Prediction, interpolation and regression for spatially misaligned data

Spatial models for point-referenced data are used for capturing spatial association and for providing spatial prediction, typically in the presence of explanatory variables. The goal of this paper is to treat the situation where there is misalignment between at least one of the explanatory variables and the response variable. In this context we formalize three inference problems. One, which we call interpolation, seeks to infer about missing response at an observed explanatory location. The second, which we call prediction, seeks to infer about a response at a location with the explanatory variable unobserved. The last, which we call regression, seeks to investigate the functional relationship between the response and explanatory variable through the conditional mean of the response. We treat both the case of Gaussian and binary spatial response. We adopt a Bayesian approach, providing full posterior inference for each of the above problems. We illustrate both cases using portions of a study of isopod burrows in the Negev desert in Israel.