Incomplete Cholesky decomposition for the kriging of large datasets
Résumé
Kriging of very large spatial datasets is challenging. Sometimes a spatial datum is expensive to obtain (e.g. drilling wells for oil reserve estimation), in which case the sample size N is typically small and kriging can be performed straightforwardly. Recently, data-base paradigms have moved from small to massive, often of the order of gigabytes per day. The size N of the dataset causes problems in computing the kriging estimate: solving the kriging equations directly involves inverting an N x N covariance matrix. This operation requires O(N3) computations and a storage of O(N2). Under these circumstances, straightforward kriging of massive datasets is not possible. Several approaches have been proposed in the literature among which two families exist: covariance tapering that is based on a sparse approximation of the covariance function and low rank approaches that are based on a functional approximation. We propose here a novel approach that is based on a low rank approximation of the covariance matrix built by incomplete Cholesky decomposition with symmetric pivoting. This algorithm requires O(Nk) storage and takes O(Nk2) arithmetic operations, where k is the rank of the approximation, whose quality is controlled by a parameter of the algorithm. We detail the main properties of this method and explore its links with existing methods. Its benefits are illustrated on simple examples and compared to existing approaches. Finally, we show that this low rank representation is also suited for inverse conditioning of Gaussian random fields.