CVA: Canonical Variate Analysis

Description

performs a Canonical Variate Analysis.

Usage

CVA(dataarray, groups, weighting = TRUE, tolinv = 1e-10, plot = TRUE,
  rounds = 0, cv = FALSE)

Arguments

dataarray

Either a k x m x n real array, where k is the number of points, m is the number of dimensions, and n is the sample size. Or alternatively a n x m Matrix where n is the numeber of observations and m the number of variables (this can be PC scores for exampl

groups

a character/factor vector containgin grouping variable.

weighting

Logical: Determines whether the between group covariance matrix and Grandmean is to be weighted according to group size.

tolinv

Threshold for the eigenvalues of the pooled within-group-covariance matrix to be taken as zero - for calculating the general inverse of the pooled withing groups covariance matrix.

plot

Logical: determins whether in the two-sample case a histogramm ist to be plotted.

rounds

integer: number of permutations if a permutation test of the Mahalanobis distances (from the pooled within-group covariance matrix) and Euclidean distance between group means is requested.If rounds = 0, no test is performed.

logical: requests a Jackknife Crossvalidation.

Value

CVA matrix containing the Canonical Variates
CVscoresA matrix containing the individual Canonical Variate scores
Grandma vector or a matrix containing the Grand Mean (depending if the input is an array or a matrix)
groupmeansa matrix or an array containing the group means (depending if the input is an array or a matrix)
VarVariance explained by the Canonical Variates
CVvisCanonical Variates projected back into the original space - to be used for visualization purposes, for details see example below
DistMahalanobis Distances between group means - if requested tested by permutation test if the input is an array it is assumed to be superimposed Landmark Data and Procrustes Distance will be calculated
CVcvA matrix containing crossvalidated CV scores
groupsfactor containing the grouping variable
classclassification results based on posteriror probabilities. If cv=TRUE, this will be done by a leaving-one-out procedure
posteriorposterior probabilities
priorprior probabilities

References

Cambell, N. A. & Atchley, W. R.. 1981 The Geometry of Canonical Variate Analysis: Syst. Zool., 30(3), 268-280.

Klingenberg, C. P. & Monteiro, L. R. 2005 Distances and directions in multidimensional shape spaces: implications for morphometric applications. Systematic Biology 54, 678-688.

Examples

Run this code

## all examples are kindly provided by Marta Rufino

library(shapes)
# perform procrustes fit on raw data
alldat<-procSym(abind(gorf.dat,gorm.dat))
# create factors
groups<-as.factor(c(rep("female",30),rep("male",29)))
# perform CVA and test Mahalanobis distance
# between groups with permutation test by 100 rounds)
cvall<-CVA(alldat$orpdata,groups,rounds=10000)
## visualize a shape change from score -5 to 5:
cvvis5 <- 5*matrix(cvall$CVvis[,1],nrow(cvall$Grandm),ncol(cvall$Grandm))+cvall$Grandm
cvvisNeg5 <- -5*matrix(cvall$CVvis[,1],nrow(cvall$Grandm),ncol(cvall$Grandm))+cvall$Grandm
plot(cvvis5,asp=1)
points(cvvisNeg5,col=2)
for (i in 1:nrow(cvvisNeg5))
  lines(rbind(cvvis5[i,],cvvisNeg5[i,]))

### Morpho CVA
data(iris)
vari <- iris[,1:4]
facto <- iris[,5]

cva.1=CVA(vari, groups=facto)
# plot the CVA
plot(cva.1$CVscores, col=facto, pch=as.numeric(facto), typ="n",asp=1,
   xlab=paste("1st canonical axis", paste(round(cva.1$Var[1,2],1),"%")),
   ylab=paste("2nd canonical axis", paste(round(cva.1$Var[2,2],1),"%")))

  text(cva.1$CVscores, as.character(facto), col=as.numeric(facto), cex=.7)

  # add chull (merge groups)
  for(jj in 1:length(levels(facto))){
        ii=levels(facto)[jj]
    kk=chull(cva.1$CVscores[facto==ii,1:2])
    lines(cva.1$CVscores[facto==ii,1][c(kk, kk[1])],
    cva.1$CVscores[facto==ii,2][c(kk, kk[1])], col=jj)
    }

  # add 80\% ellipses
  require(car)
  for(ii in 1:length(levels(facto))){
    dataEllipse(cva.1$CVscores[facto==levels(facto)[ii],1],
    cva.1$CVscores[facto==levels(facto)[ii],2],
                    add=TRUE,levels=.80, col=c(1:7)[ii])}

  # histogram per group
  require(lattice)
  histogram(~cva.1$CVscores[,1]|facto,
  layout=c(1,length(levels(facto))),
          xlab=paste("1st canonical axis", paste(round(cva.1$Var[1,2],1),"%")))
  histogram(~cva.1$CVscores[,2]|facto, layout=c(1,length(levels(facto))),
          xlab=paste("2nd canonical axis", paste(round(cva.1$Var[2,2],1),"%")))

  # plot Mahalahobis
  dendroS=hclust(cva.1$Dist$GroupdistMaha)
  dendroS$labels=levels(facto)
  par(mar=c(4,4.5,1,1))
  dendroS=as.dendrogram(dendroS)
  plot(dendroS, main='',sub='', xlab="Geographic areas",
          ylab='Mahalahobis distance')


   # Variance explained by the canonical roots:
   cva.1$Var
   # or plot it:
   barplot(cva.1$Var[,2])

# another landmark based example in 3D:
data(boneData)
groups <- name2factor(boneLM,which=3:4)
proc <- procSym(boneLM)
cvall<-CVA(proc$orpdata,groups)
#' ## visualize a shape change from score -5 to 5:
cvvis5 <- 5*matrix(cvall$CVvis[,1],nrow(cvall$Grandm),ncol(cvall$Grandm))+cvall$Grandm
cvvisNeg5 <- -5*matrix(cvall$CVvis[,1],nrow(cvall$Grandm),ncol(cvall$Grandm))+cvall$Grandm
#visualize it
deformGrid3d(cvvis5,cvvisNeg5,ngrid = 0)

#for using (e.g. the first 5) PCscores, one will do:
cvall <- CVA(proc$PCscores[,1:5],groups)
#' ## visualize a shape change from score -5 to 5:
cvvis5 <- 5*cvall$CVvis[,1]+cvall$Grandm
cvvisNeg5 <- -5*cvall$CVvis[,1]+cvall$Grandm
cvvis5 <- showPC(cvvis5,proc$PCs[,1:5],proc$mshape)
cvvisNeg5 <- showPC(cvvisNeg5,proc$PCs[,1:5],proc$mshape)
#visualize it
deformGrid3d(cvvis5,cvvisNeg5,ngrid = 0)