When trying to run the cor() function on sparse matrices (of either type dgCMatrix or dgTMatrix) I get the following error:
Error in cor(x) : supply both 'x' and 'y' or a matrix-like 'x'
Converting my matrix to be dense will be very inefficient. Is there an easy way to calculate this correlation (without an all pairs loop?).
Thanks,
EDITED ANSWER - optimized for memory use and speed.
Your error is logic, as a sparse matrix is not recognized by the cor function as a matrix, and there is -yet- no method for correlations in the Matrix package.
There is no function I am aware of that will let you calculate this, but you can easily calculate that yourself, using the matrix operators that are available in the Matrix package :
sparse.cor <- function(x){
n <- nrow(x)
m <- ncol(x)
ii <- unique(x@i)+1 # rows with a non-zero element
Ex <- colMeans(x)
nozero <- as.vector(x[ii,]) - rep(Ex,each=length(ii)) # colmeans
covmat <- ( crossprod(matrix(nozero,ncol=m)) +
crossprod(t(Ex))*(n-length(ii))
)/(n-1)
sdvec <- sqrt(diag(covmat))
covmat/crossprod(t(sdvec))
}
the covmat is your variance-covariance matrix, so you can calculate that one as well. The calculation is based on selecting the rows where at least one element is non-zero. to the cross product of this one, you add the colmeans multiplied by the number of all-zero rows. This is equivalent to
( X - E[X] ) times ( X - E[X] ) transposed
Divide by n-1 and you have your variance-covariance matrix. The rest is easy.
A test case :
X <- sample(0:10,1e8,replace=T,p=c(0.99,rep(0.001,10)))
xx <- Matrix(X,ncol=5)
> system.time(out1 <- sparse.cor(xx))
user system elapsed
0.50 0.09 0.59
> system.time(out2 <- cor(as.matrix(xx)))
user system elapsed
1.75 0.28 2.05
> all.equal(out1,out2)
[1] TRUE
This is what I ended up using. Thanks @Joris for all the help!
My x matrix is quite big. Assuming it's size is n * p, n=200k and p=10k in my case.
The trick is to maintain the sparsity of the operations and perform the calculations on p * p matrices instead of p*n x n*p.
Version 1, is more straightforward, but less efficient on time and memory, as the outer product operation is expensive:
sparse.cor2 <- function(x){
n <- nrow(x)
covmat <- (crossprod(x)-2*(colMeans(x) %o% colSums(x))
+n*colMeans(x)%o%colMeans(x))/(n-1)
sdvec <- sqrt(diag(covmat)) # standard deviations of columns
covmat/crossprod(t(sdvec)) # correlation matrix
}
Version 2 is more efficient on time (saves several operations) and on memory. Still requires huge amounts of memory for a p=10k matrix:
sparse.cor3 <- function(x){
memory.limit(size=10000)
n <- nrow(x)
cMeans <- colMeans(x)
cSums <- colSums(x)
# Calculate the population covariance matrix.
# There's no need to divide by (n-1) as the std. dev is also calculated the same way.
# The code is optimized to minize use of memory and expensive operations
covmat <- tcrossprod(cMeans, (-2*cSums+n*cMeans))
crossp <- as.matrix(crossprod(x))
covmat <- covmat+crossp
sdvec <- sqrt(diag(covmat)) # standard deviations of columns
covmat/crossprod(t(sdvec)) # correlation matrix
}
Timing comparisons (sparse.cor is @Joris latest version):
> X <- sample(0:10,1e7,replace=T,p=c(0.9,rep(0.01,10)))
> x <- Matrix(X,ncol=10)
>
> object.size(x)
11999472 bytes
>
> system.time(corx <- sparse.cor(x))
user system elapsed
0.50 0.06 0.56
> system.time(corx2 <- sparse.cor2(x))
user system elapsed
0.17 0.00 0.17
> system.time(corx3 <- sparse.cor3(x))
user system elapsed
0.13 0.00 0.12
> system.time(correg <-cor(as.matrix(x)))
user system elapsed
0.25 0.03 0.29
> all.equal(c(as.matrix(corx)),c(as.matrix(correg)))
[1] TRUE
> all.equal(c(as.matrix(corx2)),c(as.matrix(correg)))
[1] TRUE
> all.equal(c(as.matrix(corx3)),c(as.matrix(correg)))
[1] TRUE
Much larger x matrix:
> X <- sample(0:10,1e8,replace=T,p=c(0.9,rep(0.01,10)))
> x <- Matrix(X,ncol=10)
> object.size(x)
120005688 bytes
> system.time(corx2 <- sparse.cor2(x))
user system elapsed
1.47 0.07 1.53
> system.time(corx3 <- sparse.cor3(x))
user system elapsed
1.18 0.09 1.29
> system.time(corx <- sparse.cor(x))
user system elapsed
5.43 1.26 6.71
The answer was solved elegantly by @Ron but a slight modification to the solution is a little cleaner and also returns the sample covariance matrix.
sparse.cor4 <- function(x){
n <- nrow(x)
cMeans <- colMeans(x)
covmat <- (as.matrix(crossprod(x)) - n*tcrossprod(cMeans))/(n-1)
sdvec <- sqrt(diag(covmat))
cormat <- covmat/tcrossprod(sdvec)
list(cov=covmat,cor=cormat)
}
The simplification comes from this: with an n x p matrix X, and an n x p matrix M of the column means of X:
cov(X) = E[(X-M)'(X-M)] = E[X'X - M'X - X'M + M'M]
M'X = X'M = M'M, which have (i,j) elements = sum(column i) * sum(column j) / n
= n * mean(column i) * mean(column j)
or written with a row vector m of the column means,
= n * m'm
Then cov(X) = E[X'X - n m'm]
and it is now a smidge faster.
> X <- sample(0:10,1e7,replace=T,p=c(0.9,rep(0.01,10)))
> x <- Matrix(X,ncol=10)
> system.time(corx <- sparse.cor(x))
user system elapsed
1.139 0.196 1.334
> system.time(corx3 <- sparse.cor3(x))
user system elapsed
0.194 0.007 0.201
> system.time(corx4 <- sparse.cor4(x))
user system elapsed
0.187 0.007 0.194
> system.time(correg <-cor(as.matrix(x)))
user system elapsed
0.341 0.067 0.407
> system.time(covreg <- cov(as.matrix(x)))
user system elapsed
0.314 0.016 0.330
> all.equal(c(as.matrix(corx)),c(as.matrix(correg)))
[1] TRUE
> all.equal(c(as.matrix(corx3)),c(as.matrix(correg)))
[1] TRUE
> all.equal(c(as.matrix(corx4$cor)),c(as.matrix(correg)))
[1] TRUE
> all.equal(c(as.matrix(corx4$cov)),c(as.matrix(covreg)))
[1] TRUE
