PART II
CHAPTER III
ANALYSIS OF POINT PATTERNS
Assist. Prof. Dr. Mahmut Çavur
3.1. Introduction
In this chapter it is considered to investigate methods
for analysis of a set of point locations, which is often
referred as “point pattern”.
A spatial point process is any stochastic mechanism that
generates a countable set of events (si) in a plane
Basic Definitions:
Event: The location of observed occurrence of the spatial
phenomena, differentiated from other arbitrary locations in the
study region.
Mapped point pattern: All relevant events in a study area R have
been recorded
Point: Arbitrary locations or locations other than events.
Sampled point pattern: Events are recorded from a sample of
different areas
3.1. Introduction
Objectives:
 To determine if there is a tendency for points to exhibit a
systematic pattern (i.e. some form of regularity or clustering)
 If there is a systematic pattern, then to examine at what spatial
scale this pattern occurs and whether particular clusters are
associated with proximity to particular sources of some factors.
 To estimate how the intensity of points varies across the study
region
 To seek models to account for observed point patterns
3.1. Introduction
Analysis Approach:
 Events may have attributes which can be used to
distinguish types – but it is the location pattern that is
analyzed
 Patterns in event locations are the focus
 Stochastic aspect is where events are likely to occur
 Does a pattern exhibit clustering or regularity?
 Over what spatial scales do patterns exist?
E.g.
Such methods are relevant to the study of
patterns of occurrence of:





Diseases
Crime types
Earthquake epicenters
Plant distributions
Etc.
A Point pattern is simple example of spatial data, since
the data contains only the coordinates of events.
However, this does not mean that the analysis is any
easier than for other spatial data types. In fact from a
statistical perspective, point patterns can in some
ways be mathematically more complex to handle.
Usually data in point pattern analysis comprise:
1. Locations (coordinates)
1. Attributes (tree types, crime type, date of
disease notification, etc.)
A point pattern is a data set consisting of a series of
point locations (s1,s2,…) in some study region R at
which events of interest have occurred.
Basic Assumptions:
1. Data present a complete set of events in the study
region R, which is called mapped point pattern. i.e.
all relevant events occurred in R have been
recorded.
!!!Remark: Some point pattern analysis are directed
towards extracting limited information about a point
process, by recording events in a sample of different
areas of the whole region, which is called sampled
point pattern.
E.g.
Field studies in forestry, ecology or biology,
where complete enumeration is not feasible.
Basic Assumptions:
2. The study region R might be of any arbitrary shape.
Some of the methods can be applied to only to
regions, which are square or rectangle.
3. In order to eliminate edge effects, a suitable guard
area between perimeter of the original study region
and sub-region within which analysis is performed is
left.
4. In all cases, the final area selected for study is
assumed to be in some sense representative of any
larger region from which it has been selected.
From a statistical point of view spatial point pattern can
be thought of:
Number of events occurring in arbitrary sub-regions or
areas, A, of the whole study region R.
Spatial point process is defined by:
Where;
Y(A) is the number of events occurring in the area A.
First-Order Properties of Point Patterns
First-order properties are described in terms of
intensity, (s), of the process, which is the mean
number of events per unit area at the point s.
Mathematically (s) is defined by:
Where;
ds
=
AS
=
Small region around the point s
Areas of this region
For a stationary process (s) is constant over R,
expressed by .
Then;
Where;
a is the area of A.
Second-Order Properties of Point Patterns
Second-order properties relate to spatial dependence and involve
relationship between numbers of events in pairs of areas in R.
This can be formally defined as second order intensity, (sİ ,sJ) of
the process. i.e. It is the number of events in pairs of areas in R.
Mathematically  (sİ ,sJ) is defined by:
For a stationary process
.
i.e.Second-order intensity depends on the vector
difference (h), (direction and distance) between si and sj
(not on their absolute locations).
For an isotropic process
. i.e. the
dependence is purely a function of length, h, of the
vector, h, and not its orientation, in other words
dependence is purely a function of the distance
between si and sj not the direction.
3.2. Case Studies
The following cases will be of concern when studying point
patterns.
1. The locations of craters in a volcanic field in Uganda
2. The locations of granite tors in Bodmin Moor
3. The locations of redwood seedlings in a forest
4. The locations of centers of biological cells in a section of
tissue
5. The locations of the homes of juvenile offenders on a
Cardiff estate
6. Locations of “theft from property” offences in Oklahoma
City
7. Locations of cases of cancer larynx and lung in part of
Lancashire
8. Locations of Burkitt’s lymphoma in an area of Uganda
1. The locations of craters in a volcanic field in Uganda
The data set involves the locations of centers of
craters of 120 volcanoes in the Bunyaruguru volcanic
field in west Uganda. A map of the distribution shows
a broad regional trend in a north-easterly direction,
representing elongation along a major fault.
The purposes of studying this case:
 To obtain a smooth map of such broad regional
variation.
 To explore and model the distribution of craters in a
smaller scale.
 To answer the following questions:
 Is the distribution random within the study region?
 Is there evidence of clustering or regularity?
 To test the following hypotheses:
It is expected that rift faults would guide volcanic activity
to the surface, along fractures or lines of weakness. The
hypothesis is to test weather this holds true.
2. The locations of granite tors in Bodmin Moor
There are 35 locations of granite tors and on a large
scale there is clear spatial patterning.
2. The locations of granite tors in Bodmin Moor
The purposes of studying this case:
 To detect any evidence of departures from
randomness at smaller scales.
 To find if the regularity in the distribution is valid for
only small distances.
 To determine if the spatial distribution shows other
patterning at slightly longer distances.
3. The locations of redwood seedlings in a forest
There are 62 redwood seedlings distributed in a square
region of 23 m2.
The purposes of studying this case:
 To see some evidence of clustering around existing
parent trees.
4. The locations of centers of biological cells in a section
of tissue
There are centers of 42 biological cells in a section of
tissue.
The purposes of studying this case:
To know whether there is evidence for departures
from randomness in such data.
To answer the following question:
 Are such cells clustered or regular?
5. The locations of the homes of juvenile offenders on
a Cardiff estate
The data recorded in 1971. The purposes of studying
this case:
 To know whether the distribution of homes of
juvenile
offenders
exhibit
some
regularity
(clustering).
 To explore the locations of homes of juvenile
offenders
6. Locations of “theft from property” offences in
Oklahoma City
The data are taken from research done on crime in
Oklahoma City in late 1970s and comprise two distinct
categories of events. One set refers to offences
committed by whites, the other by blacks.
The purposes of studying this case:
 To see if the spatial pattern of the events differ
 To investigate if the two sub-groups have different
activity places
 To answer the following questions:
 Do the crimes committed by different groups display
different spatial patterns?
 Are those for one group clustered or aggregated in
some way, while those for the other group are more
random?
7. Locations of cases of cancer larynx and lung in part
of Lancashire
The data are for a part of Lancashire in U.K. and have
been collected over a 10 year period 1974-83. Lung
cancer is quite a common disease and there are 917
cases in the study area. Larynx cancer rare and there
are only 57 cases notified during the study period.
The purposes of studying this case:
 To investigate if the residents living near the site of
an old industrial waste incinerator that their health
had been affected by exposure to the by-products of
the incineration process.
8. Locations of Burkitt’s lymphoma in an area of
Uganda
The data comprise information on 188 cases of
Burkitt’s lymphoma (a cancer affecting usually the jaw
and abdomen, primarily in children) in the West Nile
district of Uganda for the time period of 1961-75.
The purposes of studying this case:
 To assess evidence for space-time clustering in
order to answer the following questions:
 Are the cases that are near each other in geographic
space also near each other in time? If so, this might be
evidence in support of the hypothesis that suggests an
infective etiology for the disease.
3.3. Visualizing Spatial Point Patterns
Point patterns are visualized by the use of dot map.
This gives an initial impression of the shape of the
study region and any obvious pattern present in the
distribution of events.
!!!Remark: Intuitive ideas about what constitutes as
“random pattern” can be misleading. Generally it is
hard to come to any conclusion purely on the basis of
a visual analysis.
3.3. Visualizing Spatial Point Patterns
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
Figure 4.1. Craters in Uganda
Figure 4.2. Tors on Bodmin Moor
No conclusions possible from visual inspection alone
3.3. Visualizing Spatial Point Patterns
Visualization Issues

Is there an underlying population distribution from which
events arise in a region?

If population varies we would expect events to cluster in
areas of high population.

Are they more or less clustered than we would expect on the
basis of population alone?
Can create event symbols inversely proportional to population
density in event location and look for gaps in the maps
3.4. Exploring Spatial Point Patterns
The methods of exploration of point patterns are
divided into two:
 Methods concerned with investigating the firstorder effects
 Quadrat methods
 Kernel estimation
 Methods concerned with investigating the secondorder effects
 Nearest neighbor distances
 The K function
3.4.1. Quadrat Methods
The simple way of summarizing the pattern in the
locations of events in some region R is to partition R
into sub-regions of equal area or quadrats and to use
the counts of the number of events in each of the
quadrats to summarize the spatial pattern. (i.e. creating
a 2-D histogram or frequency distribution of the
observed event occurrences).
How ?
1. Impose a regular grid over R
2. Count the number of events falling into each of grid
3. Convert this into an intensity measure by dividing the
area of each of the grid
4. Observe the behaviour of intensity over R.
3.4.1. Quadrat Methods
1. Impose a regular grid over R
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.1. Quadrat Methods
2. Count the number of events falling into each of grid
3. Convert this into an intensity measure by dividing the area of
each of the grid http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.1. Quadrat Methods
4. Observe the behaviour of intensity over R.
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
The intensity of the process, λ(s) is defined by:
The quadrats may, may be randomly scattered in R and
all events within each quadrat counted to give a crude
estimate of how intensity varies over R.
Problem of Quadrat Methods
Basic problem: Although the method gives a global idea of subregions with high or low intensity it throws away much of the
spatial detail in the observed pattern. As quadrats are made
smaller to retain most spatial information, variability of quadrat
counts gets increased.
 E.g. The variance mean ratio (or index of dispersion) varies
depending on the size and hence the number of quadrats
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
Solution: Use of counts per unit area in a “moving window” can be a
solution. A suitable window is defined and moved over a fine grid of
locations in R. The intensity at each grid point is estimated from the
event count per unity area of the window centered at that point. This
produces a more spatially smooth estimate of the way in which λ (s)
is varying.
Problem of Moving Window Approach
1. No account is taken of the relative location of
events within the particular window
2. It is difficult to decide the size of the window
3.4.1. Quadrat Methods
A windows is moved over a grid of points in R.
What should be the size of the window?
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.2. Kernel Estimation
It was originally developed to obtain a smooth estimate of a
univariate or multivariate probability density from an observed
sample of observations (i.e. smooth histogram). Estimating the
intensity of a spatial point pattern is very like estimating a bivariate
probability density .
If s represents a general location in R and s1 ,...,sn are the locations
of n observed events then the intensity, λ(s) at s is estimated by:
Where;
k( )
=

=
(s)
=
Kernel
Bandwidth
Edge correction factor
Kernel: It is a suitably chosen bivariate
probability density function, which is
symmetric about the origin.
Bandwidth: It determines the amount of
smoothing. It is the radius of a disc
centered on si within which point si will
contribute significantly to
. Note
that  > 0.
Edge correction factor: It is the volume under
the scaled kernel centered on s which lies
inside R.
For any chosen kernel and bandwidth, values of
can
be estimated at locations on a suitably chosen fine grid
over R to provide a useful visual indication of the variation
in the intensity
over the study region.
Most of the time, for reasonably possible probability
distributions of k ( ), the kernel estimate
will be very
similar for a given bandwidth . A typical choice of k ( )
might be the quadratic kernel:
When the above kernel used, ignoring the edge
correction factor,
takes the following form:
Where;
hi = Distance between the point s and the observed event location si
!!!Remark: Summation is all over the values of hi, which
do not exceed .
Figure 4.3. Kernel estimation of a point pattern
The region of influence within which observed events
contribute to
is determined by the circle with
radius  centered on s.
From a visual point of view,
kernel estimation can be
thought of a 3-D floating
function visiting each point
s on a fine grid of locations
in R. Distances to each
observed event si lying in
the region of influence are
measured and contribute to
intensity estimate according
to how close they are to s.
Figure 4.4. Slice through a quadratic kernel
The kernel function visits each s point. Events within the
bandwidth contribute to the intensity based on weighting of kernel
at that distance
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
The effect of bandwidth on kernel estimate
1. For large ,
will appear flat and local features
will be obscured.
2.
If  is small then
tends to become a collection
of spikes centered on the si.
Changing the bandwidth allows you to look at the variation in
intensity at different scales.
For exploratory purposes it is useful to test various bandwidths to
examine the change in intensity at different scales
The effect of bandwidth on kernel estimate
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
Figure 4.5. Kernel estimates of intensity of volcanic craters
( = (a) 100, (b) 220, (c) 500)
A rough choice for  has been suggested as:
for estimating the intensity, when R is unit square and n is
the number of observed events in R.
In order to avoid too much smoothing and not to obscure
details in dense areas, local adjustment of bandwidth may
be applied, which is called “adaptive kernel estimation”. In
this method  is replaced by (si), which is some function of
presence of events in the neighborhood of si. Ignoring the
edge effects,
will be:
One practical method for specifying (si) is:
1. Perform non-adaptive kernel estimation with
some reasonable bandwidth 0 and achieve a
pilot estimate of
.
2. Compute the geometric mean,
, of pilot
estimates
at each si (nth root of their
product).
3. Formulate the adaptive bandwidths as:
Where; α is the sensitivity parameter and
If α = 0

No local adjustment of τ
If α = 1

Maximum local adjustment
The choice of α = 0.5 is found to be reasonable in
practice.
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.3. Nearest Neighbor Distance
This method is designed for investigating the second
order properties of the spatial point process and focuses
on the relationship between inter-event distances. In this
method the nearest neighbor event-event distance (W)
and the nearest neighbor point-event distance (X) will
constitute the basic area of interest.
W: The distance between a randomly selected event in the
study region a nearest neighboring event.
X: The distance between a randomly selected point in the
study region an the nearest neighboring event
W
X


Mapped point pattern
Sampled point pattern
3.4.3. Nearest Neighbor Distance
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
!Remark: This method only provides information about
inter-event interactions at a small physical scale, since
by definition it uses only small inter-event distances.
The simple way of summarizing pattern is to estimate
the empirical cumulative probability distribution
function (
for W or
for X).
for W
for X
Where;
# = Number of
n = Total number of events in R
m = Total number of sampled points
The resulting
or
are plotted against values of
w and x. Then it is examined purely an exploratory way
to see the evidence of inter-event interaction.
Figure 4.6. A typical function of G
Interpretations for the plots of
or
 If the distribution function (
or
) climbs
very steeply in the early part of its range before
flattening out, then the indication would be an
observed probability of short as opposed to long
nearest neighbor distances, which suggest
clustering.
 If distribution function (
or
) climbs very
steeply in the later part of its range, then the
suggestion might be one of inter-event regularity.
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
Early sharply rising function could
indicate
clustering
–
inter-event
interaction
Late sharply rising function could
indicate a regular pattern –
repulsion
Note that a distance between 50 and 150 m
climbs up rapidly.
This implies that there are relatively a lot of short event-event
distances. (i.e. İndicating an impression of local clustering in the
data.
Figure 4.6. Nearest neighbor distribution function for volcanic craters
Another alternative would be to plot
against
.
If there is no interaction then these two distributions
should be very similar and it is expected to obtain roughly
a straight line in the plot.
In the case of positive interaction or clustering, the
point-event distances (xi) will tend to be large relative to
event-event distances (wi). Hence
will have higher
values than
.The reverse holds for regular pattern.
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
Corrections for Edge Effects
For boundary cases, because the nearest event may
be located outside R, distance to the nearest event is
unknown.
If the nearest neighbor is taken to be the closest event
within the study area, expected nearest neighbor
distances will be greater for events located near the
boundary than for events located near the center of
the study region Thus estimates based on nearest
neighbor statistics will be biased without some edge
correction applied
There are several ways of handling edge effects such as:
1. The problem can be overcome by constructing a guard area inside
the perimeter of R. The nearest neighbor distances are not used for
events within the guard area. But events in the guard area are
allowed as neighbors of any event from the rest of R.
2. Another approach to the problem can be employed when the study
region is rectangle, which is called use of toroidal edge correction.
The study region is regarded as the central region of a 3×3 grid of
rectangle regions, each identical to the study region. i.e. top of the
study region is assumed to be joined to the bottom and the left to
the right. Events in the copies are allowed to be neighbors of any
events (points) which are selected in the study region.
3.
can be approximately estimated as:
Where;
bi is the distance from event i to the nearest point on
the boundary of R. This effectively ignores wi values
for events close to the boundary.
There are several ways of handling edge effects such
as:
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.4. The K function
The nearest neighbor distances method uses
distances only closest events and therefore only
considers the smallest scales of pattern.
Information on larger scales of pattern is ignored.
An alternative approach is to use an estimate of
the reduced second moment measure or K
function of the observed process, which provides
a more effective summary of spatial dependence
over a wider range of scales.
Properties of the K function
1. The K function represents information at various
scales of pattern.
2. It involves use of precise location of events and
includes all event-event distances, not just nearest
neighbor distances.
3. The theoretical form of K(h) is not only used for
various possible spatial point pattern models, but
also suggest specific models to present it and to
estimate the parameters of such models.
3.4.4. The K function
Remark: When examining spatial dependence over small
scales in R, an implicit assumption is made, which is
assuming that the process is isotropic over such scales.
However, second order properties are not necessarily
constant over the considered scale and may be confused
with first order effects.
 E.g. If it is clear that there is large scale variation in
intensity of given point pattern over the whole of R, this is
truly a first order effect not a result of spatial dependence. In
this case it is convinient to study second order effects over
scales in R small enough for the assumption of isotropy to
hold.
If there is no variation in the intensity, it is appropriate to
study the second order effects over larger scales in the study
region.
3.4.4. The K function

The K function relates to the second order
properties of an isotropic process. However, if it is
used in a situation where there are large scale first
order effects, then any spatial dependence it may
indicate could be due to first order effects rather
than to interaction effects. In such a case, it is
better to examine smaller sub regions of R, since
isotropy can reasonably be assumed to hold.
3.4.4. The K function
The K function is defined by:
λK(h) = E (#(events within distance h of an
arbitrary event))
Where;
# = Number of
E () = Expectation operator
λ = Intensity (mean number of events / unit area)
3.4.4. The K function
http://www.spatial.maine.edu/~beard/Lectu
res/Lecture%208%202010.pdf
3.4.4. The K function
The practical value of K (h) as a summary measure of second order effects is
that it is feasible to obtain a direct estimate of it, (
) from an observed point
pattern.
How?
If A is the area of R, then the expected number of events in R is λA.
The expected number of pairs of events a distance at most h apart is λ2AK(h).
If dij is the distance between ith and jth observed events in R and Ih(dij) is an
indicator function which is 1 if dij
0 otherwise, then the observed
number of pairs is
then a suitable estimate of K(h) is :
3.4.4. The K function
The summation above excludes pairs of events for which the second event
is outside R. Therefore, above equation should be corrected for edge
effects.
Consider a circle centered on event i, passing through the point j, let wij be
the proportion of the circumference of this circle which lies within R. Then
wij is effectively the conditional probability that an event is observed in R,
given that it is a distance dij from the ith event. Thus edge corrected
estimator for K (h) is:
When the unknown λ is replaced by its estimate, which is
Graphical Representaion of the K function
Imagine that an event is visited
and that around it is constructed
a set of concentric circles at a
fine spacing. The cumulative
number of events within each of
these
distance
bands
are
counted. Every other event is
similarly
visited
and
the
cumulative number of events
within distance bands up to
radius h around all the events
becomes the estimate of K(h)
when scaled by A/n2.
Figure 4.7. Estimation of K Function
Graphical Representaion of the K function
Assume that there are 62 events in a 100
m2 study area. It is required to estimate
K(h) for h = 0.4 m.
K (0.4) = (58/62) / (62/100) = 1.508
Figure 4.8. Estimating K
Function for h =0.4 m
Table 4.1. Counts of events within 0.4 m
(Total # of events in each circle = 58)
Comparison for randomness
The random occurrences of the events implies that an event at any point
in R is independent of other events and equally likely over the whole of R.
Hence for a random process the expected number of events within a
distance of h of a randomly chosen event would be h2.
 The K function for a random event should be:
#(events within distance h of an arbitrary event))
K(h) = h2 → K(h) = h2 for a random process
If the point pattern has regularity then K(h) < h2
If the point pattern has clustering K(h) > h2
For the observed data, the
estimated
is compared with
h2 One way of doing this can be
achieved by plotting L(h) against h,
where
An alternative to the square root
transformation
is
to
use
a
logarithmic transformation, plotting
I(h) against h.
In this plot peaks in positive values
tend to indicate clustering and
troughs of negative values indicate
regularity at corresponding scales
of distance h in each case.
In this plot again peaks indicate
clustering and troughs indicate
regularity at corresponding scales
of distance h in each case.

E.g. Explore the juvenile offenders on a Cardiff estate. Visually some
form of clustering is observed on the nothern part.
There are a peaks at h = 10 and h = 20 m, suggesting clustering at these
scales.
Figure 4.9. (a) Juvenile offenders in Cardiff and (b) assocaited L function