Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T04:16:41.087Z Has data issue: false hasContentIssue false
  • English
  • Français

On Fréchet means in simplex shape spaces

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 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.
Get access Rights & Permissions [Opens in a new window]

Abstract

By making use of the geometric properties of simplex shape spaces, this paper investigates the problems relating to the estimation of the Fréchet means of the random shapes of simplices in Euclidean spaces and also, for the random shapes induced by certain normally distributed simplices, the problems relating to the location of these Fréchet means. In particular, we obtain an algorithm for computing sample mean shapes in simplex shape spaces which converges reasonably fast.

Keywords

Fréchet mean shapessimplex shape spaces

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 885 - 897
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions. Statist. Sci. 1, 181–242.Google Scholar
[2] Diaz Garcia, J. A., Jaimez, R. G. and Mardia, K. V. (1997). Wishart and pseudo-Wishart distributions and some applications to shape theory. J. Multivariate Anal. 63, 73–87.CrossRefGoogle Scholar
[3] Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis. John Wiley, Chichester.Google Scholar
[4] Goodall, C. R. and Mardia, K. V. (1992). The noncentral Bartlett decomposition and shape densities. J. Multivariate Anal. 40, 94–108.CrossRefGoogle Scholar
[5] Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541.CrossRefGoogle Scholar
[6] Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.CrossRefGoogle Scholar
[7] Kume, A. and Le, H. (2000). Estimating Fréchet means in Bookstein's shape space. Adv. Appl. Prob. 32, 663–674.CrossRefGoogle Scholar
[8] Le, H. (1998). On consistency of procrustean mean shapes. Adv. Appl. Prob. 30, 53–63.CrossRefGoogle Scholar
[9] Le, H. and Barden, D. (2001). On simplex shape spaces. J. London Math. Soc. 64, 501–512.CrossRefGoogle Scholar
[10] Le, H. and Kume, A. (2000). Fréchet mean shape and the shape of the means. Adv. Appl. Prob. 32, 101–113.CrossRefGoogle Scholar
[11] Le, H. and Small, C. G. (1999). Multidimensional scaling of simplex shapes. Pattern Recognition 32, 1601–1613.CrossRefGoogle Scholar
[12] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. John Wiley, New York.CrossRefGoogle Scholar
[13] Small, C. G. (1996). The Statistical Theory of Shape. Springer, New York.CrossRefGoogle 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. Inf. Theory, Statist. Decision Functions, Random Process. (Tech. University Prague, 1974), Vol. A, Reidel, Dordrecht, pp. 591–602.Google Scholar