library(mixOmics) # call mixOmics library
data(nutrimouse) # read in nutrimouse datasetX <- nutrimouse$lipid[, 1:10] # extract first ten lipid concentration variables
Y <- nutrimouse$gene[, 1:10] # extract first ten gene expression variables
result.cca.nutrimouse <- rcc(Y, X) # run the CCA method
# plot projection into canonical variate subspace
plotIndiv(result.cca.nutrimouse)
# plot original variables' correlation with canonical variates
plotVar(result.cca.nutrimouse) Note that the X and Y datasets have been
sliced such that each contains only ten variables. CCA does not perform
well when the sum of the number of variables from each dataset is
greater than the number of samples (ie. P + Q > N, where
P is the number of variables in the first dataset, Q
is the number of variables in the second dataset and N is the
number of samples in each dataset). Use the rCCA Quick Start if your
data does not suit the condition P + Q < N.
?rcc can be run to determine all default arguments of
this function. The default parameters of interest are as follows (when
undergoing classical CCA):
ncomp = 2): Only the first two
pairs of canonical variates are calculated .method.X <- nutrimouse$lipid # extract all lipid concentration variables
Y <- nutrimouse$gene # extract all gene expression variables
# Only one of these calls of rcc is required,
# pick depending on regularisation method
# using the ridge method
result.cca.nutrimouse <- rcc(Y, X, method = "ridge",
lambda1 = 0.5, lambda2 = 0.05)
# using the shrinkage method
result.rcca.nutrimouse <- rcc(Y, X, method = 'shrinkage')
# plot projection into canonical variate subspace
plotIndiv(result.cca.nutrimouse)
# plot original variables' correlation with canonical variates
plotVar(result.cca.nutrimouse) As rCCA is not bound by the same requirement (P + Q > N),
all features are used to construct X and
Y. Of the two datasets, the set with the smaller number
of variables should be passed as a parameter first. In this example,
Y has less variables than X and is used as the
first parameter.
The default parameters for the rcc() function are as
follows (when undergoing regularised CCA):
ncomp = 2): Only the first two
pairs of canonical variates are calculated .method = c("ridge", "shrinkage")): If regularisation is
to be done, one of these methods must be passed in.lambda1, lambda2 = 0):
Controls the degree of regularisation. These parameters are only
required if method = 'ridge'. These can be tuned using
tune.rcc.Canonical Correlation Analysis (CCA) is a multivariate approach to highlight correlations between two data sets acquired on the same experimental units. It is a dimension reduction technique that aids in exploring datasets. The components yielded by CCA (referred to as canonical variates) are linear combinations of variables from each original dataset. Canonical variates are constructed via the maximisation of the correlation between pairs of canonical variates. Each pair of canonical variates has an associated canonical correlation - the correlation between the two novel components.
This classical CCA method is only applicable when P + Q < N, where P is the number of variables in the first dataset, Q is the number of variables in the second dataset and N is the number of samples in each dataset.
The issue of high dimensionality can be by-passed by introducing
regularisation into the CCA method. Regularised Canonical Correlation
Analysis (rCCA) is able to perform on datasets of high dimensions and/or
those with high collinearities (both of which are common in biological
contexts). Ridge penalities (\(\lambda1\), \(\lambda2\)) are added to the diagonal of
X and Y respectively to make them invertible.
This method was proposed by Vinod (1976) [1], then developed by Leurgans
et al. (1993) [2].
There are two methods included in the mixOmics package
to allow the CCA method to be regularised, such that the \(\lambda1\) and \(\lambda2\) are optimised. These
include:
Cross Validation Approach: In the
tune.rcc() function, a coarse grid of possible values for
\(\lambda1\) and \(\lambda2\) is input to assess every
possible pair of parameters. As this process is computationally
intensive, it may not run for very large data sets (\(P\) or \(Q >
5,000\)). The tuning function outputs the optimal regularisation
parameters, which are then input into the rcc() function
with the argument method = 'ridge'.
Shrinkage Approach: This approach proposes an
analytical calculation of the regularisation parameters for large-scale
correlation matrices, and is implemented directly in the
rcc() function using the argument
method = 'shrinkage'. The downside of this approach is that
the (\(\lambda1\), \(\lambda2\)) values are calculated
independently, regardless of the cross-correlation between
X and Y , and thus may not be
successful in optimising the correlation between the data sets.