Advertisement

Strict positive definiteness in geostatistics

  • S. De IacoEmail author
  • D. Posa
Original Paper

Abstract

Geostatistical modeling is often based on the use of covariance functions, i.e., positive definite functions. However, when interpolation problems have to be solved, it is advisable to consider the subset of strictly positive definite functions. Indeed, it will be argued that ensuring strict positive definiteness for a covariance function is convenient from a theoretical and practical point of view. In this paper, an extensive analysis on strictly positive definite covariance functions has been given. The closure of the set of strictly positive definite functions with respect to the sum and the product of covariance functions defined on the same Euclidean dimensional space or on factor spaces, as well as on partially overlapped lower dimensional spaces, has been analyzed. These results are particularly useful (a) to extend strict positive definiteness in higher dimensional spaces starting from covariance functions which are only defined on lower dimensional spaces and/or are only strictly positive definite in lower dimensional spaces, (b) to construct strictly positive definite covariance functions in space–time as well as (c) to obtain new asymmetric and strictly positive definite covariance functions.

Keywords

Strict positive definiteness Covariance functions Spatial dimension 

Notes

Acknowledgements

The authors would like to thank the associate editor, the reviewers and prof. Giorgio Metafune for their interest in the paper and their useful comments and suggestions.

References

  1. Bernstein S (1928) Sur les fonctions absolument monotones. Acta Math 52(1):1–66CrossRefGoogle Scholar
  2. Bochner S (1959) Lectures on Fourier integrals. Princeton University Press, New JerseyGoogle Scholar
  3. Chang K (1996) Strictly positive definite functions. J Approx Theory 87(2):148–158CrossRefGoogle Scholar
  4. Chen D, Menegatto V, Sun X (2003) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc Am Math Soc 131(9):2733–2740CrossRefGoogle Scholar
  5. Cressie N, Huang H (1999) Classes of nonseparable, spatio-temporal stationary covariance functions. J Am Stat Assoc 94(448):1330–1340CrossRefGoogle Scholar
  6. Cressie N, Majure J (1997) Spatio-temporal statistical modeling of livestock waste in streams. J Agric Biol Environ Stat 2(1):24–47CrossRefGoogle Scholar
  7. De Iaco S, Posa D (2013) Positive and negative non-separability for space–time covariance models. J Stat Plan Inf 143(2):378–391CrossRefGoogle Scholar
  8. De Iaco S, Myers D, Posa D (2001) Space–time analysis using a general product–sum model. Stat Probab Lett 52(1):21–28CrossRefGoogle Scholar
  9. De Iaco S, Myers D, Posa D (2011) On strict positive definiteness of product and product–sum covariance models. J Stat Plan Inf 141:1132–1140CrossRefGoogle Scholar
  10. Gneiting T (2002) Nonseparable, stationary covariance functions for space–time data. J Am Stat Assoc 97(458):590–600CrossRefGoogle Scholar
  11. Gneiting T (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4):1327–1349CrossRefGoogle Scholar
  12. Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, New YorkCrossRefGoogle Scholar
  13. Horn RA, Johnson CR (1996) Matrix analysis. Cambridge University Press, New YorkGoogle Scholar
  14. Khinchin A (1934) Korrelations theorie der stationären stochastischen prozesse. Math Ann 109:604–615CrossRefGoogle Scholar
  15. Kolovos A, Christakos G, Hristopulos D, Serre M (2004) Methods for generating non-separable spatiotemporal covariance models with potential environmental applications. Adv Water Resour 27(8):815–830CrossRefGoogle Scholar
  16. Ma C (2002) Spatio-temporal covariance functions generated by mixtures. Math Geol 34(8):965–975CrossRefGoogle Scholar
  17. Ma C (2003) Families of spatio-temporal stationary covariance models. J Stat Plan Inf 116(2):489–501CrossRefGoogle Scholar
  18. Ma C (2005) Linear combinations of space–time covariance functions and variograms. IEEE Trans Signal Process 53(3):857–864CrossRefGoogle Scholar
  19. Martinez-Ruiz F, Mateu J, Montes F, Porcu E (2010) Mortality risk assessment through stationary space–time covariance functions. Stoch Environ Res Risk Assess 24(4):519–526CrossRefGoogle Scholar
  20. Mathias M (1923) Über positive Fourier-Integrale. Math Z 16:103–125CrossRefGoogle Scholar
  21. Menegatto VA (1994) Strictly positive definite kernels on the Hilbert sphere. Appl Anal 55:91–101CrossRefGoogle Scholar
  22. Miller K, Samko S (2001) Completely monotonic functions. Integr Transforms Spec Funct 12(4):389–402CrossRefGoogle Scholar
  23. Montero J, Fernández-Avilés G, Mateu J (2015) Spatial and spatio-temporal geostatistical modeling and kriging. Wiley, HobokenCrossRefGoogle Scholar
  24. Myers DE (1988) Interpolation with positive definite functions. Sciences de la Terre 28:251–265Google Scholar
  25. Myers DE, Journel AG (1990) Variograms with zonal anisotropies and non-invertible kriging systems. Math Geol 22(7):779–785CrossRefGoogle Scholar
  26. Pinkus A (2004a) Strictly Hermitian positive definite functions. J Anal Math 94:293–318CrossRefGoogle Scholar
  27. Pinkus A (2004b) Strictly positive definite functions on a real inner product space. Adv Comput Math 20(4):263–271CrossRefGoogle Scholar
  28. Porcu E, Schilling L (2011) From Schoenberg to Pick–Nevanlinna: toward a complete picture of the variogram class. Bernoulli 17(1):441–455CrossRefGoogle Scholar
  29. Porcu E, Gregori P, Mateu J (2006) Nonseparable stationary anisotropic space–time covariance functions. Stoch Environ Res Risk Assess 21(2):113–122CrossRefGoogle Scholar
  30. Rodrigues A, Diggle P (2010) A class of convolution-based models for spatio-temporal processes with non-separable covariance structure. Scand J Stat 37(4):553–567CrossRefGoogle Scholar
  31. Ron A, Sun X (1996) Strictly positive definite functions on spheres in euclidean spaces. Math Comput 65(216):1513–1530CrossRefGoogle Scholar
  32. Schoenberg I (1938a) Metric spaces and completely monotone functions. Ann Math 39(4):811–841CrossRefGoogle Scholar
  33. Schoenberg I (1938b) Metric spaces and positive definite functions. Trans Am Math Soc 44(3):522–536CrossRefGoogle Scholar
  34. Schreiner M (1997) On a new condition for strictly positive definite functions on spheres. Proc Am Math Soc 125:531–539CrossRefGoogle Scholar
  35. Stein ML (2005) Space-time covariance functions. J Am Stat Assoc 100(469):310–320CrossRefGoogle Scholar
  36. Strauss H (1997) On interpolation with products of positive definite functions. Numer Algorithms 15(2):153–165CrossRefGoogle Scholar
  37. Wendland H (2005) Scattered data approximation. Cambridge University Press, New YorkGoogle Scholar
  38. Xu Y, Cheney E (1992) Strictly positive definite functions on spheres. Proc Am Math Soc 116(4):977–981CrossRefGoogle Scholar
  39. Yaglom A (1962) An introduction to the theory of stationary random functions. Dover Publications Inc, New York, Translated and edited by R.A. Silverman, p 235Google Scholar
  40. zu Castell W, Filbir F, Szwarc R (2005) Strictly positive definite functions in \({{\mathbb{R}}}^d\). J Approx Theory 137(2):277–280CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Dipartimento di Scienze dell’EconomiaUniversità del SalentoLecceItaly
  2. 2.Dipartimento di Scienze Economiche e Matematico-StatisticheUniversità del SalentoLecceItaly

Personalised recommendations