Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 32

Significantly large values indicate regularity, whereas small values indicate a
clustered pattern.
Ripley s K -function (Ripley 1977 ) is an appropriate alternative that can be used

to summarize a point pattern, to test hypotheses, to estimate parameters, and to fit
models. As before, suppose that we have a spatial point pattern with n events in
some region A , and we want to test for CSR. Additionally, suppose that the process
is isotropic 3 over A . Ripley

'

s K -function is defined as '

KðÞ ¼ 1 =ð EnðÞ ð Þ; ð 1 : 60 Þ

where n ( t ) is the number of additional events within a distance t of a randomly
chosen point, and

ʻ

is the intensity of the process. The naive estimator of K ( t ) is its

empirical average

X X

n n

I ij ðÞ

i ¼1 j ¼1

^ KðÞ ¼

; ð 1 : 61 Þ

ʻ

n

= A is the estimated intensity

where ʻ ¼ n

1se t ij < t

0se t ij t

I ij ðÞ ¼ ; ð 1 : 62 Þ

and t ij is the Euclidean distance between i and j .
It has been demonstrated that KðÞ ¼ π t 2 under the CSR hypothesis. The simplest

use of Ripley s K ( t ) is to test the CSR hypothesis for a spatial point pattern.

However, it is easier to use the transformation LðÞ ¼ KðÞ=π

'

1 = 2 and its estimator

½

is nearly constant for a homoge-
neous process. Under CSR, LðÞ ¼ t . In practice, the value LðÞ ¼ t is used as a
benchmark. In fact, if ^ LðÞ>

1 = 2 . This is because Var ^ LðÞ

^ LðÞ ¼ ^ KðÞ=π

t for some t , the probability of finding a neighbor at a
distance t is greater than the probability of finding a point in the same area anywhere
in the domain. So, the points are aggregated. Conversely, if ^ LðÞ<

t for some t , the
average neighbor density is smaller than the average point density on the studied
spatial domain. In this case, the points are dispersed.
However, the boundaries of the study area are usually arbitrary, which means
that edge effects can occur. Edge effects arise because points outside the boundary
are not counted in the numerator, even if they are within a distance t of a point in the
study area. If we ignore the edge effects, we introduce a bias into the estimator ^ KðÞ ,
especially for large values of t . Many authors proposed corrections to the

3 Isotropy can be also defined as uniformity in all spatial directions; the pattern depends on the
spatial locations only through the Euclidean distance between the points.