The generalization of Latin hypercube sampling (1507.06716v1)

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.

• The paper introduces a framework that generalizes Latin Hypercube Sampling into Partially Stratified Sampling, effectively reducing variance from low-order interactions.
• It proposes Latinized Stratified Sampling, an efficient method that combines features of traditional stratified sampling and LHS to simplify sample generation.
• Theoretical derivations and high-dimensional numerical experiments validate the approach, notably enhancing uncertainty analysis in structural mechanics applications like plate buckling.

An Overview of the Generalization of Latin Hypercube Sampling

The paper under review extends the methodology of Latin Hypercube Sampling (LHS), a prominent technique in the domain of Monte Carlo simulations used extensively for uncertainty quantification. The authors propose a framework that generalizes LHS into a broader class of stratified sampling designs called Partially Stratified Sampling (PSS). This framework positions LHS and traditional stratified sampling (SS) at the extremes of a spectrum, with PSS occupying the intermediate space.

Core Contributions

The research delineates several key contributions:

1. Variance Reduction: The paper systematically derives the variance properties for PSS designs, demonstrating that while traditional LHS addresses variance reduction primarily through accounting for main effects, PSS designs effectively mitigate variance associated with low-order interactions in high-dimensional problems.
2. Latinized Stratified Sampling (LSS): An innovative approach, termed Latinized Stratified Sampling, is introduced. This method produces sample sets that fulfill both SS and LHS properties. The authors equate LSS to an Orthogonal Array-based LHS under specific conditions but emphasize the simplicity and efficiency of LSS compared to traditional methods reliant on orthogonal arrays.
3. Theoretical and Numerical Validation: Theoretical developments are supported with high-dimensional numerical experiments. These examples showcase the performance of the proposed methods, notably demonstrating improved variance reduction over classical techniques, especially when both main and interaction effects are significant.
4. Application in Structural Mechanics: The practical applicability of the proposed methodology is illustrated through its deployment in assessing uncertainty in plate buckling strength, a problem characterized by the interplay between geometry and material properties.

Implications and Future Directions

Theoretical Implications: The generalization presented in this work extends the applicability of stratified sampling designs. By constructing a spectrum of sampling strategies, researchers have tools that can be adapted based on the problem's characteristics, particularly the nature and significance of interactions and main effects within the data.

Practical Implications: From a practical standpoint, the development of LPSS provides a formidable approach for industries reliant on simulation-based analyses. This is particularly pertinent in fields such as structural engineering, finance, and reliability engineering, where interaction effects can significantly influence outcomes.

Future Research Directions: The paper opens avenues for further exploration in improving sampling efficiency. One potential area is the exploration of adaptive PSS designs that evolve based on preliminary results from simulation evaluations. Additionally, investigating the integration of these advanced sampling methods with machine learning models could enhance predictive accuracy and computational efficiency.

In conclusion, this paper presents a significant methodological advancement in the field of stratified sampling. By broadening the scope and utility of LHS through PSS and LSS, the authors provide a robust framework capable of addressing a wider spectrum of variance reduction challenges encountered in high-dimensional statistical estimation problems. This work not only elevates theoretical understanding but also offers tangible benefits for practical applications across various disciplines.