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Isotropic correlation functions on d-dimensional balls

Part of: Geophysics (general) Geophysics Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Tilmann Gneiting
Tilmann Gneiting*
Affiliation:
Universität Bayreuth
*
∗ Postal address: University of Washington, Department of Statistics, Box 354322, Seattle, WA 98195-4322, USA. Email address: tilmann@stat.washington.edu
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Abstract

A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - α ||x||, ||x|| < 1, to observed correlations at (appropriately scaled) spatial lag x in d-dimensional space. We show that such an approach is permissible if and only if

the upper bound depending on the spatial dimension d. The proof relies on Matheron's turning bands operator and an extension theorem for positive definite functions due to Rudin. Side results and examples include a general discussion of isotropic correlation functions defined on d-dimensional balls.

Keywords

Covariance functionextension theorempositive definiteradialspatial dataturning bands

MSC classification

Primary: 86A32: Geostatistics
Secondary: 60D05: Geometric probability and stochastic geometry 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 625 - 631
Copyright
Copyright © Applied Probability Trust 1999 

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