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On the conditional distributions of spatial point processes

  • François Caron (a1), Pierre Del Moral (a1), Arnaud Doucet (a2) and Michele Pace (a1)

Abstract

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

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Copyright

Corresponding author

Postal address: INRIA Sud-Ouest and Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France.
∗∗ Postal address: Department of Statistics, 333-6356 Agricultural Road, Vancouver BC, V6T 1Z2, Canada. Email address: arnaud@cs.ubc.ca

References

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[1] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer, New York.
[2] Lund, J. and Rudemo, M. (2000). Models for point processes observed with noise. Biometrika, 87, 235249.
[3] Lund, J. and Thönnes, E. (2004). Perfect simulation and inference for point processes given noisy observations. Comput. Statist. 19, 317336.
[4] Lund, J., Penttinen, A. and Rudemo, M. (1999). Bayesian analysis of spatial point patterns from noisy observations. Tech. Rep., Department of Tech. Rep.
[5] Mahler, R. P. S. (2003). Multi-target Bayes filtering via first-order multitarget moments. IEEE Trans. Aerospace Electronic Systems 39, 11521178.
[6] Singh, S. S., Vo, B.-N., Baddeley, A. and Zuyev, S. (2009). Filters for spatial point processes. SIAM J. Control Optimization 48, 22752295.
[7] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.
[8] Van Lieshout, M. N. M. and Baddeley, A. J. (2002). Extrapolating and interpolating spatial patterns. In Spatial Cluster Modelling, Chapman & Hall/CRC, Boca Raton, FL.
[9] Vo, B.-T., Vo, B.-N. and Cantoni, A. (2007). Analytic implementations of the cardinalized probability hypothesis density filter. IEEE Trans. Signal Process. 55, 35533567.

Keywords

MSC classification

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