procGPA {shapes}R Documentation

Generalised Procrustes analysis

Description

Generalised Procrustes analysis to register landmark configurations into optimal registration using translation, rotation and scaling. Reflection invariance can also be chosen, and registration without scaling is also an option. Also, obtains principal components, and some summary statistics.

Usage

procGPA(x, scale = TRUE, reflect = FALSE, eigen2d = FALSE, 
tol1 = 1e-05, tol2 = tol1, tangentresiduals = TRUE, proc.output=FALSE, 
distances=TRUE, pcaoutput=TRUE, alpha=0, affine=FALSE, expomap=FALSE)

Arguments

x Input k x m x n real array, (or k x n complex matrix for m=2 is OK), where k is the number of points, m is the number of dimensions, and n is the sample size.
scale Logical quantity indicating if scaling is required
reflect Logical quantity indicating if reflection is required
eigen2d Logical quantity indicating if complex eigenanalysis should be used to calculate Procrustes mean for the particular 2D case when scale=TRUE, reflect=FALSE
tol1 Tolerance for optimal rotation for the iterative algorithm: tolerance on the mean sum of squares (divided by size of mean squared) between successive iterations
tol2 tolerance for rescale/rotation step for the iterative algorithm: tolerance on the mean sum of squares (divided by size of mean squared) between successive iterations
tangentresiduals Logical quantity indicating if Procrustes residuals should be used for analysis. If tangentresiduals=TRUE for the shape (scale=TRUE) case these are approximate tangent space coordinates, and for the size-and-shape (scale=FALSE) case these are exact tangent space coordinates. If tangentresiduals=FALSE then the partial tangent shape coordinates (see tan below).
proc.output Logical quantity indicating if printed output during the iterations of the Procrustes GPA algorithm should be given
distances Logical quantity indicating if shape distances and sizes should be calculated
pcaoutput Logical quantity indicating if PCA should be carried out
alpha The parameter alpha used for relative warps analysis, where alpha is the power of the bending energy matrix. If alpha = 0 then standard Procrustes PCA is carried out. If alpha = 1 then large scale variations are emphasized, if alpha = -1 then small scale variations are emphasised. Requires m=2 and m=3 dimensional data if alpha $!=$ 0.
affine Logical. If TRUE then only the affine subspace of shape variability is considered.
expomap Logical. If TRUE then the exponential map tangent co-ordinates are used instead of the the partial tangent shape co-ordinates

Value

A list with components

k no of landmarks
m no of dimensions (m-D dimension configurations)
n sample size
mshape Procrustes mean shape. Note this is unit size if complex eigenanalysis used, but on the scale of the data if iterative GPA is used.
tan If tangentresiduals=TRUE this is the mk x n matrix of Procrustes residuals $X_i^P$ - Xbar , where Xbar = mean($X_i^P$). If approxtangent=FALSE this is the km-m x n matrix of partial Procrustes tangent shape coordinates with pole given by the preshape of the Procrustes mean
rotated the k x m x n array of full 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. If alpha $!=0$ then it is the percent of non-affine variation of the relative warp scores. If affine is TRUE it is the percentage of total shape variability of each affine component.
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 root mean square (r.m.s.) of rho
rmsd1 r.m.s. of full Procrustes distances to the mean shape $d_F$

Author(s)

Ian Dryden, with input from Mohammad Faghihi and Alfred Kume

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.

Gower, J.C. (1975). Generalized Procrustes analysis, Psychometrika, 40, 33–50.

Kent, J.T. (1994). The complex Bingham distribution and shape analysis, Journal of the Royal Statistical Society, Series B, 56, 285-299.

Ten Berge, J.M.F. (1977). Orthogonal Procrustes rotation for two or more matrices. Psychometrika, 42, 267-276.

See Also

procOPA,riemdist,shapepca,testmeanshapes

Examples


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

data(gorf.dat)
data(gorm.dat)

plotshapes(gorf.dat,gorm.dat)
n1<-dim(gorf.dat)[3]
n2<-dim(gorm.dat)[3]
k<-dim(gorf.dat)[1]
m<-dim(gorf.dat)[2]
gor.dat<-array(0,c(k,2,n1+n2))
gor.dat[,,1:n1]<-gorf.dat
gor.dat[,,(n1+1):(n1+n2)]<-gorm.dat

gor<-procGPA(gor.dat)
shapepca(gor,type="r",mag=3)
shapepca(gor,type="v",mag=3)

gor.gp<-c(rep("f",times=30),rep("m",times=29))
x<-cbind(gor$size,gor$rho,gor$scores[,1:3])
pairs(x,panel=function(x,y) text(x,y,gor.gp),
   label=c("s","rho","score 1","score 2","score 3"))

##########################################################
#3D example

data(macm.dat)
out<-procGPA(macm.dat,scale=FALSE)

par(mfrow=c(2,2))
plot(out$rawscores[,1],out$rawscores[,2],xlab="PC1",ylab="PC2")
title("PC scores")
plot(out$rawscores[,2],out$rawscores[,3],xlab="PC2",ylab="PC3")
plot(out$rawscores[,1],out$rawscores[,3],xlab="PC1",ylab="PC3")
plot(out$size,out$rho,xlab="size",ylab="rho")
title("Size versus shape distance")


[Package shapes version 1.1-3 Index]