Analysing spatial point patterns in R - CSIRO

20.1 Envelopes and Monte Carlo tests 133
Clearly, the K-function estimated from thecells data lies outside the typical range of values
of the K-function for a completely random pattern.
To conclude formally that there is a ‘significant’ difference between ̂K and K pois , we use
the language of hypothesis testing. Our null hypothesis is that the data point pattern is a
realisation of complete spatial randomness. The alternative hypothesis is that the data pattern
is a realisation of another, unspecified point process.
20.1.2 Monte Carlo tests
A Monte Carlo test is a test based on simulations from the null hypothesis. The principle was
originated independently by Barnard [18] and Dwass [39]. It was applied in spatial statistics
by Ripley [54, 56] and Besag [22, 23]. See also [41]. Monte Carlo tests are a special case of
randomisation tests which are commonly used in nonparametric statistics.
Suppose the reference curve is the theoretical K function for CSR. Generate M independent
simulations of CSR inside the study region W. Compute the estimated K functions for each of
these realisations, say ̂K (j) (r) for j = 1,... ,M. Obtain the pointwise upper and lower envelopes
of these simulated curves,
L(r) = min
j
U(r) = max
j
̂K(j) (r)
̂K (j) (r).
For any fixed value of r, consider the probability that ̂K(r) lies outside the envelope [L(r),U(r)]
for the simulated curves. If the data came from a uniform Poisson process, then ̂K(r) and
̂K (1) (r),... , ̂K (M) (r) are statistically equivalent and independent, so this probability is equal
to 2/(M + 1) by symmetry. That is, the test which rejects the null hypothesis of a uniform
Poisson process when ̂K(r) lies outside [L(r),U(r)], has exact significance level α = 2/(M + 1).
Instead of the pointwise maximum and minimum, one could use the pointwise order statistics
(the pointwise kth largest and k smallest values) giving a test of exact size α = 2k/(M + 1).
20.1.3 Envelopes in spatstat
In spatstat the function envelope computes the pointwise envelopes.
> data(cells)
> E E
Pointwise critical envelopes for K(r)
Edge correction: iso
Obtained from 39 simulations of CSR
Significance level of pointwise Monte Carlo test: 2/40 = 0.05
Data: cells
Function value object (class fv)
for the function r -> K(r)
Entries:
id label description
-- ----- -----------
r r distance argument r
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