procGPA: Generalised Procrustes analysis

In shapes: Statistical Shape Analysis

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, tangentcoords = "residual", proc.output=FALSE, 
distances=TRUE, pcaoutput=TRUE, alpha=0, affine=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
tangentcoords	Type of tangent coordinates. If (SCALE=TRUE) the options are "residual" (Procrustes residuals, which are approximate tangent coordinates to shape space), "partial" (Kent's partial tangent co-ordinates), "expomap" (tangent coordinates from the inverse of the exponential map, which are the similar to "partial" but scaled by (rho/sin(rho)) where rho is the Riemannian distance to the pole of the projection. If (SCALE=FALSE) then all three options give the same tangent co-ordinates to size-and-shape space, which is simply the Procrustes residual X^P - mu.
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.

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	The tangent shape (or size-and-shape) coordinates
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
stdscores	standardised PC scores (each with unit variance) using tan
rawscores	raw PC scores using tan
rho	Kendall's Riemannian shape 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 $d_F$
GSS	Minimized Procrustes sum of squares

Author(s)

Ian Dryden, with input from Mohammad Faghihi and Alfred Kume

References

Dryden, I.L. and Mardia, K.V. (2016). Statistical Shape Analysis, with applications in R (Second Edition). Wiley, Chichester. Chapter 7.

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, 2016)

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")

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