Skip to main content Accessibility help
×
Home

Pair correlation functions and limiting distributions of iterated cluster point processes

  • Jesper Møller (a1) and Andreas D. Christoffersen (a1)

Abstract

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

Copyright

Corresponding author

* Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark.
** Email address: jm@math.aau.dk

References

Hide All
[1]Andersen, I. T. et al. (2018). Double Cox cluster processes - with applications to photoactivated localization microscopy. Spatial Statist. 27, 5873.
[2]Barndorff-Nielsen, O., Kent, J. and Sørensen, M. (1982). Normal variance-mean mixtures and z distributions. Internat. Statist. Rev. 50, 145159.
[3]Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods, 2nd edn. Springer, New York.
[4]Felsenstein, J. (1975). A pain in the torus: some difficulties with models of isolation by distance. Amer. Naturalist 109, 359368.
[5]Kingman, J. F. C. (1977). Remarks on the spatial distribution of a reproducing population. J. Appl. Prob. 14, 577583.
[6]Lavancier, F., Møller, J. and Rubak, E. (2015). Determinantal point process models and statistical inference. J. R. Statist. Soc. B 77, 853877.
[7]Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83122.
[8]Matérn, B. (1960). Spatial Variation. Meddelanden från Statens Skogforskningsinstitut, Stockholm.
[9]Matérn, B. (1986). Spatial Variation (Lecture Notes Statist. 36), 2nd edn. Springer, Berlin.
[10]McCullagh, P. and Møller, J. (2006). The permanental process. Adv. Appl. Prob. 38, 873888.
[11]Møller, J. (1989). Random tessellations in Rd. Adv. Appl. Prob. 21, 3773.
[12]Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.
[13]Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614640.
[14]Møller, J. and Christoffersen, A. D. (2018). Pair correlation functions and limiting distributions of iterated cluster point processes. Preprint. Available at https://arxiv.org/abs/1711.08984.
[15]Møller, J. and Torrisi, G. L. (2005). Generalised shot noise Cox processes. Adv. Appl. Prob. 37, 4874.
[16]Møller, J. and Torrisi, G. L. (2007). The pair correlation function of spatial Hawkes processes. Statist. Prob. Lett. 77, 9951003.
[17]Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.
[18]Myllymäki, M. et al. (2017). Global envelope tests for spatial processes. J. R. Statist. Soc. B 79, 381404.
[19]Neyman, J. and Scott, E. L. (1958). Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 143.
[20]Shimatani, I. K. (2010). Spatially explicit neutral models for population genetics and community ecology: extensions of the Neyman-Scott clustering process. Theoret. Pop. Biol. 77, 3241.
[21]Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Functional Anal. 205, 414463.
[22]Thomas, M. (1949). A generalization of Poisson's binomial limit for use in ecology. Biometrika 36, 1825.
[23]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, pp. 6186.
[24]Wiegand, T., Gunatilleke, S., Gunatilleke, N. and Okuda, T. (2007). Analyzing the spatial structure of a Sri Lankan tree species with multiple scales of clustering. Ecology 88, 30883102.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed