Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-01T04:46:06.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak homogenization of point processes by space deformations

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

R. Senoussi ,
J. Chadœuf  and
D. Allard
R. Senoussi*
Affiliation:
INRA, France
J. Chadœuf*
Affiliation:
INRA, France
D. Allard*
Affiliation:
INRA, France
*
∗ Postal address: INRA, Laboratoire de Biométrie, Domaine Saint-Paul, Site Agroparc, 84 914 Avignon, Cedex 9, France.
∗ Postal address: INRA, Laboratoire de Biométrie, Domaine Saint-Paul, Site Agroparc, 84 914 Avignon, Cedex 9, France.
∗ Postal address: INRA, Laboratoire de Biométrie, Domaine Saint-Paul, Site Agroparc, 84 914 Avignon, Cedex 9, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the transformation of a non-stationary point process ξ on ℝn into a weakly stationary point process ͂ξ, with ͂ξ(B) = ξ(Φ-1(B)), where B is a Borel set, via a deformation Φ of the space ℝn. When the second-order measure is regular, Φ is uniquely determined by the homogenization equations of the second-order measure. In contrast, the first-order homogenization transformation is not unique. Several examples of point processes and transformations are investigated with a particular interest to Poisson processes.

Keywords

Spatial point processmoment measurehomogenizationnon-stationary point processspace transformation

MSC classification

Primary: 60G55: Point processes
Secondary: 60G12: General second-order processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 948 - 959
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Andersen, P. K., Borgan, O., Gill, R. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.Google Scholar
[2] Baddeley, A., Möller, J. and Waagepetersen, R. P. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statist. Neerlandica 54, 329–350.Google Scholar
[3] Brix, A. (1999). Generalized Gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929–953.Google Scholar
[4] Brix, A. and Möller, J. (1998). Space–time multitype log Gaussian Cox processes with a view to modeling weed data. Res. Rept R-98-2012, Dept Math. Sci., Aalborg University.Google Scholar
[5] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[6] Jensen, E. B. V. and Nielsen, L. S. (2000). Inhomogeneous Markov point processes by transformation. Bernoulli 6, 761–782.Google Scholar
[7] Meiring, W., Monestiez, P., Sampson, P. D. and Guttorp, P. (1997). Developments in the modelling of nonstationary spatial covariance structure from space–time monitoring data. In Geostatistics Wollongong '96, Vol. 1, eds Baafi, E. Y. and Schofield, N.. Kluwer Academic Publishers, Dordrecht, pp. 162–173.Google Scholar
[8] Möller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451–482.Google Scholar
[9] Perrin, O. and Senoussi, R. (1999). Reducing non-stationary stochastic processes to stationarity by a time deformation. Statist. Prob. Lett. 43, 393–397.CrossRefGoogle Scholar
[10] Perrin, O. and Senoussi, R. (2000). Reducing non-stationary random fields to stationarity using a space deformation. Statist. Prob. Lett. 48, 23–32.Google Scholar
[11] Schoenberg, F. (1999). Transforming spatial point processes into Poisson processes. Stoch. Proc. Appl. 81, 155–164.Google Scholar