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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2002-11
April 2002

Inhomogeneous Spatial Point Processes by Location Dependent Scaling


Eva B. Vedel Jensen

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)