Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 30

CSR Clustered Uniform

Fig. 1.3 Types of spatial point patterns

Although CSR is of limited scientific interest in itself, empirical analyses usually
begin with a test of the hypothesis of CSR. In this case, the null hypothesis ( H 0 )is
that the events follow the CSR distribution; while the alternative hypothesis ( H 1 )is
that the events are spatially clustered or dispersed. The rejection of CSR is a
prerequisite for any serious attempt to model an observed pattern.
There are several methods and algorithms that attempt to identify patterns in a
collection of points. These methods can be broadly divided into two classes:
quadrat methods and distance methods. See Diggle ( 2003 ) for an excellent and
exhaustive review of these methods.
Quadrat count analysis is a relatively easy method to implement, and it provides
several opportunities to apply basic mathematical and statistical concepts. The
quadrat count method can be simply described through a partition of the data into
n equal sized sub-regions; we call these sub-regions quadrats. We count the number
of events that occur in each quadrat, and the distribution of quadrat counts serves as
our indicator of pattern. The choice of quadrat size can greatly affect our analysis.
Large quadrats obviously produce a coarse description of the pattern. If the quadrat
is too small, then many quadrats may contain only one event, or none at all.
The techniques based on quadrat counts can be implemented using either an
exhaustive census of quadrats, or by placing quadrats randomly across the area of
interest. In both cases, the output includes the counts in each cell. When this
information is available, it is possible to compare its frequency distribution with
the expected distribution.
So, the expected probability distribution for a quadrat count of a random point
pattern can be given by the Poisson distribution

=

PðÞ ¼ e ʻ ʻ

k

k ! k ¼ 0, 1, 2, ...; ð 1 : 56 Þ

where k is the number of points in a quadrat, and is the expected number of points

per sample unit area (the intensity of the process, which can be estimated by the
mean points per quadrat in the pattern under study).

ʻ