On the condition number of covariance matrices in kriging, estimation, and simulation of random fields |
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Authors: | Rachid Ababou Amvrossios C. Bagtzoglou Eric F. Wood |
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Affiliation: | (1) Centre d'Etude Nucléaire de Saclay, Commissariat a l'Energie Atomique, 91191 Gif Sur Yvette Cedex, France;(2) Center for Nuclear Waste Regulatory Southwest Research Institute, Analyses, 78238-5166 San Antonio, Texas;(3) Department of Civil Engineering and Operations Research, Princeton University, 08540 Princeton, New Jersey |
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Abstract: | The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed. |
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Keywords: | kriging condition number random fields conditional simulation covariance matrices state-space estimation |
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