Hierarchical factor models for large spatially misaligned data: a low-rank predictive process approach

This article deals with jointly modeling a large number of geographically referenced outcomes observed over a very large number of locations. We seek to capture associations among the variables as well as the strength of spatial association for each variable. In addition, we reckon with the common setting where not all the variables have been observed over all locations, which leads to spatial misalignment. Dimension reduction is needed in two aspects: (i) the length of the vector of outcomes, and (ii) the very large number of spatial locations. Latent variable (factor) models are usually used to address the former, although low-rank spatial processes offer a rich and flexible modeling option for dealing with a large number of locations. We merge these two ideas to propose a class of hierarchical low-rank spatial factor models. Our framework pursues stochastic selection of the latent factors without resorting to complex computational strategies (such as reversible jump algorithms) by utilizing certain identifiability characterizations for the spatial factor model. A Markov chain Monte Carlo algorithm is developed for estimation that also deals with the spatial misalignment problem. We recover the full posterior distribution of the missing values (along with model parameters) in a Bayesian predictive framework. Various additional modeling and implementation issues are discussed as well. We illustrate our methodology with simulation experiments and an environmental data set involving air pollutants in California.