Dr. Elizabeth C. Mannshardt-Shamseldin (Duke University)

Title: Asymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands

Abstract: The need often arises in spatial settings to perform a data transformation to achieve a stationary process and/or variance stabilization.  The transformation may be a non-linear transformation, and the desired predictand may be multivariate in that it is necessary to interpolate predictions at multiple sites.  We assume the underlying spatial model is a Gaussian random field with a parametrically specified covariance structure, but that the predictions of interest are for multivariate nonlinear functions of the Gaussian field. This induces new complications in the spatial interpolation known as kriging. For instance, it is no longer possible to derive the predictive distribution function in closed form.  A difficulty that arises with traditional kriging methods is the fact that the standard formula for the mean squared prediction error does not take into account the estimation of the covariance parameters. This generally leads to underestimated prediction errors, even if the model is correct.  Smith and Zhu (2004) establish a second-order expansion for predictive distributions in Gaussian processes with estimated covariances. Here, we establish a similar expansion for multivariate kriging with non-linear predictands.

Bayesian methods provide a possible resolution to errors encountered through employing frequentist estimation techniques for obtaining spatial parameters. An important property of Bayesian methods is the ability to deal with the uncertainty in a particular model.  Here we explore a Laplace approximation to Bayesian techniques that provides an alternative to common iterative Bayesian methods, such as Markov Chain Monte Carlo.  The main results are asymptotic formulae for a general, non-linear predictand for the expected length of a Bayesian prediction interval, which has possible applications in network design, and for the coverage probability bias, which can lead to the
development of a matching prior.