Likelihood approximation and prediction for large spatial and spatio-temporal datasets using H-matrix approach

The Gaussian distribution is the most fundamental distribution in statistics. However, many applications of Gaussian random fields (GRFs) are limited by the computational complexity associated to the evaluation of probability density functions. Particularly, large datasets with N irregularly sited spatial (or spatio-temporal) locations are difficult to handle for several applications of GRF such as maximum likelihood estimation (MLE) and kriging prediction. This is due to the fact that computation of the inverse of the dense covariance function requires a computational complexity of O(N^3) floating points operations in spatial or spatio-temporal context. For relatively large N the exact computation becomes unfeasible and alternative methods are necessary. Several approaches have been proposed to tackle this problem. Most assume a specific form for the spatial(-temporal) covariance function and use different methods to approximate the resulting covariance matrix. We aim at approximating covariance functions in a format that facilitates the computation of MLE and kriging prediction with very large spatial and spatio-temporal datasets. For a sufficiently general class of spatial and specific class of spatio-temporal covariance functions, a methodology is developed using a hierarchical matrix approach. Since this method was originally created for the approximation of dense matrices coming from partial differential and integral equations, a theoretical framework is formulated in terms of Stochastic Partial Differential equations (SPDEs). The application of this technique is detailed for covariance functions of GRFs obtained as solutions to SPDEs. The approximation of the covariance matrix in such a low-rank format allows for computation of the matrix-vector products and matrix factorisations in a log-linear computational cost followed by an efficient MLE and kriging prediction. The numerical studies are provided for based on spatial and spatio-temporal datasets and the H-matrix approach is compared with the other methods in terms of computational and statistical efficiency.a