Approximate likelihood for large irregularly spaced spatial data

Likelihood approaches for large, irregularly spaced spatial datasets are often very difficult, if not infeasible, to implement due to computational limitations. Even when we can assume normality, exact calculations of the likelihood for a Gaussian spatial process observed at n locations requires O(n(3)) operations. We present a version of Whittle's approximation to the Gaussian log-likelihood for spatial regular lattices with missing values and for irregularly spaced datasets. This method requires O(n log(2) n) operations and does not involve calculating determinants. We present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values. We apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values.