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The Fréchet mean shape and the shape of the means

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Huiling Le  and
Alfred Kume
Huiling Le*
Affiliation:
University of Nottingham
Alfred Kume*
Affiliation:
University of Nottingham
*
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

We identify the Fréchet mean shape with respect to the Riemannian metric of a class of probability measures on Bookstein's shape space of labelled triangles and show, in contrast to the case of Kendall's shape space, that the Fréchet mean shape of the probability measure on Bookstein's shape space induced from independent normal distributions on vertices, having the same covariance matrix σ2I2, is not necessarily the shape of the means.

Keywords

Kendall's shape spaceBookstein's shape spaceFréchet mean shapeshape of the means

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 101 - 113
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Beardon, A. F. (1983). The Geometry of Discrete Groups. Springer, Berlin.CrossRefGoogle Scholar
[2] Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1, 181–242.Google Scholar
[3] Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J. R. Statist. Soc. B, 53, 285–339.Google Scholar
[4] Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121.Google Scholar
[5] Kendall, D. G. (1991). The Mardia–Dryden shape distribution for triangles: a stochastic calculus approach. J. Appl. Prob. 28, 225–230.CrossRefGoogle Scholar
[6] Kendall, W. S. (1998). A diffusion model for Bookstein triangle shape. Adv. Appl. Prob. 30, 317–334.Google Scholar
[7] Kent, J. T. (1992). New directions in shape analysis. In The Art of Statisical Science, ed. Mardia, K. V.. John Wiley, New York, pp. 115–127.Google Scholar
[8] Kent, J. T. and Mardia, K. V. (1995). Consistency of procrustes estimators. Tech. Rep. 95-05. Department of Statistics, University of Leeds, Leeds, UK.Google Scholar
[9] Le, H. (1991). On geodesics in Euclidean shape spaces. J. Lond. Math. Soc. 44, 360–372.Google Scholar
[10] Le, H. (1995). Mean size-and-shape and mean shapes: a geometric point of view. Adv. Appl. Prob. 27, 44–55.Google Scholar
[11] Le, H. (1998). On consistency of procrustean mean shapes. Adv. Appl. Prob. 30, 53–63.Google Scholar
[12] Le, H. and Small, C. G. (1999). Multidimensional scaling of simplex shapes. Pattern Recognition 32, 1601–1613.Google Scholar
[13] Mardia, K. V. and Dryden, I. L. (1989). Shape distributions for landmark data. Adv. Appl. Prob. 21, 742–755.Google Scholar
[14] Mardia, K. V. and Dryden, I. L. (1991). General shape distributions in a plane. Adv. Appl. Prob. 23, 259–276.Google Scholar
[15] Small, C. G. (1996). The Statistical Theory of Shape. Springer, New York.Google Scholar
[16] Ziezold, H. (1977). On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. Trans. 7th Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes. Vol. A. Reidel, Dordrecht, pp. 591–602.Google Scholar