The generalization of Latin hypercube sampling

TL;DR

This paper introduces a generalized framework for Latin hypercube sampling called partially stratified sample designs, which can reduce variance in estimates by balancing main effects and interactions, and proposes a new Latinized stratified sampling method to improve sampling efficiency.

Contribution

It extends LHS to a spectrum of stratified designs, derives their properties, and introduces LSS to combine advantages of both methods for improved variance reduction.

Findings

PSS reduces variance from variable interactions.

LHS reduces variance from main effects.

LSS achieves simultaneous variance reduction for effects and interactions.

Abstract

Latin hypercube sampling (LHS) is generalized in terms of a spectrum of stratified sampling (SS) designs referred to as partially stratified sample (PSS) designs. True SS and LHS are shown to represent the extremes of the PSS spectrum. The variance of PSS estimates is derived along with some asymptotic properties. PSS designs are shown to reduce variance associated with variable interactions, whereas LHS reduces variance associated with main effects. Challenges associated with the use of PSS designs and their limitations are discussed. To overcome these challenges, the PSS method is coupled with a new method called Latinized stratified sampling (LSS) that produces sample sets that are simultaneously SS and LHS. The LSS method is equivalent to an Orthogonal Array based LHS under certain conditions but is easier to obtain. Utilizing an LSS on the subspaces of a PSS provides a sampling strategy that reduces variance associated with both main effects and variable interactions and can be designed specially to minimize variance for a given problem. Several high-dimensional numerical examples highlight the strengths and limitations of the method. The Latinized partially stratified sampling method is then applied to identify the best sample strategy for uncertainty quantification on a plate buckling problem.