frechet {shapes}

Mean shape estimators

Description

Calculation of different types of Frechet mean shapes, or the isotropic offset Gaussian MLE mean shape

Usage

frechet(x, mean="intrinsic")

Arguments

x

Input k x m x n real array, where k is the number of points, m is the number of dimensions, and n is the sample size.

mean

Type of mean shape. The Frechet mean shape is obtained by minimizing sum d(x_i,mu)^2 with respect to mu. Different estimators are obtained with different choices of distance d. "intrinsic" intrinsic mean shape (d = rho = Riemannian distance); "partial.procrustes" partial Procrustes (d = 2*sin(rho/2)); "full.procrustes" full Procrustes (d = sin(rho)); h (positive real number) M-estimator (d^2 = (1 - cos^(2h)(rho))/h) Kent (1992); "mle" - isotropic offset Gaussian MLE of Mardia and Dryden (1989)

Value

A list with components

mshape

Mean shape estimate

var

Minimized Frechet variance (not available for MLE)

kappa

(if available) The estimated kappa for the MLE

code

Code from optimization, as given by function nlm - should be 1 or 2

gradient

Gradient from the optimization, as given by function nlm - should be close to zero

Author(s)

Ian Dryden

References

Dryden, I. L. (1991). Discussion to 'Procrustes methods in the statistical analysis of shape' by C.R. Goodall. Journal of the Royal Statistical Society, Series B, 53:327-328.

Dryden, I.L. and Mardia, K.V. (2016). Statistical Shape Analysis, with applications in R (Second Edition). Wiley, Chichester.

Kent, J. T. (1992). New directions in shape analysis. In Mardia, K. V., editor, The Art of Statistical Science, pages 115-127. Wiley, Chichester.

Mardia, K. V. and Dryden, I. L. (1989b). The statistical analysis of shape data. Biometrika, 76:271-282.

See Also

procGPA

Examples

#2D example : female and male Gorillas (cf. Dryden and Mardia, 2016)

data(gorf.dat)
frechet(gorf.dat[,,1:4],mean="intrinsic")