Petr,
   Thanks for the help. That certainly makes sense and solves my current
problem. But, in general, for arbitrary covariance matrix (instead of the
special equi-correlation case), it there a way to generate numbers from
 multivariate normal when the dimension is very high?
   Thanks.
         Hannah

在 2012年2月19日 上午5:01，Petr Savicky <savicky@cs.cas.cz>写道：

> 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).
> > Below is my code and the error I got from R. Sigma in the code is the
> > covariance
> > matrix. Can anyone give some idea on how to take care of this error.
> Thank
> > you.
> >                 Hannah
> >
> > > m <- 5000000
> > > m1 <- 0.5*m
> > > rho <- 0.5
> > > Sigma <- rho* matrix(1, m, m)+diag(1-rho, m)
> > Error in matrix(1, m, m) : too many elements specified
>
> Hi.
>
> The matrix of dimension m times m does not fit into memory,
> since it requires 8*m^2 = 2e+14 bytes = 2e+05 GB.
>
> Generating a multivariate normal with a covariance matrix
> with 1 on the diagonal and rho outside of the diagonal may
> be done also as follows.
>
>  m <- 10 # can be 5000000
>  rho <- 0.5
>  # single vector
>  x <- rnorm(1, sd=sqrt(rho)) + rnorm(m, sd=sqrt(1 - rho))
>
>  # several vectors
>  a <- t(replicate(10000, rnorm(1, sd=sqrt(rho)) + rnorm(m, sd=sqrt(1 -
> rho))))
>
>  # check the sample covariance matrix if m is not too large
>  sigma <- cov(a)
>  range(diag(sigma)) # elements on the diagonal
>
>  [1] 0.9963445 1.0196015
>
>  diag(sigma) <- NA
>  range(sigma, na.rm=TRUE) # elements outside of the diagonal
>
>  [1] 0.4935129 0.5162836
>
> Generating several vectors using replicate() may not be efficient.
> The following can be used instead.
>
>  n <- 10000
>  a <- matrix(rnorm(n*m, sd=sqrt(1 - rho)), nrow=n, ncol=m) + rnorm(n,
> sd=sqrt(rho))
>
> Note that the size of "a" is n times m and it should fit into the memory.
>
> Hope this helps.
>
> Petr Savicky.
>
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