On the inverse geostatistical problem of inference on missing locations

The standard geostatistical problem is to predict the values of a spatially continuous phenomenon, S(x) say, at locations x using data (y_i, x_i):i=1, ..., n where y_i is the realisation at location x_i of S(x_i), or of a random variable Y_i that is stochastically related to S(x_i). In this paper we address the inverse problem of predicting the locations of observed measurements y. We discuss how knowledge of the sampling mechanism can and should inform a prior specification, π(x) say, for the joint distribution of the measurement locations X={x_i:i=1, ..., n}, and propose an efficient Metropolis-Hastings algorithm for drawing samples from the resulting predictive distribution of the missing elements of X. An important feature in many applied settings is that this predictive distribution is multi-modal, which severely limits the usefulness of simple summary measures such as the mean or median. We present three simulated examples to demonstrate the importance of the specification for π(x) and show how a one-by-one approach can lead to substantially incorrect inferences in the case of multiple unknown locations. We also analyse rainfall data from Paraná State, Brazil to show how, under additional assumptions, an empirical estimate of π(x) can be used when no prior information on the sampling design is available.