Weighted Procrustes analysis

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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

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procWGPA(x, fixcovmatrix=FALSE, initial="Identity", maxiterations=10, scale=TRUE,
 reflect=FALSE, 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

,

reflect

Logical quantity indicating if reflection invariance 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

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

data(gorf.dat)

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

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