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The distribution of volume reductions induced by isotropic random projections

Part of: Multivariate analysis Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  01 July 2016

Jørgen Nielsen
Jørgen Nielsen*
Affiliation:
Danish Institute of Agricultural Sciences
*
∗ Postal address: Danish Institute of Agricultural Sciences, Biometry Research Unit, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark. Email address: Jorgen.Nielsen@agrsci.dk
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Abstract

In this paper, isotropic random projections of d-sets in ℝn are studied, where a d-set is a subset of a d-dimensional affine subspace which satisfies certain regularity conditions. The squared volume reduction induced by the projection of a d-set onto an isotropic random p-subspace is shown to be distributed as a product of independent beta-distributed random variables, for d ≤ p. One of the proofs of this result uses Wilks' lambda distribution from multivariate normal theory. The result is related to Cauchy's and Crofton's formulae in stochastic geometry. In particular, it can be used to give a new and quite simple proof of one of the classical Crofton intersection formulae.

Keywords

Beta distributionCauchy's formulaCrofton's formulaisotropic random projectionsmultivariate normal theorystochastic geometryWilks' lambda distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C65: Integral geometry 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 985 - 994
Copyright
Copyright © Applied Probability Trust 1999 

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References

Affentranger, F. and Schneider, R. (1992). Random projections of regular simplices. Discrete Comput. Geom. 7, 219–226.Google Scholar
Chu, D. P. T. (1993). Random r-content of an r-simplex from beta-type-2 random points. Canad. J. Statist. 21, 285–293.CrossRefGoogle Scholar
Jensen, E. B. V. (1995). Rotational versions of the Crofton formula. Adv. Appl. Prob. 27, 87–96.Google Scholar
Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.CrossRefGoogle Scholar
Jensen, E. B. V. and Kiêu, K. (1994). Unbiased stereological estimation of d-dimensional volume in Rn from an isotropic random slice through a fixed point. Adv. Appl. Prob. 26, 1–12.CrossRefGoogle Scholar
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
Nielsen, J. (1996). The volume of isotropic random projections of simplices. Res. Rep. 353, University of Aarhus.Google Scholar
Ruben, H. (1979). The volume of an isotropic random parallelotope. J. Appl. Prob. 16, 84–94.CrossRefGoogle Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.Google Scholar