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

A multi-resolution basis is developed to predict two-dimensional spatial fields based on irregularly spaced observations. 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. The other important feature of this model is that it can be applied to statistical inference for large spatial datasets because key matrices in the computations are sparse. The computational efficiency applies to both the evaluation of the likelihood and spatial predictions. Although our framework has similarities to fixed rank Kriging, the model gives a better approximation to situations where the nugget variance is small and the spatial process is close to interpolating the observations.