EigenTensorDecomposition function

Eigentensor Decomposition

This function performs eigentensor decomposition on a set of covariance matrices.

EigenTensorDecomposition(matrices, return.projection = TRUE, ...)

## S3 method for class 'list'
EigenTensorDecomposition(matrices, return.projection = TRUE, ...)

## Default S3 method:
EigenTensorDecomposition(matrices, return.projection = TRUE, ...)

Arguments

• matrices: k x k x m array of m covariance matrices with k traits;
• return.projection: Should we project covariance matrices into estimated eigentensors? Defaults to TRUE
• ...: additional arguments for methods

Returns

List with the following components:

mean mean covariance matrices used to center the sample (obtained from MeanMatrix)

mean.sqrt square root of mean matrix (saved for use in other functions, such as ProjectMatrix and RevertMatrix)

values vector of ordered eigenvalues associated with eigentensors;

matrices array of eigentensor in matrix form;

projection matrix of unstandardized projected covariance matrices over eigentensors.

Details

The number of estimated eigentensors is the minimum between the number of data points (m) and the number of independent variables (k(k + 1)/2) minus one, in a similar manner to the usual principal component analysis.

Examples

data(dentus)

dentus.vcv <- daply (dentus, .(species), function(x) cov(x[,-5]))

dentus.vcv <- aperm(dentus.vcv, c(2, 3, 1))

dentus.etd <- EigenTensorDecomposition(dentus.vcv, TRUE)

# Plot some results
oldpar <- par(mfrow = c(1,2))  
plot(dentus.etd $ values, pch = 20, type = 'b', ylab = 'Eigenvalue')
plot(dentus.etd $ projection [, 1:2], pch = 20, 
     xlab = 'Eigentensor 1', ylab = 'Eigentensor 2')
text(dentus.etd $ projection [, 1:2],
     labels = rownames (dentus.etd $ projection), pos = 2)
par(oldpar)  

# we can also deal with posterior samples of covariance matrices using plyr

dentus.models <- dlply(dentus, .(species), 
                       lm, formula = cbind(humerus, ulna, femur, tibia) ~ 1)

dentus.matrices <- llply(dentus.models, BayesianCalculateMatrix, samples = 100)

dentus.post.vcv <- laply(dentus.matrices, function (L) L $ Ps)

dentus.post.vcv <- aperm(dentus.post.vcv, c(3, 4, 1, 2))

# this will perform one eigentensor decomposition for each set of posterior samples
dentus.post.etd <- alply(dentus.post.vcv, 4, EigenTensorDecomposition)

# which would allow us to observe the posterior 
# distribution of associated eigenvalues, for example
dentus.post.eval <- laply (dentus.post.etd, function (L) L $ values)

boxplot(dentus.post.eval, xlab = 'Index', ylab = 'Value', 
        main = 'Posterior Eigenvalue Distribution')

References

Basser P. J., Pajevic S. 2007. Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI. Signal Processing. 87:220-236.

Hine E., Chenoweth S. F., Rundle H. D., Blows M. W. 2009. Characterizing the evolution of genetic variance using genetic covariance tensors. Philosophical transactions of the Royal Society of London. Series B, Biological sciences. 364:1567-78.

See Also

ProjectMatrix, RevertMatrix

Author(s)

Guilherme Garcia, Diogo Melo