Abstract
Spatial point processes play a fundamental role in spatial statistics. In the simplest case they model “small” objects that may be identified by a map of points showing stores, towns, plants, nests, or cases of a disease observed in a two dimensional region or galaxies observed in a three dimensional region. The points may be decorated with marks (such as sizes or types) whereby marked point processes are obtained. The areas of applications are manifold: astronomy, geography, ecology, forestry, spatial epidemiology, image analysis, and many more. Currently spatial point processes is an active area of research, which probably will be of increasing importance for many new applications.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adler, R. (1981). The Geometry of Random Fields, Wiley, New York.
Baddeley, A. & Gill, R. D. (1997). Kaplan-Meier estimators of distance distributions for spatial point processes, Annals of Statistics 25: 263–292.
Baddeley, A. J., Moyeed, R. A., Howard, C. V. & Boyde, A. (1993). Analysis of a three-dimensional point patttern with applications, Applied Statistics 42: 641–668.
Baddeley, A. J. & van Lieshout, M. N. M. (1993). Stochastic geometry models in high-level vision, in K. V. Mardia & G. K. Kanji (eds), Statistics and Images, Advances in Applied Statistics, a supplement to the Journal of Applied Statistics, Vol. 20, Carfax Publishing, Abingdon, chapter 11, pp. 235–256.
Baddeley, A. J. & van Lieshout, M. N. M. (1995). Area-interaction point processes, Annals of the Institute of Statistical Mathematics 46: 601–619.
Baddeley, A. & Moller, J. (1989). Nearest-neighbour Markov point pro- cesses and random sets, International Statistical Review 2: 89–121.
Baddeley, A., Moller, J. & Waagepetersen, R. (2000). Non-and semi-parametric estimation of interaction in inhomogeneous point patterns, Statistica Neerlandica 54: 329–350.
Baddeley, A. & Silverman, B. W. (1984). A cautionary example for the use of second-order methods for analysing point patterns, Biometrics 40: 1089–1094.
Baddeley, A. & Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns, Australian and New Zealand Journal of Statistics 42: 283–322.
Bartlett, M. S. (1963). The spectral analysis of point processes, Journal of the Royal Statistical Society Series B 29: 264–296.
Bartlett, M. S. (1964). The spectral analysis of two-dimensional point processes, Biometrika 51: 299–311.
Benes, V., Bodlak, K., Moller, J. & Waagepetersen, R. P. (2002). Bayesian analysis of log Gaussian Cox process models for disease mapping, Technical Report R-02–2001, Department of Mathematical Sciences, Aalborg University.
Berthelsen, K. K. & Moller, J. (2001a). Perfect simulation and inference for spatial point processes, Technical Report R-01–2009, Department of Mathematical Sciences, Aalborg University. Conditionally accepted for publication in the Scandinavian Journal of Statistics.
Berthelsen, K. K. & Moller, J. (2001b). A primer on perfect simulation for spatial point processes, Technical Report R-01–2026, Department of Mathematical Sciences, Aalborg University. To appear in Bulletin of the Brazilian Mathematical Society, 33, 2003.
Berman, M. & Turner, T. R. (1992). Approximating point process likelihoods with GLIM, Applied Statistics 41: 31–38.
Berthelsen, K. K. & Moller, J. (2002). Spatial jump processes and perfect simulation, in K. Mecke & D. Stoyan (eds), Morphology of Condensed Matter, Lecture Notes in Physics, Springer-Verlag, Heidelberg. To appear.
Besag, J. (1977a). Some methods of statistical analysis for spatial data, Bulletin of the Institute of International Statistics 47: 77–92.
Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion), Journal of the Royal Statistical Society Series B 36: 192–236.
Besag, J. E. (1975). Statistical analysis of non-lattice data, The Statistician 24: 179–195.
Besag, J. E. (1977b). Discussion on the paper by Ripley (1977), Journal of the Royal Statistical Society Series B 39: 193–195.
Besag, J. E. (1994). Discussion on the paper by Grenander and Miller, Journal of the Royal Statistical Society Series B 56: 591–592.
Besag, J., Milne, R. & Zachary, S. (1982). Point process limits of lattice processes, Journal of Applied Probability 19: 210–216.
Breyer, L. A. & Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling, Stochastic Processes and their Applications 90: 181–206.
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes, Advances in Applied Probability 31: 929–953.
Brix, A. & Chadoeuf, J. (2000). Spatio-temporal modeling of weeds and shot-noise G Cox processes. Submitted.
Brix, A. & Kendall, W. S. (2002). Simulation of cluster point processes without edge effects, Advances in Applied Probability 34: 267–280.
Brix, A. & Moller, J. (2001). Space-time multitype log Gaussian Cox processes with a view to modelling weed data, Scandinavian Journal of Statistics 28: 471–488.
Chan, K. S. & Geyer, C. J. (1994). Discussion of the paper ‘Markov chains for exploring posterior distributions’ by Luke Tierney, Annals of Statistics 22: 1747–1747.
Christensen, O. F., Moller, J. & Waagepetersen, R. P. (2001). Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalised linear mixed models, Methodology and Computing in Applied Probability 3: 309–327.
Christensen, O. F. & Waagepetersen, R. (2002). Bayesian prediction of spatial count data using generalised linear mixed models, Biometrics 58: 280–286.
Coles, P. & Jones, B. (1991). A lognormal model for the cosmological mass distribution, Monthly Notices of the Royal Astronomical Society 248: 1–13.
Cressie, N. A. C. (1993). Statistics for Spatial Data, second edn, Wiley, New York.
Daley, D. J. & Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer-Verlag, New York.
Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns, Academic Press, London.
Diggle, P. J. (1985). A kernel method for smoothing point process data, Applied Statistics 34: 138–147.
Diggle, P. J., Lange, N. & Benés, F. (1991). Analysis of variance for replicated spatial point patterns in clinical neuroanatomy, Journal of the American Statistical Association 86: 618–625.
Diggle, P. J., Mateu, L. & Clough, H. E. (2000). A comparison between parametric and non-parametric approaches to the analysis of replicated spatial point patterns, Advances of Applied Probability 32: 331–343.
Fernandez, R., Ferrari, P. A. & Garcia, N. L. (1999). Perfect simulation for interacting point processes, loss networks and Ising models. Manuscript.
Fiksel, T. (1984). Estimation of parameterized pair potentials of marked and nonmarked Gibbsian point processes, Elektronische Informationsverarbeitung and Kypernetik 20: 270–278.
Gelfand, A. E. (1996). Model determination using sampling-based methods, in W. R. Gilks, S. Richardson & D. J. Spiegelhalter (eds), Markov chain Monte Carlo in Practice, Chapman and Hall, London, pp. 145–161.
Gelman, A. & Meng, X.-L. (1998). Simulating normalizing constants: from importance sampling to bridge sampling to path sampling, Statistical Science 13: 163–185.
Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems, Communications of Mathematical Physics 48: 31–51.
Georgii, H.-O. (1988). Gibbs Measures and Phase Transition, Walter de Gruyter, Berlin.
Geyer, C. J. (1991). Markov chain Monte Carlo maximum likelihood, Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 156–163.
Geyer, C. J. (1994). On the convergence of Monte Carlo maximum likelihood calculations, Journal of the Royal Society of Statistics Series B 56: 261–274.
Geyer, C. J. (1999). Likelihood inference for spatial point processes, in O. E. Barndorff-Nielsen, W. S. Kendall & M. N. M. van Lieshout (eds), Stochastic Geometry: Likelihood and Computation, Chapman and Hall/CRC, London, Boca Raton, pp. 79–140.
Geyer, C. J. & Moller, J. (1994). Simulation procedures and likelihood inference for spatial point processes, Scandinavian Journal of Statistics 21: 359–373.
Geyer, C. J. & Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data, Journal of the Royal Society of Statistics Series B 54: 657–699.
Goulard, M., Särkkä, A. & Grabarnik, P. (1996). Parameter estimation for marked Gibbs point processes through the maximum pseudo-likelihood method, Scandinavian Journal of Statistics 23: 365–379.
Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination, Biometrika 82: 711–732.
Gu, M. G. & Zhu, H.-T. (2001). Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation, Journal of the Royal Statistical Society Series B 63: 339–355.
Häggström, O., van Lieshout, M. N. M. & Moller, J. (1999). Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes, Bernoulli 5: 641–659.
Heikkinen, J. & Arjas, E. (1998). Non-parametric Bayesian estimation of a spatial Poisson intensity, Scandinavian Journal of Statistics 25: 435–450.
Heikkinen, J. & Penttinen, A. (1999). Bayesian smoothing in the estimation of the pair potential function of Gibbs point processes, Bernoulli 5: 1119–1136.
Jensen, E. B. V. & Nielsen, L. S. (2001). A review on inhomogeneous spatial point processes, in I. V. Basawa, C. C. Heyde & R. L. Taylor (eds), Selected Proceedings of the Symposium on Inference for Stochastic Processes, Vol. 37, IMS Lecture Notes & Monographs Series, Beachwood, Ohio, pp. 297–318.
Jensen, J. L. & Künsch, H. R. (1994). On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46: 475–486.
Jensen, J. L. & Moller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes, Annals of Applied Probability 3: 445–461.
Kallenberg, O. (1975). Random Measures, Akadamie-Verlag, Berlin.
Kallenberg, O. (1984). An informal guide to the theory of conditioning in point processes, International Statistical Review 52: 151–164.
Karr, A. F. (1991). Point Processes and Their Statistical Inference, Marcel Dekker, New York.
Kelly, F. P. & Ripley, B. D. (1976). A note on Strauss’ model for clustering, Biometrika 63: 357–360.
Kendall, W. S. (1998). Perfect simulation for the area-interaction point process, in L. Accardi & C. Heyde (eds), Probability Towards 2000, Springer, pp. 218–234.
Kendall, W. S. & Moller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes, Advances in Applied Probability 32: 844–865.
Kerscher, M. (2000). Statistical analysis of large-scale structure in the Universe, in K. R. Mecke & D. Stoyan (eds), Statistical Physics and Spatial Statistics, Lecture Notes in Physics, Springer, Berlin, pp. 36–71.
Kerstan, J., Matthes, K. & Mecke, J. (1974). Unbegrenzt teilbare Punktprozesse, Akademie-Verlag, Berlin.
Kingman, J. F. C. (1993). Poisson Processes, Clarendon Press, Oxford.
Lieshout, M. N. M. van (2000). Markov Point Processes and Their Applications, Imperial College Press, London.
Lieshout, M. N. M. van & Baddeley, A. J. (1996). A nonparametric measure of spatial interaction in point patterns, Statistica Neerlandica 50: 344–361.
Loizeaux, M. A. & McKeague, I. W. (2001). Perfect sampling for posterior landmark distributions with an application to the detection of disease clusters, in I. V. Basawa, C. C. Heyde & R. L. Taylor (eds), Selected Proceedings of the Symposium on Inference for Stochastic Processes, Vol. 37, IMS Lecture Notes & Monographs Series, Beachwood, Ohio, pp. 321–331.
Lund, J., Penttinen, A. & Rudemo, M. (1999). Bayesian analysis of spatial point patterns from noisy observations. Available at http://www.math.chalmers.se/Stat/ Research/Preprints/.
Lund, J. & Rudemo, M. (2000). Models for point processes observed with noise, Biometrika 87: 235–249.
Lund, J. & Thönnes, E. (2000). Perfect simulation for point processes given noisy observations. Research Report 366, Department of Statistics, University of Warwick.
Mase, S. (1995). Consistency of the maximum pseudo-likelihood estimator of continuous state space Gibbs processes, Annals of Applied Probability 5: 603–612.
Mase, S. (1999). Marked Gibbs processes and asymptotic normality of maximum pseudo-likelihood estimators, Mathematische Nachrichten 209: 151–169.
Mase, S., Moller, J., Stoyan, D., Waagepetersen, R. P. & Döge, G. (2001). Packing densities and simulated tempering for hard core Gibbs point processes, Annals of the Institute of Statistical Mathematics 53: 661–680.
Matérn, B. (1960). Spatial Variation. Meddelanden frân Statens Skogforskningsinstitut, Band 49, No. 5.
Matérn, B. (1986). Spatial Variation, Lecture Notes in Statistics. Springer-Verlag, Berlin.
Matheron, G. (1975). Random Sets and Integral Geometry, Wiley, New York.
Mecke, J. (1967). Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 9: 36–58.
Mecke, J., Schneider, R. G., Stoyan, D. & Weil, W. R. R. (1990). Stochastische Geometrie, Birkhäuser Verlag, Basel.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics 21: 1087–1092.
Moller, J. (1989). On the rate of convergence of spatial birth-and-death processes, Annals of the Institute of Statistical Mathematics 3: 565–581.
Moller, J. (1999). Markov chain Monte Carlo and spatial point processes, in O. E. Barndorff-Nielsen, W. S. Kendall & M. N. M. van Lieshout (eds), Stochastic Geometry: Likelihood and Computation, Monographs on Statistics and Applied Probability 80, Chapman and Hall/CRC, Boca Raton, pp. 141–172.
Moller, J. (2001). A review of perfect simulation in stochastic geometry, in I. V. Basawa, C. C. Heyde & R. L. Taylor (eds), Selected Proceedings of the Symposium on Inference for Stochastic Processes, Vol. 37, IMS Lecture Notes & Monographs Series, Beachwood, Ohio, pp. 333–355.
Moller, J. (2002a). A comparison of spatial point process models in epidemiological applications, in P. J. Green, N. L. Hjort & S. Richardson (eds), Highly Structured Stochastic Systems, Oxford University Press, Oxford. To appear.
Moller, J. (2002b). Shot noise Cox processes, Technical Report R-02–2009, Department of Mathematical Sciences, Aalborg University.
Moller, J., Syversveen, A. R. & Waagepetersen, R. P. (1998). Log Gaussian Cox processes, Scandinavian Journal of Statistics 25: 451–482.
Moller, J. & Waagepetersen, R. P. (2002). Statistical inference for Cox processes, in A. B. Lawson & D. Denison (eds), Spatial Cluster Modelling, Chapman and Hall/CRC, Boca Raton.
Moller, J. & Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes, Chapman and Hall/CRC, Boca Raton. In preparation.
Neyman, J. & Scott, E. L. (1958). Statistical approach to problems of cosmology, Journal of the Royal Statistical Society Series B 20: 1–43.
Nguyen, X. X. & Zessin, H. (1979). Integral and differential characteriza- tions of Gibbs processes, Mathematische Nachrichten 88: 105–115.
Ohser, J. & Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science, Wiley, New York.
Peebles, P. J. E. (1974). The nature of the distribution of galaxies, Astronomy and Astrophysics 32: 197–202.
Peebles, P. J. E. & Groth, E. J. (1975). Statistical analysis of extragalactic objects. V. Three-point correlation function for the galaxy distribution in the Zwicky catalog, Astrophysical Journal 196: 1–11.
Penttinen, A. (1984). Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method, Number 7 in Jyväskylä Studies in Computer Science, Economics, and Statistics.
Penttinen, A., Stoyan, D. & Henttonen, H. M. (1992). Marked point processes in forest statistics, Forest Science 38: 806–824.
Preston, C. (1976). Random Fields, Lecture Notes in Mathematics, 534. Springer-Verlag, Berlin-Heidelberg.
Preston, C. J. (1977). Spatial birth-and-death processes, Bulletin of the International Statistical Institute 46: 371–391.
Propp, J. G. & Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9: 223–252.
Quine, M. P. & Watson, D. F. (1984). Radial simulation of n-dimensional Poisson processes, Journal of Applied Probability 21: 548–557.
Rathbun, S. L. (1996). Estimation of Poisson intensity using partially observed concomitant variables, Biometrics 52: 226–242.
Reiss, R.-D. (1993). A Course on Point Processes, Springer Verlag, New York.
Ripley, B. D. (1977). Modelling spatial patterns (with discussion), Journal of the Royal Statistical Society Series B 39: 172–212.
Ripley, B. D. (1979). Simulating spatial patterns: dependent samples from a multivariate density. Algorithm AS 137, Applied Statistics 28: 109–112.
Ripley, B. D. (1981). Spatial Statistics, Wiley, New York.
Ripley, B. D. (1988). Statistical Inference for Spatial Processes, Cambridge University Press, Cambridge.
Ripley, B. D. & Kelly, F. P. (1977). Markov point processes, Journal of the London Mathematical Society 15: 188–192.
Roberts, G. O., Gelman, A. & Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms, Annals of Applied Probability 7: 110–120.
Roberts, G. O. & Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2: 13–25.
Roberts, G. O. & Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions, Journal of the Royal Statistical Society Series B 60: 255–268.
Roberts, G. O. & Tweedie, R. L. (1996). Exponential convergence of Langevin diffusions and their discrete approximations, Bernoulli 2: 341–363.
Rossky, P. J., Doll, J. D. & Friedman, H. L. (1978). Brownian dynamics as smart Monte Carlo simulation, Journal of Chemical Physics 69: 4628–4633.
Ruelle, D. (1969). Statistical Mechanics: Rigorous Results, W.A. Benjamin, Reading, Massachusetts.
Schladitz, K. & Baddeley, A. J. (2000). A third-order point process characteristic, Scandinavian Journal of Statistics 27: 657–671.
Schlather, M. (2001). On the second-order characteristics of marked point processes, Bernoulli 7: 99–117.
Stoyan, D., Kendall, W. S. & Mecke, J. (1995). Stochastic Geometry and Its Applications, second edn, Wiley, Chichester.
Stoyan, D. & Stoyan, H. (1994). Fractals, Random Shapes and Point Fields, Wiley, Chichester.
Stoyan, D. & Stoyan, H. (2000). Improving ratio estimators of second order point process characteristics, Scandinavian Journal of Statistics 27: 641–656.
Strauss, D. J. (1975). A model for clustering, Biometrika 63: 467–475.
Thönnes, E. (1999). Perfect simulation of some point processes for the impatient user, Advances in Applied Probability 31: 69–87.
Waagepetersen, R. & Sorensen, S. (2001). A tutorial on reversible jump MCMC with a view toward applications in QTL-mapping, International Statistical Review 69 (1): 49–61.
Wolpert, R. L. & Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics, Biometrika 85: 251–267.
Wood, A. T. A. & Chan, G. (1994). Simulation of stationary Gaussian processes in [0,1)d, Journal of Computational and Graphical Statistics 3: 409–432.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Møller, J., Waagepetersen, R.P. (2003). An Introduction to Simulation-Based Inference for Spatial Point Processes. In: Møller, J. (eds) Spatial Statistics and Computational Methods. Lecture Notes in Statistics, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21811-3_4
Download citation
DOI: https://doi.org/10.1007/978-0-387-21811-3_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-00136-4
Online ISBN: 978-0-387-21811-3
eBook Packages: Springer Book Archive