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.

The Fréchet mean shape and the shape of the means

2000 ◽  
Vol 32 (1) ◽  
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 σ2I2, is not necessarily the shape of the means.

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.

Estimating Fréchet means in Bookstein's shape space

2000 ◽  
Vol 32 (03) ◽  
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 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.

Locating Fréchet means with application to shape spaces

2001 ◽  
Vol 33 (2) ◽  
pp. 324-338 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Riemannian Manifolds ◽  
Contraction Mapping ◽  
Riemannian Metric ◽  
Jacobi Field ◽  
Probability Measures ◽  
Contraction Mapping Theorem ◽  
Shape Spaces ◽  
Fréchet Mean ◽  
Sectional Curvatures ◽  
Frechet Mean

We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function ρ determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of ‘horizontally lifting’ to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to ρ, of a class of probability measures on Kendall's shape spaces in terms of the vertices of random shapes. This gives us, for example, an algorithm for finding Fréchet mean shapes of samples of configurations on the plane which is expressed directly in terms of the vertices.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

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