relative.eigen {vcvComp}

Relative eigenanalysis

Description

Computes the Riemanian distance between two variance-covariance matrices of same dimensions and the relative eigenvectors and eigenvalues of S1 with respect to S2

Usage

relative.eigen(S1, S2, method = 0, pa = 0)

Arguments

S1

a variance-covariance matrix

S2

a variance-covariance matrix

method

an integer for the method of matrix inversion (see function 'minv')

pa

an integer for the parameter of matrix inversion (see function 'minv')

Value

A list containing the following named components:

relValues

the vector of relative eigenvalues

relVectors

the matrix of relative eigenvectors

distCov

the distance between the two covariance matrices

relGV

the product of the nonzero relative eigenvalues = the ratio of the generalized variances. The generalized variance corresponds to the determinant of the covariance matrix.

logGV

the log ratio of the generalized variances

q

the number of nonzero eigenvalues

References

Bookstein F, Mitteroecker P (2014) Comparing covariance matrices by relative eigenanalysis, with applications to organismal biology. Evolutionary Biology 41: 336-350. https://doi.org/10.1007/s11692-013-9260-5

See Also

See minv for the method and the parameter used for the matrix inversion

Examples

# Data matrix of 2D landmark coordinates
data("Tropheus.IK.coord")
coords <- which(names(Tropheus.IK.coord) == "X1"):which(names(Tropheus.IK.coord) == "Y19")
proc.coord <- as.matrix(Tropheus.IK.coord[coords])

# Data reduction
phen.pca <- prcomp(proc.coord, rank. = 5, tol = sqrt(.Machine$double.eps))
pc.scores <- phen.pca$x

# Covariance matrix of each population
S.phen.pop <- cov.group(pc.scores, groups = Tropheus.IK.coord$POP.ID)

# Relative PCA = relative eigenanalysis between 2 covariance matrices
# (population IKA1 relative to IKS5)
relEigen.a1s5 <- relative.eigen(S.phen.pop[, , "IKA1"], S.phen.pop[, , "IKS5"])