NOTEBOOK FOR SPATIAL DATA ANALYSIS
Part I. Spatial Point Pattern Analysis
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ESE 502
I.3-1
Tony E. Smith
3. Testing Spatial Randomness
There are at least three approaches to testing the
CSR Hypothesis
: the
quadrat method
,
the
nearest-neighbor method
, and the
method of K-functions
. We shall consider each of
these in turn.
3.1 Quadrat Method
This simple method is essentially a direct test of the CSR Hypothesis as stated in
expression 2.1(6).
1
Given a realized point pattern from a point process in a
rectangular
region,
R
, one begins by partitioning
R
it into congruent rectangular subcells (quadrats)
1
,..,
m
CC
as in Figure 1 below (where
16
m
=
). Then, regardless of whether the given
pattern represents trees in a forest or beetles in a field, the CSR Hypothesis asserts that
the cell-count distribution for each
i
C
must be the same, as given by 2.1(6).
But rather
than use this Binomial distribution, it is typically assumed that
R
is large enough to use
the Poisson approximation in 2.1(9). In the present case, if there are
n
points in
R
, and if
we let
1
()
aa
C
=
, and estimate expected point density
λ
by
(1)
ˆ
n
aR
λ=
then this common
Poisson cell-count distribution
has the form
(2)
ˆ
ˆ
ˆ
Pr[
| ]
,
0,1,2,.
..
!
k
a
i
a
Nk
e
k
k
−λ
λ
=λ
=
=
Moreover, since the CSR Hypothesis also implies that each of the cell counts,
(
),
1,.
.,
ii
NN
C
i
k
==
, is
independent
, it follows that
( )
:
1,.
.,
i
Ni
k
=
must be a
independent random samples from this Poisson distribution. Hence the simplest test of
1
This refers to expression (6) in section 2.1 of Part I. All other references will follow this convention.
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Fig. 1. Quadrat Partition of
R