Faster Multivariate Normal densities with RcppArmadillo and OpenMP
Nino Hardt, Dicko Ahmadou, Benjamin Christoffersen —
written Jul 13, 2013 —
updated Feb 2, 2020 —
The Multivariate Normal density function is used frequently
for a number of problems. Especially for MCMC problems, fast
evaluation is important. Multivariate Normal Likelihoods,
Priors and mixtures of Multivariate Normals require numerous
evaluations, thus speed of computation is vital.
We show dramatic increases in speed by using efficient algorithms,
RcppArmadillo, and some extra gain by using OpenMP.
The code is based on the latest version of RcppArmadillo (0.9.800.1.0).
While the dmvnorm() function from the mvtnorm package is quite popular,
and in an earlier version of this article we demonstrated that an
Rcpp implementation would lead to faster computation.
Peter Rossi, author of bayesm, called our attention to the bayesm pure R
implementation which is much faster than dmvnorm().
The function dMvn() is used internally by the mixture of normals model in
bayesm. It is the matrix-equivalent version of lndMvn:
Translating the vectorized approach into RcppArmadillo,
we precompute the inverse of the Cholesky decomposition of the covariance
matrix ahead of the main loop over the rows of X.
The loop can easily be parallelized, and the code is easy to read and
manipulate. For instance, the inverse Cholesky decomposition can be put
inside the main loop, if varying covariance matrices are necessary.
It is worth remarking that multiplying with a precomputed inverse of the
Cholesky decomposition of
the covariance matrix is faster but less numerically stable compared to
a backsolve as in the Mahalanobis function we will define later.
The use of trimatu allows to exploit that of the Cholesky
decomposition the covariance matrix is an upper triangular matrix in the
inversion. The dmvnrm_arma_old is an older version of the function used in
a previous version of this article. The new version differs mainly by
using const & for the input parameters.
declaring z outside the loop.
using arma::dot instead of arma::sum.
other minor things.
This turns out to be quite important for the computation times. The
dmvnrm_arma_fast makes an inplace vector matrix product and exploits
that the matrix is an upper triangular matrix. One can use the
BLAS function instead. It is not available through the Armadillo library
Additionally, we can make use of the OpenMP library to use multiple
cores. For the OpenMP implementation, we need to enable OpenMP support.
One way of doing so is by adding the required compiler and linker
flags as follows:
Rcpp version 0.10.5 and later will also provide a plugin to set these
variables for us:
When used in a package, the standard src/Makevars supplied by the package
ensures the variable $(SHLIB_OPENMP_CXXFLAGS) is used. It relies on the R
configuration to automatically add OpenMP support where available.
We also need to set the number of cores to be used before running the
compiled functions. One way is to use detectCores() from the parallel
Only two additional lines are needed to enable multicore processing.
In this example, a dynamic schedule is used for OpenMP.
A static schedule might be faster in some cases. However,this is
left to further experimentation.
Likewise, it is easy to translate dmvnorm from the mvtnorm
package into Rcpp:
We use each_col, an appropriate overload of arma::solve, and for_each
to do the computations without performing additional allocations after the
copy at x_cen = x.t().