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Fractal and smoothness properties of space-time Gaussian models

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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.

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References

  1. Adler R J. The Geometry of Random Fields. New York: Wiley, 1981

    MATH  Google Scholar 

  2. Adler R J, Taylor J E. Random Fields and Geometry. New York: Springer, 2007

    MATH  Google Scholar 

  3. Anderes E B, Stein M L. Estimating deformations of isotropic Gaussian random fields on the plane. Ann Statist, 2008, 36: 719–741

    Article  MATH  MathSciNet  Google Scholar 

  4. Banerjee S, Gelfand A E. On smoothness properties of spatial processes. J Multivariate Anal, 2003, 84: 85–100

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  6. Berg C, Forst G. Potential Theory on Locally Compact Abelian Groups. New York-Heidelberg: Springer-Verlag, 1975

    MATH  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  11. Cramér H, Leadbetter M R. Stationary and Related Stochastic Processes. New York: John Wiley & Sons, Inc, 1967

    MATH  Google Scholar 

  12. Cressie N. Statistics for Spatial Data (rev ed). New York: Wiley, 1993

    Google Scholar 

  13. Cressie N, Huang H -C. Classes of nonseparable, spatiotemporal stationary covariance functions. J Amer Statist Assoc, 1999, 94: 1330–1340

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  Google Scholar 

  16. de Iaco S, Myers D E, Posa D. Nonseparable space-time covariance models: some parametric families. Math Geology, 2002, 34: 23–42

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  18. Falconer K J. Fractal Geometry—Mathematical Foundations and Applications. New York: Wiley & Sons, 1990

    MATH  Google Scholar 

  19. Fuentes M. Spectral methods for nonstationary spatial processes. 2002, 89: 197–210

    MATH  MathSciNet  Google Scholar 

  20. Fuentes M. A formal test for nonstationarity of spatial stochastic processes. J Multivariate Anal, 2005, 96: 30–54

    Article  MATH  MathSciNet  Google Scholar 

  21. Gneiting T. Nonseparable, stationary covariance functions for space-time data. J Amer Statist Assoc, 2002, 97: 590–600

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  25. 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

    Google Scholar 

  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

    Chapter  Google Scholar 

  27. Kahane J -P. Some Random Series of Functions. 2nd ed. Cambridge: Cambridge University Press, 1985

    MATH  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  30. Kyriakidis P C, Journe A G. Geostatistical space-time models: a review. Math Geology, 1999, 31: 651–684

    Article  MATH  Google Scholar 

  31. Ma C. Families of spatio-temporal stationary covariance models. J Statist Plan Infer, 2003, 116: 489–501

    Article  MATH  Google Scholar 

  32. Ma C. Spatio-temporal stationary covariance models. J Multivariate Anal, 2003, 86: 97–107

    Article  MATH  MathSciNet  Google Scholar 

  33. Ma C. Spatial autoregression and related spatio-temporal models. J Multivariate Anal, 2004, 88: 152–162

    Article  MATH  MathSciNet  Google Scholar 

  34. Ma C. Spatio-temporal variograms and covariance models. Adv Appl Probab, 2005, 37: 706–725

    Article  MATH  Google Scholar 

  35. Ma C. A class of stationary random fields with a simple correlation structure. J Multivariate Anal, 2005, 94: 313–327

    Article  MATH  MathSciNet  Google Scholar 

  36. Ma C. Stationary random fields in space and time with rational spectral densities. IEEE Trans Inform Th, 2007, 53: 1019–1029

    Article  Google Scholar 

  37. Ma C. Recent developments on the construction of spatio-temporal covariance models. Stoch Environ Res Risk Assess, 2008, 22(suppl 1): 39–47

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  41. Stein M L. Interpolation of Spatial Data: Some Theory for Kriging. New York: Springer, 1999

    Book  MATH  Google Scholar 

  42. Stein M L. Space-time covariance functions. J Amer Statist Assoc, 2005, 100: 310–321

    Article  MATH  MathSciNet  Google Scholar 

  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

    Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  47. Zhu Z, Stein M L. Parameter estimation for fractional Brownian surfaces. Statist Sinica, 2002, 12: 863–883

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yun Xue & Yimin Xiao

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  1. Yun Xue
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  2. Yimin Xiao
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Correspondence to Yimin Xiao.

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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

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  • DOI: https://doi.org/10.1007/s11464-011-0126-9

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