On The Inverse Geostatistical Problem of Inference on Missing Locations

TL;DR

This paper tackles the inverse geostatistical problem of predicting missing measurement locations using a Bayesian approach, emphasizing the importance of sampling mechanism knowledge and demonstrating methods through simulations and real rainfall data analysis.

Contribution

It introduces an efficient Metropolis-Hastings algorithm for sampling from the predictive distribution of missing locations, incorporating sampling design knowledge into prior specification.

Findings

The predictive distribution of missing locations can be multi-modal, affecting summary measures.

Proper prior specification significantly influences location inference.

Empirical estimation of sampling distribution is feasible when no prior info is available.

Abstract

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 realization 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, $\pi(x)$ 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 two simulated examples to demonstrate the importance of the specification for $\pi(x)$, and analyze rainfall data from Paran\'a State, Brazil to show how, under additional assumptions, an empirical of estimate of $\pi(x)$ can be used when no prior information on the sampling design is available.