Abstract
Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts, interests are often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in a given direction. In this paper, we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models with stationary increments, establish bounds on the prediction errors, and determine the smoothness properties and fractal properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary spacetime models introduced by Cressie and Huang (1999), Gneiting (2002), and Stein (2005), respectively.
Similar content being viewed by others
References
Adler R J. The Geometry of Random Fields. New York: Wiley, 1981
Adler R J, Taylor J E. Random Fields and Geometry. New York: Springer, 2007
Anderes E B, Stein M L. Estimating deformations of isotropic Gaussian random fields on the plane. Ann Statist, 2008, 36: 719–741
Banerjee S, Gelfand A E. On smoothness properties of spatial processes. J Multivariate Anal, 2003, 84: 85–100
Banerjee S, Gelfand A E, Sirmans C F. Directional rates of change under spatial process models. J Amer Statistical Assoc, 2003, 98: 946–954
Berg C, Forst G. Potential Theory on Locally Compact Abelian Groups. New York-Heidelberg: Springer-Verlag, 1975
Calder C A, Cressie N. Some topics in convolution-based spatial modeling. In: Proceedings of the 56th Session of the International Statistics Institute, Lisbon, Portugal. 2007
Chan G, Wood A T A. Increment-based estimators of fractal dimension for twodimensional surface data. Statist Sinica, 2000, 10: 343–376
Chan G, Wood A T A. Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields. Ann Statist, 2004, 32: 1222–1260
Constantine A G, Hall P. Characterizing surface smoothness via estimation of effective fractal dimension. J Roy Statist Soc Ser B, 1994, 56: 97–113
Cramér H, Leadbetter M R. Stationary and Related Stochastic Processes. New York: John Wiley & Sons, Inc, 1967
Cressie N. Statistics for Spatial Data (rev ed). New York: Wiley, 1993
Cressie N, Huang H -C. Classes of nonseparable, spatiotemporal stationary covariance functions. J Amer Statist Assoc, 1999, 94: 1330–1340
Davies S, Hall P. Fractal analysis of surface roughness by using spatial data (with discussion). J Roy Statist Soc Ser B, 1999, 61: 3–37
de Iaco S, Myers D E, Posa D. Space-Time analysis using a general product-sum model. Statist Probab Letters, 2001, 52: 21–28
de Iaco S, Myers D E, Posa D. Nonseparable space-time covariance models: some parametric families. Math Geology, 2002, 34: 23–42
de Iaco S, Myers D E, Posa D. The linear coregionalization model and the product-sum space-time variogram. Math Geology, 2003, 35: 25–38
Falconer K J. Fractal Geometry—Mathematical Foundations and Applications. New York: Wiley & Sons, 1990
Fuentes M. Spectral methods for nonstationary spatial processes. 2002, 89: 197–210
Fuentes M. A formal test for nonstationarity of spatial stochastic processes. J Multivariate Anal, 2005, 96: 30–54
Gneiting T. Nonseparable, stationary covariance functions for space-time data. J Amer Statist Assoc, 2002, 97: 590–600
Gneiting T, Kleiber W, Schlather M. Matérn cross-covariance functions for multivariate random fields. Preprint, 2009
Hall P, Wood A T A. On the performance of box-counting estimators of fractal dimension. Biometrika, 1993, 80: 246–252
Higdon D. Space and space-time modeling using process convolutions. In: Anderson C, Barnett V, Chatwin P C, El-Shaarawi A H, eds. Quantitative Methods for Current Environmental Issues. New York: Springer-Verlag, 2002, 37–56
Higdon D, Swall J, Kern J. Nonstationary spatial modeling. In: Bernardo J M, et al, eds. Bayesian Statistics, Vol 6. Oxford: Oxford University Press, 1999, 761–768
Jones R H, Zhang Y. Models for continuous stationary space-time processes. In: Gregoire T G, Brillinger D R, Diggle P J, Russek-Cohen E, Warren W G, Wolfinger R D, eds. Modelling Longitudinal and Spatially Correlated Data. Lecture Notes in Statist, No 122. New York: Springer, 1997, 289–298
Kahane J -P. Some Random Series of Functions. 2nd ed. Cambridge: Cambridge University Press, 1985
Kent J T, Wood A T A. Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J Roy Statist Soc Ser B, 1997, 59: 679–699
Kolovos A, Christakos G, Hristopulos D T, Serre M L. Methods for generating nonseparable spatiotemporal covariance models with potential environmental applications. Adv Water Resour, 2004, 27: 815–830
Kyriakidis P C, Journe A G. Geostatistical space-time models: a review. Math Geology, 1999, 31: 651–684
Ma C. Families of spatio-temporal stationary covariance models. J Statist Plan Infer, 2003, 116: 489–501
Ma C. Spatio-temporal stationary covariance models. J Multivariate Anal, 2003, 86: 97–107
Ma C. Spatial autoregression and related spatio-temporal models. J Multivariate Anal, 2004, 88: 152–162
Ma C. Spatio-temporal variograms and covariance models. Adv Appl Probab, 2005, 37: 706–725
Ma C. A class of stationary random fields with a simple correlation structure. J Multivariate Anal, 2005, 94: 313–327
Ma C. Stationary random fields in space and time with rational spectral densities. IEEE Trans Inform Th, 2007, 53: 1019–1029
Ma C. Recent developments on the construction of spatio-temporal covariance models. Stoch Environ Res Risk Assess, 2008, 22(suppl 1): 39–47
Meerschaert M M, Wang W, Xiao Y. Fernique-type inequalities and moduli of continuity of anisotropic Gaussian random fields. Trans Amer Math Soc (to appear)
Paciorek C J, Schervish M J. Spatial modelling using a new class of nonstationary covariance functions. Environmetrics, 2006, 17: 483–506
Schmidt A, O’Hagan A. Bayesian inference for nonstationary spatial covariance structure via spatial deformation. J Roy Statist Soc Ser B, 2003, 65: 745–758
Stein M L. Interpolation of Spatial Data: Some Theory for Kriging. New York: Springer, 1999
Stein M L. Space-time covariance functions. J Amer Statist Assoc, 2005, 100: 310–321
Xiao Y. Strong local nondeterminism of Gaussian random fields and its applications. In: Lai T-L, Shao Q-M, Qian L, eds. Asymptotic Theory in Probability and Statistics with Applications. Beijing: Higher Education Press, 2007, 136–176
Xiao Y. Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan D, Rassoul-Agha F, eds. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math, Vol 1962. New York: Springer, 2009, 145–212
Xiao Y. Properties of strong local nondeterminism and local times of stable random fields. In: Dalang R, Dozzi M, Russo F, eds. Stochastic Analysis, Random Fields and Applications VI. Progress in Probability 63. Basel: Birkhäuser, 2011, 279–310
Yaglom A M. Some classes of random fields in n-dimensional space, related to stationary random processes. Th Probab Appl, 1957, 2: 273–320
Zhu Z, Stein M L. Parameter estimation for fractional Brownian surfaces. Statist Sinica, 2002, 12: 863–883
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xue, Y., Xiao, Y. Fractal and smoothness properties of space-time Gaussian models. Front. Math. China 6, 1217–1248 (2011). https://doi.org/10.1007/s11464-011-0126-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-011-0126-9
Keywords
- Space-time model
- anisotropic Gaussian field
- prediction error
- mean square differentiability
- sample path differentiability
- Hausdorff dimension