Skip to main content

Stationary and Isotropic Vector Random Fields on Spheres

Abstract

This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer’s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a χ 2 or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres.

This is a preview of subscription content, access via your institution.

References

  1. Bapat RB, Raghavan TES (1997) Nonnegative matrices and applications. Cambridge University Press, Cambridge

    Book  Google Scholar 

  2. Bingham NH (1973) Positive definite functions on spheres. Proc Camb Philos Soc 73:145–156

    Article  Google Scholar 

  3. Calder CA (2007) Dynamic factor process convolution models for multivariate space–time data with application to air quality assessment. Environ Ecol Stat 14:229–247

    Article  Google Scholar 

  4. Chen D, Menegatto A, Sun X (2003) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc Am Math Soc 131:2733–2740

    Article  Google Scholar 

  5. Chilés, JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York

    Book  Google Scholar 

  6. Choi J, Reich B, Fuentes M, Davis J (2009) Multivariate spatial-temporal modeling and predication of speciated fine particles. J Statist Theory Pract 3:407–418

    Article  Google Scholar 

  7. Christensen JPR, Ressel P (1978) Functions operating on positive definite matrices and a theorem of Schoenberg. Trans Am Math Soc 243:89–95

    Article  Google Scholar 

  8. Cohen A, Johns RH (1969) Regression on a random field. J Am Stat Assoc 64:1172–1182

    Google Scholar 

  9. Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York

    Google Scholar 

  10. 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

    Article  Google Scholar 

  11. Du J, Ma C (2011) Spherically invariant vector random fields in space and time. IEEE Trans Signal Process 59:5921–5929

    Article  Google Scholar 

  12. Du J, Ma C (2012) Variogram matrix functions for vector random fields with second-order increments. Math Geosci 44:411–425

    Article  Google Scholar 

  13. Feller W (1971) An introduction to probability theory and its applications, vol. II, 2nd edn. Wiley, New York

    Google Scholar 

  14. Gaspari G, Cohn SE (1999) Construction of correlations in two and three dimensions. Q J R Meteorol Soc 125:723–757

    Article  Google Scholar 

  15. Gaspari G, Cohn E, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Q J R Meteorol Soc 132:1815–1838

    Article  Google Scholar 

  16. Haas TC (1998) Multivariate spatial prediction in the presence of non-linear trend and covariance non-stationarity. Environmetrics 7:145–165

    Article  Google Scholar 

  17. Hannan EJ (1966) Spectral analysis for geophysical data. Geophys J R Astron Soc 11:225–236

    Article  Google Scholar 

  18. Hannan EJ (1969) Fourier methods and random processes. Bull Internat Statist Inst 42:475–496

    Google Scholar 

  19. Hannan EJ (1970) Multiple time series. Wiley, New York

    Book  Google Scholar 

  20. Huang C, Yao Y, Cressie N, Hsing T (2009) Multivariate intrinsic random functions for cokriging. Math Geosci 41:887–904

    Article  Google Scholar 

  21. Huang C, Zhang H, Robeson SM (2011) On the validity of commonly used covariance and variogram functions on the sphere. Math Geosci 43:721–733

    Article  Google Scholar 

  22. Johns RH (1963a) Stochastic processes on a sphere. Ann Math Stat 34:213–218

    Article  Google Scholar 

  23. Johns RH (1963b) Stochastic processes on a sphere as applied to meteorological 500-millibar forecasts. Proceedings of the symposium of time series analysis, vol 119. Wiley, New York

    Google Scholar 

  24. Jun M (2011) Non-stationary cross-covariance models for multivariate processes on a globe. Scand J Stat 38:726–747

    Article  Google Scholar 

  25. Le ND, Zidek JV (2006) Statistical analysis of environmental space–time processes. Springer, New York

    Google Scholar 

  26. Ma C (2011a) Vector random fields with second-order moments or second-order increments. Stoch Anal Appl 29:197–215

    Article  Google Scholar 

  27. Ma C (2011b) Covariance matrices for second-order vector random fields in space and time. IEEE Trans Signal Process 59:2160–2168

    Article  Google Scholar 

  28. Ma C (2011c) Covariance matrix functions of vector χ 2 random fields in space and time. IEEE Trans Commun 59:2254–2561

    Article  Google Scholar 

  29. Mangulis V (1965) Handbook of series for scientists and engineers. Academic Press, New York

    Google Scholar 

  30. Mardia KV (1988) Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. J Multivar Anal 24:265–284

    Article  Google Scholar 

  31. Matérn B (1986) Spatial variation, 2nd edn. Springer, New York

    Google Scholar 

  32. 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 

  33. McLeod MG (1986) Stochastic processes on a sphere. Phys Earth Planet Inter 43:283–299

    Article  Google Scholar 

  34. Roy R (1972) Spectral analysis for random field on the circle. J Appl Probab 9:745–757

    Article  Google Scholar 

  35. Roy R (1973) Estimation of the covariance function of a homogeneous process on the sphere. Ann Stat 1:780–785

    Article  Google Scholar 

  36. Roy R (1976) Spectral analysis for a random process on the sphere. Ann Inst Stat Math 28:91–97

    Article  Google Scholar 

  37. Sain R, Cressie N (2007) A spatial model for multivariate lattice data. J Econom 140:226–259

    Article  Google Scholar 

  38. Sain R, Furrer R, Cressie N (2011) A spatial analysis of multivariate output from regional climate models. Ann Appl Stat 5:150–175

    Article  Google Scholar 

  39. Schoenberg I (1942) Positive definite functions on spheres. Duke Math J 9:96–108

    Article  Google Scholar 

  40. Szegö G (1959) Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ., vol 23. Amer. Math. Soc., Providence

    Google Scholar 

  41. 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

    Article  Google Scholar 

  42. Trenberth KE, Shea DJ (2005) Relationships between precipitation and surface temperature. Geophys Res Lett 32:L14703. doi:10.1029/2005GL022760

    Article  Google Scholar 

  43. Watson GN (1944) A treatise on the theory of Bessel functions, 2nd edn. Cambridge University Press, London

    Google Scholar 

  44. Weaver A, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815–1846

    Article  Google Scholar 

  45. Widder DV (1946) The Laplace transform. Princeton Univ. Press, Princeton

    Google Scholar 

  46. Yadrenko AM (1983) Spectral theory of random fields, Optimization Software, New York.

  47. Yaglom AM (1987) Correlation theory of stationary and related random functions, vol. I. Springer, New York

    Google Scholar 

  48. 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

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported in part by U.S. Department of Energy under Grant DE-SC0005359, in part by the Kansas NSF EPSCoR under Grant EPS0903806, and in part by a Kansas Technology Enterprise Corporation grant. The author would like to thank an associate editor and two anonymous reviewers for their valuable comments and suggestions which helped to improve the presentation of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Chunsheng Ma.

Appendix: Proof of Theorem 2

Appendix: Proof of Theorem 2

Suppose that C(θ) is the covariance matrix function of an m-variate stationary and isotropic random field \(\{ \mathbf {Z}(\mathbf{x}), \mathbf{x}\in \mathbb{S}^{\infty}\}\). Then C ij (θ)=C ji (θ),θ∈[0,π], i,j=1,…,m. According to Theorem 2 of Schoenberg (1942), for each k∈{1,…,m}, the direct covariance function C kk (θ) of a component random field \(\{ Z_{k} (\mathbf{x}), \mathbf{x}\in\mathbb{S}^{\infty}\}\) possesses an ultraspherical expansion

(11)

where {b n (k,k),n=0,1,…} is a summable sequence of nonnegative numbers. Similarly, for ij, a scalar random field \(\{ Z_{i} (\mathbf {x})+Z_{j} (\mathbf{x} ), \mathbf{x}\in\mathbb{S}^{\infty}\}\) has the covariance function C ii (θ)+C jj (θ)+2C ij (θ) that possesses an ultraspherical expansion

(12)

and a scalar random field \(\{ Z_{i} (\mathbf{x})-Z_{j} (\mathbf{x}), \mathbf {x}\in\mathbb{S}^{\infty}\}\) has the covariance function C ii (θ)+C jj (θ)−2C ij (θ) that possesses an ultraspherical expansion

(13)

Taking the difference of Eqs. (12) and (13), we obtain

(14)

where \(b_{n} (i, j) = \frac{1}{4} (e^{+}_{n} (i, j)-e^{-}_{n} (i, j)), n = 0, 1, \ldots, i, j =1, 2, \ldots, m\). It follows from Eqs. (11) and (14) that C(θ) adopts an expansion like Eq. (6). Therefore, it remains to show that B n =(b n (i,j)) m×m is positive definite, for each nonnegative integer n. To this end, suppose that a 1,…,a m are arbitrary real numbers. Using these constants, we formulate a scalar random field \(\{ \sum_{k=1}^{m} a_{k} Z_{k} (\mathbf{x}), \mathbf{x}\in\mathbb {S}^{\infty}\}\). This is a second-order random field on \(\mathbb{S}^{\infty}\) with covariance function

where the second equality follows from Eqs. (11) and (14). Applying Theorem 2 of Schoenberg (1942) to the last expansion, we obtain that all its coefficients aB n a are nonnegative, that is, each B n =(b n (i,j)) m×m is positive definite.

On the other hand, let C(θ) be an m×m matrix function with expansion of Eq. (6). For each n≥0, it is known (Schoenberg 1942) that cosn θ is positive definite on \(\mathbb{S}^{\infty}\), and thus 1cosn θ is a covariance matrix function, where 1 is an m×m matrix with all entries equal 1. By Theorem 6 of Ma (2011b), the Hadamard product of B n and 1cosn θ, which is the same as B n cosn θ, is also a covariance matrix function on \(\mathbb{S}^{\infty}\). So is C(θ), the sum of the sequence {B n cosn θ,n=0,1,2,…}.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ma, C. Stationary and Isotropic Vector Random Fields on Spheres. Math Geosci 44, 765–778 (2012). https://doi.org/10.1007/s11004-012-9411-8

Download citation

Keywords

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