Norm-dependent covariance permissibility of weakly homogeneous spatial random fields and its consequences in spatial statistics |
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Authors: | G Christakos V Papanicolaou |
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Institution: | (1) Center for the Advanced Study of the Environment, Department of Environmental Sciences & Engineering, 111 Rosenau Hall, CB#7400, University of North Carolina, Chapel Hill, NC 27599-7400, USA Phone: (919)966-7911 e-mail: george_christakos@unc.edu, US;(2) Department of Mathematics, National Technical University of Athens, Athens, Greece, GR |
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Abstract: | Permissibility of a covariance function (in the sense of Bochner) depends on the norm (or metric) that determines spatial
distance in several dimensions. A covariance function that is permissible for one norm may not be so for another. We prove
that for a certain class of covariances of weakly homogeneous random fields, the spatial distance can be defined only in terms
of the Euclidean norm. This class includes commonly used covariance functions. Functions that do not belong to this class
may be permissible covariances for some non-Euclidean metric. Thus, a different class of covariances, for which non-Euclidean
norms are valid spatial distances, is also discussed. The choice of a coordinate system and associated norm to describe a
physical phenomenon depends on the nature of the properties being described. Norm-dependent permissibility analysis has important
consequences in spatial statistics applications (e.g., spatial estimation or mapping), in which one is concerned about the
validity of covariance functions associated with a physically meaningful norm (Euclidean or non-Euclidean). |
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Keywords: | : Random fields norm covariance permissibility estimation |
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