Analysing spatial point patterns in R - CSIRO

149
In data sharpening [27] the points effectively exert a force of attraction on each other, and
are allowed to move in the direction of the resultant force. This tends to enhance tight linear
concentrations of points.
> Y plot(Y, pch = ".", main = "data sharpened")
data sharpened
25 Adjusting for inhomogeneity
If a point pattern is known or suspected to be spatially inhomogeneous, then our statistical
analysis of the pattern should take account of this inhomogeneity.
25.1 Inhomogeneous K function
There is a modification of the K function that applies to inhomogeneous processes [7]. If λ(u)
is the true intensity function of the point process X, then the idea is that each point x i will be
weighted by w i = 1/λ(x i ).
The inhomogeneous K-function is defined as
⎡
K inhom (r) = E ⎣ ∑
⎤
1
λ(x j ) 1 {0 < ||u − x j|| ≤ r}
∣ u ∈ X ⎦ (28)
x j ∈X
assuming that this does not depend on location u. Thus, λ(u)K(r) is the expected total ‘weight’
of all random points within a distance r of the point u, where the ‘weight’ of a point x i is 1/λ(x i ).
If the process is actually homogeneous, then λ(u) is constant and K inhom (r) reduces to the
usual K function (21).
It turns out that, for an inhomogeneous Poisson process with intensity function λ(u), the
inhomogeneous K function is
K inhom, pois (r) = πr 2 (29)
exactly as for the homogeneous case.
The standard estimators of K can be extended to the inhomogeneous K function:
̂K inhom (r) = 1 D
∑∑
i
j≠i
1 {||x i − x j || ≤ r}
e(x i ,x j ;r) (30)
̂λ(x i )̂λ(x j )
where e(u,v,r) is an edge correction weight as before, and ̂λ(u) is an estimate of the intensity
function λ(u).
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