Assessing the estimation of nearly singular covariance matrices for modeling spatial variables

Abstract

Spatial analysis commonly relies on the estimation of a covariance matrix associated with a random field. This estimation strongly impacts the prediction where the process has not been observed, which in turn influences the construction of more sophisticated models. If some of the distances between all the possible pairs of observations in the plane are small, then we may have an ill-conditioned problem that results in a nearly singular covariance matrix. In this paper, we suggest a covariance matrix estimation method that works well even when there are very close pairs of locations on the plane. Our method is an extension to a spatial case of a method that is based on the estimation of eigenvalues of the unitary matrix decomposition of the covariance matrix. Several numerical examples are conducted to provide evidence of good performance in estimating the range parameter of the correlation structure of a spatial regression process. In addition, an application to macroalgae estimation in a restricted area of the Pacific Ocean is developed to determine a suitable estimation of the effective sample size associated with the transect sampling scheme.