A multi-resolution Gaussian process model for the analysis of large spatial data sets

The basis functions at each level of resolution are constructed as radial basis functions using a Wendland compactly supported correlation function with the nodes arranged on a rectangular grid. The grid at each finer level increases by a factor of two and the basis functions are scaled to have a constant overlap.

The coefficients associated with the basis functions at each level of resolution are distributed according to a Gaussian Markov random field (GMRF) and take advantage of the fact that the basis is organized as a lattice. Several numerical examples and analytical results establish that this scheme gives a good approximation to standard covariance functions such as the Matern and also has flexibility to fit more complicated shapes.