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Gaussian Random Particles with Flexible Hausdorff Dimension

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

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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 307 - 327
Copyright
© Applied Probability Trust 

References

Adler, R. J. (2010). The Geometry of Random Fields. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Cressie, N. and Pavlicová, M. (2002). Calibrated spatial moving average simulations. Statist. Modelling 2, 267–279.CrossRefGoogle Scholar
Digital Library of Mathematical Functions (2011). Release 2011-07-01. Available at http://dlmf.nist.gov.Google Scholar
Do Carmo, M. P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Estrade, A. and Istas, J. (2010). Ball throwing on spheres. Bernoulli 16, 953–970.Google Scholar
Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. John Wiley, Chichester.Google Scholar
Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data. Cambridge University Press.Google Scholar
Gagnon, J.-S., Lovejoy, S. and Schertzer, D. (2006). Multifractal earth topography. Nonlinear Process. Geophys. 13, 541–570.Google Scholar
Gangolli, R. (1967). Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters. Ann. Inst. H. Poincaré B 3, 121–226.Google Scholar
Gneiting, T. (1999). Radial positive definite functions generated by Euclid's hat. J. Multivariate Anal. 69, 88–119.Google Scholar
Grenander, U. and Miller, M. I. (1994). Representations of knowledge in complex systems. J. R. Statist. Soc. B 56, 549–603.Google Scholar
Hall, P. and Roy, R. (1994). On the relationship between fractal dimension and fractal index for stationary stochastic processes. Ann. Appl. Prob. 4, 241–253.Google Scholar
Hammersley, J. M. and Nelder, J. A. (1955). Sampling from an isotropic Gaussian process. Proc. Camb. Phil. Soc. 51, 652–662.Google Scholar
Hansen, L. V. and Thorarinsdottir, T. L. (2013). A note on moving average models for Gaussian random fields. Statist. Prob. Lett. 83, 850–855.Google Scholar
Hausdorff, F. (1918). Dimension und äusseres Mass. Math. Ann. 79, 157–179.Google Scholar
Hobolth, A. (2003). The spherical deformation model. Biostatistics 4, 583–595.Google Scholar
Hobolth, A., Kent, J. T. and Dryden, I. L. (2002). On the relation between edge and vertex modelling in shape analysis. Scand. J. Statist. 29, 355–374.Google Scholar
Hobolth, A., Pedersen, J. and Jensen, E. B. V. (2003). A continuous parametric shape model. Ann. Inst. Statist. Math. 55, 227–242.Google Scholar
Jones, R. H. (1963). Stochastic processes on a sphere. Ann. Math. Statist. 34, 213–218.Google Scholar
Jones, T. and Stofan, E. (2008). Planetology: Unlocking the Secrets of the Solar System. National Geographic, Washington, D.C..Google Scholar
Jónsdóttir, K. Y., Schmiegel, J. and Jensen, E. B. V. (2008). Lévy-based growth models. Bernoulli 14, 62–90.CrossRefGoogle Scholar
Kent, J. T., Dryden, I. L. and Anderson, C. R. (2000). Using circulant symmetry to model featureless objects. Biometrika 87, 527–544.Google Scholar
Kucinskas, A. B. et al. (1992). Fractal analysis of Venus topography in Tinatin Planitia and Ovda Regio. J. Geophys. Res. Planets 97, 13635–13641.Google Scholar
Leopardi, P. (2006). A partition of the unit sphere into regions of equal area and small diameter. Electron. Trans. Numer. Anal. 25, 309–327.Google Scholar
Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman, San Francisco, CA.Google Scholar
Miller, M. I. et al. (1994). Membranes, mitochondria and amoebae: shape models. J. Appl. Statist. 21, 141–163.Google Scholar
Muinonen, K. et al. (1996). Light scattering by Gaussian random particles: ray optics approximation. J. Quant. Spectrosc. Radiat. Transfer 55, 577–601.CrossRefGoogle Scholar
Muñoz, O. et al. (2007). Scattering matrix of large Saharan dust particles: experiments and computations. J. Geophys. Res. Atmospheres 112, D13215.Google Scholar
Oliver, D. S. (1995). Moving averages for Gaussian simulation in two and three dimensions. Math. Geology 27, 939–960.Google Scholar
Orford, J. D. and Whalley, W. B. (1983). The use of the fractal dimension to quantify the morphology of irregular-shaped particles. Sedimentology 30, 655–668.CrossRefGoogle Scholar
Price, F. W. (1988). The Moon Observer's Handbook. Cambridge University Press.Google Scholar
R Development Core Team (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.r-project,orgGoogle Scholar
Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields. John Wiley, Chichester.Google Scholar
Tovchigrechko, A. and Vakser, I. A. (2001). How common is the funnel-like energy landscape in protein-protein interactions? Protein Sci. 10, 1572–1583.CrossRefGoogle ScholarPubMed
Turcotte, D. L. (1987). A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars. J. Geophys. Res. Solid Earth 92, E597–E601.Google Scholar
Wicksell, S. D. (1925). The corpuscle problem. A mathematical study of a biometric problem. Biometrika 17, 84–99.Google Scholar
Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85, 251–267.Google Scholar
Wood, A. T. A. (1995). When is a truncated covariance function on the line a covariance function on the circle? Statist. Prob. Lett. 24, 157–164.CrossRefGoogle Scholar
Xue, Y. and Xiao, Y. (2011). Fractal and smoothness properties of space–time Gaussian models. Front. Math. China 6, 1217–1248.Google Scholar
Ziegel, J. (2013). Stereological modelling of random particles. Commun. Statist. Theory Meth. 42, 1428–1442.CrossRefGoogle Scholar
Ziegel, J. (2014). Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Amer. Math. Soc. 142, 2063–2077.CrossRefGoogle Scholar