Maximum expected entropy transformed Latin hypercube designs

Abstract

Existing projection designs (e.g. maximum projection designs) attempt to achieve good space-filling properties in all projections. However, when using a Gaussian process (GP), model-based design criteria such as the entropy criterion is more appropriate. We employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We show that maximum EEC designs are invariant to monotonic transformations of the response, i.e. they are optimal for a wide class of stochastic process models. We also demonstrate that transformation of each column of a Latin hypercube design (LHD) based on a monotonic function can substantially improve the EEC. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Numerical studies show that the proposed transformations of the LHD are efficient and effective for building robust maximum EEC designs. These designs give projections with markedly higher entropies and lower maximum prediction variances (MPV's) at the cost of small increases in average prediction variances (APV's) compared to state-of-the-art space-filling designs over wide ranges of covariance parameter values.

Keywords:

• Space-filling designs
• entropy criterion
• input transformation
• factor sparsity
• projection

2010 Mathematics Subject Classifications:

• 62K05
• 60G15
• 94A17
• 62B15

Acknowledgments

We thank an associate editor and two referees for comments that helped improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by General Research Fund projects GRF 11201117, GRF 11205118 and GRF 11201519 funded by the Research Grants Council, University Grants Committee, Hong Kong