Michael Stein: Spatial Interpolation with Estimated Covariance Functions
Abstract
Gaussian process models are frequently used for continuously varying environmental quantities. Based on some set of observations, if we know the mean and covariance functions of the Gaussian process over some domain of interest, we can write down the conditional Gaussian distribution of the process at any unobserved location. In applications, the mean and/or covariance functions are generally estimated from the same observations used for prediction. Common practice is to ignore the uncertainties in the estimates of the mean and covariance functions when making predictive inferences, which is known as plug-in prediction. By studying the properties of spatial predictions in the frequency domain, we can gain useful insights into how using estimated covariance functions affect point predictions and uncertainty quantification. This approach is explored via simulations for some examples with known mean functions and simple parametric models for the covariance function that include a parameter controlling the smoothness of the process. The location of the predictand relative to the observations plays an essential role in the quality of plug-in predictions in a way that can be readily explained by thinking in the frequency domain. I will briefly consider extension of these ideas to joint prediction uncertainties at multiple unobserved locations, in particular, explaining how a bivariate posterior predictive density can deviate substantially from having elliptical contours.