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Gaussian Random Particles with Flexible Hausdorff Dimension

Part of: Stochastic processes Complex dynamical systems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Linda V. Hansen ,
Thordis L. Thorarinsdottir ,
Evgeni Ovcharov ,
Tilmann Gneiting  and
Donald Richards
Linda V. Hansen*
Affiliation:
Varde College
Thordis L. Thorarinsdottir*
Affiliation:
Norwegian Computing Center
Evgeni Ovcharov*
Affiliation:
Heidelberg Institute for Theoretical Studies
Tilmann Gneiting*
Affiliation:
Heidelberg Institute for Theoretical Studies and Karlsruhe Institute of Technology
Donald Richards*
Affiliation:
Penn State University
*
∗ Postal address: Varde College, Frisvadvej 72, 6800 Varde, Denmark. Email address: lv@varde-gym.dk
∗∗ Postal address: Norwegian Computing Center, PO Box 114, Blindern, 0314 Oslo, Norway. Email address: thordis@nr.no
∗∗∗ Postal address: Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany.
∗∗∗ Postal address: Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany.
∗∗∗∗∗∗ Postal address: Department of Statistics, Penn State University, 326 Thomas Building, University Park, PA 16802, USA. Email address: richards@stat.psu.edu
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Abstract

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Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises-Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.

Keywords

Celestial bodycorrelation functionfractal dimensionLévy basisrandom field on a spheresimulation of star-shaped random set

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G60: Random fields 37F35: Conformal densities and Hausdorff dimension
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 307 - 327
Copyright
© Applied Probability Trust 

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