Skip to main content
SpringerLink
Log in

Hybrid Parametric Classes of Isotropic Covariance Functions for Spatial Random Fields

  • Published:
Mathematical Geosciences Aims and scope Submit manuscript

Abstract

Covariance functions are the core of spatial statistics, stochastic processes, machine learning, and many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Covariance functions that allow one to specify both local and global properties are certainly in demand. This paper provides a method for finding new classes of covariance functions having such properties. We refer to these models as hybrid, as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combinations of parametric families of measures. To illustrate our methodology, we provide new families of covariance functions that are proved to be richer than other well-known families proposed in earlier literature. More precisely, we derive a hybrid Cauchy–Matérn model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid hole-effect–Matérn model which is capable of attaining negative values (hole effect) while preserving the local attributes of the traditional Matérn model. Our findings are illustrated through numerical studies with both simulated and real data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, Mineola

    Google Scholar 

  • Adler RJ (2010) The geometry of random fields. SIAM, University City

    Book  Google Scholar 

  • Alegría A (2020) Cross-dimple in the cross-covariance functions of bivariate isotropic random fields on spheres. Stat 9(1):e301

    Article  Google Scholar 

  • Alegría A, Emery X, Porcu E (2021) Bivariate Matérn covariances with cross-dimple for modeling coregionalized variables. Spat Stat 41:100491

    Article  Google Scholar 

  • Barndorff-Nielsen O (1978) Hyperbolic distributions and distributions on hyperbolae. Scand J Stat 5(3):151–157

    Google Scholar 

  • Barp A, Oates CJ, Porcu E, Girolami M (2022) A Riemann-Stein kernel method. Bernoulli 28(4):2181–2208

    Article  Google Scholar 

  • Berg C, Mateu J, Porcu E (2008) The Dagum family of isotropic correlation functions. Bernoulli 14(4):1134–1149

    Article  Google Scholar 

  • Bevilacqua M, Caamaño-Carrillo C, Porcu E (2022) Unifying compactly supported and Matérn covariance functions in spatial statistics. J Multivar Anal 189:104949

    Article  Google Scholar 

  • Bevilacqua M, Gaetan C (2015) Comparing composite likelihood methods based on pairs for spatial Gaussian random fields. Stat Comput 25(5):877–892

    Article  Google Scholar 

  • Bingham NH (1973) Positive definite functions on spheres. In: Proceedings of the Cambridge Philosophical Society, vol. 73, pp. 145–156

  • Chaudhry MA, Zubair SM (1994) Generalized incomplete gamma functions with applications. J Comput Appl Math 55(1):99–123

    Article  Google Scholar 

  • Chen W, Castruccio S, Genton MG, Crippa P (2018) Current and future estimates of wind energy potential over Saudi Arabia. J Geophys Res Atmos 123(12):6443–6459

    Article  Google Scholar 

  • Chilés JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. John Wiley & Sons, Hoboken

    Book  Google Scholar 

  • Cockayne J, Oates CJ, Sullivan TJ, Girolami M (2019) Bayesian probabilistic numerical methods. SIAM Rev 61(4):756–789

    Article  Google Scholar 

  • Cressie N (1993) Statistics for spatial data. Wiley, New York

    Book  Google Scholar 

  • Cressie N, Kornak J (2003) Spatial statistics in the presence of location error with an application to remote sensing of the environment. Stat Sci 18(4):436–456

    Article  Google Scholar 

  • Daley D, Porcu E (2014) Dimension walks and Schoenberg spectral measures. Proc Am Math Soc 142(5):1813–1824

    Article  Google Scholar 

  • Di Lorenzo E, Combes V, Keister J, Strub P, Andrew T, Peter F, Marck O, Furtado J, Bracco A, Bograd S, Peterson W, Schwing F, Taguchi B, Hormázabal S, Parada C (2014) Synthesis of Pacific Ocean climate and ecosystem dynamics. Oceanography 26(4):68–81

    Article  Google Scholar 

  • Edwards M, Castruccio S, Hammerling D (2019) A multivariate global spatio-temporal stochastic generator for climate ensembles. J Agric Biol Environ Sci. https://doi.org/10.1007/s13253-019-00352-8

    Article  Google Scholar 

  • Emery X, Lantuéjoul C (2006) Tbsim: a computer program for conditional simulation of three-dimensional gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628

    Article  Google Scholar 

  • Emery X, Séguret SA (2020) Geostatistics for the mining industry: applications to porphyry copper deposits. CRC Press, Boca Raton

    Book  Google Scholar 

  • Furrer R, Genton MG, Nychka D (2006) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15(3):502–523

    Article  Google Scholar 

  • Furrer R, Sain SR, Nychka D, Meehl GA (2007) Multivariate Bayesian analysis of atmosphere-ocean general circulation models. Environ Ecol Stat 14(3):249–266

    Article  Google Scholar 

  • Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177

    Article  Google Scholar 

  • Gneiting T, Schlather M (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev 46(2):269–282

    Article  Google Scholar 

  • Gradshteyn I, Ryzhik I (2007) Table of integrals, series, and products, 7th edn. Academic Press, Amsterdam

    Google Scholar 

  • Guinness J, Fuentes M (2016) Isotropic covariance functions on spheres: some properties and modeling considerations. J Multivar Anal 143:143–152

    Article  Google Scholar 

  • Guinness J, Hammerling D (2018) Compression and conditional emulation of climate model output. J Am Stat Assoc 113(521):56–67

    Article  Google Scholar 

  • Hristopulos DT (2020) Random fields for spatial data modeling. Springer, Berlin

    Book  Google Scholar 

  • James G, Witten D, Hastie T, Tibshirani R (2013) An introduction to statistical learning, vol 112. Springer, Berlin

    Book  Google Scholar 

  • Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103(484):1545–1555

    Article  Google Scholar 

  • Laga I, Kleiber W (2017) The modified Matérn process. Stat 6(1):241–247

    Article  Google Scholar 

  • Ma P, Bhadra A (2022) Beyond matérn: on a class of interpretable confluent hypergeometric covariance functions. J Am Stat Assoc. https://doi.org/10.1080/01621459.2022.2027775

    Article  Google Scholar 

  • Matérn B (1986) Spatial variation—stochastic models and their application to some problems in forest surveys and other sampling investigations. Springer, Berlin

    Google Scholar 

  • Moreva O, Schlather M (2022) Bivariate covariance functions of Pólya type. J Multivar Anal. https://doi.org/10.1016/j.jmva.2022.105099

    Article  Google Scholar 

  • Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7(4):308–313

    Article  Google Scholar 

  • Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132

    Article  Google Scholar 

  • Pazouki M, Schaback R (2011) Bases for kernel-based spaces. J Comput Appl Math 236(4):575–588

    Article  Google Scholar 

  • Porcu E, Alegria A, Furrer R (2018) Modeling temporally evolving and spatially globally dependent data. Int Stat Rev 86(2):344–377

    Article  Google Scholar 

  • Porcu E, Bevilacqua M, Hering AS (2018) The Shkarofsky-Gneiting class of covariance models for bivariate Gaussian random fields. Stat 7(1):e207

    Article  Google Scholar 

  • Porcu E, Zastavnyi V (2011) Characterization theorems for some classes of covariance functions associated to vector valued random fields. J Multivar Anal 102(9):1293–1301

    Article  Google Scholar 

  • Posa D (2023) Special classes of isotropic covariance functions. Stoch Environ Res Risk Assess 37:1615–1633

    Article  Google Scholar 

  • Schaback R, Wendland H (2006) Kernel techniques: from machine learning to meshless methods. Acta Numer 15:543–639

    Article  Google Scholar 

  • Schlather M (2010) Some covariance models based on normal scale mixtures. Bernoulli 16(3):780–797

    Article  Google Scholar 

  • Schlather M, Moreva O (2017) A parametric model bridging between bounded and unbounded variograms. Stat 6(1):47–52

    Article  Google Scholar 

  • Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39(4):811–841

    Article  Google Scholar 

  • Schoenberg IJ (1942) Positive definite functions on spheres. Duke Math J 9(1):96–108

    Article  Google Scholar 

  • Stein ML (1999) Statistical interpolation of spatial data: some theory for kriging. Springer, New York

    Book  Google Scholar 

  • Stein ML (2007) Spatial variation of total column ozone on a global scale. Ann Appl Stat 1(1):191–210

    Article  Google Scholar 

  • Yaglom AM (1987) Correlation theory of stationary and related random functions, volume I: basic results. Springer, Berlin

    Book  Google Scholar 

  • Zhang H, Wang Y (2010) Kriging and cross-validation for massive spatial data. Environmetrics 21(3–4):290–304

    Article  Google Scholar 

Download references

Acknowledgements

Alfredo Alegría was supported in part by the National Agency for Research and Development of Chile, through grant ANID/FONDECYT/INICIACIÓN/No. 11190686. Fabián Ramirez was supported in part by the Dirección de Postgrados y Programas (DPP) of the Universidad Técnica Federico Santa María. Emilio Porcu is supported by the Khalifa University of Science and Technology under Award No. FSU-2021-016. We thank an anonymous reviewer and the associate editor for constructive comments.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaiso, Chile

    Alfredo Alegría & Fabián Ramírez

  2. Department of Mathematics, Khalifa University, Abu Dhabi, United Arab Emirates

    Emilio Porcu

  3. ADIA Lab, Abu Dhabi, United Arab Emirates

    Emilio Porcu

Authors
  1. Alfredo Alegría
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabián Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilio Porcu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfredo Alegría.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alegría, A., Ramírez, F. & Porcu, E. Hybrid Parametric Classes of Isotropic Covariance Functions for Spatial Random Fields. Math Geosci (2024). https://doi.org/10.1007/s11004-023-10123-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11004-023-10123-4

Keywords

Search

Navigation