Numerical studies of space-filling designs: optimization of Latin Hypercube Samples and subprojection properties

Abstract

Quantitative assessment of the uncertainties tainting the results of computer simulations is nowadays a major topic of interest in both industrial and scientific communities. One of the key issues in such studies is to get information about the output when the numerical simulations are expensive to run. This paper considers the problem of exploring the whole space of variations of the computer model input variables in the context of a large dimensional exploration space. Various properties of space-filling designs are justified: interpoint-distance, discrepancy, minimum spanning tree criteria. A specific class of design, the optimized Latin Hypercube Sample, is considered. Several optimization algorithms, coming from the literature, are studied in terms of convergence speed, robustness to subprojection and space-filling properties of the resulting design. Some recommendations for building such designs are given. Finally, another contribution of this paper is the deep analysis of the space-filling properties of the design 2D-subprojections.

Keywords:

• discrepancy
• optimal design
• Latin Hypercube Sampling
• computer experiment

Acknowledgements

Part of this work has been backed by French National Research Agency (ANR) through the COSINUS programme (project COSTA BRAVA noANR-09-COSI-015). All our numerical tests were performed within the R statistical software environment with the DiceDesign package. We thank Luc Pronzato for helpful discussions and Catalina Ciric for providing the prey-predator model example. Finally, we are grateful to both of the reviewers for their valuable comments and their help with the English.

Notes

1 In the following, LHS may refer to Latin Hypercube Sampling as well.

2 Which includes boolean ones, that is properties like ‘having a Latin Hypercube structure’.

3 Modelling MEsocosm structure and functioning for representing LOtic DYnamic ecosystems

4 In the following, a ‘point’ corresponds to an ‘experiment’ (at least a subset of experimental conditions).

5 The range is the support of the distribution.

6 A permutation π of {1, …, N} is a bijective function from {1, …, N} to {1, …, N}.