Hostname: page-component-89b8bd64d-4ws75 Total loading time: 0 Render date: 2026-05-08T10:25:58.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Lévy-based Cox point processes

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

Published online by Cambridge University Press:  01 July 2016

Gunnar Hellmund ,
Michaela Prokešová  and
Eva B. Vedel Jensen
Gunnar Hellmund*
Affiliation:
University of Aarhus
Michaela Prokešová*
Affiliation:
University of Aarhus
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
∗ Postal address: T. N. Thiele Centre for Applied Mathematics in Natural Science, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark.
∗∗ Current address: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 18675 Praha 8, Czech Republic.
∗ Postal address: T. N. Thiele Centre for Applied Mathematics in Natural Science, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

Keywords

Cox processinfinitely divisible distributioninhomogeneitykernel smoothingLévy basislog Gaussian Cox processmixingproduct densityshot noise Cox process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 603 - 629
Copyright
Copyright © Applied Probability Trust 2008 

References

Baddeley, A. J., Møller, J. and Waagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statist. Neerlandica 54, 329–350.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). Lévy based tempo-spatial modeling; with applications to turbulence. Uspekhi Mat. Nauk 159, 63–90.Google Scholar
Barndorff-Nielsen, O. E., Blæsild, P. and Schmiegel, J. (2004). A parsimonious and universal description of turbulent velocity increments. Europ. Phys. J. B 41, 345–363.Google Scholar
Best, N. G., Ickstadt, K. and Wolpert, R. L. (2000). Spatial Poisson regression for health and exposure data measured at disparate resolutions. J. Amer. Statist. Assoc. 95, 1076–1088.Google Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929–953.Google Scholar
Brix, A. and Chadœuf, J. (2002). Spatio-temporal modeling of weeds by shot-noise G Cox processes. Biom. 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.CrossRefGoogle Scholar
Brix, A. and Møller, J. (2001). Space-time multitype log Gaussian Cox processes with a view to modeling weed data. Scand. J. Statist. 28, 471–488.CrossRefGoogle Scholar
Coles, P. and Jones, B. (1991). A lognormal model for the cosmological mass distribution. Monthly Notes R. Astronomical Soc. 248, 1–13.Google Scholar
Comtet, L. (1970). Analyse Combinatoire. Tomes I, II. Presses Universitaires de France, Paris.Google Scholar
Cox, D. R. (1955). Some statistical models related with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edn. Hodder Arnold, London.Google Scholar
Diggle, P., Rowlingson, B. and Su, T. (2005). Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics 16, 423–434.Google Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.Google Scholar
Ferguson, T. S. and Klass, M. J. (1972). A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43, 1634–43.Google Scholar
Hahn, U. (2007). Global and Local Scaling in the Statistics of Spatial Point Processes. Habilitationsschrift, Institut für Mathematik, Universität Augsburg.Google Scholar
Hahn, U., Jensen, E. B. V., van Lieshout, M. N. M. and Nielsen, L. S. (2003). Inhomogeneous spatial point processes by location dependent scaling. Adv. Appl. Prob. 35, 319–336.CrossRefGoogle Scholar
Heikkinen, J. and Arjas, E. (1998). Non-parametric Bayesian estimation of a spatial Poisson intensity. Scand. J. Statist. 25, 435–450.Google Scholar
Hellmund, G. (2005). Lévy driven Cox processes with a view towards modelling tropical rain forest. , Department of Mathematical Sciences, University of Aarhus.Google Scholar
Higdon, D. (2002). Space and space-time modeling using process convolutions. In Quantitative Methods for Current Environmental Issues, eds Anderson, C. W. et al., Springer, London, pp. 37–56.Google Scholar
Jensen, E. B. V., Jónsdóttir, K. Y., Schmiegel, J. and Barndorff-Nielsen, O. E. (2007). Spatio-temporal modeling — with a view to biological growth. In Statistics of Spatio-Temporal Systems, eds Finkenstadt, B. F. et al., Chapman & Hall/CRC, Boca Raton, FL, pp. 47–75.Google Scholar
Jónsdóttir, K. Y., Schmiegel, J. and Jensen, E. B. V. (2008). Lévy-based growth models. Bernoulli 14, 62–90.CrossRefGoogle Scholar
Khoshnevisan, D. (2002). Multiparameter Processes. An Introduction to Random Fields. Springer, New York.Google Scholar
Matérn, B. (1986). Spatial Variation. (Lecture Notes Statist. 36), 2nd edn. Springer, Berlin.Google Scholar
McCullagh, P. and Møller, J. (2006). The permanental process. Adv. Appl. Prob. 38, 873–888.Google Scholar
Møller, J. (2003). A comparison of spatial point process models in epidemiological applications. In Highly Structured Stochastic Systems, eds Green, P. J. et al., Oxford University Press, pp. 264–268.CrossRefGoogle Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614–640.CrossRefGoogle Scholar
Møller, J. and Torrisi, G. L. (2005). Generalised shot noise Cox processes. Adv. Appl. Prob. 37, 48–74.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 & Hall/CRC, Boca Raton, FL, pp. 37–60.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Møller, J. and Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scand. J. Statist. 34, 643–684.Google Scholar
Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451–482.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
Prokešová, M., Hellmund, G. and Jensen, E. B. V. (2006). On spatio-temporal Lévy based Cox processes. In Proc. S4G, Internat. Conf. Stereology, Spatial Statist. Stoch. Geometry (Prague, June 2006), eds Lechnerová, R., et al., Union Czech Mathematicians and Physicists, Prague, pp. 111–116.Google Scholar
Rajput, R. S. and Rosinski, J. (1989). Spectral representation of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.Google Scholar
Rathburn, S. R. and Cressie, N. (1994). A space-time survival point process for longleaf pine forest in southern Georgia. J. Amer. Statist. Assoc. 89, 1164–1174.Google Scholar
Richardson, S. (2003). Spatial models in epidemiological applications. In Highly Structured Stochastic Systems, eds Green, P. J., et al., Oxford University Press, pp. 237–259.Google Scholar
Rosinski, J. (2008). Simulation of Lévy processes. In Encyclopedia of Statistics in Quality and Reliability: Computational Intensive Methods and Simulation. John Wiley.Google Scholar
Sato, K. -I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.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. and Baddeley, A. J. (1996). A nonparametric measure of spatial interaction in point patterns. Statist. Neerlandica 50, 344–361.CrossRefGoogle Scholar
Waagepetersen, R. P. (2005). Discussion of the paper by Baddeley, Turner, Møller and Hazelton (2005). J. R. Statist. Soc. B 67, 662.Google Scholar
Waagepetersen, R. P. (2007). Personal communication.Google Scholar
Walker, S. and Damien, P. (2000). Representations of Lévy processes without Gaussian components. Biometrika 87, 477–483.Google Scholar
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, 3088–3102.Google Scholar
Wolpert, R. L. (2001). Lévy moving averages and spatial statistics. Lecture. Slides available at http://citeseer.ist.psu.edu/wolpert01leacutevy.html.Google Scholar
Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random fields models for spatial statistics. Biometrika 85, 251–267.Google Scholar
Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random fields models for spatial statistics. Simulation of Lévy random fields. In Practical Nonparametric and Semiparametric Bayesian Statistics, (Lecture Notes Statist. 133), eds Dey, D. et al., Springer, New York, pp. 227–242.Google Scholar