Markov Point Processes and Their Applications

182 Pages·2000·8.153 MB·English

by  M N M Van Lieshout

MARKOV POINT PROCESSES
AND
THEIR APPLICATIONS
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MARKOV POINT PROCESSES
AND
THEIR APPLICATIONS
M. N. M. van Lieshout
Centrum voor Wiskunde en Informatica
The Netherlands
Imperial College Press
Published by
Imperial College Press
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British Library Cataloguing-in-Publication Data
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MARKOV POINT PROCESSES AND THEIR APPLICATIONS
Copyright®2000 by Imperial College Press
All rights reserved . This book,or parts thereof may not be reproduced in any form or by any means,
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ISBN 1-86094-071-4
Thisbook is printed on acid-free paper.
Printed in Singapore by Uto-Print
In memory of
Nic. van Lieshout
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Contents
Chapter 1 Point Processes 1
1.1 Introduction . . . . . . . . . .. . . . . . . . .. .. . . . .. . . 1
1.2 Definitions and notation ......... ......... .... 4
1.3 Simple point processes .. .............. ....... 9
1.4 Finite point processes ................... .... 13
1.5 Poisson point processes . . . . . . . . . . . .. .. . . . .. . . 15
1.6 Finite point processes specified by a density . . . . . . . . . . . 23
1.7 Campbell and moment measures . .. .............. 29
1.8 Interior and exterior conditioning ....... .. ..... .. 35
1.8.1 A review of Palm theory ..... ....... ...... 35
1.8.2 A review of conditional intensities ...... ...... 39
Chapter 2 Markov Point Processes 43
2.1 Ripley-Kelly Markov point processes . ..... ... .. ... . 43
2.2 The Hammersley-Clifford theorem ... .. .. ....... .. 48
2.3 Markov marked point processes .. .......... .... 52
2.4 Nearest-neighbour Markov point processes ..... ..... .. 62
2.5 Connected component Markov point processes . . . . . .. . . 68
Chapter 3 Statistical Inference 75
3.1 Introduction . . . . . . . . . . . . . . . . .. . . . . . . . .. . . 75
3.2 The Metropolis-Hastings algorithm .. ........ ..... . 76
3.3 Conditional simulation ......... . .. ..... ...... 83
3.4 Spatial birth-and-death processes .... .. ....... .... 85
3.5 Exact simulation . .. .. ..... ....... .. ....... 89
vii
viii Contents
3.6 Auxiliary variables and the Gibbs sampler ............ 96
3.7 Maximum likelihood estimation .................. 100
3.8 Estimation based on the conditional intensity ...... .... 105
3.8.1 Takacs-Fiksel estimation ...... ....... .... 105
3.8.2 Maximum pseudo-likelihood estimation .. ... .... 108
3.9 Goodness of fit testing ................ ... .. .. 110
3.10 Discussion ..... ......... ......... ... .... 113
Chapter 4 Applications 115
4.1 Modelling spatial patterns ... .. ....... ... ...... 115
4.2 Pairwise interaction processes ... ... .... ....... .. 116
4.3 Area-interaction processes ........ .... ....... .. 124
4.4 Shot noise and quermass-interaction processes .... ...... 131
4.5 Morphologically smoothed area-interaction processes ...... 135
4.6 Hierarchical and transformed processes ..... ......... 138
4.7 Cluster processes .......................... 140
4.8 Case study ......... ...... .............. 143
4.8.1 Exploratory analysis .................... 143
4.8.2 Model fitting ........................ 146
4.9 Interpolation and extrapolation .................. 148
4.9.1 Model ............................ 149
4.9.2 Posterior sampling ..................... 151
4.9.3 Monotonicity properties and coupling from the past .. 154
Bibliography 157
Index 173
Chapter 1
Point Processes
1.1 Introduction
Forest data on the location of trees, a sample of cells under a microscope,
an object scene to be interpreted by computer vision experts, interacting
particles of interest to physicists and a sketch of line segments representing
geological faults all share a particular trait: they are presented in the form
of a map of geometric objects scattered over someregion. Such data by their
very nature are spatial, and any analysis has to take this into account. For
instance, when the region mapped is large, lack of spatial homogeneity may
well result in data that are dense in some regions, while sparse in others.
Moreover, interaction between the objects (generically called points in the
sequel) influences the appearance of the mapped pattern. In biological
applications for example, competition for food or space may cause repulsion
between the points; on the other hand, if the observed pattern can be
seen as the descendants of some unobserved predecessors, the map may
look aggregated due to a clustering of the offspring around their ancestor.
However, it is important to realise that apparently clustered patterns may
also arise from spatial inhomogeneity rather than interpoint interactions,
especially at larger scales.
As a motivating example, suppose we want to implement sustainable
farming techniques that apply fertilisers and pesticides only there where
it is needed, thus causing minimal damage to the environment [15; 215].
Since pesticides seep to the ground water through pores and cracks in the
soil, studying the stain patterns in horizontal soil slices treated with a dye
tracer provides useful environmental information [26; 51; 87; 206]. One
1
2 Point Processes
such pattern, collected by Hatano and Booltink [87] in reclaimed land in
the Netherlands and further studied in [87; 206; 207], is given in Figure 1.1.
Here, the soil stains are represented by the coordinates of their center of
gravity. The stain pattern has a clustered appearance : most points are
close to other points, and groups of stains are separated by open spaces
that are large in comparison to the typical distance between a point and
its nearest neighbour. A closer investigation also reveals repulsion at short
range as a result of the non-negligible size of the stains.
Fig. 1.1 Stain centres in methylene-blue coloured clay observed in a square horizontal
slice of side 20 cm sampled at 4 cm depth (Hatano and Booltink, 1992).
A more detailed analysis divides the stains in different types. Thus,
elongated stains may be labelled as `cracks', and roughly circular ones as
`pores'. Those stains that are neither cracks nor pores are called `vughs'.
Distinguishing between these types naturally leads to a trivariate point pat-
tern,the three components representing the cracks, vughs and pores respec-
tively (see Figure 1.2). This approach allows one to investigate each stain
type separately as well as to study the correlations between the component
patterns. For the data in Figure 1.2, except at short range, both the vugh