Spatial models for point and areal data using Markov random fields on a fine grid

TL;DR

This paper explores the use of Markov random fields on fine grids for modeling spatial data, comparing different neighborhood structures and demonstrating that certain MRFs can outperform traditional models like CAR and kriging in various scenarios.

Contribution

It provides a comprehensive review and comparison of MRF neighborhood structures for spatial modeling, highlighting the advantages of certain approaches over standard CAR models.

Findings

Standard CAR models can behave strangely for point data.

Higher-order neighborhood approaches do not produce smooth fields.

MRFs approximating thin plate splines often outperform CAR and kriging.

Abstract

I consider the use of Markov random fields (MRFs) on a fine grid to represent latent spatial processes when modeling point-level and areal data, including situations with spatial misalignment. Point observations are related to the grid cell in which they reside, while areal observations are related to the (approximate) integral over the latent process within the area of interest. I review several approaches to specifying the neighborhood structure for constructing the MRF precision matrix, presenting results comparing these MRF representations analytically, in simulations, and in two examples. The results provide practical guidance for choosing a spatial process representation and highlight the importance of this choice. In particular, the results demonstrate that, and explain why, standard CAR models can behave strangely for point-level data. They show that various neighborhood weighting approaches based on higher-order neighbors that have been suggested for MRF models do not produce smooth fields, which raises doubts about their utility. Finally, they indicate that an MRF that approximates a thin plate spline compares favorably to standard CAR models and to kriging under many circumstances.