All three processes were derived from the same homogeneous Thomas process but different types of inhomogeneity were used.
The simplest example of a shot noise Cox process is the so-called modified Thomas process (Thomas, 1949) (in the sequel we will call it shortly a Thomas process).
Inhomogeneous Thomas process with log-linear intensity function (Waagepetersen, 2007): Let X [subset or equal to] [R.sup.2], [X.sub.[theta]] be a Thomas process with mother intensity [mu], mean number of daughter points in a cluster equal to v and k the bivariate normal kernel with scale parameter [sigma].
and [g.sub.i](u - v) = [g.sub.[theta]](u - v) is equal to the g-function of the homogeneous Thomas process
For the inhomogeneous Thomas process with log-linear intensity function the (vector) first order estimation equation (Eq.
This is a realization of the inhomogeneous Thomas process model from the preceding chapter with [beta] = ([[beta].sub.0], [[beta].sub.1]), z(u) = [u.sub.2] and [f.sub.[beta]] = exp([[beta].sub.1] [u.sub.2]).
Let us consider the inhomogeneous Thomas process with [zeta](dr, dw) = [f.sub.[beta]](w)[mu] [[delta].sub.v](dr) dw, i.e., the mothers are distributed inhomogeneously according to the intensity function [f.sub.[beta]][mu].