PMA (version 1.0.4)

CCA: Perform sparse canonical correlation analysis using the penalized matrix decomposition.

Description

Given matrices X and Z, which represent two sets of features on the same set of samples, find sparse u and v such that u'X'Zv is large. If the columns of Z are ordered (and type="ordered") then v will also be smooth. For X and Z, the samples are on the rows and the features are on the columns. X and Z must have same number of rows, but may (and usually will) have different numbers of columns.

Usage

CCA(x, z, type=c("standard", "ordered"), sumabs=.4, sumabsu=4,
sumabsv=NULL,
lambda=NULL, K=1, niter=25,v=NULL, trace=TRUE, standardize=TRUE, xnames=NULL,
znames=NULL, chrom=NULL, upos=FALSE, uneg=FALSE, vpos=FALSE, vneg=FALSE,
outcome=NULL,y=NULL,cens=NULL)

Arguments

x

Data matrix; samples are rows and columns are features. Cannot contain missing values.

z

Data matrix; samples are rows and columns are features. Note that x and z must have the same number of rows, but may (and generally will) have different numbers of columns. Cannot contain missing values.

type

Are the columns of z 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 appli

sumabs

Used only for type "standard". Controls both sumabsu and sumabsv at once. Must be between 0 and 1. If either sumabsu or sumabsv is NULL, then the value of sumabs will be used to determine values for sumabsu and sumabsv, as follows: sumabsu

sumabsu

How sparse do you want u to be? This is the sum of absolute values of elements of u. It must be between 1 and the square root of the number of columns of x. The smaller it is, the sparser u will be. Default is NULL (sumabs is used instead).

sumabsv

Used only for type "standard". How sparse do you want v to be? This is the sum of absolute values of elements of v. It must be between 1 and square root of number of columns of z. The smaller it is, the sparser v will be. Default is NULL (suma

lambda

Used only for type "ordered", controls fused lasso penalty on v, which takes the form $lambda( sum_j |v_j| + sum_j |v_j - v_(j-1)|)$.

K

How many factors u and v do you want? Default is 1.

niter

How many iterations should be performed? Default is 25.

v

The first K columns of the v matrix of the SVD of X'Z. If NULL, then the SVD of X'Z will be computed inside the CCA function. However, if you plan to run this function multiple times, then save a copy of this argument so that it does not need

trace

Print out progress?

standardize

Should the columns of x and z be centered (to have mean zero) and scaled (to have standard deviation 1)? Default is TRUE.

xnames

An optional vector of column names for x.

znames

An optional vector of column names for z.

chrom

Used only if type is "ordered"; allows user to specify a vector of length ncol(z) giving the chromosomal location of each CGH spot. This is so that smoothness will be enforced within each chromosome, but not between chromosomes.

upos

If TRUE, then require elements of u to be positive. FALSE by default.

uneg

If TRUE, then require elements of u to be negative. FALSE by default.

vpos

Only applicable if type="standard". If TRUE, require elements of v to be positive. FALSE by default.

vneg

Only applicable if type="standard". If TRUE, require elements of v to be negative. FALSE by default.

outcome

If you would like to incorporate a phenotype into CCA analysis - that is, you wish to find features that are correlated across the two data sets and also correlated with a phenotype - then use one of "survival", "multiclass", or "quantitat

y

If outcome is not NULL, then this is a vector of phenotypes - one for each row of x and z. If outcome is "survival" then these are survival times; must be non-negative. If outcome is "multiclass" then these are class labels (1,2,3,...). Defaul

cens

If outcome is "survival" then these are censoring statuses for each observation. 1 is complete, 0 is censored. Default NULL.

Value

• u
  u is output. If you asked for multiple factors then each column of u is a factor. u has dimension nxK if you asked for K factors.
• v
  v is output. If you asked for multiple factors then each column of v is a factor. v has dimension pxK if you asked for K factors.
• d
  A vector of length K, which can alternatively be computed as the diagonal of the matrix $u'X'Zv$.
• v.init
  The first K factors of the v matrix of the SVD of x'z. This is saved in case this function will be re-run later.

Details

This function is useful for performing an integrative analysis of two data sets taken on the same set of samples: for instance, gene expression and CGH measurements on the same set of patients. It takes in two data sets, called x and z, each of which have (the same set of) samples on the rows. If z is a matrix of CGH data with *ordered* CGH spots on the columns, then use type="ordered". If z consists of unordered columns, then use type="standard". This function performs the penalized matrix decomposition on the data matrix $X'Z$. Therefore, the results should be the same as running the PMD function on t(x)%*%z. However, when ncol(x)>>nrow(x) and ncol(z)>>nrow(z) then using the CCA function is much faster because it avoids computation of $X'Z$. (It is easy to verify that PMD(t(x)%*%z, sumabs=.5, type="standard", center=FALSE) gives same results as CCA(x, z, sumabs=.5, type="standard", standardize=FALSE) for some matrices x and z).

The CCA criterion is as follows: find unit vectors $u$ and $v$ such that $u'X'Zv$ is maximized subject to constraints on $u$ and $v$. If type="standard" then the constraints on $u$ and $v$ are lasso ($L_1$). If type="ordered" then the constraint on $u$ is a lasso constraint, and there is a fused lasso constraint on $v$ (promoting sparsity and smoothness).

When type is "standard": If either sumabsu or sumabsv is NULL, then sumabs must be non-NULL. In this case, sumabs will be used to set values for both sumabsu and sumabsv, as follows: sumabsu will be set to sumabs*sqrt(ncol(x)) and sumabsv will be set to sumabs*sqrt(ncol(z)). The sumabs argument is ignored if sumabsu and sumabsv both are non-NULL.

When type is "ordered": lambda controls the amount of sparsity and smoothness in v, via the fused lasso penalty: $lambda sum_j |v_j| + lambda sum_j |v_j - v_(j-1)|$. If NULL, then it will be chosen adaptively from the data. Sumabsu is the bound on the sum of absolute values of elements of u - the greater it is, the sparser u will be.

When running CCA on gene expression and CGH data (type="ordered"), there are multiple ways to do it:

(1) using all gene expression data and all CGH data, regardless of chromosome. (2) using CGH data on chromosome j and gene expression data on all chromosomes. This will allow for discovery of cis/trans interactions relating to copy number change on chrom j. Then, repeat for other chromosomes. (3) using CGH data on chromosome j and gene expression data on chromosome j. Cis interactions can be found this way. (4) using CGH data on chromosome j and gene expression data NOT on chromosome j; one can find trans interactions this way...

References

Witten, DM and Tibshirani, R and T Hastie (2008) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Submitted.

See Also

PMD,CCA.permute

Examples

# first, do CCA with type="standard"
# A simple simulated example
u <- matrix(c(rep(1,25),rep(0,75)),ncol=1)
v1 <- matrix(c(rep(1,50),rep(0,450)),ncol=1)
v2 <- matrix(c(rep(0,50),rep(1,50),rep(0,900)),ncol=1)
x <- u%*%t(v1) + matrix(rnorm(100*500),ncol=500)
z <- u%*%t(v2) + matrix(rnorm(100*1000),ncol=1000)
# Can run CCA with default settings, and can get e.g. 3 components
out <- CCA(x,z,type="standard",K=3)
print(out,verbose=TRUE) # To get less output, just print(out)
# Or can use CCA.permute to choose optimal parameter values
perm.out <- CCA.permute(x,z,type="standard",nperms=7,sumabss=seq(0.1,.75,len=12))
print(perm.out)
plot(perm.out)
out <- CCA(x,z,type="standard",K=1,sumabs=perm.out$bestsumabs, v=perm.out$v.init)
print(out)
# Now try CCA with a constraint that elements of u must be negative and
# elements of v must be positive:
perm.out <- CCA.permute(x,z,type="standard",nperms=7,
sumabss=seq(0.1,.75,len=12), uneg=TRUE, vpos=TRUE)
print(perm.out)
plot(perm.out)
out <- CCA(x,z,type="standard",K=1,sumabs=perm.out$bestsumabs,
v=perm.out$v.init, uneg=TRUE, vpos=TRUE)
print(out)

# Suppose we also have a quantitative outcome, y, and we want to find
# features in x and z that are correlated with each other and with the
# outcome:
y <- rnorm(nrow(x))
perm.out <- CCA.permute(x,z,type="standard",outcome="quantitative",y=y)
print(perm.out)
out<-CCA(x,z,type="standard",outcome="quantitative",y=y,sumabs=perm.out$bestsumabs)
print(out)


# now, do CCA with type="ordered"
# Example involving the breast cancer data: gene expression + CGH
set.seed(22)
data(breastdata)
attach(breastdata)
dna <- t(dna)
rna <- t(rna)
perm.out <- CCA.permute(rna,dna[,chrom==1],type="ordered",nperms=5)
# We run CCA using all gene exp. data, but CGH data on chrom 1 only.
print(perm.out)
plot(perm.out)
out <- CCA(rna,dna[,chrom==1], type="ordered",sumabsu=perm.out$bestsumabsu,
v=perm.out$v.init, lambda=perm.out$lambda, xnames=substr(genedesc,1,20),
znames=paste("Pos", sep="", nuc[chrom==1])) # Save time by inputting  lambda and v
print(out) # could do print(out,verbose=TRUE)
print(genechr[out$u!=0]) # Cool! The genes associated w/ gain or loss
# on chrom 1 are located on chrom 1!!
par(mfrow=c(1,1))
PlotCGH(out$v, nuc=nuc[chrom==1], chrom=chrom[chrom==1],
main="Regions of gain/loss on Chrom 1 assoc'd with gene expression")
detach(breastdata)