Analysing spatial point patterns in R - CSIRO

88 Tests of Complete Spatial Randomness
14 Tests of Complete Spatial Randomness
The basic ‘reference’ or ‘benchmark’ model of a random point pattern is the uniform Poisson
point process in the plane with intensity λ, sometimes called Complete Spatial Randomness
(CSR). Its basic properties are
the number of points falling in any region A has a Poisson distribution with mean λ ×
area(A)
given that there are n points inside region A, the locations of these points are i.i.d. and
uniformly distributed inside A
the contents of two disjoint regions A and B are independent.
For historical reasons, many researchers are focussed on establishing that their data do not
conform to this model. The logic is that, if a point pattern is completely random, then there
is nothing “interesting” happening (because the points are completely unpredictable, and have
no trend or association with anything else). Statisticians would say that the uniform Poisson
process often serves as the ‘null model’ in a statistical analysis.
14.1 Definition
The homogeneous Poisson process of intensity λ > 0 has the properties
(PP1): the number N(X ∩ B) of points falling in any region B is a Poisson random variable;
(PP2): the expected number of points falling in B is E[N(X ∩ B)] = λ · area(B);
(PP3): if B 1 ,B 2 are disjoint sets then N(X∩B 1 ) and N(X∩B 2 ) are independent random variables;
(PP4): given that N(X ∩ B) = n, the n points are independent and uniformly distributed in B.
The list is redundant; (PP2) and (PP3) are sufficient.
This process is often called “Complete Spatial Randomness” (CSR) especially in biological
science. Under CSR, points are independent of each other and have the same propensity to be
found at any location.
It is easy to simulate the Poisson process directly by following the properties (PP1)–(PP4).
In spatstat, use the command rpoispp (by convention, random data generators have names
beginning with r).
> plot(rpoispp(100))
rpoispp(100)
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