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.
Highlights
Markov random field (MRF) models ( called conditional autoregressive (CAR) models) are the dominant approach to analyzing areally-aggregated spatial data, such as disease counts in administrative units
We see that with uniformly distributed locations, in general the thin plate spline (TPS)-MRF either matches the sum of squared errors (SSE) of the intrinsic CAR model (ICAR) or improves upon it
For ν = 0.5, which produces locally heterogeneous surfaces for which we would expect the ICAR model to perform well, we see that the two MRF models perform fairly
Summary
Markov random field (MRF) models ( called conditional autoregressive (CAR) models) are the dominant approach to analyzing areally-aggregated spatial data, such as disease counts in administrative units. The results are consistent with Besag and Mondal (2005), who show that the intrinsic (i.e., improper) first-order MRF on a two-dimensional regular grid produces spatial fields whose distribution approaches two-dimensional Brownian motion (the de Wijs process) asymptotically as the grid resolution increases Given this continuous but non-differentiable representation of the underlying surface, the local heterogeneity of the surface estimate in Fig. 1 is not surprising. First-order MRF representations are widely-used for areal data, generally with little consideration of the properties of the latent process or resulting suitability for a given application In light of this and of the lack of smoothness of the popular first-order MRF seen, I compare several existing approaches for the MRF neighborhood structure (Section 2.2) using both analytic calculations and simulations (Section 3) in the context of the general model presented here. Online supplementary material (Paciorek, 2013) contains the R code for all analyses and figures as well as the data for the pollution data only, as the breast cancer data are not publicly available
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