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Analysis of Spatial Point Patterns in
Microscopic and Macroscopic Biological Image
Data
Frank Fleischer1, Michael Beil2, Marian Kazda3, and Volker Schmidt4
1Department of Applied Information Processing and Department of Stochastics,
University of Ulm, D-89069 Ulm, Germany
frank.fleischer@mathematik.uni-ulm.de
2Department of Internal Medicine I, University Hospital Ulm, D-89070 Ulm,
Germany
3Department of Systematic Botany and Ecology, University of Ulm, D-89069
Ulm, Germany
4Department of Stochastics, University of Ulm, D-89069 Ulm, Germany
1 Introduction
The analysis of spatial point patterns by means of estimated point process
characteristics like for example Ripley’s K-function ([37]), the L-function or
Baddeley’s J-function ([7], [24]) has proven to be a very useful tool in Stochas-
tic Geometry during the last years (see e.g. [10], [38], [43], and [44]). They
offer the possibility to get not only qualitative knowledge about the observed
spatial structures of such point patterns, but to quantify them for specific re-
gions of point-pair distances. Other advantages of these methods compared to
alternative techniques of spatial analysis like e.g. Voronoi tessellations ([26],
[31]) or minimum spanning trees ([13], [14]) is their independence of underly-
ing point process intensities, in other words of the average number of points
per unit square. Therefore they are especially useful for cases of data with sig-
nificant differences with respect to intensities. We will discuss two examples in
the fields of cell biology and ecology were these methods can be applied. They
have been chosen, because they demonstrate the wide range of applications
for the described techniques and because both examples have specific interest-
ing properties. While the point patterns regarded in the first application are
three dimensional, the second application reveals planar point patterns having
a vertically inhomogeneous structure. For a more extensive description of the
studied cases, the reader is referred to [4], [5], and [16], [41], [42], respectively.
2 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
1.1 Analysis of Centromeric Heterochromatin Structures
The first example deals with the structure of the cell nucleus, notably the dis-
tribution of centromeres, during differentiation (maturation) of myeloid cells.
These are the precursors of white blood cells and are normally found in the
bone marrow. During differentiation, myeloid cells acquire specialized func-
tions by activating a strictly defined set of genes and producing new proteins.
In addition, other genes whose function are not needed in maturated cells
become silenced. The mechanisms regulating the activity of genes during dif-
ferentiation remain to be defined in detail.
The architecture of the cell nucleus during the interphase, i.e. the time between
consecutive cell divisions, is determined by the packaging of the DNA molecule
at various levels of organisation (chromatin structure). The open state of DNA
is referred to as euchromatin, whereas heterochromatin is the condensed form
of DNA. The production of gene transcripts (mRNA) requires the molecules
of the transcriptional machinery to access the DNA molecule. Thus, the regu-
lation of DNA packaging represents an important process for controlling gene
activity, i.e. the synthesis of mRNA ([8]). In general, transcriptional activity
appears to be impeded by a restrictive (compacted) packaging of DNA ([35]).
This way, the hetrochromatin compartment is an important regulator of gene
transcription and, hence, influences the biological function of cells.
There has been a great interest in investigating the processes governing the or-
ganisation of chromatin. Whereas the regulation at the level of nucleosomes,
e.g. through biochemical modifications of histones, is now becoming eluci-
dated, the long-range remodelling of large portions of the DNA molecule
(higher-order chromatin structure) proceeds by yet unknown rules. The con-
densed form of DNA, i.e. heterochromatin, is generally associated with the
telomeres and centromeres of chromosomes ([36]). These regions can also in-
duce transcriptional repression of nearby genes ([35]). Consequently, the dis-
tribution of these regions should be changed during cellular differentiation
that is associated with a marked alteration of the profile of activated genes.
In fact, previous studies described a progressive clustering of interphase cen-
tromeres during cellular differentiation of lymphocytes and Purkinje neurons
([1], [32]). However, the overall structural characteristics of the centromeric
heterochromatin compartment, e.g. with respect to spatial randomness, re-
main to be determined.
Leukaemias are malignant neoplasias of white blood cells. They represent an
interesting biological model to study cellular differentiation since they can
develop at every level of myeloid differentiation. A particular type, acute
promyelocytic leukaemia (APL), is characterized by a unique chromosomal
translocation fusing the PML gene on chromosome 15 with the gene of the
retinoic acid receptor alpha on chromosome 17 ([47]). Due to the function
Analysis of Spatial Point Patterns 3
of the resulting fusion protein, cellular differentiation is arrested at the level
of promyelocytes. However, pharmacological doses of all-trans retinoic acid
(ATRA) can induce further differentiation of these cells along the neutrophil
pathway ([15]).
In a recent study, the three-dimensional (3D) structure of the centromeric
heterochromatin was studied in the NB4 cell line which was established from
a patient with APL ([23]). The 3D positions of centromeres served as a sur-
rogate marker for the structure of centromeric heterochromatin. Due to the
diffraction-limited resolution of optical microscopy, the notion chromocenter
was used to define clusters of centromeres with a distance below the limit of
optical resolution. During differentiation of NB4 cells as induced by ATRA,
a progressive clustering of centromeres was implied from a decreased number
of detectable chromocenters. The 3D distribution of chromocenters was eval-
uated by analysing the minimal spanning tree (MST) constructed from the
3D coordinates of the chromocenters. The results obtained by this method
suggested that a large-scale remodelling of higher order chromatin structure
occurs during differentiation of NB4 cells.
1.2 Planar Sections of Root Systems in Tree Stands
In our second example we examine the spatial distribution of tree root pat-
terns in pure stands of Norway Spruce (Picea Abies) and European Beech
(Fagus sylvatica (L.) Karst.). While there exists knowledge about the vertical
root distribution which can usually be described by one-dimensional depth
functions ([20], [34]), early studies assumed that rooting zones are completely
and almost homogeneously exploited by roots ([22]). In recent studies it is
however shown that fine roots concentrate in distinct soil patches and that
they proliferate into zones of nutrient enrichment and water availability ([6],
[39]). Hence a horizontal heterogeneity of the spatial distribution of fine roots
might be expected. Trench soil profile walls can be used for the assessment of
two-dimensional root distributions, regarding the roots on the wall as points
of different diameter. A new method provided (x,y)-coordinates of each root
greater than 2 mm ([41]). Using this method, small roots with a diameter
between 2 mm and 5 mm were examined in 19 pits on altogether 72 m2of
soil profiles on monospecific stands of European Beech and of Norway Spruce.
1.3 Detection of Structural Differences
So to summarize, our aims in both studied applications were quite similar from
a mathematical point of view. First, as it is generally the case in such a spatial
analysis, to compare our data sets with the null model of the homogeneous
Poisson process, otherwise described as ’complete spatial randomness’ (CSR)
([10]). If such a hypothesis can be rejected, a goal is to detect structural
differences between the two regarded groups in the data, namely nuclei from
4 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
non-differentiated and differentiated NB4 cells in the first application and
beech roots and spruce roots in the second one. For the cell nuclei example
we were especially interested in an explanation for the decreasing number of
chromocenters during differentation, while for the tree roots a main aim was
to quantify the degree of root aggregation, i.e. the degree of intensity of the
exploitation of the soil resources by each tree species. Another question of
interest is to provide a suitable and not too complicated mathematical model
for underlying generating point processes. Finally, of course it is a necessity
to obtain an interpretation of the results from the biological standpoint.
2 Image Data
As it has been mentioned in Section 1, there were two different data sets
considered, three dimensional point patterns in cell nuclei of a NB4 cell line
and two dimensional point patterns in profile walls of European Beech and
Norway Spruce.
2.1 NB4 Cell Nuclei
The procedures for cell culture of NB4 cells, specimen preparation, immunoflu-
orescence confocal microscopy and image analysis are described in detail in
[4]. In the following two paragraphs the applied techniques are summarized.
Sample Preparation and Image Acquisition
Differentiation of NB4 cells was induced by incubating cells with 5 mol/l
ATRA (Sigma, St.Louis, MO) for 4 days. Visualization of centromeres was
based on immunofluorescence staining of centromere-associated proteins with
CREST serum (Euroimmun Corp., Gross Groenau, Germany). Nuclear DNA
was stained with YoPro-3 (Molecular Probes). Two channel acquisition of
3D images was performed by confocal scanning laser microscopy (voxel size:
98 nm in lateral and 168 nm in axial direction).
Image Segmentation
Segmentation of chromocenters as stained by CREST serum was performed
in two steps. First, objects at each confocal section were segmented by edge
detection followed by a conglomerate cutting procedure. In a second step,
3D chromocenters were reconstructed by analyzing series of 2D profiles. The
center of gravity was used to define the 3D coordinates for each chromocen-
ter. The final analysis included 28 cell nuclei from untreated controls with 68
chromocenters on average and 27 cell nuclei from ATRA-differentiated NB4
cells with 57 chromocenters on average (see Figure 1).
Analysis of Spatial Point Patterns 5
Fig. 1. Projections of the three dimensional chromocenter location patterns of an
undifferentiated NB4 cell (left) and a differentiated NB4 cell (right) onto the xy-
plane
2.2 Profile Walls
For details of site description, pit excavation and root mapping and already
attained results, see [41]. Our investigations are based upon this article and
thus only a short summary of the most important facts is given.
Site Description
Data collection took place near Wilhelmsburg, Austria (4805′51′′ N, 1539′48′′E)
in adjoining pure stands of Fagus sylvatica and of planted Picea abies. One
experimental plot of about 0.5ha was selected within each stand. The sites
were similar in aspect (NNE), inclination (10%) and altitude (480 m). The
characteristics of the spruce and beech stands, e.g. the age (55 and 65 years),
the dominant tree height (27 mand 28 m) and the stand density (57.3 and
46.6 trees /ha), also were similar to each other. The soils with only thin or-
ganic layer (about 4 cm) can be classified as Stagnic Cambisols developed
from Flysch sediments. Annual rainfall in Wilhelmsburg averages 843 mm
with a mean summer precipitation from May to September of 433 mm. The
mean annual temperature is 8.4◦C, and the mean summer temperature is
15.7◦C.
Pit Excavation and Root Mapping
In every stand 10 soil pits with a size of 2 ×1mwere excavated. Thus up to 20
profile walls could be obtained in each stand. In most cases 13−19 trees were
within a radius of 10 maround the pit centre. The minimum distance from
the pit centre to the nearest tree ranged from 0.5mto 2.8m. On each wall
all coarse roots were identified and divided into living and dead. All living
small roots (2 −5mm) were marked with pins and digitally photographed.
These pictures were evaluated and a coordinate plane was drawn over each
profile wall W, so that every root corresponds to a point xnin the plane.
Thus, for each profile wall W, a point pattern {xn} ⊂ Wof root locations
6 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
was determined.
Data Description
Root mapping was performed on 20 profile walls of Fagus sylvatica and on
16 profile walls of Picea abies (see Figure 3). The profile walls Bwith area
νd(B) = 200 cm ×100 cm are regarded as sampling windows of stochastic
point processes in 2.
Fig. 2. Data collection
3 Statistical Methods and Results
Data analysis for all data groups was performed using the GeoStoch li-
brary system. GeoStoch is a Java-based open-library system developed by
the Department of Applied Information Processing and the Department
of Stochastics of the University of Ulm which can be used for stochastic-
geometric modelling and spatial statistical analysis of image data ([27],[28],
http://www.geostoch.de).
For both study cases it is important to notice that, considering estimated
point process characteristics, means for each group were regarded. This is due
to the fact that variability inside a single group (European Beech and Norway
Spruce or NB4 cell nuclei, respectively) was large compared to the differences
between the two groups for each studied case (tree roots and cell nuclei). For
functions, these means were taken in a pointwise sense.
Analysis of Spatial Point Patterns 7
Fig. 3. a) an original sample of roots for Picea abies b) the transformed sample
for Picea abies c) an original sample of roots for Fagus sylvatica d) the transformed
sample for Fagus sylvatica
3.1 3D Point Patterns of Chromocenters in NB4 Cells
The real sampling regions for the cell nuclei are not known, therefore as-
sumed sampling regions were constructed as follows: For all three coordinates
the smallest and largest values appearing in a sample were determined and de-
noted as xmin,xmax ,ymin,ymax ,zmin and zmax respectively. Then the 8 ver-
tices of the assumed sampling cuboid were given by all possible combinations
of the three coordinate pairs {xmin, xmax },{ymin, ymax }and {zmin, zmax }.
Although we performed statistical tests on stationarity and isotropy of the
regarded point fields which showed results in favor of such assumptions, we
would like to consider stationarity and isotropy as prior assumptions that are
not under investigation. This is due to the fact that the numbers of points
per sample do not seem to be large enough to provide reliable information
on both properties, especially with regard to the three dimensionality of the
data. Therefore formal tests can be only hints that such assumptions might
not be badly chosen.
Intensities and Volumes
It is already known from [4] that the average number of detected chromocen-
ters was significantly decreased during differentiation. Regarding the volumes
of the assumed sampling cuboids, the hypothesis of having the same volume
before and after differentiation could not be rejected (α= 0.05), observing
mean volumes of 429.507 µm3before differentiation and 470.929 µm3after-
wards. Hence the intensity estimate b
λof detected chromocenters of NB4 cells,
that means the average number per unit volume (see Formula (3) in Ap-
pendix A), is significantly decreased as well.
8 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
Averaged Estimated Pair Correlation Function
Figure 4 shows estimations bg(r) of the pair correlation function g(r) for
c= 0.06, where cdetermines the bandwith of the Epanechnikov kernel used
in the definition of bg(r); see Formula (11) in Appendix A. It is clearly visible
that the frequency of point-pair distances for a distance between 350 nm and
800 nm is higher before than after differentiation of NB4 cells. Also a hard-
core distance r0of about 350 nm can be recognized which is determined by
the diffraction-limited spatial resolution of the microscopic imaging method.
This means that all point pairs have a distance bigger than r0. Note that the
smaller hardcore values for larger values of care due to the increased band-
withs of the Epanechnikov kernel in these cases. The results for the estimated
pair correlation functions do not depend on the fact that the two groups have
different numbers of detectable chromocenters.
Averaged Estimated L-Function
We consider the estimator b
L(r) for L(r) given in Formula (17) of Appendix A.
Figure 5 shows the estimated averaged L-function b
L(r) while Figure 6 shows
b
L(r)−rwhere the theoretical value rfor Poisson point processes has been
subtracted; see Formula (16) in Appendix A.
A similar scenario as for the pair correlation function is observed. Especially
for small point-pair distances between 350 nm and 500 nm, there is a higher
r in nm
0.2
0.4
0.6
0.8
1
1.2
1.4
500 1000 1500 2000
Fig. 4. Averaged estimated pair correlation functions bg(r) using Epanechnikov ker-
nel and parameter c= 0.06. The group of undifferentiated NB4 cells is denoted by
+, while the group of differentiated NB4 cells is denoted by o
Analysis of Spatial Point Patterns 9
r in nm
0
500
1000
1500
2000
500 1000 1500 2000
Fig. 5. Averaged estimated functions
b
L(r), where + denotes the group of undiffer-
entiated NB4 cells and o denotes the group of NB4 differentiated cells
percentage of point pairs before than after ATRA-induced differentiation of
NB4 cells. While for the group of undifferentiated cells the graph b
L(r)−r
has a mostly positive slope in this region, which is an indicator for attraction,
the group of differentiated NB4 cells shows a negative slope which is a sign
for rejection. The same hardcore distance r0≈350 nm is visible. Again the
results do not depend on the different numbers of detectable choromocenters.
Performing a Wilcoxon-Mann-Whitney test for the two group samples for
fixed radii shows a significant difference in the values of L-functions before
and after differentiation for all radii between 350 nm and 1300 nm, especially
for the region between 500 nm and 700 nm (α= 0.05).
Averaged Estimated Nearest-Neighbor Distance Distribution and
Averaged Estimated J-Function
The structural conclusions obtained from the results for the estimated point
field characteristics d
DH(r) and b
J(r) were very similar compared to the av-
eraged estimated pair correlation function bg(r) and the averaged estimated
L-Function b
L(r), where d
DH(r) and b
J(r) are given by Formulae (18) and (22)
in Appendix A. Therefore the averaged estimates d
DH(r) and b
J(r) are not
displayed here. Again a hardcore distance of 350 nm can be recognized and
the two different groups show strong differences in their behavior especially
for a range between 350 nm and about 800 nm.
10 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
r in nm
–300
–200
–100
0
100
200
500 1000 1500 2000
Fig. 6. Averaged estimated functions
b
L(r)−r, where + denotes the group of NB4
undifferentiated cells and o denotes the group of NB4 differentiated cells
3.2 2D Point Patterns in Planar Sections of Root Systems
From [41] it was already known that the depth densities of the roots of Nor-
wegian Spruce and European Beech can be approximated by exponential and
gamma distributions respectively. The data has been homogenized with re-
spect to the vertical axis in oder to allow the assumptions of stationarity
and isotropy for models of generating point processes. A suitable homoge-
nization can be based on the well-known fact that each random variable Y
with a continuous distribution function FYcan be transformed to a uniformly
distributed random variable Uon the interval [0,1] by
U=FY(Y).(1)
Therefore, by denoting the original depths, the total depth of the sampling
window and the transformed depths as horig ,htot and htran respectively, we
get
htran =F∗(horig )
F∗(htot)htot ,(2)
where F∗(x) symbolizes the suitable distribution function, i.e. the exponential
distribution function in the case of Norway Spruce and the gamma distribu-
tion function in the case of European Beech. The total depth was given as
htot = 100 cm. For each sampling window parameters of the distribution func-
tions F∗(x) are estimated individually using maximum-likelihood estimators.
Analysis of Spatial Point Patterns 11
Notice that in the following, first only vertically homogenized data is regarded
(see Figure 3), that means considering the vertical coordinate a uniform dis-
tribution on [0, htot] can be assumed. Later on, an inverse transformation is
applied to obtain inference for the original data.
Intensities
The average number of points for the samples of Picea abies is significantly
higher than for the samples of Fagus sylvatica (α= 0.05). Since sampling win-
dows have the same sizes, the same result is obtained regarding the estimated
intensities per cm2(b
λspruce = 0.00403 vs. b
λbeech = 0.00262). Notice that the
following results for the considered point process characteristics are indepen-
dent of this fact since the functions are scaled with respect to the intensities.
Isotropy and Complete Spatial Randomness
Isotropy was tested by determining the directional distribution of the angles
of point pairs to the axes in a quadratic sampling window and testing them
for uniform distribution. The hypothesis of isotropy could not be rejected
(α= 0.05), hence in the following isotropy is assumed. The quadrat count
method ([44]) was used to test on complete spatial randomness. Here, using
a 4 ×4 grid, the hypothesis that the given point patterns are extracts of re-
alizations of homogeneous Poisson processes was rejected (α= 0.05).
Averaged Estimated J-Function
In Figure 7 the averaged estimated J-functions b
J(r) for both groups are dis-
played. There is a clear indication for attraction between point pairs of a
distance less than 12 cm, since both functions are below 1 in this region and
have a negative slope. A second observation is that the graph of Picea abies
lies beneath the graph of Fagus sylvatica, which means that the point pairs of
spruces are more attracted to each other than the point pairs of beeches for
such distances. Notice that for radii larger than 20 cm the estimator becomes
numerically unstable, and therefore should not be taken into further consid-
eration. Also one should keep in mind that the J-function is a cumulative
quantity.
Averaged Estimated L-Function
In Figure 8 the graphs for the averaged estimated values of b
L(r)−rare
shown. Since in the Poisson case L(r)≡ra positive slope means that there
is an attraction, while a negative slope indicates repulsion. Again there are
signs of attraction for small point-pair distances, less than 9.5cm and less
than 13.5cm respectively, and the attraction seems to be stronger for Picea
abies compared to Fagus sylvatica since the slope of b
L(r)−ris bigger. The
negative values for very small distances might indicate a slight hardcore effect
between the points.
Averaged Estimated Pair Correlation Function
12 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
0
1
2
3
4
5
510 15 20 25 30
Radius r
Fig. 7. Averaged estimated J-functions for Picea abies (·) and Fagus sylvatica (♦)
0
0.5
1.0
1.5
2.0
2.5
10 20 30 40 50
Radius r
Fig. 8. Averaged estimated functions
b
L(r)−rfor Picea abies (·) and Fagus sylvatica
(♦)
Further indication for an attraction between point pairs of distances less than
14 cm is provided by the averaged estimated pair correlation functions bg(r)
displayed in Figure 9. Again a stronger attraction is observed for the spruces
since the function runs above the function for beeches for rless than 9 cm.
Both functions are above 1 for r < 14 cm.
3.3 Model Fitting for the Root Data
Homogeneous Matern-Cluster Model
Regarding the results of the estimated point process characteristics described
before and because of its simplicity, Matern-cluster processes are chosen as
a model for the underlying point processes; see Appendix B for a definition.
Using once more the natural intensity estimator given in Formula (3) of Ap-
pendix A, the parameter λmc was estimated as d
λmc
spruce = 0.00403 and
Analysis of Spatial Point Patterns 13
1
2
3
4
5
6
7
10 20 30 40 50
Radius r
Fig. 9. Averaged estimated pair correlation functions bg(r) for Picea abies (·) and
Fagus sylvatica (♦), estimated using Epanechnikov kernel with parameter c= 0.15
d
λmc
beech = 0.00262, respectively. Concerning the regarded point-pair dis-
tances a range from 0 cm to rmax = 50 cm has been chosen, where rmax
equals half the minimum of the given depth and width. The parameters R
and λeare estimated by minimum-contrast estimators, which are computed
by numerical minimization of the integral
Zrmax
0
(bg(r)−gtheo(r))2dr, (3)
where bg(r) is the averaged estimated pair correlation function given in For-
mula (11) of Appendix A for the Epanechnikov kernel with c= 0.15, and
gtheo(r) is the theoretical value for the pair correlation function of the Matern-
cluster process with parameters λe,Rand λmc . Since the parameter λmc has
already been estimated, the minimization of the integral in (3) yields an es-
timation for the pair of parameters Rand λe. The obtained estimates are
b
Rspruce = 4.9cm and c
λe
spruce = 0.00690 for spruce roots, while for beech
roots b
Rbeech = 7.4cm and c
λe
beech = 0.00603 are obtained.
Model Conclusions
The given point patterns are modeled as extracts of realizations of station-
ary Matern-cluster processes with estimated intensities b
λspruce
mc = 0.00403 and
b
λbeech
mc = 0.00262, with cluster radii b
Rspruce = 4.9cm and b
Rbeech = 7.4cm,
and with parent-process intensities c
λe
spruce = 0.00690 and c
λe
beech = 0.00603.
In order to get an idea for the degree of clustering, the quantity
b
λt=d
λmc
c
λeπR2(4)
14 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
was evaluated. For Picea abies one gets b
λt
spruce = 0.00774, while for Fagus
sylvatica b
λt
beech = 0.00253 is obtained. From the estimated parameters one
can conclude that there is stronger clustering within a smaller cluster radius
for spruce roots, while for the beech roots the clustering is weaker, but the
cluster radius is slightly larger.
Inhomogeneous Matern-Cluster Model
For the original data, which shows a vertical distribution property, the Matern-
cluster model fitted for the homogeneous case has to be retransformed, where
the inverse transformation
horig = (F∗)−1(F∗(htot)
htot
htran) (5)
of the depth is considered, with (F∗)−1(y) representing the generalized inverse
function of F∗(x). Then, in the retransformed model, the parent process is
given by an inhomogeneous Poisson process with intensity function
λe(x, y) = λe(y) = λe
f∗(y)
F∗(htot)htot ,(6)
where xand yrepresent the horizontal and vertical coordinate, f∗(x) is the
density function of the suitable distribution function F∗(x) (exponential dis-
tribution for spruces and gamma distribution for beeches), λeis the intensity
of the parent process of the homogeneous model and htot represents the to-
tal depth of the sampling window. The cluster regions are no longer circles,
but the images of these circles under the mapping given in (5). They can be
written as
{(x, y) : (x−xp)2+ (F∗(y)−F∗(yp))2(htot
F∗(htot))2≤R2},(7)
where the corresponding parent point is denoted as (xp, yp). Since the mean
total number of points in the given window as well as the mean total number
of points in a cluster stay the same compared to the homogeneous model, the
intensity function for the inhomogeneous matern cluster point process is given
as
λmc(x, y) = λmc(y) = λmc
f∗(y)
F∗(htot)htot ,(8)
where λmc is the corresponding intensity of the homogeneous model. Fig-
ure 11 shows a realization of the inhomogeneous Matern-cluster model, which
corresponds to the homogeneous realization displayed in Figure 10 using an
exponential depth distribution. Note that only those simulated data shown in
the upper part of Figure 11 should be used for interpretation purposes. In the
lower part of Figure 11 the influence of transformation and retransformation
of data clearly dominates the original spatial structure of those (sparse) root
data with larger vertical depths.
Analysis of Spatial Point Patterns 15
Fig. 10. Realization of the homogeneous Matern-cluster model fitted in the case of
Picea abies
Fig. 11. Realization of the retransformed inhomogeneous Matern-cluster model
4 Discussion
In both studied cases clear differences between two biologically distinct groups
can be recognized using estimated point process characteristics. Apart from re-
jecting in all cases the null hypotheses of having homogeneous Poisson process
as generating processes for the observed point patterns, it has been possible
in the first example to detect a distance region where the number of chromo-
centers differ strongly between the non-differentiated and the differentiated
state of NB4 cell nuclei. In the second example differences in the clustering
behavior of the fine roots for European Beech compared to Norway Spruce
have become visible. Apart from that a simple point process model has been
fitted to the tree root data.
4.1 NB4 Cell Nuclei
The centromeric regions of chromosomes represent an important part of the
heterochromatin compartment in interphase nuclei. A previous study was fo-
cused on the quantitative description of three dimensional distribution pat-
terns of centromeric hetrochromatin in NB4 cells using features of the MST
([4]). From a mathematical point of view, this approach has several disad-
vantages. Quantities like the MST edge lengths or their variance are strongly
dependent on the mean number of points per volume unit. Apart from that,
16 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
the methods applied in the present study allow to get inference about dif-
ferent specific regions of point pair distances. Thus, this approach provides
the opportunity for a more detailed analysis of three dimensional centromere
distributions.
Notice that, although the observed point patterns are finite and bounded, it
can be assumed that they are realizations of stationary point processes re-
stricted to a bounded sampling region. This notional step is supported by the
fact that tests for isotropy and stationarity do not show any significant rejec-
tions and that the volumes of the assumed sampling regions before and after
differentiation are of comparable sizes. The method of assuming unbounded
stationary point processes as sources for observed realizations restricted to a
bounded sampling region is a quite common practice since very often data is
given in finite sampling regions and behave in a rather different non-stationary
way outside of these regions ([11], [40]).
Due to the assumption that the observed point samples in the bounded sam-
pling regions are extracts of unbounded realizations of stationary point pro-
cesses it is necessary, although having only bounded sampling regions, to
perform edge-corrections in order to insure compatibility with the applied
methods. We want to emphasize that this procedure has its statistical justi-
fication in the facts that tests for isotropy and stationarity do not show any
significant rejections and that the sampling regions have similar volumes.
Other types of estimators apart from spatial Horvitz-Thompson style estima-
tors, e.g. of Kaplan-Meier type ([3]) and other techniques of edge corrections
might also be applicable.
Clustering of chromosomal regions in interphase cell nuclei is supposed to be
an important mechanism regulating the functional architecture of chromatin.
In our previous study, we observed a progressive clustering of centromeric
heterochromatin after differentiation of NB4 cells with ATRA ([4]). These
clusters (chromocenters) represent groups of centromeres with a distance be-
low the limit of spatial resolution of optical microscopy. In the present study,
we have analyzed the distance of these chromocenters and found a higher fre-
quency of distances between 350 nm and 800 nm for undifferentiated cells
in comparison to ATRA-differentiated NB4 cells (Figures 4 and 6). These
new data imply the existence of heterochromatin regions with a range of
350 nm to 800 nm containing functionally related centromeric zones. The
centromeres in these regions cluster during ATRA-induced differentiation of
NB4 cells as demonstrated by the decreased number of detectable chromocen-
ters, i.e. groups of centromeres within a sphere with a diameter of less than
350 nm. The existence of heterochromatin regions containing centromeres of
specific chromosomes would imply that the restructuring of these chromo-
some territories has to proceed in a coordinated nonrandom way during the
differentiation-induced ”collapse” of these heterochromatin zones. This model
is in accordance with a topological model for gene regulation based on the
structural remodeling of chromosome territories during modulation of tran-
scription ([8], [33]).
Analysis of Spatial Point Patterns 17
Another important result of the present study is the finding that the 3D distri-
bution of chromocenters is not completely random in undifferentiated as well
as in ATRA-differentiated NB4 cells. These findings, thus, rejects a previous
hypothesis which was based on the comparison of centromere distributions
in NB4 cells with simulated completely random patterns using MST features
([4]). The result of the present study is in accordance with other studies, which
suggested that interphase centromeres are not arranged in a completely ran-
dom way ([17], [21]). Importantly, investigations of interphase chromosome
positions indicate that a strictly maintained structure of chromatin appears
to be necessary for a normal function of cells even in tumours ([9]).
4.2 Planar Sections of Root Systems
The point process characteristics using the transformed data described the
two dimensional distribution of small roots in pure stands of Picea abies and
Fagus sylvatica. The results for such homogenised data using the averaged
estimated pair correlation function (Fig. 9) show that attraction can be ob-
served for point pairs with distances less than approximately 14 cm. It means
that roots of both species tend to cluster in areas up to this diameter. As
roots react to nutrient enriched soil patches by enhanced growth and greater
biomass in these areas ([12], [30]), this attraction of roots within this diam-
eter could also be a direct link to a local occurrence of soil resources. On
the other hand, for very small distances of less than 0.5cm there is a hard-
core property in the homogeneous case, which can possibly be explained by
the thickness of the regarded roots. Also, as the small roots are associated
to the uptake-oriented fine roots, concentration of the small roots in clusters
of smaller diameters (i.e. less than 0.5cm) is not reasonable. In this context
it is important to notice that these structural differences are independent of
the observed significant difference in the average number of detected points
(roots) for the samples of Picea abies and Fagus sylvatica.
The homogenized point patterns were modeled as Matern-cluster processes
with estimated parameters described in Section 3.2. The Matern-cluster model
chosen has some serious advantages. First the model is of a certain simplicity
and theoretical values for point process characteristics are known. Even more
important is that the sample data is fitted well by this model. The estimated
point process characteristics using the Matern-cluster processes further differ-
entiated between the species. The results show for spruces a stronger cluster-
ing in a smaller range of attraction ( b
Rspruce = 4.9cm), while the clustering is
weaker for beeches, but the range of attraction ( b
Rbeech = 7.4cm) seems to be
slightly larger. This finding is in accordance with another investigation ([42])
calculating influence areas for each root. Their results indicated that the root
system of spruce requires more roots to achieve a similar degree of space acqui-
sition and thus beech exploits patchily distributed soil resources at lower root
numbers. In summary there is a combination of two effects, the depth distri-
bution already described in [41] and the cluster effects analyzed in the present
18 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
paper. Structural differences between spruce and beech indicated in [42] have
been mathematically described. Stronger clustering in the case of spruce than
in the case of beech can be seen also regarding the characteristics mentioned
above as well as by a comparison of the estimated parameters for the ho-
mogeneous model b
λt
spruce = 0.00774 vs. b
λt
beech = 0.00253, keeping in mind
that the estimated parent intensities are almost equal (c
λp
spruce = 0.00690 vs.
c
λp
beech = 0.00603). The previous GIS-based investigation of root distribution
([42]) was not able to quantify the differences of clustering between the two
species so precisely as the applied modelling by point processes.
Finally a non-homogeneous Matern-cluster model has been constructed by a
retransformation of the homogeneous model, thereby reflecting the observed
depth distribution of the tree roots. The visualisation of the retransformed
data suggests a depth-dependent size and shape of root clusters. Close to the
soil surface, roots form clusters along the horizontal axis. This shape agrees
also with the horizontally distributed root points in the original samples (c.f.
Fig. 3). Horizontally distributed roots as well as the shape of generated clus-
ters may reflect the attractive soil patches in the nutrient-rich topsoil layers.
Deeper, the real size of clusters is larger and more circular. However, because
the transformation and retransformation of root data at low intensities in the
deep parts of the soil profile makes the results unstable, the lower third of
Figure 11 is not really useful for interpretation of spatial structures. The in-
vestigated small roots were also described regarding water and nutrient uptake
([25]) and mediates to the most active fine roots (<2mm). Thus, clusters of
small roots reflect the presence of nutrient patches or zones of better water
availability ([19], [34], [39]). As the number of small roots and their clustering
was independent of the distance to the surrounding trees and of their diameter
([42]), the root clusters are suggested as an inherent property of below-ground
space acquisition.
5 Acknowledgement
Data collection of the profile walls was financially supported by the Austrian
Science Foundation within the Special Research Program ”Restoration of For-
est Ecosystems”, F008-08.
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22 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
A Point Process Characteristics and their Estimators
In the following let x={xn}be a random point process in d, where
d∈ {2,3,...}and let N(B) = #{n:xn∈B}denote the number of points
xnof xlocated in a sampling window B
Intensity Measure
The intensity measure Λis defined as
Λ(B) = EN (B) (1)
for a given set B. Hence Λ(B) is the mean number of points in B. In the
homogeneous case it suffices to regard an intensity λsince then
Λ(B) = λνd(B) (2)
where νd(B) denotes the volume of B. A natural estimator for λis given by
b
λ=N(B)
νd(B).(3)
However, for the estimation of the nearest-neighbor distance distribution a
different estimator
b
λH=X
xn∈B
1B⊖b(o,s(xn))(xn)
νd(B⊖b(o, s(xn))) (4)
is recommended ([46]), where s(xn) denotes the distance of xnto its nearest
neighbour and b(x, r) is the ball with radius rand midpoint x.
Notice that, following the recommendation in [45], λ2has been estimated by
c
λ2=N(B)(N(B)−1)
(νd(B))2,(5)
since even in the Poisson case (b
λ)2is not an unbiased estimator for λ2.
Moment Measure and Product Density
Let B1and B2be two sets. The second factorial moment measure α(2) of x
is defined by
α(2)(B1×B2) = E( X
x1,x2∈N
x16=x2
1B1(x1)1B2(x2)).(6)
Often α(2) can be expressed using a density function ̺(2) as follows
α(2)(B1×B2) = ZB1ZB2
̺(2)(xi, xj)dxidxj.(7)
Analysis of Spatial Point Patterns 23
The density function ̺(2) is called the second product density. If one takes
two balls C1and C2with infinitesimal volumes dV1and dV2and midpoints x1
and x2respectively, the probability for having in each ball at least one point
of xis approximately equal to ̺(2) (x1, x2)dV1dV2. In the homogeneous and
isotropic case ̺(2)(x1, x2) can be replaced by ̺(2)(r), where r=||x1−x2||.
As an estimator
d
̺(2)(r) = 1
dbdrd−1X
xi,xj∈B
i6=j
kh(r− ||xi−xj||)
νd(Bi∩Bj)(8)
has been used ([45]), where kh(x) denotes the Epanechnikov kernel
kh(x) = 3
4h(1 −x2
h2)1(−h,h)(x),(9)
Bxj={x+xj:x∈B}is the set Btranslated by the point xj, and the sum
in (8) extends over all pairs of points xi,xj∈Bwith i6=j. The bandwith h
has been chosen as h=cb
λ−1/d with a fixed parameter c.
Pair Correlation Function
The product density ̺(2)(r) is used to obtain the pair correlation function
g(r) as
g(r) = ̺(2) (r)
λ2.(10)
The pair correlation function at a certain value rcan be regarded as the
frequency of point pairs with distance r, where g(r) = 1 is a base value.
The pair correlation function can be estimated by the usage of estimators for
̺(2)(r) and λ2respectively. In particular, we consider the estimator
bg(r) = d
̺(2)(r)
c
λ2,(11)
where c
λ2and d
̺(2) are given by (5) and (8), respectively. Note that g(r)≥0
for all distances r. In the Poisson case gP o i(r)≡1, therefore g(r)>1 indi-
cates that there are more point pairs having distance rthan in the Poisson
case, while g(r)<1 indicates that there are less point pairs of such a distance.
K-Function
Ripley’s K-function ([37]) is defined such that λK(r) is the expected num-
ber of points of the stationary point process x={xn}within a ball b(xn, r)
centered at a randomly chosen point xnwhich itself is not counted. Formally
λK(r) = EX
xn∈B
N(b(xn, r)) −1
λνd(B).(12)
24 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
The K-function has been estimated by
b
K(r) = κ(r)
c
λ2,(13)
where
κ(r) = X
xi,xj∈B
i6=j
1b(o,r)(xj−xi)
|Bxj∩Bxi|,(14)
For Poisson processes it is easy to see that KP oi (r) = bdrd.
L-Function
Often it is more convenient to scale the K(r) in order to get a function equal
to rfor the Poisson case. Hence L(r) is defined as
L(r) = d
sK(r)
bd
,(15)
where bddenotes the volume of the d-dimensional unit sphere. Thus, in the
Poisson case, we have
L(r)−r= 0.(16)
A natural estimator for L(r) is given by
b
L(r) = d
sb
K(r)
bd
.(17)
Nearest-Neighbor Distance Distribution
The nearest-neighbor distance distribution Dis the distribution function of
the distance from a randomly chosen point xnof the given stationary point
process xto its nearest neighbor. Hence D(r) is the probability that a ran-
domly chosen point xnof xhas a neighbor with a distance less than or equal
to r. According to [46] we used the Hanisch estimator b
DH(r) = DH(r)/b
λH
([2], [18]) with
d
DH(r) = X
xn∈B
1B⊖b(o,s(xn))(xn)1(0,r](s(xn))
νd((B⊖b(o, s(xn))) .(18)
A useful property of the nearest-neighbor distance distribution is that in the
case of stationary Poisson processes we have
DP oi(r) = 1 −exp (−λbdrd).(19)
Therefore one can conclude that D(r)< DP oi(r) indicates rejection between
points, on the other hand D(r)> DP oi (r) indicates attraction, keeping in
Analysis of Spatial Point Patterns 25
mind that the nearest-neighbor distance distribution function is a cumulated
quantity.
Spherical Contact Distribution Function
The spherical contact distribution function Hs(r) is the distribution function
of the distance from an arbitrary point, chosen independently of the point
process x, to the nearest point belonging to x. Notice that the value Hs(r)
can be interpreted as the probability that at least one point xnof xis in the
sphere of radius rcentered at the origin. As an estimator for Hs(r),
b
Hs(r) = νd((B⊖b(0, r)) Sxn∈Bb(xn, r))
νd(B⊖b(0, r)) (20)
is used.
J-Function
Based on Hs(r) and on D(r), Baddeley’s J-function is defined by
J(r) = 1−Hs(r)
1−D(r).(21)
where
b
J(r) = 1−b
Hs(r)
1−b
D(r),(22)
is a natural estimator for J(r). In the case of Poisson point processes JP oi (r)≡
1 and therefore if J(r)>1 one can conclude that there is repulsion between
point pairs of distance r. On the other hand if J(r)<1 there is attraction
between point pairs compared to the case of complete spatial randomness.
B Matern-Cluster Model
The Matern-cluster point process xmc is based on a Poisson process with
intensity λewhose points are called parent points. Around each parent point
a sphere with radius Ris taken in which the points of the Matern-cluster
process are scattered uniformly where the number of points in such a sphere
is Poisson distributed with parameter Rdbdλt. Notice that λtis the mean
number of points per unit area generated by a single parent point in a sphere
of radius R. Since the parent points themselves are not part of the Matern-
cluster process, its intensity is given as
λmc =Rdbdλtλe.(23)
Thus, the Matern-cluster point process xmc is uniquely determined by three
of the four parameters λe,λt,Rand λmc Obviously, for small distances, points
of the Matern-cluster process are attracted to each other, in other words there
26 Frank Fleischer, Michael Beil, Marian Kazda, and Volker Schmidt
is a bigger expected number of points of xmc in a sphere around an arbitrarily
chosen point of xmc than for Poisson processes of comparable intensity λP oi =
λmc. For xmc, closed formulae for the point process characteristics described
in Appendix A are known ([44]).
