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Isotropic correlation functions on d-dimensional balls

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

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

Information

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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References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Armstrong, M. and Jabin, R. (1981). Variogram models must be positive-definite. Math. Geol. 13, 455–459.Google Scholar
Benchimol, N. and Hay, J. E. (1986). An assessment of the ability of a geostationary satellite-based model to characterize the mesoscale variability of solar irradiance over the Lower Fraser Valley. Atmos.–Ocean 24, 128–144.Google Scholar
Bouka Biona, C. and Boutin, C. (1985). Évolution saisonnière de l'échelle spatiale du rayonnement solaire global en zone sahélienne. Atmos.–Ocean 23, 67–79.Google Scholar
Christakos, G. (1984). On the problem of permissible covariance and variogram models. Water Resources Res. 20, 251–265.Google Scholar
Christensen, R. (1991). Linear Models for Multivariate, Time Series, and Spatial Data. Springer, New York.Google Scholar
Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.Google Scholar
Cressie, N. and Davis, R. W. (1981). The supremum distribution of another Gaussian process. J. Appl. Prob. 18, 131–138.Google Scholar
Gneiting, T. (1998). Simple tests for the validity of correlation function models on the circle. Statist. Prob. Lett. 39, 119–122.Google Scholar
Gneiting, T. and Sasvári, Z. (1999). The characterization problem for isotropic covariance functions. Math. Geol. 31, 105–111.Google Scholar
Matérn, B., (1986). Spatial Variation, 2nd edn (Lecture Notes in Statist. 36). Springer, Berlin.Google Scholar
Matheron, G. (1973). The intrinsic random functions and their applications. Adv. Appl. Prob. 5, 439–468.Google Scholar
Nussbaum, A. E. (1973). Integral representations of functions and distributions positive definite relative to the orthogonal group. Trans. Amer. Math. Soc. 175, 355–387.Google Scholar
Onn, S.-C., Wang, K.-H. and Tseng, C.-Y. (1994). Numerical simulation for the climatic statistical structure with the new correlation functions of the extended polynomial form and the extended exponential form. Adv. Space Res. 14, 77–81.Google Scholar
Rudin, W. (1970). An extension theorem for positive-definite functions. Duke Math. J. 37, 49–53.Google Scholar
Schlather, M. and Stoyan, D. (1997). The covariance of the Stienen model. In Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets, ed. Jeulin, D.. World Scientific, Singapore, pp. 157–174.Google Scholar
Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. Math. 39, 811–841.Google Scholar
Stewart, J. (1976). Positive definite functions and generalizations, an historical survey. Rocky Mountain J. Math. 6, 409–434.Google Scholar
Stol, P. T. (1981). Rainfall interstation correlation functions. I. An analytic approach.. 50, 45–71.Google Scholar
Weber, R. O. and Talkner, P. (1993). Some remarks on spatial correlation function models. Mon. Wea. Rev. 121, 2611–2617.Google Scholar
Wiencek, K. and Stoyan, D. (1993). Spatial correlations in metal structures and their analysis, II: The covariance. Materials Charact. 31, 47–53.Google Scholar