MultiCCA: Perform sparse multiple canonical correlation analysis.

In PMA2: Penalized Multivariate Analysis

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
)

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.

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.

References

Ali Mahzarnia, Alexander Badea (2022), Joint Estimation of Vulnerable Brain Networks and Alzheimer’s Disease Risk Via Novel Extension of Sparse Canonical Correlation at bioRxiv.

See Also

MultiCCA.permute,CCA, CCA.permute

Examples

# 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)
oldpar <- par(mfrow = c(1,1))
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)))
par(oldpar)

PMA2 documentation built on May 12, 2022, 9:06 a.m.