Transformation and Additivity in Gaussian Processes

We discuss the problem of approximating a deterministic function using Gaussian processes (GPs). The role of transformation in GP modeling is not well understood. We argue that transformation of the response can be used for making the deterministic function approximately additive, which can then be easily estimated using an additive GP. We call such a GP a transformed additive Gaussian (TAG) process. To capture possible interactions which are unaccounted for in an additive model, we propose an extension of the TAG process called transformed approximately additive Gaussian (TAAG) process. We develop efficient techniques for fitting a TAAG process. In fact, we show that it can be fitted to high-dimensional data much more efficiently than a standard GP. Furthermore, we show that the use of the TAAG process leads to better estimation, interpretation, visualization, and prediction. The proposed methods are implemented in the R package TAG.我们探讨了使用高斯过程（Gaussian processes, GPs）近似确定性函数的问题。目前，变换在高斯过程建模中的作用尚未得到充分阐明。我们提出，可通过对响应变量进行变换，使确定性函数近似满足可加性，进而可利用加性高斯过程轻松地对其进行估计。我们将此类高斯过程称为变换加性高斯（TAG）过程。为了捕捉加性模型中未被考虑的潜在交互作用，我们进一步提出了变换加性高斯过程的扩展版本，即变换近似加性高斯（TAAG）过程。我们开发了适配TAAG过程的高效方法，事实上，相较于标准高斯过程，该方法可更高效地应用于高维数据的拟合。此外，我们证实，采用TAAG过程能够提升估计、解释、可视化以及预测的效果。本文所提出的方法已在R包TAG中实现。