Computationally efficient methods for block average models

Abstract

In many applications, spatial data are typically collected at areal levels (i.e., block data), while inferences and predictions are desired about the variable at points or blocks different from those at which the variable has been observed. These inferences and predictions typically depend on integrals of the moment functions of an underlying continuous spatial process. These integrals are often analytically intractable, and numerically expensive to approximate to high levels of accuracy. In this dissertation I describe a method based on Fourier transforms by which multiple integrals of covariance functions over irregular data regions may be numerically approximated with the same level of accuracy as traditional methods, but at a greatly reduced computational expense. This method also provides convenient values used in spatial prediction. Examples are provided that demonstrate the accuracy and speed of the method in both simulation and real data contexts. Additionally, I examine a model misspecification used with block average observations that avoids expensive covariance integrals. I assess the extent to which both the block design and underlying process properties can affect the accuracy and stability of estimation and prediction tasks performed using this misspecification (relative to when these tasks are performed using a correctly specified observational model), and provide guidance for practitioners as to when this misspecification is inappropriate. Finally, as classic geostatistical approaches (both in point-referenced and block average contexts) do not scale well to big data problems (largely due to costly matrix inversion calculations), several alternative Gaussian process models have been developed for use with big point-referenced data. I describe and demonstrate the adaption of one such model for use with big block average data.