MPS-RR 2002-11

April 2002

# Inhomogeneous Spatial Point Processes by Location Dependent Scaling

by:

### Ute Hahn, Marie-Colette Van Lieshout, Linda Stougaard Nielsen

A new class of models for inhomogeneous spatial point processes is
introduced. These locally scaled point processes are modifications of
homogeneous template point processes, having the property that regions
with different intensity differ only by a scale factor, i.e. appear to be scaled
versions of the template point process. This is achieved by replacing volume
measures used in the density with locally scaled analogues defined by a
location dependent scaling function. If the scaling function is constant, then
local scaling coincides with global scaling by a constant factor. The new
approach is particularly appealing for modelling inhomogeneous Markov
point processes. Distance-interaction and shot noise Markov point
processes are discussed in detail. It is shown that the locally scaled
versions are again Markov and that locally the Papangelou conditional
intensity of the new process behaves like that of a global scaling of the
homogeneous process. Approximations are suggested that simplify
calculation of the density e.g. in simulation. For sequential point processes,
an alternative and simpler definition of local scaling is proposed.

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This paper has now been published in *Adv. Appl. Prob. (SGSA) 35, 319--336 (2003)*