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Admissible nested covariance models over spheres cross time

  • Ana Peron
  • Emilio Porcu
  • Xavier Emery
Original Paper
  • 58 Downloads

Abstract

Nested covariance models, defined as linear combinations of basic covariance functions, are very popular in many branches of applied statistics, and in particular in geostatistics. A notorious limit of nested models is that the constants in the linear combination are bound to be nonnegative in order to preserve positive definiteness (admissibility). This paper studies nested models on d-dimensional spheres and spheres cross time. We show the exact interval of admissibility for the constants involved in the linear combinations. In particular, we show that at least one constant can be negative. One of the implications is that one can obtain a nested model attaining negative correlations. We provide characterization theorems for arbitrary linear combinations as well as for nonconvex combinations involving two covariance functions. We illustrate our findings through several examples involving nonconvex combinations of well-known parametric families of covariance functions.

Keywords

Covariance functions Nested models Negative covariance Spheres 

Notes

Acknowledgements

Ana Peron was partially supported by São Paulo Research Foundation (FAPESP) under Grants 2016/03015-7 and 2016/09906-0. Emilio Porcu and Xavier Emery acknowledge the support of Grant CONICYT/FONDECYT/REGULAR/1170290 from the Chilean Commission for Scientific and Technological Research.

References

  1. Arafat A, Porcu E, Bevilacqua M, Mateu J (2018) Equivalence and orthogonality of Gaussian measures on spheres. J Multivar Anal 167:306–318CrossRefGoogle Scholar
  2. Atkinson K, Han W (2012) Spherical harmonics and approximations on the unit sphere: an introduction, volume 2044 of lecture notes in mathematics. Springer, HeidelbergCrossRefGoogle Scholar
  3. Berg C, Porcu E (2017) From Schoenberg coefficients to Schoenberg functions. Constr Approx 45:217–241CrossRefGoogle Scholar
  4. Berg C, Peron AP, Porcu E (2018) Schoenberg’s theorem for real and complex Hilbert spheres revisited. J Approx Theory 228:58–78CrossRefGoogle Scholar
  5. Bevilacqua M, Gaetan C, Mateu J, Porcu E (2012) Estimating space and space–time covariance functions: a weighted composite likelihood approach. J Am Stat Assoc 107:268–280CrossRefGoogle Scholar
  6. Bonat WH, Jørgensen B (2016) Multivariate covariance generalized linear models. J R Stat Soc Ser C Appl Stat 65(5):649–675CrossRefGoogle Scholar
  7. Chen D, Menegatto VA, Sun X (2003) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc Am Math Soc 131(9):2733–2740 (electronic)CrossRefGoogle Scholar
  8. Chilès J, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  9. Clarke J, Alegría A, Porcu E (2018) Regularity properties and simulations of Gaussian random fields on the sphere cross time. Electron J Stat 12:399–426. arXiv:1611.02851
  10. Dai F, Xu Y (2013) Approximation theory and harmonic analysis on spheres and balls. Springer monographs in mathematics. Springer, New YorkGoogle Scholar
  11. Daley DJ, Porcu E (2014) Dimension walks and Schoenberg spectral measures. Proc Am Math Soc 142(5):1813–1824CrossRefGoogle Scholar
  12. Daley DJ, Porcu E, Bevilacqua M (2015) Classes of compactly supported covariance functions for multivariate random fields. Stoch Environ Res Risk Assess 29(4):1249–1263CrossRefGoogle Scholar
  13. De Iaco S, Posa D (2018) Strict positive definiteness in geostatistics. Stoch Environ Res Risk Assess 32(3):577–590CrossRefGoogle Scholar
  14. De Iaco S, Myers DE, Posa D (2001) Space–time analysis using a general product–sum model. Stat Probab Lett 52(1):21–28CrossRefGoogle Scholar
  15. Estrade A, Fariñas A, Porcu E (2017) Characterization theorems for covariance functions on the n-dimensional sphere across time. Technical report, University Federico Santa Maria, MAP5 2016-34 [hal-01417668]Google Scholar
  16. Gneiting T (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4):1327–1349CrossRefGoogle Scholar
  17. Gregori P, Porcu E, Mateu J, Sasvári Z (2008) On potentially negative space time covariances obtained as sum of products of marginal ones. Ann Inst Stat Math 60(4):865–882CrossRefGoogle Scholar
  18. Jones RH (1963) Stochastic processes on a sphere. Ann Math Stat 34:213–218CrossRefGoogle Scholar
  19. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, CambridgeGoogle Scholar
  20. Kleiber W, Porcu E (2015) Nonstationary matrix covariances: compact support, long range dependence and quasi-arithmetic constructions. Stoch Environ Res Risk Assess 29(1):193–204CrossRefGoogle Scholar
  21. Lang A, Schwab C (2015) Isotropic random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann Appl Probab 25:3047–3094CrossRefGoogle Scholar
  22. Marinucci D, Peccati G (2011) Random fields on the sphere, representation, limit theorems and cosmological applications. Cambridge University Press, New YorkCrossRefGoogle Scholar
  23. Menegatto VA (1995) Strictly positive definite kernels on the circle. Rocky Mt J Math 25(3):1149–1163CrossRefGoogle Scholar
  24. Menegatto VA, Oliveira CP, Peron AP (2006) Strictly positive definite kernels on subsets of the complex plane. Comput Math Appl 51(8):1233–1250CrossRefGoogle Scholar
  25. Møller J, Nielsen M, Porcu E, Rubak E (2018) Determinantal point process models on the sphere. Bernoulli 24(2):1171–1201. arXiv:1607.03675
  26. Pan J, Mackenzie G (2003) On modelling mean-covariance structures in longitudinal studies. Biometrika 90(1):239–244CrossRefGoogle Scholar
  27. Porcu E, Gregori P, Mateu J (2006) Nonseparable stationary anisotropic space–time covariance functions. Stoch Environ Res Risk Assess 21(2):113–122CrossRefGoogle Scholar
  28. Porcu E, Daley DJ, Buhmann M, Bevilacqua M (2013) Radial basis functions with compact support for multivariate geostatistics. Stoch Environ Res Risk Assess 27(4):909–922CrossRefGoogle Scholar
  29. Porcu E, Bevilacqua M, Genton MG (2016) Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J Am Stat Assoc 111(514):888–898CrossRefGoogle Scholar
  30. Porcu E, Alegria A, Furrer R (2017) Modeling temporally evolving and spatially globally dependent data. Int Stat Rev. arXiv:1706.09233
  31. Pourahmadi M (1999) Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86(3):677–690CrossRefGoogle Scholar
  32. Pourahmadi M (2011) Covariance estimation: the GLM and regularization perspectives. Stat Sci 26(3):369–387CrossRefGoogle Scholar
  33. Schoenberg IJ (1942) Positive definite functions on spheres. Duke Math J 9:96–108CrossRefGoogle Scholar
  34. Serra J (1968) Les structures gigognes: morphologie mathématique et interprétation métallogénique. Miner Depos 3:135–154CrossRefGoogle Scholar
  35. Soubeyrand S, Enjalbert J, Sache I (2008) Accounting for roughness of circular processes: using Gaussian random process to model the anisotropic spread of airbone plant disease. Theor Popul Biol 73(1):92–103CrossRefGoogle Scholar
  36. Wackernagel H (2003) Multivariate geostatistics: an introduction with applications. Springer, BerlinCrossRefGoogle Scholar
  37. Yakhot V, Orszag SA, She Z-S (1989) Space–time correlations in turbulence: kinematical versus dynamical effects. Phys Fluids A 1(2):184–186CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaICMC-USPSão CarlosBrazil
  2. 2.School of Mathematics and StatisticsUniversity of NewcastleNewcastle upon TyneUK
  3. 3.Department of MathematicsUniversity of AtacamaCopiapóChile
  4. 4.Department of Mining EngineeringUniversity of ChileSantiagoChile
  5. 5.Advanced Mining Technology CenterUniversity of ChileSantiagoChile

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