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Inhomogeneous spatial point processes by location-dependent scaling

Part of: Inference from stochastic processes Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Ute Hahn ,
Eva B. Vedel Jensen ,
Marie-Colette van Lieshout  and
Linda Stougaard Nielsen
Ute Hahn*
Affiliation:
University of Aarhus
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
Marie-Colette van Lieshout*
Affiliation:
CWI, Amsterdam
Linda Stougaard Nielsen*
Affiliation:
University of Aarhus
*
∗ Current address: Department of Mathematics, Augsburg University, D-86135 Augsburg, Germany. Email address: hahn@math.uni-augsburg.de
∗∗ Postal address: Laboratory for Computational Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗∗∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands.
∗∗ Postal address: Laboratory for Computational Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
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Abstract

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 intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted 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, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Keywords

Point processinhomogeneitylocal scalingsequential point process

MSC classification

Primary: 60G55: Point processes
Secondary: 62M30: Spatial processes 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 319 - 336
Copyright
Copyright © Applied Probability Trust 2003 

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References

[1] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
[2] Baddeley, A. J., Möller, J. and Waagepetersen, R. (2000). Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statist. Neerlandica 54, 329–350.Google Scholar
[3] Clausen, W. H. O., Rasmussen, H. H. and Rasmussen, M. H. (2000). Inhomogeneous point processes. Doctoral Thesis, Aalborg University.Google Scholar
[4] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[5] Geyer, C. (1999). Likelihood inference for spatial point patterns. In Stochastic Geometry—Likelihood and Computation, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M., Chapman and Hall, London, pp. 79–140.Google Scholar
[6] Hahn, U. et al. (1999). Stereological analysis and modelling of gradient structures. J. Microscopy 195, 113–124.Google Scholar
[7] Harary, F. (1969). Graph Theory. Addison-Wesley, Reading, MA.Google Scholar
[8] Jensen, E. B. V. and Nielsen, L. S. (2000). Inhomogeneous Markov point processes by transformation. Bernoulli 6, 761–782.Google Scholar
[9] Jensen, E. B. V. and Nielsen, L. S. (2001). A review on inhomogeneous spatial point processes. In Selected Proc. Symp. Inference for Stoch. Process. (IMS Lecture Notes Monogr. Ser. 37), eds Basawa, I. V., Heyde, C. C. and Taylor, R. L., Institute of Mathematical Statistics, Beachwood, OH, pp. 297–318.CrossRefGoogle Scholar
[10] Nielsen, L. S. (2001). Point process models allowing for interaction and inhomogeneity. Doctoral Thesis, University of Aarhus.Google Scholar
[11] Nielsen, L. S. and Jensen, E. B. V. (2003). Statistical inference for transformation inhomogeneous point processes. To appear in Scand. J. Statist.Google Scholar
[12] Ogata, Y. and Tanemura, M. (1986). Likelihood estimation of interaction potentials and external fields of inhomogeneous spatial point patterns. In Proc. Pacific Statist. Congress 1985, eds Francis, I. S., Manly, B. F. J. and Lam, F. C., Elsevier, Amsterdam, pp. 150–154.Google Scholar
[13] Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. 15, 188–192.Google Scholar
[14] Stoyan, D. and Stoyan, H. (1998). Non homogeneous Gibbs process models for forestry—a case study. Biometrical J. 40, 521–531.3.0.CO;2-R>CrossRefGoogle Scholar
[15] Strauss, D. J. (1975). A model for clustering. Biometrika 63, 467–475.Google Scholar
[16] Van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications. World Scientific, Singapore.Google Scholar
[17] Van Lieshout, M. N. M. and Molchanov, I. S. (1998). Shot noise weighted processes: a new family of spatial point processes. Commun. Statist. Stoch. Models 14, 715–734.Google Scholar