
552 Spatial Point Patterns: Methodology and Applications with R
As an examp le , the J-f unction of a stationary p oint process (Section 8.6) ca n be written explicitly
in terms of the c onditional intensity: since
P
0
{d(0, X \0) > r}= P
!0
{d(0, X) > r}= E
λ
(0 | X)
λ
1{d(0, X) > r}
where
λ
is the intensity, we have
J(r) =
P
0
{d(0, X \0) > r}
P {d(0,X) > r}
=
E[
λ
(0|X)
λ
1{d(0,X) > r}]
P {d(0, X) > r}
= E
λ
(0 | X)
λ
d(0,X) > r
. (13 .72)
This representation can often be evaluated, while F and G often cannot be evaluated explicitly.
For a Cox process with random driving intensity Λ(u), assuming E
R
W
Λ(u)du < ∞, the condi-
tional intensity has the implicit representation ...