Extended Latin Hypercube Sampling for Integration and Simulation

Abstract

We analyze an extended form of Latin hypercube sampling technique that can be used for numerical quadrature and for Monte Carlo simulation. The technique utilizes random point sets with enhanced uniformity over the s-dimensional unit hypercube. A sample of N = n ^s points is generated in the hypercube. If we project the N points onto their ith coordinates, the resulting set of values forms a stratified sample from the unit interval, with one point in each subinterval $[(k - 1)/N,k/N)$. The scheme has the additional property that when we partition the hypercube into N subcubes $\prod _{i=1}^{s}[(\ell_{i} - 1)/n,\ell_{i}/n)$, each one contains exactly one point. We establish an upper bound for the variance, when we approximate the volume of a subset of the hypercube, with a regular boundary. Numerical experiments assess that the bound is tight. It is possible to employ the extended Latin hypercube samples for Monte Carlo simulation. We focus on the random walk method for diffusion and we show that the variance is reduced when compared with classical random walk using ordinary pseudo-random numbers. The numerical comparisons include stratified sampling and Latin hypercube sampling.