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Gaussian Random Particles with Flexible Hausdorff Dimension

  • Linda V. Hansen (a1), Thordis L. Thorarinsdottir (a2), Evgeni Ovcharov (a3), Tilmann Gneiting (a4) and Donald Richards (a5)...

Abstract

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.

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Corresponding author

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.
∗∗∗∗ Email address: evgeni.ovcharov@h-its.org
∗∗∗∗∗ Email address: tilmann.gneiting@h-its.org
∗∗∗∗∗∗ 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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