Geoscience Reference
In-Depth Information
Ripley's (
1976
) estimator
K
(
r
) can be used as an example of how undesirable
edge effects can be avoided in a map-based statistical application. Statistical theory
of spatial point processes is used mainly for the study of patterns of points in the
plane (see, e.g., Diggle
1983
). The first-order property of a point process is its
intensity (
ʻ
) which is independent of location for stationary processes. It is esti-
mated by dividing the total number (
n
) of points within the study area (
A
) by its total
area, or Ave (
n
/|
A
|. The second-order property of an isotropic, stationary spatial
point process can be described by the function
K
(
r
) which is proportional to the
expected number of points within distance
r
of an arbitrary point. If the points are
distributed randomly according to a Poisson process, the expected number of
points within distance
r
of an arbitrary point is equal to
ʻ
¼
)
r
2
. This model provides
a convenient benchmark for complete randomness tests. Clustering of points
occurs if there are more points in the vicinity than predicted by the Poisson
model; anti-clustering or “regularity” arises if there are fewer points than expected
in the vicinity of an arbitrary point. For graphical representation, the function
L
(
r
)
ˀʻ
]
0.5
may be used. The Poisson model simply gives
L
(
r
)
r
.
Estimators of
K
(
r
) should account for edge effects related to the shape of the
study area. There are two reasons why edge effects are important: (1) probability of
occurrence of a point cannot be measured directly at a point but only indirectly with
respect to neighborhoods that are more strongly affected by boundaries than single
points, and (2) the proportion of study area located within distance
r
from the
boundary tends to be large even for small
r
. It may be possible to work with a guard
area inside the boundary of the study area in order to obtain an approximately
unbiased estimator but, in general, too much information is lost by doing this.
Ripley's estimator
K
(
r
) is applicable to regions that have relatively simple
geometrical shapes (e.g., rectangles or circles) or for regions with irregular shapes
bounded by polygons. Digitizing the boundary of an irregularly shaped study region
in vector mode generally results in a polygon with many sides which are so short
that the boundary cannot be distinguished from a smooth curve when it is replotted
on the map. An irregular boundary can be compared with a coastline with fjords and
peninsulas. An algorithm for obtaining Ripley's estimator originally was developed
by Rowlingson and Diggle (
1991
,
1993
). The algorithm of Agterberg (
1994
) also
can be used when there are islands, more than a single enclosed study area, or even
lakes within islands. The underlying geometrical rationale is as follows.
Suppose that the second-order properties of an isotropic, stationary point process
are characterized by the function
K
(
r
)
¼
[
K
(
r
)/
ˀ
¼
¼ ʻ
1
E
[number of further events within
distance
r
of an arbitrary event] where
E
denotes mathematical expectation. It
follows that the expected number of ordered pairs of events within distance
r
from each other is
2
|
A
|
K
(
r
) if the first event of each ordered pair falls in area
A
. Suppose that
r
ij
denotes distance between events
i
and
j
in
A
, and that
I
r
(
r
ij
)isan
indicator function assuming the value 1 if
r
ij
<
ʻ
r
; 0 otherwise, then the observed
number of these ordered pairs is
j
where the double sum denotes
summation over both
i
and
j
. This summation excludes pairs of events for which the
second event is outside of
A
. Let the weight
w
ij
represent the proportion of the
circumference of the circle around event
i
with radius
r
ij
that lies within
A
. Then, as
Σ
i
Σ
j
I
r
(
r
ij
),
i
6¼
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