Frontiers of Mathematics in China >
Fractal and smoothness properties of space-time Gaussian models
Received date: 07 Aug 2010
Accepted date: 01 Mar 2011
Published date: 01 Dec 2011
Copyright
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.
Yun XUE , Yimin XIAO . Fractal and smoothness properties of space-time Gaussian models[J]. Frontiers of Mathematics in China, 2011 , 6(6) : 1217 -1248 . DOI: 10.1007/s11464-011-0126-9
1 |
Adler R J. The Geometry of Random Fields. New York: Wiley, 1981
|
2 |
Adler R J, Taylor J E. Random Fields and Geometry. New York: Springer, 2007
|
3 |
Anderes E B, Stein M L. Estimating deformations of isotropic Gaussian random fields on the plane. Ann Statist, 2008, 36: 719-741
|
4 |
Banerjee S, Gelfand A E. On smoothness properties of spatial processes. J Multivariate Anal, 2003, 84: 85-100
|
5 |
Banerjee S, Gelfand A E, Sirmans C F. Directional rates of change under spatial process models. J Amer Statistical Assoc, 2003, 98: 946-954
|
6 |
Berg C, Forst G. Potential Theory on Locally Compact Abelian Groups. New York-Heidelberg: Springer-Verlag, 1975
|
7 |
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
|
8 |
Chan G, Wood A T A. Increment-based estimators of fractal dimension for twodimensional surface data. Statist Sinica, 2000, 10: 343-376
|
9 |
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
|
10 |
Constantine A G, Hall P. Characterizing surface smoothness via estimation of effective fractal dimension. J Roy Statist Soc Ser B, 1994, 56: 97-113
|
11 |
Cramér H, Leadbetter M R. Stationary and Related Stochastic Processes. New York: John Wiley & Sons, Inc, 1967
|
12 |
Cressie N. Statistics for Spatial Data (rev ed). New York: Wiley, 1993
|
13 |
Cressie N, Huang H-C. Classes of nonseparable, spatiotemporal stationary covariance functions. J Amer Statist Assoc, 1999, 94: 1330-1340
|
14 |
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
|
15 |
de Iaco S, Myers D E, Posa D. Space-Time analysis using a general product-sum model. Statist Probab Letters, 2001, 52: 21-28
|
16 |
de Iaco S, Myers D E, Posa D. Nonseparable space-time covariance models: some parametric families. Math Geology, 2002, 34: 23-42
|
17 |
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
|
18 |
Falconer K J. Fractal Geometry—Mathematical Foundations and Applications. New York: Wiley & Sons, 1990
|
19 |
Fuentes M. Spectral methods for nonstationary spatial processes. 2002, 89: 197-210
|
20 |
Fuentes M. A formal test for nonstationarity of spatial stochastic processes. J Multivariate Anal, 2005, 96: 30-54
|
21 |
Gneiting T. Nonseparable, stationary covariance functions for space-time data. J Amer Statist Assoc, 2002, 97: 590-600
|
22 |
Gneiting T, Kleiber W, Schlather M. Matérn cross-covariance functions for multivariate random fields. Preprint, 2009
|
23 |
Hall P, Wood A T A. On the performance of box-counting estimators of fractal dimension. Biometrika, 1993, 80: 246-252
|
24 |
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
|
25 |
Higdon D, Swall J, Kern J. Nonstationary spatial modeling. In: Bernardo J M,
|
26 |
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
|
27 |
Kahane J-P. Some Random Series of Functions. 2nd ed. Cambridge: Cambridge University Press, 1985
|
28 |
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
|
29 |
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
|
30 |
Kyriakidis P C, Journe A G. Geostatistical space-time models: a review. Math Geology, 1999, 31: 651-684
|
31 |
Ma C. Families of spatio-temporal stationary covariance models. J Statist Plan Infer, 2003, 116: 489-501
|
32 |
Ma C. Spatio-temporal stationary covariance models. J Multivariate Anal, 2003, 86: 97-107
|
33 |
Ma C. Spatial autoregression and related spatio-temporal models. J Multivariate Anal, 2004, 88: 152-162
|
34 |
Ma C. Spatio-temporal variograms and covariance models. Adv Appl Probab, 2005, 37: 706-725
|
35 |
Ma C. A class of stationary random fields with a simple correlation structure. J Multivariate Anal, 2005, 94: 313-327
|
36 |
Ma C. Stationary random fields in space and time with rational spectral densities. IEEE Trans Inform Th, 2007, 53: 1019-1029
|
37 |
Ma C. Recent developments on the construction of spatio-temporal covariance models. Stoch Environ Res Risk Assess, 2008, 22(suppl 1): 39-47
|
38 |
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)
|
39 |
Paciorek C J, Schervish M J. Spatial modelling using a new class of nonstationary covariance functions. Environmetrics, 2006, 17: 483-506
|
40 |
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
|
41 |
Stein M L. Interpolation of Spatial Data: Some Theory for Kriging. New York: Springer, 1999
|
42 |
Stein M L. Space-time covariance functions. J Amer Statist Assoc, 2005, 100: 310-321
|
43 |
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
|
44 |
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
|
45 |
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
|
46 |
Yaglom A M. Some classes of random fields in n-dimensional space, related to stationary random processes. Th Probab Appl, 1957, 2: 273-320
|
47 |
Zhu Z, Stein M L. Parameter estimation for fractional Brownian surfaces. Statist Sinica, 2002, 12: 863-883
|
/
〈 |
|
〉 |