Monotonic Metamodels for Deterministic Computer Experiments

In deterministic computer experiments, it is often known that the output is a monotonic function of some of the inputs. In these cases, a monotonic metamodel will tend to give more accurate and interpretable predictions with less prediction uncertainty than a nonmonotonic metamodel. The widely used Gaussian process (GP) models are not monotonic. A recent article in <i>Biometrika</i> offers a modification that projects GP sample paths onto the cone of monotonic functions. However, their approach does not account for the fact that the GP model is more informative about the true function at locations near design points than at locations far away. Moreover, a grid-based method is used, which is memory intensive and gives predictions only at grid points. This article proposes the weighted projection approach that more effectively uses information in the GP model together with two computational implementations. The first is isotonic regression on a grid while the second is projection onto a cone of monotone splines, which alleviates problems faced by a grid-based approach. Simulations show that the monotone B-spline metamodel gives particularly good results. Supplementary materials for this article are available online.

在确定性计算机实验中，通常已知模型输出为部分输入变量的单调函数。在此类场景中，相较于非单调元模型，单调元模型往往可生成精度更高、可解释性更强的预测结果，且预测不确定性更低。广泛使用的高斯过程（Gaussian Process, GP）模型并不具备单调性。近日发表于《Biometrika》的一项研究提出了一种修正方法，将GP的样本路径投影至单调函数锥域内。然而该方法未考虑到：相较于远离设计点的位置，GP模型在设计点附近区域对真实函数的信息表征更为充分。此外，其采用的基于网格的方法存在内存占用过高的问题，且仅能在网格点上生成预测结果。本文提出了加权投影方法，可更高效地利用GP模型中的信息，并配套提出两种计算实现方案：其一为基于网格的保序回归，其二为向单调样条锥域的投影，二者均可缓解基于网格方法的固有问题。仿真实验结果表明，单调B样条元模型可取得尤为优异的预测效果。本文的补充材料可在线获取。