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Mathematical Geosciences

, Volume 45, Issue 3, pp 341–357 | Cite as

Isotropic Variogram Matrix Functions on Spheres

  • Juan DuEmail author
  • Chunsheng Ma
  • Yang Li
Article

Abstract

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ 2, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization.

Keywords

Absolutely monotone function Cross variogram Direct variogram Elliptically contoured random field Gaussian random field Gegenbauer’s polynomials Positive definite matrix 

Notes

Acknowledgements

The authors wish to thank the Editor-in-Chief, an associate editor, two reviewers and Dr. James Stapleton for their valuable comments and suggestions which helped to improve the presentation of this paper. Ma’s work is supported in part by US Department of Energy under Grant DE-SC0005359.

References

  1. Advieto-Borbe MAA, Haddix ML, Binder DL, Walters DT, Dobermann A (2007) Soil greenhouse gas fluxes and global warming potential in four high-yielding maize systems. Glob Change Biol 13:1972–1988 CrossRefGoogle Scholar
  2. Alkhaled AA, Michalak AM, Kawa SR, Olsen SC, Wang JW (2008) A global evaluation of the regional spatial variability of column integrated CO2 distributions. J Geophys Res 113:D20303. doi: 10.1029/2007JD009693 CrossRefGoogle Scholar
  3. Bochner S (1941) Hilbert distances and positive definite functions. Ann Math 42:647–656 CrossRefGoogle Scholar
  4. Carmona-Moreno C, Belward A, Malingreau JP, Hartley A, Garcia-Algere M, Antonovsky M, Buchshtaber V, Pivovarov V (2005) Characterizing interannual variations in global fire calendar using data from Earth observing satellites. Glob Change Biol 11:1537–1555 CrossRefGoogle Scholar
  5. Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York. Google Scholar
  6. Cressie N, Wikle CK (1998) The variance-based cross-variogram: you can add apples and oranges. Math Geol 30:789–800 CrossRefGoogle Scholar
  7. Cressie N, Johannesson G (2008) Fixed rank kriging for very large spatial data sets. J R Stat Soc B 70:209226 CrossRefGoogle Scholar
  8. de Boyer Montégut C, Madec G, Fischer AS, Lazar A, Iudicone D (2004) Mixed layer depth over the global ocean: an examination of profile data and a profile-based climatology. J Geophys Res 109:C12003. doi: 10.1029/2004JC002378 CrossRefGoogle Scholar
  9. Dimitrakopoulos RD, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42:65–99 CrossRefGoogle Scholar
  10. Du J, Leonenko N, Ma C, Shu H (2012) Hyperbolic vector random fields with hyperbolic direct and cross covariance functions. Stoch Anal Appl 30:662–674 CrossRefGoogle Scholar
  11. Du J, Ma C (2011) Spherically invariant vector random fields in space and time. IEEE Trans Signal Process 59:5921–5929 CrossRefGoogle Scholar
  12. Du J, Ma C (2012) Variogram matrix functions for vector random fields with second-order increments. Math Geosci 44:411–425 CrossRefGoogle Scholar
  13. Feller W (1971) An introduction to probability theory and its applications, vol. II, 2nd edn. Wiley, New York Google Scholar
  14. Gangolli R (1967a) Abstract harmonic analysis and Lévy’s Brownian motion of several parameters. In: Proc fifth Berkeley symp math statist prob, vol II. University of California Press, Berkeley, pt. 1, pp 13–30 Google Scholar
  15. Gangolli R (1967b) Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann Inst Henri Poincaré B, Probab Stat 3:121–226 Google Scholar
  16. Gaspari G, Cohn SE (1999) Construction of correlations in two and three dimensions. Q J R Meteorol Soc 125:723–757 CrossRefGoogle Scholar
  17. Gaspari G, Cohn SE, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Q J R Meteorol Soc 132:1815–1838 CrossRefGoogle Scholar
  18. Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177 CrossRefGoogle Scholar
  19. Haas TC (1998) Multivariate spatial prediction in the presence of non-linear trend and covariance non-stationarity. Environmetrics 7:145–165 CrossRefGoogle Scholar
  20. Hannan EJ (1970) Multiple time series. Wiley, New York CrossRefGoogle Scholar
  21. Huang C, Yao Y, Cressie N, Hsing T (2009) Multivariate intrinsic random functions for cokriging. Math Geosci 41:887–904 CrossRefGoogle Scholar
  22. Huang C, Hsing T, Cressie N (2011a) Nonparametric estimation of the variogram and its spectrum. Biometrika 98:775–789 CrossRefGoogle Scholar
  23. Huang C, Zhang H, Robeson SM (2011b) On the validity of commonly used covariance and variogram functions on the sphere. Math Geosci 43:721–733 CrossRefGoogle Scholar
  24. Im HK, Stein ML, Zhu Z (2007) Semiparametric estimation of spectral density with irregular observations. J Am Stat Assoc 102:726–735 CrossRefGoogle Scholar
  25. Istas J (2005) Spherical and hyperbolic fractional Brownian motion. Electron Commun Probab 10:254–262 CrossRefGoogle Scholar
  26. Johns RH (1963a) Stochastic processes on a sphere. Ann Math Stat 34:213–218 CrossRefGoogle Scholar
  27. Johns RH (1963b) Stochastic processes on a sphere as applied to meteorological 500-millibar forecasts. In: Proc symp time series analysis. Wiley, New York, pp 119–124 Google Scholar
  28. Jun M, Stein ML (2007) An approach to producing space-time covariance functions on spheres. Technometrics 49:468–479 CrossRefGoogle Scholar
  29. Jun M, Stein ML (2008) Nonstationary covariance models for global data. Ann Appl Stat 2:1271–1289 CrossRefGoogle Scholar
  30. Le ND, Zidek JV (2006) Statistical analysis of environmental space-time processes. Springer, New York Google Scholar
  31. Leonenko N, Sakhno L (2012) On spectral representations of tensor random fields on the sphere. Stoch Anal Appl 31:167–182 Google Scholar
  32. Ma C (2011a) Vector random fields with second-order moments or second-order increments. Stoch Anal Appl 29:197–215 CrossRefGoogle Scholar
  33. Ma C (2011b) Covariance matrix functions of vector χ 2 random fields in space and time. IEEE Trans Commun 59:2254–2561 CrossRefGoogle Scholar
  34. Ma C (2012) Stationary and isotropic vector random fields on spheres. Math Geosci 44:765–778 CrossRefGoogle Scholar
  35. Ma C (2013) Student’s t vector random fields with power-law and log-law decaying direct and cross covariances. Stoch Anal Appl 31:167–182 CrossRefGoogle Scholar
  36. Mangulis V (1965) Handbook of series for scientists and engineers. Academic Press, New York Google Scholar
  37. Matheron G (1989) The internal consistency of models in geostatistics. In: Armstrong M (ed) Geostatistics, vol 1. Kluwer Academic, Dordrecht, pp 21–38 Google Scholar
  38. McLeod MG (1986) Stochastic processes on a sphere. Phys Earth Planet Inter 43:283–299 CrossRefGoogle Scholar
  39. Minozzo M, Ferracuti L (2012) On the existence of some skew-normal stationary processes. Chilean J Stat 3:159–172 Google Scholar
  40. Myers DE (1991) Pseudo-cross variograms, positive-definiteness, and cokriging. Math Geol 23:805–816 CrossRefGoogle Scholar
  41. Pollard H (1946) The representation of \(e^{-x^{\lambda}}\) as a Laplace integral. Bull Am Math Soc 52:908 CrossRefGoogle Scholar
  42. Røislien J, Omre H (2006) T-distributed random fields: a parametric model for heavy-tailed well-log data. Math Geol 38:821–849 CrossRefGoogle Scholar
  43. Roy R (1973) Estimation of the covariance function of a homogeneous process on the sphere. Ann Stat 1:780–785 CrossRefGoogle Scholar
  44. Roy R (1976) Spectral analysis for a random process on the sphere. Ann Inst Stat Math 28:91–97 CrossRefGoogle Scholar
  45. Sain SR, Cressie N (2007) A spatial model for multivariate lattice data. J Econom 140:226–259 CrossRefGoogle Scholar
  46. Sain SR, Furrer R, Cressie N (2011) A spatial analysis of multivariate output from regional climate models. Ann Appl Stat 5:150–175 CrossRefGoogle Scholar
  47. Schoenberg I (1942) Positive definite functions on spheres. Duke Math J 9:96–108 CrossRefGoogle Scholar
  48. Szegö G (1959) Orthogonal polynomials. Amer Math Soc Colloq Publ, vol 23. Amer. Math. Soc., Providence Google Scholar
  49. Tebaldi C, Lobell DB (2008) Towards probabilistic projections of climate change impacts on global crop yields. Geophys Res Lett 35:L08705. doi: 10.1029/2008GL033423 CrossRefGoogle Scholar
  50. Trenberth KE, Shea DJ (2005) Relationships between precipitation and surface temperature. Geophys Res Lett 32:L14703. doi: 10.1029/2005GL022760 CrossRefGoogle Scholar
  51. Ver Hoef, JM, Cressie N (1993) Multivariate spatial prediction. Math Geol 25:219–239 CrossRefGoogle Scholar
  52. Wackernagel H (2003) Multivariate geostatistics: an introduction with applications, 3rd edn. Springer, New York CrossRefGoogle Scholar
  53. Weaver A, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815–1846 CrossRefGoogle Scholar
  54. Widder DV (1946) The Laplace transform. Princeton University Press, Princeton Google Scholar
  55. Yadrenko AM (1983) Spectral theory of random fields. Optimization Software, New York Google Scholar
  56. Yaglom AM (1987) Correlation theory of stationary and related random functions, vol. I. Springer, New York Google Scholar
  57. Zidek JV, Sun W, Le ND (2000) Designing and integrating composite networks for monitoring multivariate Gaussian pollution fields. J R Stat Soc, Ser C, Appl Stat 49:63–79 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  1. 1.Department of StatisticsKansas State UniversityManhattanUSA
  2. 2.Department of Mathematics, Statistics, and PhysicsWichita State UniversityWichitaUSA
  3. 3.School of EconomicsWuhan University of TechnologyWuhanChina
  4. 4.Department of StatisticsIowa State UniversityAmesUSA

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