shapes (version 1.0-2)

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 = TRUE, 
tol1 = 1e-05, tol2 = tol1, approxtangent = TRUE, proc.output=FALSE, 
distances=TRUE, pcaoutput=TRUE)

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 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

approxtangent

Logical quantity indicating if the approximate tangent coordinates (Procrustes residuals) should be given if TRUE, or the partial tangent coordinates (see tan below) if FALSE

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

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 approxtangent=TRUE this is the mk x n matrix of Procrustes APPROXIMATE tangent coordinates $X_i^P$ - Xbar , where Xbar = mean($X_i^P$), which are also known as the Procrustes residuals. If approxtangent=FALSE this is the km-m x n matrix of partial Procrustes tangent 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
• 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 $$ to the mean shape rmsrho{r.m.s. of rho} rmsd1{r.m.s. of full Procrustes distances to the mean shape dF}
• 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.

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)

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")