Deep Compositional Spatial Models

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.

具有非平稳且各向异性协方差结构的空间过程，常被应用于复杂环境现象的建模、分析与预测。这类过程通常可被表述为：在经扭曲的空间域上具备平稳且各向同性协方差结构的过程。然而，此类扭曲函数往往难以拟合，且未被约束为单射函数，常引发“空间折叠”问题。为此，本文提出在深度学习框架下，通过多个基础单射函数的复合来构建单射扭曲函数模型。本文考虑两种情形：其一，函数形式已知，但存在若干待估计的权重参数；其二，每一层的权重均为随机变量。受深度学习与深度高斯过程（deep Gaussian process）近期在方法学与技术领域的进展启发，本文采用近似贝叶斯方法，并借助图形处理器（graphics processing unit, GPU）完成此类模型的推断任务。通过一维与二维仿真实验，本文证明：深度复合空间模型不仅拟合速度快捷，且相较于同类复杂度的其他深度随机模型，能够提供更精准的预测结果与更可靠的不确定性量化（uncertainty quantification）表现。此外，本文还证实，借助搭载于Aqua卫星上的MODIS（Moderate Resolution Imaging Spectroradiometer，中分辨率成像光谱仪）仪器获取的辐射数据，该模型具备出色的非平稳、各向异性空间数据建模能力。