Residual-based specification of a hidden random field included in a hierarchical spatial model

A method for specifying a hidden random field (HRF) included in a hierarchical spatial model is proposed. In hierarchical models of interest the first stage describes, conditional on a realization of the HRF, a response variable which is observable on a continuous spatial domain; the second stage models the HRF which reflects unobserved spatial heterogeneity. The question which is investigated is how can the HRF be modeled, i.e. specified. The method developed to address this question is based on residuals obtained when the base model, i.e. the hierarchical model in which the HRF is assumed constant, is fitted to data. It is shown that the residuals are linked with the HRF, and the link is used to specify the HRF. The method is applied to simulated data in order to assess its performance, and then to real data on radionuclide concentrations on Rongelap Island.