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On the consistency of procrustean mean shapes

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

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

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

Keywords

Fréchet mean shapesprocrustean mean shapesshape ofthe mean

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochatic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 53 - 63
Copyright
Copyright © Applied Probability Trust 1998 

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References

[1] Artstein, Z. and Vitale, R. A. (1975). A strong law of large numbers for random compact sets. Ann. Prob. 5, 879–882.Google Scholar
[2] Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J. R. Statist. Soc. B 53, 285–339.Google Scholar
[3] Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541.CrossRefGoogle Scholar
[4] Kendall, D. G. (1984). Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[5] Kent, J. T. (1992). New directions in shape analysis. In The Art of Statisical Science. ed. Mardia, K. V.. New York, Wiley.Google Scholar
[6] Kent, J. T. and Mardia, K. V. (1997). Consistency of procrustes estimators. J. R. Statist. Soc. B59, 281–290.CrossRefGoogle Scholar
[7] Le, H. (1991). A stochastic calculus approach to the shape distribution induced by a complex normal model. Math. Proc. Camb. Phil. Soc. 109, 221–228.Google Scholar
[8] Le, H. (1995). Mean size-and-shapes and mean shapes: a geometric point of view. Adv. Appl. Prob. 27, 44–55.Google Scholar
[9] Le, H. (1995). The mean shape and the shape of the means. In Proc. Int. Conf. on Current Issues in Statistical Shape Analysis. ed. Mardia, K. V. and Gill, C. A.. Leeds University Press, Leeds.Google Scholar
[10] Lele, S. (1993). Euclidean distance matrix analysis (EDMA): estimation of mean form and mean form difference. Math. Geology 25, 573–602.Google Scholar
[11] Vitale, R. A. (1988). An alternate formulation of mean value for random geometric figures. J. Microsc. 151, 197–204.CrossRefGoogle Scholar
[12] 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