Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T02:32:15.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Identifiability for non-stationary spatial structure

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Olivier Perrin  and
Wendy Meiring
Olivier Perrin*
Affiliation:
INRA
Wendy Meiring*
Affiliation:
University of California, Santa Barbara
*
∗Postal address: GREMAQ-UMR CNRS 5604, Université des Sciences Sociales, Manufacture des Tabacs, Bâtiment F-2ème Étage, 21 Allée de Brienne, 31000 Toulouse, France. Email address: [email protected]
∗∗Postal address: Statistics and Applied Probability, University of California, Santa Barbara, CA 93106–3110, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

For modelling non-stationary spatial random fields Z = {Z(x) : x∊ℝn, n≥2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x,y) = var[Z(x)-Z(y)], (x,y)∊ℝnxℝn, depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ(u) is differentiable for u > 0.

Keywords

Bijective space deformationdispersion functionisotropyuniquenessvariogram

MSC classification

Primary: 60G60: Random fields
Secondary: 60G12: General second-order processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1244 - 1250
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blumenthal, L. M. (1936). Remarks concerning the Euclidean four-point property. Ergebnisse eines Mathematische Kolloquium 7, 8–10.Google Scholar
Brown, P. J., Le, N. D., and Zidek, J. V. (1994). Multivariate spatial interpolation and exposure to air pollutants. Canad. J. Statist. 22, 489–505.Google Scholar
Cressie, N. A. C. (1993). Statistics for Spatial Data, Revised edn. Wiley-Interscience, New York, p. 900.CrossRefGoogle Scholar
Crum, M. M. (1956). On positive-definite functions. Proc. London Math. Soc. 6, 548–560.Google Scholar
Gneiting, T. and Sasvári, Z. The characterization problem for isotropic covariance functions. To appear in Mathematical Geology.Google Scholar
Guttorp, P., and Sampson, P. D. (1994). Methods for estimating heterogeneous spatial covariance functions with environmental applications. In Handbook of Statistics XII: Environmental Statistics, eds. Patil, G. P. and Rao, C. R., Elsevier/North Holland, New York, pp. 663–690.Google Scholar
Guttorp, P., Meiring, W., and Sampson, P. D. (1994). A space–time analysis of ground-level ozone data. Environmetrics 5, 241–254.Google Scholar
Guttorp, P., Sampson, P. D., and Newman, K. (1992). Nonparametric estimation of non-stationary spatial covariance structure with application to monitoring network design. In Statistics in Environmental and Earth Sciences, eds. Walden, A. and Guttorp, P., Edward Arnold, London, pp. 39–51.Google Scholar
Meiring, W. (1995). Estimation of heterogeneous space-time covariance. . University of Washington, Seattle.Google Scholar
Meiring, W., Guttorp, P., and Sampson, P. D. (1997). On the validity and identifiability of spatial deformation models for heterogeneous spatial correlation structure. Unpublished manuscript.Google Scholar
Meiring, W., Monestiez, P., Sampson, P. D., and Guttorp, P. (1997). Developments in the modelling of nonstationary spatial covariance structure from space-time monitoring data. In Geostatistics Wollongong '96, Vol 1, eds. Baafi, E. Y. and Schofield, N. Kluwer, Dordrecht, pp. 162–173.Google Scholar
Perrin, O. (1997). Modèle de covariance d'un processus non-stationnaire par déformation de l'espace et statistique. . Université de Paris I Panthéon-Sorbonne, Paris.Google Scholar
Perrin, O., and Senoussi, R. (1998). Reducing non-stationary stochastic processes to stationarity by a time deformation. Statistics & Probability Letters, 43, 393–397.Google Scholar
Perrin, O., and Senoussi, R. (1999). Reducing non-stationary random fields to stationarity and isotropy using a space deformation. To appear in Statistics & Probability Letters.Google Scholar
Sampson, P., and Guttorp, P. D. (1992). Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Soc. 87, 108–119.Google Scholar
Schœnberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. Math. 39, 811–841.Google Scholar
Von Neumann, J. and Schœnberg, I. J. (1941). Fourier integrals and metric geometry. Trans. Amer. Math. Soc. 59, 226–251.Google Scholar