Simulation of super high-dimensional distribution

[zxjroger]

I am working on simulated method of moments which requires to draw random variables from a multivariate normal distribution for numerical integration. However, the dimension of this multivariate normal distribution is super high, something like 250000. I would like to use Halton sequence. Since the dimension is so high, I believe I need to use the so called scrambled Halton sequence to avoid highly correlated draws. I have three questions:

1. Given the super high dimension, is there a better way of sampling?
2. If I have to use scrambled Halton sequence, is there a Julia package for it?
3. Given the super high dimension, is there a rule of thumb for the number of simulations for each observation?

I noticed the following post about Sobol sequences, but it seems that there is already some problem with a dimension of 100. So I am guessing it will be the same for Halton.

Generating high-dimensional Sobol sequences

Hi there, I am using Sobol sequences to generate say 10,000 draws from the Multivariate Normal with dimension 100, mean vector 0 and covariance matrix equal to the identity matrix. Generating the sequence is fine, but I find that the distribution of the resulting points is highly concentrated. That is, the density of each draw, relative to that of the other draws, is almost zero for 99% of points, then several orders of magnitude higher for just a few points. I’d expect the relative densities …

I noticed the following two Halton packages, but I don’t think they offer the option of scrambled Halton. Thank you.

GitHub - npfrazier/FastHalton.jl: Implements Halton Sequences and Halton...

GitHub - tobydriscoll/HaltonSequences.jl: Halton sequences and points in Julia

[Tamas_Papp]

Alternatively, you could just set a random seed and use plain old rand(). Either your problem is “nice” (insensitive to issues of non-uniform coverage), or both methods will be horrible anyway for 2.5e5 dimensions for a feasible number of draws.

[zxjroger]

I see. Thank you.

[phrb]

Regarding your second point, I don’t know what you’ve used before, but I use the randtoolbox R package to generate scrambled low discrepancy sequences, you can try to use it with RCall.