Sample Mean Versus Sample Fréchet Mean for Combining Complex Wishart Matrices: A Statistical Study

2017 ◽  
Vol 65 (17) ◽  
pp. 4551-4561 ◽  
Author(s):  
L. Zhuang ◽  
A. T. Walden
Keyword(s):  
Statistical Study ◽  
Sample Mean ◽  
Wishart Matrices ◽  
Fréchet Mean ◽  
Versus Sample ◽  
Frechet Mean

On the consistency of procrustean mean shapes

1998 ◽  
Vol 30 (1) ◽  
pp. 53-63 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Mild Condition ◽  
Shape Space ◽  
Entire Space ◽  
Fréchet Mean ◽  
Mean Shape ◽  
Frechet Mean

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.

On the consistency of procrustean mean shapes

1998 ◽  
Vol 30 (01) ◽  
pp. 53-63 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Mild Condition ◽  
Shape Space ◽  
Entire Space ◽  
Fréchet Mean ◽  
Mean Shape ◽  
Frechet Mean

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.

scholarly journals Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees

Biometrika ◽  
2017 ◽  
Vol 104 (4) ◽  
pp. 901-922 ◽  
Author(s):  
Tom M W Nye ◽  
Xiaoxian Tang ◽  
Grady Weyenberg ◽  
Ruriko Yoshida
Keyword(s):  
Principal Component Analysis ◽  
Phylogenetic Trees ◽  
Principal Component ◽  
Component Analysis ◽  
Fréchet Mean ◽  
Frechet Mean

Estimating Fréchet means in Bookstein's shape space

2000 ◽  
Vol 32 (3) ◽  
pp. 663-674 ◽  
Author(s):  
Alfred Kume ◽  
Huiling Le
Keyword(s):  
Probability Measure ◽  
Shape Space ◽  
Probability Measures ◽  
Consistent Estimate ◽  
Shape Spaces ◽  
Fréchet Mean ◽  
Frechet Mean

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.

On the measure of the cut locus of a Fréchet mean

2014 ◽  
Vol 46 (4) ◽  
pp. 698-708 ◽  
Author(s):  
H. Le ◽  
D. Barden
Keyword(s):  
Cut Locus ◽  
Fréchet Mean ◽  
Frechet Mean

The Fréchet mean shape and the shape of the means

2000 ◽  
Vol 32 (01) ◽  
pp. 101-113 ◽  
Author(s):  
Huiling Le ◽  
Alfred Kume
Keyword(s):  
Probability Measure ◽  
Covariance Matrix ◽  
Shape Space ◽  
Riemannian Metric ◽  
Probability Measures ◽  
Normal Distributions ◽  
Fréchet Mean ◽  
Mean Shape ◽  
Frechet Mean

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 σ2 I 2, is not necessarily the shape of the means.

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