Bernoulli

On the second-order characteristics of marked point processes

Martin Schlather

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Abstract

New definitions for the second-order characteristics of marked point processes are presented. Non-reduced moment measures are involved in obtaining consistency with ana\-logous definitions in the random field context. A certain σ-algebra replaces the current assumptions on stationarity and isotropy so that the characteristics are still well defined even if such assumptions do not hold. The new definitions and the present ones coincide for positive arguments if the marked point process is simple, stationary and isotropic, and if the second-order product density exists. The different second-order characteristics given in the literature are discussed. A renaming of one of the characteristics is suggested since distinct characteristics have been known under identical names. Furthermore, it is shown that arbitrary measurable functions with compact support can appear as second-order characteristics of marked point processes.

Article information

Source
Bernoulli, Volume 7, Number 1 (2001), 99-117.

Dates
First available in Project Euclid: 29 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080572341

Mathematical Reviews number (MathSciNet)
MR1811746

Zentralblatt MATH identifier
0978.60045

Keywords
mark correlation function mark covariance function mark variogram marked point process moment measure second-order characteristic

Citation

Schlather, Martin. On the second-order characteristics of marked point processes. Bernoulli 7 (2001), no. 1, 99--117. https://projecteuclid.org/euclid.bj/1080572341


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References

  • [1] Baddeley, A.J. (1999) Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds), Stochastic Geometry, Likelihood and Computation, pp. 37-78. Boca Raton, FL: Chapman & Hall/CRC.
  • [2] Baddeley, A.J., M¢ller, J.M. and Waagepetersen, R. (1998) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Technical Report 12, Biometry Research Unit, Danish Institute of Agricultural Sciences, Tjele. Abstract can also be found in the ISI/STMA publication
  • [3] Capobianco, R. and Renshaw, E. (1998) The autocovariance function for marked point processes: a comparison between two different approaches. Biometrical J., 40, 431-446. Abstract can also be found in the ISI/STMA publication
  • [4] Cressie, N.A.C. (1993) Statistics for Spatial Data. New York: Wiley.
  • [5] Diggle, P.J. (1983) Statistical Analysis of Point Processes. London: Chapman & Hall.
  • [6] Diggle, P.J., Lange, N. and Benes, F. (1991) Analysis of variance for replicated spatial point patterns in clinical neuroanatomy. J. Amer. Statist. Assoc., 86, 618-625. Abstract can also be found in the ISI/STMA publication
  • [7] Gavrikov, V.L. and Stoyan, D. (1995) The use of marked point processes in ecological and environmental forest studies. Environ. Ecological Statist., 2, 331-344.
  • [8] Goulard, M., Pages, L. and Cananettes, A. (1995) Marked point processes - using correlation functions to explore a spatial data set. Biometrical J., 37, 837-853.
  • [9] Isham, V. (1985) Marked point processes and their correlations. In F. Droesbeke (ed.), Spatial Processes and Spatial Time Series Analysis, pp. 63-75. Brussels: Publications des Facultés Universitaires Saint-Louis.
  • [10] Mase, S. (1996) The threshold method for estimating total rainfall. Ann. Inst. Statist. Math., 48, 201- 213. Abstract can also be found in the ISI/STMA publication
  • [11] Penttinen, A.K. and Stoyan, D. (1989) Statistical analysis for a class of line segment processes. Scand. J. Statist., 16, 153-168. Abstract can also be found in the ISI/STMA publication
  • [12] Penttinen, A.K., Stoyan, D. and Henttonen, H.M. (1992) Marked point processes in forest statistics. Forest Sci., 38, 806-824.
  • [13] Ripley, B.D. (1977) Modelling spatial patterns. J. Roy. Stat. Soc. Ser. B, 39, 172-192.
  • [14] Ripley, B.D. (1981) Spatial Statistics. New York: Wiley.
  • [15] Sasvári, Z. (1994) Positive Definite and Definitizable Functions. Berlin: Akademie Verlag.
  • [16] Stoyan, D. (1984a) Correlations of the marks of marked point processes - statistical inference and simple models. Elektron. Informationsverarbeitung Kybernetik, 20, 285-294.
  • [17] Stoyan, D. (1984b) On correlations of marked point processes. Math. Nachr., 116, 197-207.
  • [18] Stoyan, D. (1990) Stereological formulae for a random system of nonintersecting spheres. Statistics, 21, 131-236. Abstract can also be found in the ISI/STMA publication
  • [19] Stoyan, D. and Stoyan, H. (1994) Fractals, Random Shapes and Point Fields. Chichester: Wiley.
  • [20] Stoyan, D., Kendall, W.S. and Mecke, J. (1995) Stochastic Geometry and its Applications, 2nd edition. Chichester: Wiley.
  • [21] Wackernagel, H. (1995) Multivariate Geostatistics. Berlin: Springer-Verlag.
  • [22] Wälder, O. and Stoyan, D. (1996) On variograms in point process statistics. Biometrical J., 38, 895- 905.
  • [23] Wälder, O. and Stoyan, D. (1997) Models of marking and thinning of Poisson processes. Statistics, 29, 179-202.
  • [24] Wälder, O. and Stoyan, D. (1998) On variograms in point process statistics: Erratum. Biometrical J., 40, 109.
  • [25] Wen, R. and Sinding-Larsen, R. (1997) Stochastic modelling and simulation of small faults by marked point processes and kriging. In E.Y. Baafi and N.A. Schofield (eds), Geostatistics Wollongong '96, Vol. 1, pp. 398-414. Dordrecht: Kluwer.