procGPA {shapes} | R Documentation |
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.
procGPA(x, scale = TRUE, reflect = FALSE, eigen2d = TRUE, tol1 = 1e-05, tol2 = tol1, approxtangent = TRUE, proc.output=FALSE, distances=TRUE, pcaoutput=TRUE)
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 |
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 $rho$ to the mean shape |
rmsrho |
r.m.s. of rho |
rmsd1 |
r.m.s. of full Procrustes distances to the mean shape $d_F$ |
Ian Dryden, with input from Mohammad Faghihi and Alfred Kume
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.
procOPA,riemdist,shapepca,testmeanshapes
#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")