cca {yacca}  R Documentation 
Canonical Correlation Analysis
Description
Performs a canonical correlation (and canonical redundancy) analysis on two sets of variables.
Usage
cca(x, y, xlab = colnames(x), ylab = colnames(y), xcenter = TRUE,
ycenter = TRUE, xscale = FALSE, yscale = FALSE,
standardize.scores = TRUE, use = "complete.obs", na.rm = TRUE,
use.eigs = FALSE, max.dim = Inf, reg.param = NULL)
## S3 method for class 'cca'
plot(x, ...)
## S3 method for class 'cca'
print(x, ...)
## S3 method for class 'cca'
summary(object, ...)
Arguments
x 
for 
y 
a single vector or a matrix whose columns contain the 
xlab 
an optional vector of 
ylab 
an optional vector of 
xcenter 
boolean; demean the 
ycenter 
boolean; demean the 
xscale 
boolean; scale the 
yscale 
boolean; scale the 
standardize.scores 
boolean; rescale scores (and coefficients) to produce scores of unit variance? 
use 

na.rm 
boolean; remove missing values during redundancy analysis? 
use.eigs 

max.dim 
maximum number of canonical variates to extract (only relevant if less than the minimum of the number of columns of 
reg.param 
an optional L2 regularization parameter (or vector thereof). 
object 
a 
... 
additional arguments. 
Details
Canonical correlation analysis (CCA) is a form of linear subspace analysis, and involves the projection of two sets of vectors (here, the variable sets x
and y
) onto a joint subspace. The goal of (CCA) is to find a squence of linear transformations of each variable set, such that the correlations between the transformed variables are maximized (under the proviso that each transformed variable must be orthogonal to those preceding it). These transformed variables – known as “canonical variates” (CVs) – can be thought of as expressing the common variation across the data sets, in a manner analogous to the role of principal components in withinset analysis (see, e.g., princomp
). Since the rank of the joint subspace is equal to the minimum of the ranks of the two spaces spanned by the initial data vectors, it follows that the number of CVs will usually be equal to the minimum of the number of x
and y
variables (perhaps fewer, if the sets are not of full rank or if max.dim
is used to constrain the number of variables extracted).
Formally, we may describe the CCA solution as follows. Given data matrices X
and Y
, let \Sigma_{XX}
, \Sigma_{XY}
, \Sigma_{YX}
and \Sigma_{YY}
be the respective sample covariance matrices for X
versus itself, X
versus Y
, Y
versus X
, and Y
versus itself. Now, for some i
less than or equal to the minimum rank of X
and Y
, let u_i
be the i
th eigenvector of \Sigma_{XX}^{1} \Sigma_{XY} \Sigma_{YY}^{1} \Sigma_{YX}
, with corresponding eigenvalue \lambda_i
. Then the vector u_i
contains the coefficients projecting X
onto the i
th canonical variate; the corresponding scores are given by X u_i
. Similarly, let v_i
be the i
th eigenvector of \Sigma_{YY}^{1} \Sigma_{YX} \Sigma_{XX}^{1} \Sigma_{XY}
. Then v_i
contains the coefficients projecting Y
onto the i
th canonical variate (with scores Y v_i
). The eigenvalue in the second case will be the same as the first, and corresponds to the square of the i
th canonical correlation for the CCA solution – that is, the correlation between the X
and Y
scores on the i
th canonical variate. Since the canonical correlation structure is unaffected by rescaling of the canonical variate scores, it is common to adjust the coefficients u_i
and v_i
to ensure that the resulting scores have unit variance; this option is controlled here via the standardize.scores
argument.
CCA output can be fairly complex. Quantities of particular interest include the correlations between the original variables in each set and their respective canonical variates (structural correlations or loadings), the coefficients which take the original variables into the CVs, and of course the correlations between the CV scores in one set and their corresponding scores in the opposite set (the canonical correlations). The canonical correlations provide a basic measure of concordance between the transformed variables, but are surprisingly uninformative by themselves; canonical redundancies (see below) are of more typical interest. Interpretation of CVs is usually performed by inspection of loadings, which reveal the extent to which each CV is associated with particular variables in each set. The squared loadings, in particular, convey the fraction of variance in each original variable which is accounted for by a given CV (though not necessarily by the variables in the opposite set!).
A common interest in the context of CCA is the extent to which the variance of one set of variables can be accounted for by the other (in the usual least squares sense). While it is tempting to interpret the squared canonical correlations in this manner, this is incorrect: the squared canonical correlations convey the fraction of variance in the CV scores from one variable set which can be accounted for by scores from the other, but say nothing about the extent to which the CVs themselves account for variation in the original variables. The variance in one set explainable by the other is instead expressed via the socalled redundancy index, which combines the squared canonical correlations with the canonical adequacy (withinset variance accounted for) for each CV. The use of the redundancy index in this way is sometimes called “(canonical) redundancy analysis”, although it is simply an alternate means of presenting CCA results.
As the name of the technique implies, CCA is a symmetric procedure: the designation of one variable set as x
and another as y
is arbitrary, and may be reversed without incident. (Note, however, that the coefficients and redundancies are setspecific, and will also be reversed in this case.) CCA with one x
or y
variable is equivalent to OLS regression (with the squared canonical correlation corresponding to the R^2
), and CCA on one variable pair yields the familiar Pearson productmoment correlation. Centering and scaling data prior to analysis is equivalent to working with correlation matrices in the underlying analysis (with interpretation/effects analogous to the principal components case).
Finding the CCA solution can pose numerical challenges, ironically more so when the degree of potential dimension reduction is highest. In recalcitrant cases, it can be useful to apply regularization to the solution for purposes of stabilization. The optional reg.param
can be used for this purpose: if given as a single numeric value, it adds an L2 (aka “ridge”) penalty to each variable set with the corresponding multiplier value. reg.param
can also be given as a vector of length 2, in which case the first value is applied to the x
variables and the second is applied to the y
variables. Relatedly, in highdimension/lowrank problems it can be useful to extract a much smaller number of canonical variates than the nominal maximum. This can be controlled by max.dim
, though the default diagonalization method computes the entire eigendecomposition prior to canonical variate extraction. In such cases, it can be helpful to employ the alternative diagonalization method controlled by the use.eigs
argument to compute only those dimensions that are actually required. Experience suggests that this method (eigs
) is less stable than the base eigen
, but it can be much faster in highdimensional settings.
Value
An object of class cca
, whose elements are as follows:
corr 
Canonical correlations. 
corrsq 
Squared canonical correlations (shared variance across canonical variates). 
xcoef 
Coefficients for the 
ycoef 
Coefficients for the 
canvarx 
Canonical variate scores for the 
canvary 
Canonical variate scores for the 
xstructcorr 
Structural correlations (loadings) for 
ystructcorr 
Structural correlations (loadings) for 
xstructcorrsq 
Squared structural correlations for 
ystructcorrsq 
Squared structural correlations for 
xcrosscorr 
Canonical crossloadings for 
ycrosscorr 
Canonical crossloadings for 
xcrosscorrsq 
Squared canonical crossloadings for 
ycrosscorrsq 
Squared canonical crossloadings for 
xcancom 
Canonical communalities for 
ycancom 
Canonical communalities for 
xcanvad 
Canonical variate adequacies for 
ycanvad 
Canonical variate adequacies for 
xvrd 
Canonical redundancies for 
yvrd 
Canonical redundancies for 
xrd 
Total canonical redundancy for 
yrd 
Total canonical redundancy for 
chisq 
Sequential 
df 
Degrees of freedom for Bartlett's test. 
xlab 
Variable names for 
ylab 
Variable names for 
reg.param 
Regularization parameter (if any). 
Author(s)
Carter T. Butts <buttsc@uci.edu>
References
Mardia, K. V.; Kent, J. T.; and Bibby, J. M. 1979. Multivariate Analysis. London: Academic Press.
See Also
Examples
#Example parallels the R builtin cancor example
data(LifeCycleSavings)
pop < LifeCycleSavings[, 2:3]
oec < LifeCycleSavings[, (2:3)]
cca.fit < cca(pop, oec)
cca.regfit < cca(pop, oec, reg.param=1) # Some minimal regularization
#View the results
cca.fit
summary(cca.fit)
plot(cca.fit)
cca.regfit #Not a vast difference, usually....