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.

现有的投影设计（例如最大投影设计）试图在所有投影中实现良好的空间填充特性。但是，当使用高斯过程 (GP) 时，基于模型的设计标准（例如熵标准）更合适。我们使用在一组投影上平均的熵标准，称为预期熵标准 (EEC)，来生成投影设计。我们表明最大 EEC 设计对于响应的单调变换是不变的，即它们对于广泛的随机过程模型是最优的。我们还证明了基于单调函数的拉丁超立方体设计 (LHD) 的每一列的变换可以显着提高 EEC。考虑了两种类型的输入转换：选择用于优化 EEC 的对称 Beta 分布的分位数函数，以及对应于选择用于优化 EEC 的对称密度的分位数函数的非参数变换。数值研究表明，提出的 LHD 变换对于构建稳健的最大 EEC 设计是有效的。与在宽协方差参数值范围内的最先进的空间填充设计相比，这些设计以平均预测方差 (APV) 的小幅增加为代价提供具有显着更高熵和更低最大预测方差 (MPV) 的预测。