Analysing spatial point patterns in R - CSIRO

19.4 Pairwise distances and the K function 125
19.4 Pairwise distances and the K function
The observed pairwise distances s ij = ||x i −x j || in the data pattern x constitute a biased sample
of pairwise distances in the point process, with a bias in favour of smaller distances. For example,
we can never observe a pairwise distance greater than the diameter of the window.
Ripley [54] defined the K-function for a stationary point process so that λK(r) is the expected
number of other points of the process within a distance r of a typical point of the process.
Formally
K(r) = 1 E [n(X ∩ b(u,r) \ {u}) | u ∈ X]. (21)
λ
For a homogeneous Poisson process, intuitively, the knowledge that u is a point of X does
not affect the other points of the process, so that X\{u} is conditionally a Poisson process. The
expected number of points falling in b(u,r) is λπr 2 . Thus for a homogeneous Poisson process
K pois (r) = πr 2 (22)
regardless of the intensity.
Numerous estimators of K have been proposed. Most of them are weighted and renormalised
empirical distribution functions of the pairwise distances, of the general form
̂K(r) =
1
̂λ 2 area(W)
∑∑
1 {||x i − x j || ≤ r}e(x i ,x j ;r) (23)
i
j≠i
where e(u,v,r) is an edge correction weight. The choice of estimator does not seem to be very
important, as long as some edge correction is applied.
Again we usually compare the estimate ̂K(r) with the Poisson K function. Values ̂K(r) >
πr 2 suggest clustering, while ̂K(r) < πr 2 suggests a regular pattern.
In spatstat the function Kest computes several estimates of the K-function.
> Gc Gc
Function value object (class fv)
for the function r -> K(r)
Entries:
id label description
-- ----- -----------
r r distance argument r
theo K[pois](r) theoretical Poisson K(r)
border K[bord](r) border-corrected estimate of K(r)
trans K[trans](r) translation-corrected estimate of K(r)
iso K[iso](r) Ripley isotropic correction estimate of K(r)
--------------------------------------
Default plot formula:
. ~ r
Recommended range of argument r: [0, 0.25]
Available range of argument r: [0, 0.25]
CopyrightCSIRO 2010