# Review: Nonstationary Spatial Modeling, with Emphasis on Process Convolution and Covariate-Driven Approaches

arXiv: https://arxiv.org/abs/1610.02447 · HTML: https://mlsanity.com/paper/1610.02447/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2016.

## Built on by (most cited first, 8 of 8)

- [Nonparametric Multi-shape Modeling with Uncertainty Quantification](https://mlsanity.com/paper/2206.09127/) (2022)
- [Bayesian Optimization with Informative Covariance](https://mlsanity.com/paper/2208.02704/) (2022)
- [Efficient Large-scale Nonstationary Spatial Covariance Function Estimation Using Convolutional Neural Networks](https://mlsanity.com/paper/2306.11487/) (2023)
- [Rooftop Wind Field Reconstruction Using Sparse Sensors: From Deterministic to Generative Learning Methods](https://mlsanity.com/paper/2603.13077/) (2026)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)
- [Predicting Covariate-Driven Spatial Deformation for Nonstationary Gaussian Processes](https://mlsanity.com/paper/2604.27280/) (2026)
- [Visual Spatial Learning: Single-Field Spatial Interpolation Using Convolutional Neural Networks](https://mlsanity.com/paper/2605.30167/) (2026)
- [Can Deep Generative Models Reproduce Non-Stationary Gaussian Random Fields?](https://mlsanity.com/paper/2607.25929/) (2026)

## Abstract (arXiv)

In many environmental applications involving spatially-referenced data, limitations on the number and locations of observations motivate the need for practical and efficient models for spatial interpolation, or kriging. A key component of models for continuously-indexed spatial data is the covariance function, which is traditionally assumed to belong to a parametric class of stationary models. While convenient, the assumption of stationarity is rarely realistic; as a result, there is a rich literature on alternative methodologies which capture and model the nonstationarity present in most environmental processes. This review document provides a rigorous and concise description of the existing literature on nonstationary methods, paying particular attention to process convolution (also called kernel smoothing or moving average) approaches. A summary is also provided of more recent methods which leverage covariate information and yield both interpretational and computational benefits. Note: the article is borrowed from Chapters 1 and 2 of the author's Ph.D. dissertation, joint with Catherine A. Calder.