Morpho (version 2.1)

covDist: calculates distances and PC-coordinates of covariance matrices

Description

calculates PC-coordinates of covariance matrices by using the Riemannian metric in their respective space.

Usage

covDist(s1, s2)
covPCA(data, groups, rounds = 1000, bootrounds = 0, lower.bound = 0.05,
  upper.bound = 0.95)

Arguments

s1

m x m covariance matrix

s2

m x m covariance matrix

data

matrix containing data with one row per observation

groups

factor: group assignment for each specimen

rounds

integer: rounds to run permutation of distances by randomly assigning group membership

bootrounds

integer: perform bootstrapping to generate confidence intervals (lower boundary, median and upper boundary) for PC-scores.

lower.bound

numeric: set probability (quantile) for lower boundary estimate from bootstrapping.

upper.bound

numeric: set probability (quantile) for upper boundary estimate from bootstrapping.

Value

• covDist returns the distance between s1 and s2

  covPCA returns a list containing:

  if scores = TRUE

• PCscores
  PCscores
• eigen
  eigen decomposition of the centered inner product
• if rounds > 0
• dist
  distance matrix
• p.matrix
  p-values for pairwise distances from permutation testing
• if bootrounds > 0
• bootstrap
  list containing the lower and upper bound of the confidence intervals of PC-scores as well as the median of bootstrapped values.
• boot.data
  array containing all results generated from bootstrapping.

Details

covDist calculates the Distance between covariance matrices while covPCA uses a MDS (multidimensional scaling) approach to obtain PC-coordinates from a distance matrix derived from multiple groups. P-values for pairwise distances can be computed by permuting group membership and comparing actual distances to those obtained from random resampling. To calculate confidence intervals for PC-scores, within-group bootstrapping can be performed.

References

Mitteroecker P, Bookstein F. 2009. The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution 63:727-737.

Hastie T, Tibshirani R, Friedman JJH. 2013. The elements of statistical learning. Springer New York.

See Also

prcomp

Examples

cpca <- covPCA(iris[,1:4],iris[,5])
cpca$p.matrix #show pairwise p-values for equal covariance matrices
require(car)
sp(cpca$PCscores[,1],cpca$PCscores[,2],groups=levels(iris[,5]),
   smooth=FALSE,xlim=range(cpca$PCscores),ylim=range(cpca$PCscores))

data(boneData)
proc <- procSym(boneLM)
pop <- name2factor(boneLM, which=3)
## compare covariance matrices for PCscores of Procrustes fitted data
cpca1 <- covPCA(proc$PCscores, groups=pop, rounds = 1000)
## view p-values:
cpca1$p.matrix # differences between covariance matrices
# are significant
## visualize covariance ellipses of first 5 PCs of shape
spm(proc$PCscores[,1:5], groups=pop, smooth=FALSE,ellipse=TRUE, by.groups=TRUE)
## covariance seems to differ between 1st and 5th PC
## for demonstration purposes, try only first 4 PCs
cpca2 <- covPCA(proc$PCscores[,1:4], groups=pop, rounds = 1000)
## view p-values:
cpca2$p.matrix # significance is gone
#do some bootstrapping 1000 rounds
cpca <- covPCA(iris[,1:4],iris[,5],rounds=0, bootrounds=1000)
#plot bootstrapped data of PC1 and PC2 for first group
plot(t(cpca$boot.data[1,1:2,]),xlim=range(cpca$boot.data[,1,]),
                               ylim=range(cpca$boot.data[,2,]))
points(t(cpca$PCscores[1,]),col="white",pch=8,cex=1.5)##plot actual values

for (i in 2:3) {
  points(t(cpca$boot.data[i,1:2,]),col=i)##plot other groups
  points(t(cpca$PCscores[i,]),col=1,pch=8,cex=1.5)##plot actual values
}