Statistically and Computationally Efficient Estimating Equations for Large Spatial Datasets

For Gaussian process models, likelihood based methods are often difficult to use with large irregularly spaced spatial datasets, because exact calculations of the likelihood for <i>n</i> observations require <i>O</i>(<i>n</i>³) operations and <i>O</i>(<i>n</i>²) memory. Various approximation methods have been developed to address the computational difficulties. In this paper, we propose new unbiased estimating equations based on score equation approximations that are both computationally and statistically efficient. We replace the inverse covariance matrix that appears in the score equations by a sparse matrix to approximate the quadratic forms, then set the resulting quadratic forms equal to their expected values to obtain unbiased estimating equations. The sparse matrix is constructed by a sparse inverse Cholesky approach to approximate the inverse covariance matrix. The statistical efficiency of the resulting unbiased estimating equations are evaluated both in theory and by numerical studies. Our methods are applied to nearly 90,000 satellite-based measurements of water vapor levels over a region in the Southeast Pacific Ocean.