Category tree Gaussian process for computer experiments with many-category qualitative factors and application to cooling system design

In computer experiments, Gaussian process (GP) models are commonly used for emulation. However, when both qualitative and quantitative factors are in the experiments, emulation using GP models becomes challenging. In particular, when the qualitative factors contain many categories in the experiments, existing methods in the literature become cumbersome due to the curse of dimensionality. Motivated by the computer experiments for the design of a cooling system, a new tree-based GP is proposed that emulates computer models with many-category qualitative factors, which we call <i>category tree GP</i>. The proposed method incorporates a tree structure to split the categories of the qualitative factors, and GP or mixed-input GP models are employed for modeling the simulation outputs in the leaf nodes. The splitting rule takes into account the cross-correlations between the categories of the qualitative factors, which have been shown by a recent theoretical study to be a crucial element for improving the prediction accuracy. In addition, a pruning procedure based on the cross-validation error is proposed to ensure the prediction accuracy. The application to the design of a cooling system indicates that the proposed method not only enjoys marked computational advantages and produces accurate predictions, but also provides valuable insights into the cooling system by discovering the tree structure.

在计算机实验领域，高斯过程（Gaussian Process, GP）模型常被用于替代建模。然而，当实验中同时包含定性因子与定量因子时，采用GP模型开展替代建模便颇具挑战。尤其当定性因子包含大量类别时，现有文献中的方法会因维数灾难而变得异常繁琐。本研究受冷却系统设计相关计算机实验的启发，提出了一种全新的基于树结构的GP模型，用于对含多类别定性因子的计算机模型进行替代建模，我们将其命名为类别树高斯过程（category tree GP）。所提方法引入树结构对定性因子的类别进行划分，并在叶节点处采用GP或混合输入GP模型对仿真输出进行建模。其划分规则考虑了定性因子各类别间的互相关——近期一项理论研究已证实，该要素是提升预测精度的关键所在。此外，本文还提出了一种基于交叉验证误差的剪枝流程，以保障预测精度。将所提方法应用于冷却系统设计场景的实验结果表明，该方法不仅具备显著的计算优势且能生成精准的预测结果，还可通过挖掘得到的树结构，为冷却系统设计提供极具价值的见解。