A Design Utility Approach for Preferentially Sampled Spatial Data

Spatial preferential sampling refers to the situation in which the choice of sampling locations is stochastically dependent on the values of the spatial process of interest. Traditional geostatistical methods ignore this dependence, leading to potentially inaccurate inferences. We present a general framework for modelling the preferences of the experimenter jointly with the spatial process of interest in order to adjust for this bias. We dispense with the unrealistic assumption (required by existing methods) of conditional independence of the sampling locations by defining a whole design utility function on the space of sampling designs, to which the sampling design distribution is proportional, and which may encapsulate an arbitrarily wide range of preferences. The likelihood of the proposed model is generally intractable, and we provide fitting techniques based on importance sampling and noisy Markov chain Monte Carlo before demonstrating their usage on a dataset of spatially distributed ammonia concentrations.