# Effectiveness of Latin Hypercube Sampling

I have a question regarding the disccusion on the effectiveness of LHS in multiple dimensions, linked below.

According to the accepted answer, it appears that LHS starts to lose its effectiveness in variance reduction for problems with a large number of parameters. I was wondering if there is any consensus on the upper bound for the number of parameters that LHS is proven to be effective.

I agree with the answer by R Carnell, there is no upper bound on the number of parameters/dimensions for which LHS is proven to be effective, though in many settings I've noticed that the relative benefits of LHS compared to simple random sampling tend to decrease as the number of dimensions increases. In practice, this behaviour doesn't actually matter. LHS is essentially never worse than simple random sampling, so you can always use LHS as a default sampling method and this decision won't cost you anything.

There's an interesting blog post by David Vose in which he explains why he doesn't implement LHS in his ModelRisk software. He seems to be considering the situation where it is trivial (by modern computing standards) to evaluate the output function at each sampled point in parameter space, so I don't think this article is a reason to avoid LHS. Indeed, many researchers continue to use LHS regularly as a default sampling option. I also note this blog post by Lonnie Chrisman which argues in favour of LHS as a default for sampling. This latter article also suggests a rule of thumb that LHS is most effective when at most 3 inputs/dimensions contribute most of the variation in the output. It also contains a number of references to the literature: some researchers have found that LHS substantially outperforms simple random sampling, whereas others have noted minimal improvements. Once you move outside the realm of additive functions, it's very hard to predict how much of an improvement you'll get.

• I see, thank you both for your comments! Jun 3 '20 at 2:58
• @ss_19 I suggest you upvote both answers if you found them both to be useful. Jun 3 '20 at 8:47

I interpret the literature cited in the accepted answer differently. The original poster was looking for an amount of "variance reduction" in the Latin hypercube. The plots they showed were the confidence intervals for the mean of their cost function with increasing sample size for 1 dimension and 2 dimension. If you read the chapter cited by the accepted answer here, they talk about effectiveness of variance reduction or efficiency being measured relative to some base algorithm like simple random sampling. The conclusions in the literature are clear:

For estimating the variance in functions which are "additive" in the margins of the Latin hypercube, then the variance in the estimate of the function is always less than the equivalent sample size of simple random sample, regardless of the number of dimensions and regardless of sample size. See here from the accepted answer, and also Stein 1987 and Owen 1997.

For non-additive functions, the Latin hypercube sample may still provide benefit, but it is less certain to provide benefit in all cases. A LHS of size $$n > 1$$ has variance in the non-additive estimator less than or equal to a simple random sample of size $$(n-1)$$. Owen 1997 says this is "not much worse than" simple random sampling.

These conclusions are all irrespective of the number of dimensions in the sample. There is no upper bound in dimensions for which LHS is proven to be effective.