procWGPA: Weighted Procrustes analysis

In shapes: Statistical Shape Analysis

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,
 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

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reflect

Logical quantity indicating if reflection invariance is required

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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. (2016). Statistical Shape Analysis, with applications in R (Second Edition). 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, 2016)

data(gorf.dat)

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

shapes documentation built on Feb. 16, 2023, 8:16 p.m.