Multivariate spatial fields are of interest in many applications, including climate model emulation. Not only can the marginal spatial fields be subject to nonstationarity, but the dependence structure among the marginal fields and between the fields might also differ substantially. Extending a recently proposed Bayesian approach to describe the distribution of a nonstationary univariate spatial field using a triangular transport map, we cast the inference problem for a multivariate spatial field for a small number of replicates into a series of independent Gaussian process (GP) regression tasks with Gaussian errors. Due to the potential nonlinearity in the conditional means, the joint distribution modeled can be non-Gaussian. The resulting nonparametric Bayesian methodology scales well to high-dimensional spatial fields. It is especially useful when only a few training samples are available, because it employs regularization priors and quantifies uncertainty. Inference is conducted in an empirical Bayes setting by a highly scalable stochastic gradient approach. The implementation benefits from mini-batching and could be accelerated with parallel computing. We illustrate the extended transport-map model by studying hydrological variables from non-Gaussian climate-model output.

翻译：多变量空间场在包括气候模型仿真在内的诸多应用中具有重要价值。不仅边缘空间场可能呈现非平稳性，各边缘场之间以及场际间的依赖结构也可能存在显著差异。通过扩展近期提出的利用三角传输映射描述非平稳单变量空间场分布的贝叶斯方法，我们将少量重复测量下多变量空间场的推断问题转化为一系列具有高斯误差的独立高斯过程回归任务。由于条件均值可能存在非线性特征，所建模的联合分布可呈现非高斯性。由此产生的非参数贝叶斯方法能很好地扩展到高维空间场，尤其适用于仅可获得少量训练样本的情况，因其采用了正则化先验并量化了不确定性。通过高度可扩展的随机梯度方法在经验贝叶斯框架下进行推断，该实现受益于小批量处理，并可通过并行计算加速。我们通过研究非高斯气候模式输出的水文变量，对扩展后的传输映射模型进行了验证。