Fast simulation of multivariate Student's t random variables

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Description

Fast simulation of multivariate Student's t random variables

Usage

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rmvt(n, mu, sigma, df, ncores = 1, isChol = FALSE, A = NULL)

Arguments

n

number of random vectors to be simulated.

mu

vector of length d, representing the mean of the distribution.

sigma

scale matrix (d x d). Alternatively it can be the cholesky decomposition of the scale matrix. In that case isChol should be set to TRUE. Notice that ff the degrees of freedom (the argument df) is larger than 2, the Cov(X)=sigma*df/(df-2).

df

a positive scalar representing the degrees of freedom.

ncores

Number of cores used. The parallelization will take place only if OpenMP is supported.

isChol

boolean set to true is sigma is the cholesky decomposition of the covariance matrix.

A

an (optional) numeric matrix of dimension (n x d), which will be used to store the output random variables. It is useful when n and d are large and one wants to call rmvn() several times, without reallocating memory for the whole matrix each time. NB: the element of A must be of class "numeric".

Details

There are in fact many candidates for the multivariate generalization of Student's t-distribution, here we use the parametrization described here https://en.wikipedia.org/wiki/Multivariate_t-distribution.

Notice that rmvt() does not use one of the Random Number Generators (RNGs) provided by R, but one of the parallel cryptographic RNGs described in (Salmon et al., 2011). It is important to point out that this RNG can safely be used in parallel, without risk of collisions between parallel sequence of random numbers. The initialization of the RNG depends on R's seed, hence the set.seed() function can be used to obtain reproducible results. Notice though that changing ncores causes most of the generated numbers to be different even if R's seed is the same (see example below). NB: at the moment the RNG does not work properly on Solaris OS when ncores>1. Hence, rmvt() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

If A==NULL (default) the output is an (n x d) matrix where the i-th row is the i-th simulated vector. If A!=NULL then the random vector are store in A, which is provided by the user, and the function returns NULL.

Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>, C++ RNG engine by Thijs van den Berg <http://sitmo.com/>.

References

John K. Salmon, Mark A. Moraes, Ron O. Dror, and David E. Shaw (2011). Parallel Random Numbers: As Easy as 1, 2, 3. D. E. Shaw Research, New York, NY 10036, USA.

Examples

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d <- 5
mu <- 1:d
df <- 4

# Creating covariance matrix
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp) + diag(0.5, d)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)  
rmvt(4, 1:d, mcov, df = df, ncores = 2) # These will not match the r.v. generated on a single core.

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, d)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmvt(4, 1:d, mcov, df = df, ncores = 2, A = A) # This returns NULL ...
A                                     # ... but the result is here