MultiCCA: Perform sparse multiple canonical correlation analysis.

Description

Given matrices $X1,...,XK$, which represent K sets of features on the same set of samples, find sparse $w1,...,wK$ such that $sum_(i<j) (wi' Xi' Xj wj)$ is large. If the columns of Xk are ordered (and type="ordered") then wk will also be smooth. For $X1,...,XK$, the samples are on the rows and the features are on the columns. $X1,...,XK$ must have same number of rows, but may (and usually will) have different numbers of columns.

Usage

MultiCCA(
  xlist,
  penalty = NULL,
  ws = NULL,
  niter = 25,
  type = "standard",
  ncomponents = 1,
  trace = TRUE,
  standardize = TRUE
)

Value

ws: A list of length K, containg the sparse canonical variates found (element k is a $p_k x ncomponents$ matrix).
ws.init: A list of length K containing the initial values of ws used, by default these are the v vector of the svd of matrix Xk.

Arguments

xlist: A list of length K, where K is the number of data sets on which to perform sparse multiple CCA. Data set k should be a matrix of dimension $n x p_k$ where $p_k$ is the number of features in data set k.
penalty: The penalty terms to be used. Can be a single value (if the same penalty term is to be applied to each data set) or a K-vector, indicating a different penalty term for each data set. There are 2 possible interpretations for the penalty terms: If type="standard" then this is an L1 bound on wk, and it must be between 1 and $sqrt(p_k)$ ($p_k$ is the number of features in matrix Xk). If type="ordered" then this is the parameter for the fused lasso penalty on wk.
ws: A list of length K. The kth element contains the first ncomponents columns of the v matrix of the SVD of Xk. If NULL, then the SVD of $X1,...,XK$ will be computed inside the MultiCCA function. However, if you plan to run this function multiple times, then save a copy of this argument so that it does not need to be re-computed.
niter: How many iterations should be performed? Default is 25.
type: Are the columns of $x1,...,xK$ unordered (type="standard") or ordered (type="ordered")? If "standard", then a lasso penalty is applied to v, to enforce sparsity. If "ordered" (generally used for CGH data), then a fused lasso penalty is applied, to enforce both sparsity and smoothness. This argument can be a vector of length K (if different data sets are of different types) or it can be a single value "ordered"/"standard" (if all data sets are of the same type).
ncomponents: How many factors do you want? Default is 1.
trace: Print out progress?
standardize: Should the columns of $X1,...,XK$ be centered (to have mean zero) and scaled (to have standard deviation 1)? Default is TRUE.

References

Witten D. M., Tibshirani R., and Hastie, T. (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics, Gol 10 (3), 515-534, Jul 2009

Examples

Run this code


# Generate 3 data sets so that first 25 features are correlated across
# the data sets...
set.seed(123)
u <- matrix(rnorm(50),ncol=1)
v1 <- matrix(c(rep(.5,25),rep(0,75)),ncol=1)
v2 <- matrix(c(rep(1,25),rep(0,25)),ncol=1)
v3 <- matrix(c(rep(.5,25),rep(0,175)),ncol=1)

x1 <- u%*%t(v1) + matrix(rnorm(50*100),ncol=100)
x2 <- u%*%t(v2) + matrix(rnorm(50*50),ncol=50)
x3 <- u%*%t(v3) + matrix(rnorm(50*200),ncol=200)

xlist <- list(x1, x2, x3)

# Run MultiCCA.permute w/o specifying values of tuning parameters to
# try.
# The function will choose the lambda for the ordered data set.
# Then permutations will be used to select optimal sum(abs(w)) for
# standard data sets.
# We assume that x1 is standard, x2 is ordered, x3 is standard:
perm.out <- MultiCCA.permute(xlist, type=c("standard", "ordered",
"standard"))
print(perm.out)
plot(perm.out)
out <- MultiCCA(xlist, type=c("standard", "ordered", "standard"),
penalty=perm.out$bestpenalties, ncomponents=2, ws=perm.out$ws.init)
print(out)
# Or if you want to specify tuning parameters by hand:
# this time, assume all data sets are standard:
perm.out <- MultiCCA.permute(xlist, type="standard",
penalties=cbind(c(1.1,1.1,1.1),c(2,3,4),c(5,7,10)), ws=perm.out$ws.init)
print(perm.out)
plot(perm.out)

# Making use of the fact that the features are ordered:
out <- MultiCCA(xlist, type="ordered", penalty=.6)
par(mfrow=c(3,1))
PlotCGH(out$ws[[1]], chrom=rep(1,ncol(x1)))
PlotCGH(out$ws[[2]], chrom=rep(2,ncol(x2)))
PlotCGH(out$ws[[3]], chrom=rep(3,ncol(x3)))