# Convergence of random sample vs. Latin Hypercube Sampling

Latin Hypercube Sampling, by concept, should be able to yield convergence of an estimate of output at a lower number of samples than random sampling. With the model I am working on, I'm continuously seeing convergence happen at a smaller sample size for random sampling than LHS. I was wondering if there would be any possible explanation for this observation. Thank you.

This behaviour is technically possible but very unusual. I would check your code very carefully to make sure there are no errors in your implementation of the sampling, or perhaps do some more reps to get a fuller understanding of the situation.

Proposition 10.4 in this book chapter, and the discussion below the proposition, might be relevant. The proposition states that, if $$f$$ is a real-valued function on $$[0,1]^d$$, $$\mu=\int f(x) \,dx$$, $$\sigma^2=\int (f(x)-\mu)^2\, dx<\infty$$, then the variance of the associated LHS estimator $$\hat{\mu}_{LHS}=\frac{1}{n}\sum_{i=1}^n f(X_i)$$ satisfies $$\text{Var}(\hat{\mu}_{LHS})\leq \frac{\sigma^2}{n-1}$$. This means that LHS cannot be worse than IID sampling with a sample size of one less (in terms of the variance of the estimator). LHS will tend to do less well for functions $$f$$ that are far from being additive (in a sense defined in section 10.3 of the book chapter).

There's also a theorem in the paper A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code by McKay, Beckman and Conover (Technometrics 21:239-245) which states that, if $$f$$ is monotonic in each of its arguments, then $$\text{Var}(\hat{\mu}_{LHS})\leq \frac{\sigma^2}{n}$$ i.e. LHS is never worse than IID sampling.

• Thanks for your detailed response and the reference. I was wondering if you would know a good way to test for additivity of the function when we don't have a priori information about the relationship between the output and the input parameters. LHS is commonly used in multivariate uncertainty & sensitivty analyses, for which we begin without much information about output qualities like additivity. – ss_19 May 19 at 2:02
• I'm not aware of an easy way to measure the extent to which a function is additive, though see stats.stackexchange.com/questions/84029/…. In practice, for a function that is not easy to evaluate and has no closed form expression, it is common to use LHS as a default sampling method. Prop 10.4 above says that we're not losing anything by doing this and we potentially have much to gain - but exactly how much is hard to quantify. – S. Catterall Reinstate Monica May 19 at 10:57