Deep Compositional Spatial Models

TL;DR

This paper introduces deep compositional spatial models that use deep learning to create injective warping functions for nonstationary spatial data, improving prediction accuracy and uncertainty quantification.

Contribution

It proposes a novel deep-learning framework for modeling injective warping functions in spatial processes, enabling better handling of nonstationarity and anisotropy.

Findings

Models are quick to fit in simulations.

They outperform similar deep stochastic models in prediction and uncertainty.

Successfully model nonstationary satellite radiance data.

Abstract

Spatial processes with nonstationary and anisotropic covariance structure are often used when modelling, analysing 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 modelling 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.