# Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations

arXiv: https://arxiv.org/abs/2004.08724 · HTML: https://mlsanity.com/paper/2004.08724/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2020.

## Built on by (most cited first, 5 of 5)

- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Modeling massive highly-multivariate nonstationary spatial data with the basis graphical lasso](https://mlsanity.com/paper/2101.02404/) (2021)
- [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)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)

## Abstract (arXiv)

Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments in the univariate case, we propose modeling the warping function as a composition of a number of simple injective warping functions in a deep-learning framework. Importantly, covariance-model validity is guaranteed by construction. We establish the types of warpings that allow for cross-covariance symmetry and asymmetry, and we use likelihood-based methods for inference that are computationally efficient. The utility of this new class of models is shown through two data illustrations: a simulation study on nonstationary data and an application on ocean temperatures at two different depths.