Spectral methods to approximate the likelihood for irregularly spaced spatial data

Likelihood approaches for large irregularly spaced spatial datasets are often very difficult, if not infeasible, to use 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) 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(nlog2n) operations and does not involve calculating determinants. Due to the edge effect the estimated covariance parameters using this approximated likelihood method are efficient only in one dimension. To remove this edge effect, we introduce data tapers. In spatial statistics, data tapers are often the tensor product of two one-dimensional tapers. However, we generally need more tapering for the corner observations. Thus, we introduce here a new spatial data taper, a circular taper, that gives more tapering to the corner observations. Therefore, with less overall tapering, we get the amount of smoothing that we need without losing so much information. We present simulations and theoretical results to show the benefits and the performance of the data taper and the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values.