# Deep Compositional Spatial Models

arXiv: https://arxiv.org/abs/1906.02840 · HTML: https://mlsanity.com/paper/1906.02840/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2019.

## Built on by (most cited first, 20 of 20)

- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [SPATE-GAN: Improved Generative Modeling of Dynamic Spatio-Temporal Patterns with an Autoregressive Embedding Loss](https://mlsanity.com/paper/2109.15044/) (2021)
- [Neural networks for geospatial data](https://mlsanity.com/paper/2304.09157/) (2023)
- [Compositional Modeling of Nonlinear Dynamical Systems with ODE-based Random Features](https://mlsanity.com/paper/2106.05960/) (2021)
- [A Unifying Perspective on Non-Stationary Kernels for Deeper Gaussian Processes](https://mlsanity.com/paper/2309.10068/) (2023)
- [Thin and Deep Gaussian Processes](https://mlsanity.com/paper/2310.11527/) (2023)
- [Semiparametric Regression for Spatial Data via Deep Learning](https://mlsanity.com/paper/2301.03747/) (2023)
- [Bivariate DeepKriging for Large-scale Spatial Interpolation of Wind Fields](https://mlsanity.com/paper/2307.08038/) (2023)
- [Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport](https://mlsanity.com/paper/2201.09485/) (2022)
- [Spatial Bayesian Neural Networks](https://mlsanity.com/paper/2311.09491/) (2023)
- [Probabilistic Spatiotemporal Modeling of Day-Ahead Wind Power Generation with Input-Warped Gaussian Processes](https://mlsanity.com/paper/2409.16308/) (2024)
- [A class of modular and flexible covariate-based covariance functions for nonstationary spatial modeling](https://mlsanity.com/paper/2410.16716/) (2024)
- [Modeling Nonstationary Extremal Dependence via Deep Spatial Deformations](https://mlsanity.com/paper/2505.12548/) (2025)
- [Pedestrian Volume Prediction Using a Diffusion Convolutional Gated Recurrent Unit Model](https://mlsanity.com/paper/2411.03360/) (2024)
- [A Framework for Nonstationary Gaussian Processes with Neural Network Parameters](https://mlsanity.com/paper/2507.12262/) (2025)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)
- [Distributional Random Forests for Complex Survey Designs](https://mlsanity.com/paper/2512.08179/) (2025)
- [Nonlocal Bayesian Modeling of Continuous Spatio-Temporal Dynamics](https://mlsanity.com/paper/2606.14313/) (2026)
- [A Deep Learning Model for Spatially Clustered Data via Differentiable Cluster Assignment](https://mlsanity.com/paper/2608.14968/) (2026)

## Abstract (arXiv)

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.