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On the induced distribution of the shape of the projection of a randomly rotated configuration

  • H. Le (a1) and D. Barden (a2)

Abstract

Using the geometry of the Kendall shape space, in this paper we study the shape, as well as the size-and-shape, of the projection of a configuration after it has been rotated and, when the given configuration lies in a Euclidean space of an arbitrary dimension, we obtain expressions for the induced distributions of such shapes when the rotation is uniformly distributed.

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Copyright

Corresponding author

Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: huiling.le@nottingham.ac.uk
∗∗ Postal address: DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWB, UK.

References

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[1] Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81121.
[2] Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.
[3] Kendall, W. S. and Le, H. (2009). Statistical shape theory. In New Perspectives in Stochastic Geometry, eds Kendall, W. S. and Molchanov, I., Oxford University Press, pp. 384–373.
[4] O'Neill, B. (1983). Semi-Riemannian Geometry. Academic Press, New York.
[5] Panaretos, V. M. (2006). The diffusion of Radon shape. Adv. Appl. Prob. 38, 320335.
[6] Panaretos, V. M. (2008). Representation of Radon shape diffusions via hyperspherical Brownian motion. Math. Proc. Camb. Phil. Soc. 145, 457470.
[7] Panaretos, V. M. (2009). On random tomography with unobservable projection angles. Ann. Statist. 37, 32723306.

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On the induced distribution of the shape of the projection of a randomly rotated configuration

  • H. Le (a1) and D. Barden (a2)

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