Computationally exploitable structure of covariance matrices and generalized convariance matrices in spatial models

Many spatial analyses based on random field models, as varied as kriging, likelihood-based estimation of autocovariance functions, and optimal design of spatial experiments, require the repeated evaluation of a covariance matrix V or a kth-order generalized covariance matrix K and its subsequent inversion. This is generally a formidable computing problem for moderate and large data sets. In this article, however, it is shown that under certain model assumptions, V (or K) possesses one of several types of patterned structure that, if exploited, can significantly reduce the computational burden of the analysis. These patterned structures are characterized, and their implications for matrix evaluation and inversion are considered. The usefulness of the results is illustrated with a soil pH data set