Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T08:49:53.184Z Has data issue: false hasContentIssue false
  • English
  • Français

Palm representation and approximation of the covariance of random closed sets

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Stephan Böhm  and
Volker Schmidt
Stephan Böhm*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
∗ Postal address: Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C(1)(0), and in many cases also limr→∞C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C(1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C(2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C(2)(0).

Keywords

Stochastic geometryrandom closed setcovariancePalm calculus, exponential approximationBoolean model

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 295 - 302
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[2] Franken, P., König, D., Arndt, U. and Schmidt, V. (1982). Queues and Point Processes. John Wiley, Chichester.Google Scholar
[3] Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1999). First contact distributions for spatial patterns: regularity and estimation. Adv. Appl. Prob. 31, 15–33.Google Scholar
[4] Last, G. and Holtmann, M. (1999). On the empty space function of some germ-grain models. Pattern Recognition 32, 1587–1600.Google Scholar
[5] Last, G. and Schassberger, R. (2001). On the second derivative of the spherical contact distribution function of smooth grain models. Prob. Theory Relat. Fields 121, 49–72.CrossRefGoogle Scholar
[6] Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
[7] Ohser, J. and Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley, Chichester.Google Scholar
[8] Stoyan, D. and Stoyan, H. (1995). Fractals, Random Shapes and Point Fields. John Wiley, Chichester.Google Scholar
[9] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar