Folks: Perhaps I am naive, ignorant, or foolish, but this whole discussion seems rather ridiculous. What possible relation to reality could a multivariate normal of the size requested have? Even for simulation, it seems like nonsense.
Cheers, Bert On Sun, Feb 19, 2012 at 11:35 AM, Petr Savicky <savi...@cs.cas.cz> wrote: > On Sat, Feb 18, 2012 at 06:00:53PM -0500, li li wrote: >> Dear all, >> I need to generate numbers from multivariate normal with large dimensions >> (5,000,000). > > Hi. > > I am replying to your first email, since the other did not arrive > to my folder, possibly filtered out by a spam filter. I see them > at the web interface. > > 1. Error: cannot allocate vector of size 381.5 Mb > > The error message makes sense. The matrix requires m*n*8/2^20 MB, > which is in your case > > m <- 100000 > n <- 500 > m*n*8/2^20 > > [1] 381.4697 > > May be, you already have other large objects in the memory. Try to > minimize the number and size of objects, which you need simultaneously > in an R session. > > 2. Generating a multivariate normal distribution. > > As Peter Dalgaard pointed out, a speed up is possible only > for special types of the covariance matrix Sigma. A general > way is to find a matrix A such that A A^t = Sigma. Then, > the vector A X, where X is from N(0,I) and I is an identity > matrix of an appropriate dimension, has covariance Sigma. > This is also the way, how mvtnorm package works. > > A speed up is possible, if computing the product A X does not > require to have matrix A explicitly represented in memory. > > The matrix A need not be a square matrix. In particular, the > previous case may be understood as using the matrix A, which > for a small m is as follows. > > m <- 5 > rho <- 0.5 > A <- cbind(sqrt(rho), sqrt(1 - rho)*diag(m)) > A > > > [,1] [,2] [,3] [,4] [,5] [,6] > [1,] 0.7071068 0.7071068 0.0000000 0.0000000 0.0000000 0.0000000 > [2,] 0.7071068 0.0000000 0.7071068 0.0000000 0.0000000 0.0000000 > [3,] 0.7071068 0.0000000 0.0000000 0.7071068 0.0000000 0.0000000 > [4,] 0.7071068 0.0000000 0.0000000 0.0000000 0.7071068 0.0000000 > [5,] 0.7071068 0.0000000 0.0000000 0.0000000 0.0000000 0.7071068 > > A %*% t(A) > > [,1] [,2] [,3] [,4] [,5] > [1,] 1.0 0.5 0.5 0.5 0.5 > [2,] 0.5 1.0 0.5 0.5 0.5 > [3,] 0.5 0.5 1.0 0.5 0.5 > [4,] 0.5 0.5 0.5 1.0 0.5 > [5,] 0.5 0.5 0.5 0.5 1.0 > > This construction is conceptually possible also for a large m because > the structure of A allows to compute A X by simpler operations with > the vector X than an explicit matrix product. Namely, the expression > > rnorm(1, sd=sqrt(rho)) + rnorm(m, sd=sqrt(1 - rho)) > > or, more clearly, > > sqrt(rho) * rnorm(1) + sqrt(1 - rho) * rnorm(m) > > is equivalent to the required A X, where X consists of rnorm(1) > and rnorm(m) together. > > If you have a specific Sigma, describe it and we can discuss, > whether an appropriate A can be found. > > Hope this helps. > > Petr Savicky. > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. -- Bert Gunter Genentech Nonclinical Biostatistics Internal Contact Info: Phone: 467-7374 Website: http://pharmadevelopment.roche.com/index/pdb/pdb-functional-groups/pdb-biostatistics/pdb-ncb-home.htm ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.