Myriam Elizabeth Saavedra López - Repositorio Digital USFQ ...

3.4.2 The nearest neighbour distance function G
According to Baddeley et al (2004, 51-52) the nearest neighbour distance distribution is a related con-
cept to the Empty Space Function F . Then, let X be a stationary point process in the d-dimensional
Euclidean space. The nearest neighbour function G is the cumulative distribution function of the
distance d (x, X\x) from a typical point x ∈ X to the nearest other point of X. That is:
G (r) = P x (d (x, X\ {x}) ≤ r) = P x (NX (b (x, r) \ {x}) > 0) (12)
where NX (b (x, r) \ {x}) is the numbers of event inside the ball b with radius r and center x and P x
a probability. Using the Campbell-Mecke formula, the formula (9) is written as
G (r) =
E x∈X
W I {d (x, X\ {x}) ≤ r}
E [X (W )]
where W is a window which contains the spatial points. See the folowing illustration:
12
(13)