Analysing spatial point patterns in R - CSIRO

134 Simulation envelopes and goodness-of-fit tests
obs obs(r) observed value of K(r) for data pattern
theo theo(r) theoretical value of K(r) for CSR
lo lo(r) lower pointwise envelope of K(r) from simulations
hi hi(r) upper pointwise envelope of K(r) from simulations
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Default plot formula:
. ~ r
Recommended range of argument r: [0, 0.25]
Available range of argument r: [0, 0.25]
> plot(E, main = "pointwise envelopes")
pointwise envelopes
K(r)
0.00 0.05 0.10 0.15 0.20
obs
theo
hi
lo
0.00 0.05 0.10 0.15 0.20 0.25
r
For example if r had been fixed at r = 0.10 we would have rejected the null hypothesis of
CSR at the 5% level. The value M = 39 is the smallest to yield a two-sided test with significance
level 5%.
Tip: A common and dangerous mistake is to misinterpret the simulation envelopes
as “confidence intervals” around ˆK. They cannot be interpreted as a measure of
accuracy of the estimated K function! They are the critical values for a test of the
hypothesis that K(r) = πr 2 . They assume that the pattern is completely
random. [See Section 21 for ways of making confidence intervals for K(r).]
The value returned by envelope is an object of class "fv" that can be manipulated in
the usual way: you can plot it, transform it, extract columns, and so on (see Section 19.6 on
page 128).
20.1.4 Simultaneous Monte Carlo test
Note that the theory of the Monte Carlo test, as presented above, requires that r be fixed in
advance. If we plot the envelope and check whether the empirical K function ever wanders
CopyrightCSIRO 2010