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.

多元空间场在许多应用中都很有趣，包括气候模型仿真。不仅边缘空间场会受到非平稳性的影响，而且边缘场之间以及场之间的依赖结构也可能存在很大差异。扩展最近提出的贝叶斯方法，使用三角形传输图来描述非平稳单变量空间场的分布，我们将少量重复的多元空间场的推理问题转化为一系列独立的高斯过程（GP）回归任务具有高斯误差。由于条件均值中潜在的非线性，建模的联合分布可能是非高斯的。由此产生的非参数贝叶斯方法可以很好地扩展到高维空间领域。当只有少数训练样本可用时，它特别有用，因为它采用正则化先验并量化不确定性。推理是通过高度可扩展的随机梯度方法在经验贝叶斯设置中进行的。该实现受益于小批量处理，并且可以通过并行计算来加速。我们通过研究非高斯气候模型输出的水文变量来说明扩展的传输地图模型。