Introducing a Tensor Product Cubic B-Spline Basis Function to the Gaussian Process Model LatticeKrig

Remotely sensed observations of the atmosphere play an important role in climate research since they have more extensive spatial coverage than surface measurements. However, multiple challenges arise from the large quantities of data needed to provide the necessary spatial coverage. A useful framework for spatial models involves expanding the field using basis functions and making distributional assumptions about the basis coefficients. This approach forms the foundation of the successful fixed rank kriging methodology, which has been adopted and extended by models such as LatticeKrig. We introduce a tensor product cubic B-spline basis for representing a multi-resolution Gaussian process model. Surprisingly, the use of B-splines as spatial basis functions has not been extensively explored, despite their several advantages. The cubic B-spline basis function presented here is compactly supported, preserving the efficient sparse linear algebra used in LatticeKrig. In addition, we leverage the partition of unity property of B-splines to reduce basis function artifacts and develop more accurate numerical integration over irregular spatial regions for change-of-support methods.

Gaussian process model

Basis function

Multi-resolution

Cubic B-splines

Change-of-support 

Section on Statistics and the Environment