Analysing spatial point patterns in R - CSIRO

150 Adjusting for inhomogeneity
The denominator D in (30) is either the area of the window D 1 = area(W), or
D 2 = ∑ i
1
̂λ(x i )
which is an unbiased estimator of area(W) if the intensity is correctly estimated. The denominator
D 2 is often preferred on grounds of statistical performance, because it introduces a
data-dependent normalisation.
The inhomogeneous K function is computed by the command Kinhom(X, lambda) where
X is the point pattern and lambda is the estimated intensity function. Here lambda may be a
pixel image, a function(x,y) in the R language, a numeric vector giving the values ̂λ(x i ) at
the data points x i only, or it may be omitted (and will then be estimated from X). By default,
the data-dependent denominator D 2 is used.
There remains the question of how to estimate the intensity function λ(u). It is usually
advisable to obtain the intensity estimate ̂λ(u) by fitting a parametric model, to avoid overfitting.
Here is an example for the tropical rainforest data, using the covariate data to suggest a model
for the intensity.
> data(bei)
> fit lam Ki plot(Ki, main = "Inhomogeneous K function")
Inhomogeneous K function
K(r)
0 10000 20000 30000 40000 50000
bord.modif
border
theo
0 20 40 60 80 100 120
The plot suggests that, even after accounting for dependence on altitude and slope, the trees
still appear to be clustered.
The intensity function λ(u) could also be estimated by kernel smoothing the point pattern
data. However, notice that the estimator (30) of the inhomogeneous K function depends on
the estimated intensity values at the data points, ̂λ(x i ). These are positively biased estimates
of the true values λ(x i ). In order to avoid bias, the value ̂λ(x i ) should be estimated by kernel
smoothing of the point pattern with the point x i deleted. This “leave-one-out” estimator is
implemented in Kinhom and is invoked when the argument lambda is not given:
> Ki2 plot(Ki2, main = "Kinhom using leave-one-out")
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