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Why is isotropy so prevalent in spatial statistics?
Author(s):
Chunsheng
Ma
Journal:
Proc. Amer. Math. Soc.
135
(2007),
865-871.
MSC (2000):
Primary 62M30, 60G10;
Secondary 60G60, 43A35, 86A32
Posted:
September 11, 2006
MathSciNet review:
2262884
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Additional information
Abstract:
There are many reasons for the popular use of the isotropic or geometrically anisotropic covariance function and variogram in spatial statistics. A less known reason demonstrated in this paper is that an isotropic or geometrically anisotropic model would be the only choice in certain circumstances, for instance, when the underlying random field is smooth enough.
References:
-
- 1.
- Banerjee, S. and Gelfand, A.E., On smoothness properties of spatial processes, J. Multivariate Anal. 84 (2003), 85-100. MR 1965824 (2003k:62164)
- 2.
- Berg, C., Christensen, J.P.R. and Ressel, P., Harmonic analysis on semigroups: Theory of positive definite and related functions, Springer, New York, 1984. MR 0747302 (86b:43001)
- 3.
- Berg, C. and Forst, G., Potential Theory on Locally Compact Abelian Groups, Springer, New York, 1975. MR 0481057 (58:1204)
- 4.
- Christakos, G., Random fields models in earth sciences, Academic Press, San Diego, 1992.
- 5.
- Christakos, G. and Papanicolaou, V., Norm-dependent covariance permissibility of weakly homogeneous spatial random fields and its consequences in spatial statistics, Stoch. Envir. Res. Risk Assess. 14 (2000), 471-478.
- 6.
- Cressie, N., Statistics for spatial data, revised ed., Wiley, New York, 1993. MR 1239641 (94h:62155)
- 7.
- Doob, J.L., The elementary Gaussian processes, Ann. Math. Statist. 15 (1944), 229-282. MR 0010931 (6:89a)
- 8.
- Ferguson, T.S., A representation of the symmetric bivariate Cauchy distribution, Ann. Math. Statist. 33 (1962), 1256-1266. MR 0143281 (26:840)
- 9.
- Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters, Ann. Inst. H. Poincaré B 3 (1967), 121-226. MR 0215331 (35:6172)
- 10.
- Herz, C.S., A class of negative-definite functions, Proc. Amer. Math. Soc. 14 (1963), 670-676. MR 0158251 (28:1477)
- 11.
- Lantuéjoul, C., Geostatistical simulation, Springer, New York, 2002.
- 12.
- Ma, C., Spatio-temporal stationary covariance models, J. Multivariate Anal. 86 (2003), 97-107. MR 1994723 (2004d:62329)
- 13.
- Ma, C., Spatial autoregression and related spatio-temporal models, J. Multivariate Anal. 88 (2004), 152-162. MR 2021867 (2004k:62206)
- 14.
- Ma, C., Spatio-temporal variograms and covariance models, Adv. Appl. Prob. 37 (2005), 706-725. MR 2156556 (2006b:60068)
- 15.
- Ma, C., Linear combinations of space-time covariance functions and variograms, IEEE Trans. Signal Proc. 53 (2005), 857-864. MR 2123904 (2005k:62173)
- 16.
- Mandelbrot, B.B., A fast fractional Gaussian noise generator, Water Resources Res. 7 (1971), 543-553.
- 17.
- Mandelbrot, B.B. and Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422-437. MR 0242239 (39:3572)
- 18.
- Matheron, G., The internal consistency of models in geostatistics, In Geostatistics, Vol. 1 (1989), 21-38, edited by M. Armstrong, Kluwer Academic Publishers, The Netherlands.
- 19.
- Ossiander, M. and Waymire, E.C., Certain positive-definite kernels, Proc. Amer. Math. Soc. 107 (1989), 487-492. MR 1011824 (91a:60132)
- 20.
- Sampson, P. and Guttorp, P., Nonparametric estimation of nonstationary spatial covariance structure, J. Amer. Statist. Ass. 87 (1992), 108-119.
- 21.
- Schoenberg, I.J., Metric spaces and completely monotone functions, Ann. Math. 39 (1938), 811-841. MR 1503439
- 22.
- Weber, R.O. and Talkner, P., Some remarks on spatial correlation function model, Mon. Weather Rev. 121 (1993), 2611-2617. [Correction: 127 (1999), 576].
- 23.
- Yaglom, A.M., Correlation theory of stationary and related random functions, Vol. 1, Springer, New York, 1987. MR 0893393 (89a:60105)
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Additional Information:
Chunsheng
Ma
Affiliation:
Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033
Email:
cma@math.twsu.edu
DOI:
10.1090/S0002-9939-06-08592-3
PII:
S 0002-9939(06)08592-3
Keywords:
Covariance,
geometrically anisotropic,
isotropic,
negative definite,
norm,
positive definite,
variogram.
Received by editor(s):
November 1, 2003
Received by editor(s) in revised form:
October 1, 2005
Posted:
September 11, 2006
Communicated by:
Richard A. Davis
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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