Abstract

Spatial processes with nonstationary and anisotropic covariance structure are often used when modeling, analyzing, and predicting complex environmental phenomena. Such processes may often be expressed as ones that have stationary and isotropic covariance structure on a warped spatial domain. However, the warping function is generally difficult to fit and not constrained to be injective, often resulting in “space-folding.” Here, we propose modeling an injective warping function through a composition of multiple elemental injective functions in a deep-learning framework. We consider two cases; first, when these functions are known up to some weights that need to be estimated, and, second, when the weights in each layer are random. Inspired by recent methodological and technological advances in deep learning and deep Gaussian processes, we employ approximate Bayesian methods to make inference with these models using graphics processing units. Through simulation studies in one and two dimensions we show that the deep compositional spatial models are quick to fit, and are able to provide better predictions and uncertainty quantification than other deep stochastic models of similar complexity. We also show their remarkable capacity to model nonstationary, anisotropic spatial data using radiances from the MODIS instrument aboard the Aqua satellite.

摘要

在建模、分析和预测复杂的环境现象时，经常使用具有非平稳和各向异性协方差结构的空间过程。这些过程通常可以表示为在扭曲空间域上具有平稳和各向同性协方差结构的过程。然而，翘曲函数通常难以拟合并且不受限于单射，经常导致“空间折叠”。在这里，我们建议通过深度学习框架中的多个元素单射函数的组合来建模单射翘曲函数。我们考虑两种情况；首先，当这些函数知道一些需要估计的权重时，其次，当每层中的权重是随机的时。受到深度学习和深度高斯过程的最新方法和技术进步的启发，我们采用近似贝叶斯方法使用图形处理单元对这些模型进行推理。通过一维和二维的模拟研究，我们表明深度成分空间模型可以快速拟合，并且能够提供比类似复杂度的其他深度随机模型更好的预测和不确定性量化。我们还展示了他们使用 Aqua 卫星上 MODIS 仪器的辐射对非平稳、各向异性空间数据进行建模的非凡能力。并且能够提供比其他类似复杂性的深度随机模型更好的预测和不确定性量化。我们还展示了他们使用 Aqua 卫星上 MODIS 仪器的辐射对非平稳、各向异性空间数据进行建模的非凡能力。并且能够提供比其他类似复杂性的深度随机模型更好的预测和不确定性量化。我们还展示了他们使用 Aqua 卫星上 MODIS 仪器的辐射对非平稳、各向异性空间数据进行建模的非凡能力。