
436 Spatial Point Patterns: Methodology and Applications with R
The term in brackets is a co mmonly used measure of departure from CSR, and is a sensible diag-
nostic because K(r) =
π
r
2
under CSR. Similarly for the residual G-function, when the fitted model
is CSR,
R G
x
(r) ≈ n(x)(
b
G
x
(r) −
b
F
x
(r)) (11.3 2)
where
b
F
x
(r) is the em pirical estimate of the empty-space function F. The right-hand side of (11.32)
is a reaso nable diagnostic for departure from CSR, since F ≡ G under CSR. This argument lends
support to Diggle’s [218, eq. (5.7)] pro posal to judge departure from CSR using the quantity
max
r
|
ˆ
G(r) −
ˆ
F(r)|.
Residual summary functions can be justified