SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 2, Page 775-806, January 2020.
Gaussian processes (GPs) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number $L$ of levels (typically several tens), partitioned in $G \ll L$ groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices $T$ with constant covariances between pairs of blocks and within blocks. We study the positive definiteness of such matrices to encourage their practical use. The hierarchical group/level structure, equivalent to a nested Bayesian linear model, provides a parameterization of valid block matrices $T$. The same model can then be used when the assumption within blocks is relaxed, giving a flexible parametric family of valid covariance matrices with constant covariances between pairs of blocks. The positive definiteness of ${T}$ is equivalent to the positive definiteness of a smaller matrix of size $G$, obtained by averaging each block. The model is applied to a problem in nuclear waste analysis, where one of the categorical inputs is atomic number, which has more than 90 levels.

中文翻译：

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具有分类输入的高斯过程元模型的分组核

SIAM / ASA不确定性量化期刊，第8卷，第2期，第775-806页，2020年1月。
高斯过程（GPs）被广泛用作模拟费时的计算机代码的元模型。我们专注于涉及分类输入的问题，这些分类输入可能具有大量的$ L $级别（通常为数十个），并划分为各种大小的$ G \ ll L $组。然后，可以通过块协方差矩阵$ T $定义简约协方差函数或内核，其中块对之间以及块内具有恒定协方差。我们研究此类矩阵的正定性，以鼓励其实际使用。等效于嵌套贝叶斯线性模型的分层组/级别结构提供了有效块矩阵$ T $的参数化。当放松块内的假设时，可以使用相同的模型，给出有效的协方差矩阵的灵活参数族，在成对的块之间具有恒定的协方差。$ {T} $的正定性等效于通过对每个块求平均而获得的大小为$ G $的较小矩阵的正定性。该模型适用于核废料分析中的一个问题，其中分类输入之一是原子序数，其原子序数超过90。