I am computing a complex multidimensional integral (4 dimensions) using
Gauss-legendre and am looking to optimize one piece of code that is currently
very expensive. The integral is evaluated for K individuals in my data and once
this piece is computed it can be stored and recycled over all K individuals.
Once this piece is stored, the integral can be computed in about 1.3 minutes
using R for each individual.
Because the integral has multiple dimensions, the number of nodes grows
exponentially such that I require q^4 total nodes and experimentation is
showing I need no fewer than 40 per dimension. At each of these, I need to
compute the density of the multivariate normal at all q^4 nodes and store it.
This is very expensive and I wonder if there is a way to improve the
computational time to be faster?
Below is just a reproducible toy example (not using legendre nodes) to
illustrate the issue. The final line is what I am wondering about to see if it
can be improved in terms of computational time.
Thank you
Harold
library(mvtnorm)
### Create parameters for MVN
mu <- c(0,0,0,0)
cov <- matrix(.2, ncol= 4,nrow=4)
diag(cov) <- 1
sigma <- as.matrix(cov)
### Create nodes and expand to 4 dimensions for quadrature
dm <- length(mu)
gh <- seq(-4, 4, length.out = 10)
n <- length(gh)
idx <- as.matrix(expand.grid(rep(list(1:n),dm)))
### Compute density, this is expensive
adjFactor <- sapply(1:nrow(idx), function(i) dmvnorm(gh[idx[i,]], mean = mu,
sigma = sigma))
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