Understanding Kriging with three reproducing kernels
Doug Nychka, National Center for Atmospheric Research, Boulder, CO
Monday, October 11, 2010
4:00 p.m., room 223, Weber Bldg
Kriging is a nonparametric regression method used in geostatistics for estimating curves and surfaces for spatial data. Very practical but largely lacking a statistical large sample theory. This disparity is in contrast to the well developed mathematical analysis of kernel smoothers smoothers another family of useful smoothing methods. This talk outlines an approach to understand the mathematical properties of Kriging. It may come as a surprise that the Kriging estimate, normally derived as the best linear unbiased estimator, is also the solution of a particular penalized least squares problem. Thus, Kriging estimators can also be interpreted as generalized smoothing splines where the roughness penalty is determined by the covariance function of a spatial process. We build off the early work by Silverman and the analysis by D.D. Cox, K. Messer and others and develop an equivalent kernel interpretation of geostatistical estimators. Given this connection we show how a non-stationary covariance influences the bias and variance of the Kriging estimate. Along the way we also show how to analyze the point-wise mean squared error properties of thin-plate splines. Some applications to climate temperature fields are also given to illustrate the practical value of these methods.