scca {whitening}

Perform Canonical Correlation Analysis

Description

scca computes canonical correlations and directions using a shrinkage estimate of the joint correlation matrix of X and Y.

cca computes canonical correlations and directions based on empirical correlations.

Usage

scca(X, Y, lambda.cor, scale=TRUE, verbose=TRUE)
cca(X, Y, scale=TRUE)

Arguments

X

First data matrix, with samples in rows and variables in columns.

Y

Second data matrix, with samples in rows and variables in columns.

lambda.cor

Shrinkage intensity for estimating the joint correlation matrix - see cor.shrink. If not specified this will be estimated from the data.

scale

Determines whether canonical directions are computed for standardized or raw data. Note that if data are not standardized the canonical directions contain the scale of the variables.

verbose

Report shrinkage intensities.

Details

The canonical directions in this function are scaled in such a way that they correspond to whitening matrices - see Jendoubi and Strimmer (2019) for details. Note that the sign convention for the canonical directions employed here allows purposely for both positive and negative canonical correlations.

The function scca uses some clever matrix algebra to avoid computation of full correlation matrices, and hence can be applied to high-dimensional data sets - see Jendoubi and Strimmer (2019) for details.

cca it is a shortcut for running scca with lambda.cor=0 and verbose=FALSE.

If scale=FALSE the standard deviations needed for the canonical directions are estimated by apply(X, 2, sd) and apply(X, 2, sd).

If X or Y contains only a single variable the correlation-adjusted cross-correlations K reduce to the CAR score (see function carscore in the care package) described in Strimmer and Zuber (2011).

Value

scca and cca return a list with the following components:

K - the correlation-adjusted cross-correlations.

lambda - the canonical correlations.

WX and WY - the whitening matrices for X and Y, with canonical directions in the rows. If scale=FALSE then canonical directions include scale of the data, if scale=TRUE then only correlations are needed to compute the canonical directions.

PhiX and PhiY - the loadings for X and Y. If scale=TRUE then these are the correlation loadings, i.e. the correlations between the whitened variables and the original variables.

scale - whether data was standardized (if scale=FALSE then canonical directions include scale of the data).

lambda.cor - shrinkage intensity used for estimating the correlations (0 for empirical estimator)

lambda.cor.estimated - indicates whether shrinkage intenstiy was specified or estimated.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io) with Takoua Jendoubi.

References

Jendoubi, T., and K. Strimmer 2019. A whitening approach to probabilistic canonical correlation analysis for omics data integration. BMC Bioinformatics 20: 15. <DOI:10.1186/s12859-018-2572-9>

Zuber, V., and K. Strimmer. 2011. High-dimensional regression and variable selection using CAR scores. Statist. Appl. Genet. Mol. Biol. 10: 34. <DOI:10.2202/1544-6115.1730>

See Also

cancor and whiteningMatrix.

Examples

# load whitening library
library("whitening")

# example data set
data(LifeCycleSavings)
X = as.matrix( LifeCycleSavings[, 2:3] )
Y = as.matrix( LifeCycleSavings[, -(2:3)] )
n = nrow(X)
colnames(X) # "pop15" "pop75"
colnames(Y) # "sr"   "dpi"  "ddpi" 

# CCA

cca.out = cca(X, Y, scale=TRUE)
cca.out$lambda  # canonical correlations
cca.out$WX      # whitening matrix / canonical directions X
cca.out$WY      # whitening matrix / canonical directions Y
cca.out$K       # correlation-adjusted cross-correlations
cca.out$PhiX    # correlation loadings X
cca.out$PhiX    # correlation loadings Y

corplot(cca.out, X, Y)
loadplot(cca.out, 2)
# column sums of squared correlation loadings add to 1
colSums(cca.out$PhiX^2) 

# CCA whitened data
CCAX = tcrossprod( scale(X), cca.out$WX )
CCAY = tcrossprod( scale(Y), cca.out$WY )
zapsmall(cov(CCAX))
zapsmall(cov(CCAY))
zapsmall(cov(CCAX,CCAY)) # canonical correlations


# compare with built-in function cancor 
# note different signs in correlations and directions!
cancor.out = cancor(scale(X), scale(Y))
cancor.out$cor                  # canonical correlations
t(cancor.out$xcoef)*sqrt(n-1)   # canonical directions X
t(cancor.out$ycoef)*sqrt(n-1)   # canonical directions Y


## see "User guides, package vignettes and other documentation"
## for examples with high-dimensional data using the scca function