Bayesian Multivariate Spatial Interpolation with Data Missing by Design

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1997 · Volume 59 · Issue 2

Abstract

In a network ofsgsites, responses like levels of airborne pollutant concentrations may be monitored over time. The sites need not all measure the same set of response items and unmeasured items are considered as datamissing by design. We propose a hierarchical Bayesian approach to interpolate the levels of, say,kresponses atsuother locations called ungauged sites and also the unmeasured levels of thekresponses at the gauged sites. Our method involves two steps. First, when all hyperparameters are assumed to be known, a predictive distribution is derived. In turn, an interpolator, its variance and a simultaneous interpolation region are obtained. In step two, we propose the use of an empirical Bayesian approach to estimate the hyperparameters through an EM algorithm. We base our theory on a linear Gaussian model and the relationship between a multivariate normal and matrixT-distribution. Our theory allows us to pool data from several existing networks that measure different subsets of response items for interpolation.

Keywords

Bayesian Interpolation, Co-Kriging, MatrixT-Distribution, Predictive Distribution, Spatial Interpolation

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