344 Spatial Point Patterns: Methodology and Applications with R
9.7.4 Ma ximum likelihood for loglinear Poisson models
9.7.4.1 Loglinear models
For the vast majority of Poisson models treated in th is book, the intensity is a loglinear function of
the parameters:
λ
θ
(u) = exp(B(u) + θ
⊤
Z(u)) (9. 44)
where B(u) is a known function (the ‘offset’ or ‘log baseline’), θ = (
θ
1
,... ,
θ
p
) is the vector of
parameters, Z(u) = (Z
1
(u),... , Z
p
(u)) is a vector of covariate f unctions, and
θ
⊤
Z(u) =
θ
1
Z
1
(u) + ···+
θ
p
Z
p
(u).
Note that th e model implies that the logarithm of the intensity is a linear function of the parameters:
log
λ
θ
(u) = B(u) + θ
⊤
Z(u). (9.45)
The functions B and Z
1
,... ,Z
p
could be spa tially varyin g in any fashion, so this is a very wide and
flexible class ...