Andrew B. Lawson PhD thesis

574 Pages·2015·14.47 MB·English

THE STATISTICAL ANALYSIS OF POINT EVENTS
ASSOCIATED WITH A FIXED POINT
Andrew B. Lawson
A Thesis Submitted for the Degree of PhD
at the
University of St Andrews
1991
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THE STATISTICAL ANALYSIS OF POINT
EVENTS ASSOCIATED WITH A FIXED POINT.
Andrew B Lawson
Ph.D. degree
IMAGING SERVICES NORTH
Boston Spa, Wetherby
West Yorkshire, LS23 7BQ
www.bl.uk
TEXT CUT OFF IN THE
ORIGINAL
i ..
II
DECLARATION
a) I, Andrew B Lawson, hereby certify that this thesis has been composed by myself,
that it is a record of my own work, and that it has not been accepted in partial or
complete fulfilment of any other degree or professional qualification.
2 ~ 1\ \ ) q
Signed Date 6
/
b) I was admitted to the Faculty of Science of the University of St Andrews, under
Ordinance General No 12 on October 1986, and as a candidate for the degree of
Ph.D. on October 1987.
I \\ \' {
Signed Date "2- 7> 0
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c) I hereby certify that the candidate has fulfilled the conditions of the Resolution and
Regulations appropriate to the Degree of Ph.D.
Signature of Supervisor
III
COPYRIGHT DECLARATION
A UNRESTRICfED
In submitting this thesis to the University of St Andrews I understand that I am
giving permission for it to be made available for use in accordance with the regulations of
the University Library for the time being in force, subject to any copyright vested in the
work not being affected thereby. I also understand that the title and abstract will be
published, and that a copy of the work may be made and supplied to any bona fide library
or research worker.
IV
ABSTRACT
This work concerns the analysis of point events which are distributed on a planar
region and are thought to be related to a fixed point. Data examples are considered from
Epidemiology, where morbidity events are thought to be related to a pollution source, and
Ecology and Geology where events associated with a central point are to be modelled.
We have developed a variety of Heterogeneous Poisson Process (HEPP) models
for the above examples. In particular, I have developed interaction and 8-dependence
models for angular-linear correlation, with their ML estimation and associated score/W aId
tests. In the Epidemiological case we have developed case-control models and tests.
The possibility of second-order effects being important has also led to the
development of Bayesian Spatial Prior (BSP) models.
In addition, we have developed a new deviance residual for HEPP models and
explored the use of GLIM for modelling purposes.
A variety of results were found in data analysis. In some cases HEPP models
provide adequate descriptions of the process. In others, BSP models yield better fits. In
general, the discrete case admits a simple spatial Poisson model for counts and does not
require BSP model extensions.
v
ACKNOWLEDGEMENTS
First, I should like to thank my main supervisor, Professor R M Cormack for
constant support and encouragement during the execution of this work. A number of lights
have been turned on during our many discussions and I am deeply indebted to him for such
illumination. I should also like to thank Dr P E Jupp who has acted as supervisor during
Professor Cormack's absence. As my research lies on the border between Spatial and
Directional data analysis, the combination of supervisors has provided a balance between
these areas.
Second, I should like to thank Jeremy Warnes for useful discussion concerning
Empirical Bayes methods and Kriging, and Dr T Rolf Turner (University of New
Brunswick) for use of his DELDIR program.
Third, I should like to thank Dr P Mason of the Institute of Terrestial Ecology,
Bush Estate for use of the Hebeloma Data Set, and General Register Office, Edinburgh for
Mortality and population data for Buckhaven -Methil and Bonnybridge. I should also like
to thank Dr Owen Lloyd and Dr Fiona Williams of the Environmental Epidemiology Unit,
University of Dundee, for access to the Armadale data set.
Fourth, I would like to acknowledge the excellent typographical work of Shiela
Wilson and Valerie Cobb, who have both provided speedy and accurate copy.
Finally, I should like to thank Pat for all her long-suffering patience and
encouragement during the ups and downs.
VI
TABLE OF CONTENTS
GLOSSARY
LIST OF FIGURES
LIST OF TABLES
LIST OF APPENDICES
INTRODUCTION
1 . Exploratoty Data Analysis
1.1 Kernel Density Estimation
1.2 Special methods
1.2.1 Kernel Interpolation
1.2.2 Covariate Extraction
1.3 Preliminary Testing of Mapped Patterns
1.3.1 Radial Trend
1.3.2 Angular anisotropy
1.3.3 Radial-angular interaction
1.3.4 Example
2. Point Process Modellin" (Continuous)
2.1 HEPP model definition
2.2 Fixed-point HEPP models
2.2.1 The polar model and sampling window
2.2.2 Asymptotic Methods in Likelihood Analysis
2.2.3 Types of Intensity Model
2.2.3.1 The Radial component
2.2.3.2 The Angular component
2.2.3.3 Radial-Angular Interaction
2.2.4 Evaluation of the Normalising Constant (A(A»
2.3 Likelihood Methods for HEPP models
2.3.1 Single factor intensity models
2.3.2 Composite intensity models
:'
VII
2.3.2.1 ML estimation
2.3.2.2 Hypothesis tests
3. Continuous Model: Extensions
3.1 Observed Heterogeneity
3.1.1 Case - Control Score Tests
3.1.1.1 Case a) Radial Trend
3.1.1.2 Case b) Angular Concentration
3.1.1.3 Case c) Peaked-Radial Effect
3.1.1.4 Case d) Interaction Effect
3.2 Unobserved Heterogeneity
3.2.1 Harmonic Intensity Terms
3.2.2 Spatial Prior Structure for Intensities
3.2.2.1 Cox Process Model
3.2.2.2 A Bayesian Spatial Prior (BSP) Model
4. Continuous Model: Goodness-of-Fit of Residual Analysis
4.1 Global Goodness-of-Fit
4.2 Residual Analysis
4.2.1 Binned Residuals
4.2.2 Individual Residuals
4.2.2.1 Deviance Residuals
4.2.2.2 Examples
4.2.2.3 Autocorrelation and Deviance Residuals
4.2.2.4 BSP Individual Residuals
5. Discrete Model: Introduction
5.1 The Human Morbidity Pattern
5.1.1 Functional forms for E ~
5.1.2 Likelihood Methods
5.1.2.1 Maximum Likelihood Estimation
5.1.2.1.1 Exponential Trend
VIII
5.1.2.1.2 Angular Concentration
5.1.2.1.3 Radial-Angular Interaction
5.1.2.2 Hypothesis Testing
5.1.2.2.1 Exponential Trend
5.1.2.2.2 Angular Concentration
5.1.2.2.3 Radial-Angular Interaction
5.1.2.3 General Testing with Observed Heterogeneity
5.1.2.4 Unobserved Heterogeneity
5.1.2.4.1 Harmonic Hazard Tenus
5.1.2.4.2 Spatial prior structure for hazards
5.1.3 Discrete Model: Goodness-of-Fit and Residual Analysis
5.1.3.1 Global Goodness-of-Fit (OOF)
5.1.3.2 Residual Analysis
6. Variants of the Continuous/Discrete Models and Hazards
6.1 Model Variants
6.2 Hazard Variants
7. HEPP Models on OLIM
7.1 Normalising Constant Models
7.2 Numerical Approximation
7.3 Probabilistic Approximation
7.4 Integral Approximation Accuracy
7.5 One Dimensional Models
7.5.1 The von Mises Distribution
7.5.1.1 Numerical Comparison
7.5.2 A Test for 'von Misesness'
7.5.3 The Fisher Distribution
7.5.3.1 Data Example
7.5.4 Spatial HEPP Model
8. Model Simulation and Test Statistic Behaviour