Skip to main content
SpringerLink
Log in

Parametric variogram matrices incorporating both bounded and unbounded functions

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

We construct a flexible class of parametric models for both traditional and pseudo variogram matrix (valued functions), where the off-diagonal elements are the traditional cross variograms and pseudo cross variograms, respectively, and the diagonal elements are the direct variograms, based on the method of latent dimensions and the linear model of coregionalization. The entries in the parametric variogram matrix allow for a smooth transition between boundedness and unboundedness by changing the values of parameters, and thus between joint second-order and intrinsically stationary vector random fields, or between multivariate geometric Gaussian processes and multivariate Brown–Resnick processes in spatial extreme analysis.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Apanasovich TV, Genton MG (2010) Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97(1):15–30

    Article  Google Scholar 

  • Arroyo D, Emery X (2017) Spectral simulation of vector random fields with stationary gaussian increments in d-dimensional euclidean spaces. Stoch Environ Res Risk Assess 31(7):1583–1592

    Article  Google Scholar 

  • Bochner S (2005) Harmonic analysis and the theory of probability. Courier Corporation, Chelmsford

    Google Scholar 

  • Bourgault G, Marcotte D (1991) Multivariable variogram and its application to the linear model of coregionalization. Math Geol 23(7):899–928

    Article  Google Scholar 

  • Brown BM, Resnick SI (1977) Extreme values of independent stochastic processes. J Appl Probab 14(4):732–739

    Article  Google Scholar 

  • Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New York

    Book  Google Scholar 

  • Clark I, Basinger K, Harper W (1989) MUCK-a novel approach to co-kriging. In: Buxton BE (ed) Proceedings of the conference on geostatistical, sensitivity, and uncertainty: methods for ground-water flow and radionuclide transport modeling. Batelle Press, Columnbus, pp 473–494

  • Cressie NA (1991) Statistics for spatial data: Wiley series in probability and mathematical statistics. Wiley, New York

    Google Scholar 

  • Cressie N, Wikle CK (1998) The variance-based cross-variogram: you can add apples and oranges. Math Geol 30(7):789–799

    Article  Google Scholar 

  • Danudirdjo D, Hirose A (2011) Synthesis of two-dimensional fractional Brownian motion via circulant embedding. In: 18th IEEE international conference on image processing, pp 1085–1088

  • Davison AC, Gholamrezaee MM (2012) Geostatistics of extremes. Proc R Soc A 468(2138):581–608

    Article  Google Scholar 

  • Emery X (2008) A turning bands program for conditional co-simulation of cross-correlated Gaussian random fields. Comput Geosci 34(12):1850–1862

    Article  CAS  Google Scholar 

  • Genton MG, Kleiber W (2015) Cross-covariance functions for multivariate geostatistics. Stat Sci 30(2):147–163

    Article  Google Scholar 

  • Genton MG, Castruccio S, Crippa P, Dutta S, Huser R, Sun Y, Vettori S (2015a) Visuanimation in statistics. Stat 4(1):81–96

    Article  Google Scholar 

  • Genton MG, Padoan SA, Sang H (2015b) Multivariate max-stable spatial processes. Biometrika 102(1):215–230

    Article  Google Scholar 

  • Huang C, Yao Y, Cressie N, Hsing T (2009) Multivariate intrinsic random functions for cokriging. Math Geosci 41(8):887–904

    Article  Google Scholar 

  • Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, San Diego

    Google Scholar 

  • Kabluchko Z, Schlather M (2010) Ergodic properties of max-infinitely divisible processes. Stoch Process Appl 120(3):281–295

    Article  Google Scholar 

  • Kabluchko Z, Schlather M, de Haan L (2009) Stationary max-stable fields associated to negative definite functions. Ann Probab 37(5):2042–2065

    Article  Google Scholar 

  • Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, Berlin

    Book  Google Scholar 

  • Ma C (2005) Linear combinations of space-time covariance functions and variograms. IEEE Trans Signal Process 53(3):857–864

    Article  Google Scholar 

  • Ma C (2009) Intrinsically stationary variograms in space and time. Theory Probab Appl 53(1):145–155

    Article  Google Scholar 

  • Ma C (2011a) A class of variogram matrices for vector random fields in space and/or time. Math Geosci 43(2):229–242

    Article  Google Scholar 

  • Ma C (2011b) Vector random fields with second-order moments or second-order increments. Stoch Anal Appl 29(2):197–215

    Article  Google Scholar 

  • Maleki M, Emery X (2017) Joint simulation of stationary grade and non-stationary rock type for quantifying geological uncertainty in a copper deposit. Comput Geosci 109:258–267

    Article  Google Scholar 

  • Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246–1266

    Article  CAS  Google Scholar 

  • Matheron G (1965) Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature. Masson et CIE

  • Molchanov I, Stucki K (2013) Stationarity of multivariate particle systems. Stoch Process Appl 123(6):2272–2285

    Article  Google Scholar 

  • Moreva O, Schlather M (2018) Fast and exact simulation of univariate and bivariate Gaussian random fields. Stat 7(1):e188

    Article  Google Scholar 

  • Myers DE (1982) Matrix formulation of co-kriging. Math Geol 14(3):249–257

    Article  Google Scholar 

  • Myers DE (1991) Pseudo-cross variograms, positive-definiteness, and cokriging. Math Geol 23(6):805–816

    Article  Google Scholar 

  • Myers DE (1992) Kriging, cokriging, radial basis functions and the role of positive definiteness. Comput Math Appl 24(12):139–148

    Article  Google Scholar 

  • Porcu E, Schilling RL (2011) From Schoenberg to Pick–Nevanlinna: toward a complete picture of the variogram class. Bernoulli 17(1):441–455

    Article  Google Scholar 

  • Porcu E, Mateu J, Zini A, Pini R (2007) Modelling spatio-temporal data: a new variogram and covariance structure proposal. Stat Probab Lett 77(1):83–89

    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 

  • Schlather M, Tawn JA (2003) A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90(1):139–156

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Stein ML (2002) Fast and exact simulation of fractional Brownian surfaces. J Comput Graph Stat 11(3):587–599

    Article  Google Scholar 

  • Thibaud E, Mutzner R, Davison AC (2013) Threshold modeling of extreme spatial rainfall. Water Resour Res 49(8):4633–4644

    Article  Google Scholar 

  • Wackernagel H (1988) Geostatistical techniques for interpreting multivariate spatial information. In: Chung CF, Fabbri AG, Sinding-Larsen R (eds) Quantitative analysis of mineral and energy resources. Springer, Dordrecht, pp 393–409

    Chapter  Google Scholar 

  • Wackernagel H (2003) Multivariate geostatistics: an introduction with applications, 3rd edn. Springer, Berlin

    Book  Google Scholar 

  • Wadsworth JL, Tawn JA (2013) Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101(1):1–15

    Article  Google Scholar 

  • Wang Y, Stoev SA (2010) On the structure and representations of max-stable processes. Adv Appl Probab 42(3):855–877

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Martin Schlather for providing the R code used in Schlather and Moreva (2017), based on which the visuanimations of direct and cross variograms in Movies 1 and 2 in the electronic supplementary material were produced. This research was supported by King Abdullah University of Science and Technology (KAUST).

Author information

Authors and Affiliations

  1. Statistics Program, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia

    Wanfang Chen & Marc G. Genton

Authors
  1. Wanfang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc G. Genton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wanfang Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 21594 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Genton, M.G. Parametric variogram matrices incorporating both bounded and unbounded functions. Stoch Environ Res Risk Assess 33, 1669–1679 (2019). https://doi.org/10.1007/s00477-019-01710-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-019-01710-1

Keywords

Search

Navigation