procWGPA {shapes}R Documentation

Weighted Procrustes analysis

Description

Weighted Procrustes analysis to register landmark configurations into optimal registration using translation, rotation and scaling. Registration without scaling is also an option. Also, obtains principal components, and some summary statistics.

Usage

procWGPA(x, fixcovmatrix=FALSE, initial="Identity", maxiterations=10, scale=TRUE, prior="Exponential",diagonal=TRUE,sampleweights="Equal")

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.
fixcovmatrix If FALSE then the landmark covariance matrix is estimated. If a fixed covariance matrix is desired then the value should be given here, e.g. fixcovmatrix=diag(8) for the identity matrix with 8 landmarks.
initial The initial value of the estimated covariance matrix. "Identity" - identity matrix, "Rawdata" - based on sample variance of the raw landmarks. Also, could be a k x k symmetric positive definite matrix.
maxiterations The maximum number of iterations for estimating the covariance matrix
scale Logical quantity indicating if scaling is required
prior Indicates the type of prior. "Exponential" is exponential for the inverse eigenvalues. "Identity" is an inverse Wishart with the identity matrix as parameters.
diagonal Logical. Indicates if the diagonal of the landmark covariance matrix (only) should be used. Diagonal matrices can lead to some landmarks having very small variability, which may or may not be desirable.
sampleweights Gives the weights of the observations in the sample, rather than the landmarks. This is a fixed quatity. "Equal" indicates that all observations in the sample have equal weight. The weights do not need to sum to 1.

Details

The factored covariance model is assumed: $Sigma_k x I_m$ with $Sigma_k$ being the covariance matrix of the landmarks, and the cov matrix at each landmark is the identity matrix.

Value

A list with components

k no of landmarks
m no of dimensions (m-D dimension configurations)
n sample size
mshape Weighted Procrustes mean shape.
tan This is the mk x n matrix of Procrustes residuals $X_i^P$ - Xbar.
rotated the k x m x n array of weighted Procrustes rotated data
pcar the columns are eigenvectors (PCs) of the sample covariance Sv of tan
pcasd the square roots of eigenvalues of Sv using tan (s.d.'s of PCs)
percent the percentage of variability explained by the PCs using tan.
size the centroid sizes of the configurations
scores standardised PC scores (each with unit variance) using tan
rawscores raw PC scores using tan
rho Kendall's Riemannian distance rho to the mean shape
rmsrho r.m.s. of rho
rmsd1 r.m.s. of full Procrustes distances to the mean shape $d_F$
Sigmak Estimate of the sample covariance matrix of the landmarks

Author(s)

Ian Dryden

References

Dryden, I.L. and Mardia, K.V. (1998). Statistical Shape Analysis, Wiley, Chichester.

Goodall, C.R. (1991). Procrustes methods in the statistical analysis of shape (with discussion). Journal of the Royal Statistical Society, Series B, 53: 285-339.

See Also

procGPA

Examples


#2D example : female Gorillas (cf. Dryden and Mardia, 1998)

data(gorf.dat)

gor<-procWGPA(gorf.dat,maxiterations=3)


[Package shapes version 1.1-3 Index]