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, 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.

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)
n2<-dim(gorm.dat)
k<-dim(gorf.dat)
m<-dim(gorf.dat)
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 1.2.7 Index]