Nino Hardt, Dicko Ahmadou — written Jul 13, 2013 — source
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 (0.3.910.0).
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
dMvn() is used internally by the mixture of normals model in
bayesm. It is the matrix-equivalent version of
Translating the vectorized approach into RcppArmadillo,
we precompute the inverse root of the covariance matrix ahead
of the main loop over the rows of
The loop can easily be parallelized, and the code is easy to read and
manipulate. For instance, the inverse root can be put inside the main
loop, if varying covariance matrices are necessary.
The use of
trimatu allows to exploit the diagonality of the Cholesky
root of the covariance matrix.
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:
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
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:
Now we simulate some data for benchmarking:
And run the benchmark:
 "Using 8 cores for _mc versions"
Loading required package: rbenchmark
test replications elapsed relative 4 dmvnrm_arma_mc(X, means, sigma, F, cores) 100 9.636 1.000 3 dmvnrm_arma(X, means, sigma, F) 100 18.848 1.956 2 dmvnorm_arma(X, means, sigma, F) 100 23.459 2.435 5 dMvn(X, means, sigma) 100 29.501 3.062 1 mvtnorm::dmvnorm(X, means, sigma, log = F) 100 44.556 4.624
Lastly, we show that the functions yield the same results:
The use of RcppArmadillo brings about a significant increase in speed. The addition of OpenMP leads to only little additional performance.
This example also illustrates that Rcpp does not completely substitute the need to look for faster algorithms. Basing the code of ‘lndMvn’ instead of ‘dmvnorm’ leads to a significantly faster function.Tweet