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 = FALSE, tol1 = 1e-05, tol2 = tol1, tangentresiduals = TRUE, proc.output=FALSE, distances=TRUE, pcaoutput=TRUE, alpha=0, affine=FALSE, expomap=FALSE)
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 |
tangentresiduals |
Logical quantity indicating if Procrustes residuals should be used for analysis. If tangentresiduals=TRUE for the shape (scale=TRUE) case these are approximate tangent space coordinates, and for the size-and-shape (scale=FALSE) case these are exact tangent space coordinates. If tangentresiduals=FALSE then the partial tangent shape coordinates (see tan below). |
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. |
expomap |
Logical. If TRUE then the exponential map tangent co-ordinates are used instead of the the partial tangent shape co-ordinates |
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 tangentresiduals=TRUE this is the mk x n matrix of Procrustes residuals $X_i^P$ - Xbar , where Xbar = mean($X_i^P$). If approxtangent=FALSE this is the km-m x n matrix of partial Procrustes tangent shape 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. 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 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 $d_F$ |
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")