modification. To summarize
things, the purposes of this thesis are:
• toshowthatpointprocesses aresuitableandpracticalstochasticmodelsforpoint
patterns,
• to illustrate the procedure of a descriptive statistical analysis of point patterns,
• to explain the fitting methods of specific point process models to point patterns,1.3 Outline 9
• to show that simulation–based tests are necessary to confirm the results, and
• to perform a statistical analysis for several examples from different fields, i.e. to
show that the applied methods are universal.
1.2 Software Aspects
The software developed in thecourse ofthisthesis isembedded in theGeoStoch library
which is a joint project between the Institute of Applied Information Processing and
theInstituteofStochasticsatUlmUniversity ([95]). ThisJAVA–based softwarelibrary
comprises methodsfromstochastic geometry, spatial statistics and imageanalysis. The
basicideabehindthissoftwareprojectistoofferacoreofgeneralmethodsthatareuse-
ful for different kinds of applications thereby ensuring a high degree of reusability. Ad-
ditionally, the library is constantly extended in the course of various research projects.
Further information about the GeoStoch library and some of the projects in which the
software is applied can be found on its internet domain http://www.geostoch.de.
In particular, efficient algorithmsfor the following statistical methods have been imple-
mented in the course of this thesis and included in the GeoStoch library:
• Estimation of point process characteristics such as intensity, bivariate pair corre-
lation function, bivariate K–function, J–function, mark correlation function and
(distance–dependent) Simpson indices. Most of the estimators are implemented
foranarbitrary2–dimensionalandarectangular3–dimensionalsamplingwindow.
• Fittingofpointprocessestoreplicatedpointpatterns, wherethefittingofMatern
cluster point processes uses the minimum contrast method and the fitting of
inhomogeneous(bivariate)saturationpointprocessesappliesthepseudolikelihood
method.
• Several tests on the null–hypothesis of complete spatial randomness (CSR), e.g.
Monte Carlo rank tests and the quadrat count method, as well as a Monte Carlo
rank test on the null–hypothesis of independent labelling.
This thesis therefore has strong interdisciplinary traits, i.e. it resulted from an interac-
tion of mathematics and computer science with economics, biomedicine and ecology.10 1 Introduction
1.3 Outline
We start by introducing some basic concepts of random (marked) point processes. In
Chapter 2foundationsofthisparticulardisciplineofstochasticgeometryareexplained.
dIn Section 2.1we define randompoint processes in IR and discuss basic properties, e.g.
stationarity, isotropy and ergodicity. Palm distributions are introduced which become
important for the definition of point process characteristics. Moreover, several use-
ful point process models are explained which can be used to model point patterns, e.g.
Poisson pointprocesses, Materncluster pointprocessesandGibbspointprocesses. The
section ends by defining point process characteristics which are useful tools for statis-
tical point pattern analysis. Examples of such characteristics are the pair correlation
function, Ripley’s K–function and the nearest neighbor distance distribution function.
For some of the introduced point process models theoretical formulae for these charac-
teristics are provided. Section 2.2 is arranged similarly to Section 2.1, but introducing
a markto each point. Thus, wehave additionallyinformationforeach location, e.g. for
the location of a tree the tree species is known. We discuss basic properties and Palm
distributions for random marked point processes. Then, we introduce marked point
process models. Furthermore, in themarked case, the pointprocess characteristics pro-
vide information about the distribution of the points taking the marks into account.
Some of them are an extension to the characteristics of unmarked point processes,
e.g. the bivariate pair correlation function, but there also exist characteristics which
mainly focus on the marks, i.e. the mark correlation function and the Simpson indices.
The mark correlation function deals with distance–dependent correlations between the
marks and the Simpson indices allow conclusions about the diversity. Furthermore,
those characteristics can be used to detect deviation from independent labelling.
The aim of Chapter 3 is to discuss toolsfor statistical point pattern analysis, which are
applied later in Chapters 4 to 6. We start by defining the estimators for the introduced
point process characteristics in Section 3.1. Some numerical examples round up the
discussion of estimators of such distance–dependent characteristics. In the second part
of this chapter (see Section 3.2), we describe methods to fit a specific point process
model to a given point pattern. Here we focus on two possibilities for fitting, i.e. the
minimum contrast method and the pseudolikelihood method. In Section 3.3 some effi-
cient algorithmsfor thesimulation ofpoint processes areintroduced. The simulation of
Poisson point processes is used in several parts of this thesis, e.g. for simulation–based
tests on complete spatial randomness (CSR), and is often the basis for the simulation
of more complex stochastic models. We distinguish between the simulation of (homo-
geneous) Poisson point processes in a bounded and in an unbounded sampling window
and the simulation of inhomogeneous Poisson point processes. Furthermore, we discuss
two Metropolis–Hastings algorithms for the simulation of Gibbs point processes. In
the first algorithm the number of points is fixed, i.e. the updates can only be shifts.

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