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Generalised shot noise Cox processes

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

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Giovanni Luca Torrisi
Jesper Møller*
Affiliation:
Aalborg University
Giovanni Luca Torrisi*
Affiliation:
CNR-Istituto per le Applicazioni del Calcolo ‘M. Picone’
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark. Email address: jm@math.auc.dk
∗ Postal address: CNR-Istituto per le Applicazioni del Calcolo ‘M. Picone’, Viale del Policlinico 137, I-00161 Rome, Italy. Email address: torrisi@iac.rm.cnr.it
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Abstract

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We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

Keywords

Cluster processconditional simulationCox processedge effectgeneralised shot noise G Cox processgeometric ergodicityMarkov point processMetropolis-Hastings algorithmPalm distributionperfect simulationshot noise Cox processspatial point processsummary statistic

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 62M30: Spatial processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 68U20: Simulation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 48 - 74
Copyright
Copyright © Applied Probability Trust 2005 

References

Allen, M. P. and Tildesley, D. J. (1987). Computer Simulation of Liquids. Oxford University Press.Google Scholar
Baddeley, A. and Møller, J. (1989). Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 2, 89–121.Google Scholar
Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 46, 601–619.Google Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929–953.CrossRefGoogle Scholar
Brix, A. and Chadœuf, J. (2002). Spatio-temporal modeling of weeds by shot-noise G Cox processes. Biometrical J.. 44, 83–99.3.0.CO;2-W>CrossRefGoogle Scholar
Brix, A. and Diggle, P. J. (2001). Spatio-temporal prediction for log-Gaussian Cox processes. J. R. Statist. Soc. B 63, 823–841.Google Scholar
Brix, A. and Kendall, W. S. (2002). Simulation of cluster point processes without edge effects. Adv. Appl. Prob. 34, 267–280.Google Scholar
Brix, A. and Møller, J. (2001). Space-time multi type log Gaussian Cox processes with a view to modelling weeds. Scand. J. Statist. 28, 471–488.Google Scholar
Cox, D. R. and Isham, V. (1980). Point Processes. Chapman and Hall, London.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Elementary Theory and Methods, Vol. 1, 2nd edn. Springer, New York.Google Scholar
Dellaportas, P. and Roberts, G. O. (2003). An introduction to MCMC. In Spatial Statistics and Computational Methods (Lecture Notes Statist. 173), ed. Møller, J.. Springer, New York, pp. 1–41.Google Scholar
Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edn. Arnold, London.Google Scholar
Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51.Google Scholar
Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry: Likelihood and Computation, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van, M. N. M. Lieshout, Chapman and Hall/CRC, Boca Raton, FL, pp. 79–140.Google Scholar
Geyer, C. J. and Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
Green, P. J. (1995). Reversible Jump MCMC computation and Bayesian model determination. Biometrika 82, 711–732.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
Karr, A. F. (1991). Point Processes and Their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kelly, F. P. and Ripley, B. D. (1976). A note on Strauss's model for clustering. Biometrika 63, 357–360.Google Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.Google Scholar
Kingman, J. F. C. (1977). Remarks on the spatial distribution of a reproducing population. J. Appl. Prob. 14, 577–583.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
McKeague, I. W. and Loizeaux, M. A. (2002). Perfect sampling for point process cluster modelling. In Spatial Cluster Modelling, eds Lawson, A. B. and Denison, D., Chapman and Hall/CRC, Boca Raton, FL, pp. 87–107.Google Scholar
Matérn, B. (1960). Spatial Variation, Meddelanden från Statens Skogforskningsinstitut, Stockholm.Google Scholar
Matérn, B. (1986). Spatial Variation (Lecture Notes Statist. 36), 2nd edn. Springer, Berlin.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Møller, J. (1994). Contribution to the discussion of N. L. Hjort and H. Omre (1994): Topics in spatial statistics. Scand. J. Statist. 21, 346–349.Google Scholar
Møller, J. (1999). Markov chain Monte Carlo and spatial point processes. In Stochastic Geometry: Likelihood and Computation, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M. (Monogr. Statist. Appl. Prob. 80), Chapman and Hall/CRC, Boca Raton, FL, pp. 141–172.Google Scholar
Møller, J. (2003). A comparison of spatial point process models in epidemiological applications. In Highly Structured Stochastic Systems, eds Green, P. J., Hjort, N. L. and Richardson, S., Oxford University Press, pp. 264–268.Google Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614–640.Google Scholar
Møller, J. and Waagepetersen, R. P. (2002). Statistical inference for Cox processes. In Spatial Cluster Modelling, eds Lawson, A. B. and Denison, D., Chapman and Hall/CRC, Boca Raton, FL, pp. 37–60.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). An introduction to simulation-based inference for spatial point processes. In Spatial Statistics and Computational Methods (Lecture Notes Statist. 173), ed. Møller, J., Springer, New York, pp. 143–198.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451–482.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Neyman, J. and Scott, E. L. (1958). Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
Nguyen, X. X. and Zessin, H. (1979). Integral and differential characterizations of Gibbs processes. Math. Nachr. 88, 105–115.Google Scholar
Preston, C. (1976). Random Fields (Lecture Notes Math. 534). Springer, Berlin.Google Scholar
Richardson, S. (2003). Spatial models in epidemiological applications. In Highly Structured Stochastic Systems, eds Green, P. J., Hjort, N. L. and Richardson, S., Oxford University Press, pp. 237–259.CrossRefGoogle Scholar
Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172–212.Google Scholar
Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. Lond. Math. Soc. 15, 188–192.Google Scholar
Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. W. A. Benjamin, Reading, MA.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Strauss, D. J. (1975). A model for clustering. Biometrika 63, 467–475.Google Scholar
Thomas, M. (1949). A generalization of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.Google Scholar
Van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications. Imperial College Press, London.CrossRefGoogle Scholar
Van Lieshout, M. N. M. and Baddeley, A. J. (1996). A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361.Google Scholar
Van Lieshout, M. N. M. and Baddeley, A. J. (2002). Extrapolating and interpolating spatial patterns. In Spatial Cluster Modelling, eds Lawson, A. B. and Denison, D., Chapman and Hall/CRC, Boca Raton, FL, pp. 61–86.Google Scholar
Widom, B. and Rowlinson, J. S. (1970). A new model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52, 1670–1684.CrossRefGoogle Scholar
Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85, 251–267.CrossRefGoogle Scholar