# Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

arXiv: https://arxiv.org/abs/1304.6949 · HTML: https://mlsanity.com/paper/1304.6949/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2013.

## Built on by (most cited first, 4 of 4)

- [Stochastic Modeling of Inhomogeneities in the Aortic Wall and Uncertainty Quantification using a Bayesian Encoder-Decoder Surrogate](https://mlsanity.com/paper/2202.10244/) (2022)
- [A Novel Gaussian Process Based Ground Segmentation Algorithm with Local-Smoothness Estimation](https://mlsanity.com/paper/2112.05847/) (2021)
- [Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds](https://mlsanity.com/paper/2512.10407/) (2025)
- [Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds](https://mlsanity.com/paper/2606.13827/) (2026)

## Abstract (arXiv)

Gaussian random fields (GRFs) constitute an important part of spatial modelling, but can be computationally infeasible for general covariance structures. An efficient approach is to specify GRFs via stochastic partial differential equations (SPDEs) and derive Gaussian Markov random field (GMRF) approximations of the solutions. We consider the construction of a class of non-stationary GRFs with varying local anisotropy, where the local anisotropy is introduced by allowing the coefficients in the SPDE to vary with position. This is done by using a form of diffusion equation driven by Gaussian white noise with a spatially varying diffusion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diffusion matrix. These parameters and the GRF may be considered to be part of a hierarchical model and the parameters estimated in a Bayesian framework. The results show that the use of an SPDE with non-constant coefficients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.