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Estimating Fréchet means in Bookstein's shape space

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  19 February 2016

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

In [8], Le showed that procrustean mean shapes of samples are consistent estimates of Fréchet means for a class of probability measures in Kendall's shape spaces. In this paper, we investigate the analogous case in Bookstein's shape space for labelled triangles and propose an estimator that is easy to compute and is a consistent estimate of the Fréchet mean, with respect to sinh(δ/√2), of any probability measure for which such a mean exists. Furthermore, for a certain class of probability measures, this estimate also tends almost surely to the Fréchet mean calculated with respect to the Riemannian distance δ.

Keywords

Global symmetryFréchet mean shape

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 663 - 674
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1, 181–242.Google Scholar
[2] Bookstein, F. L. (1991). Morphometric Tools For Landmark Data: Geometry and Biology. Cambridge University Press.Google Scholar
[3] Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J. R. Statist. Soc. Ser. B 53, 285–339.Google Scholar
[4] Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[5] Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.Google 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. and Mardia, K. V. (1997). Consistency of procrustes estimators. J. R. Statist. Soc. Ser. B 59, 281–290.Google Scholar
[8] Le, H. (1998). On consistency of procrustean mean shapes. Adv. Appl. Prob. 30, 53–63.Google Scholar
[9] Le, H. and Barden, D. (1999). On simplex shape spaces. Submitted.Google Scholar
[10] Le, H. and Kume, A. (2000). Fréchet mean shape and the shape of the means. Adv. Appl. Prob. 32, 101–113.Google Scholar
[11] Le, H. and Small, C. G. (1999). Multidimensional scaling of simplex shapes. Pattern Recognition 32, 1601–1613.Google Scholar
[12] O'Neill, B. (1983). Semi-Riemannian Geometry. Academic Press, Orlando.Google Scholar
[13] Small, C. G. (1996). The Statistical Theory of Shape. Springer, New York.Google Scholar
[14] Ziezold, H. (1977). On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Trans. 7th Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes, Vol. A. Reidel, Dordrecht, pp. 591–602.Google Scholar