Deep Compositional Spatial Models

Andrew Zammit-Mangion*, Tin Lok James Ng, Quan Vu, Maurizio Filippone

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1787-1808
Number of pages22
Issue number540
StatePublished - 2022

Bibliographical note

Publisher Copyright:
© 2021 American Statistical Association.


  • Deep models
  • Nonstationarity
  • Spatial statistics
  • Stochastic processes
  • Variational Bayes

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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