shapes (version 1.2.7)

procGPA: 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)

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

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.

Author

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")