Multivariate Spatial Process Models

  • Alan E. GelfandEmail author
Living reference work entry


Spatially-referenced multivariate data is becoming increasingly common. Here, we focus on data in the form of vectors observed at a finite set of spatial locations. Regression models are of interest in order to explain the response vectors as well as to predict response at unobserved locations. Such models need to capture both dependence among the components of the vectors as well as spatial dependence across the locations of the vectors. The objective of this chapter is to develop classes of models which provide the desired dependence. We consider this both formally and constructively. Constructive development is supplied through four different strategies. An example, using soil nutrient data from Costa Rica is presented.


Convolution Co-regionalization Cross-covariance Cross-variogram Gausspian process Matérn covariance Separability 


  1. Apanasovich T, Genton M (2010) Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97(1):15–30CrossRefGoogle Scholar
  2. Apanasovich T, Genton M, Sun Y (2012) A valid matérn class of cross-covariance functions for multivariate random fields with any number of components. J Am Stat Assoc 107(497):180–193CrossRefGoogle Scholar
  3. Banerjee S, Carlin B, Gelfand AE (2014) Hierarchical modeling and analysis for spatial data, 2nd edn. Chapman and Hall CRC, Boca RatonGoogle Scholar
  4. Clark I, Basinger K, Harper S (1989) Muck: a novel approach to co-kriging. In: Proceedings of the conference on geostatistical, sensitivity, and uncertainty methods for ground-water flow and radionuclide transport modeling, pp 473–493Google Scholar
  5. Cramér H (1940) On the theory of stationary random processes. Ann Math 41(1):215–230CrossRefGoogle Scholar
  6. Cressie N, Wikle C (2011) Statistics for Spatio-temporal data. Wiley, HobokenGoogle Scholar
  7. Finley A, Banerjee S, Carlin B (2007) spBayes: an R package for univariate and multivariate hierarchical point-referenced spatial models. J Stat Softw 19(4):1–24CrossRefGoogle Scholar
  8. Gelfand A, Kim H-J, Sirmans C, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98(462):387–396CrossRefGoogle Scholar
  9. Gelfand A, Schmidt A, Banerjee S, Sirmans CF (2004) Nonstationary multivariate process modelling through spatially varying coregionalization (with discussion). TEST 13(2):1–50CrossRefGoogle Scholar
  10. Genton M, Kleiber W (2015) Cross-covariance functions for multivariate geostatistics. Stat Sci 30(2):147–163CrossRefGoogle Scholar
  11. Gneiting T (2002) Nonseparable, stationary covariance functions for space-time data. J Am Stat Assoc 97(458):590–600CrossRefGoogle Scholar
  12. Gneiting T, Kleiber W, Schlather M (2010) Geostatistical space-time models, stationarity, separability, and full symmetry. J Am Stat Assoc 105(491):1167–1177CrossRefGoogle Scholar
  13. Higdon D, Swall J, Kern J (1999) Non-stationary spatial modeling. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 6. Oxford University Press, Oxford, pp 761–768Google Scholar
  14. Le N, Sun W, Zidek J (2002) Bayesian multivariate spatial interpolation with data missing by design. J R Stat Soc-Ser B 59(2):501–510CrossRefGoogle Scholar
  15. Majumdar A, Gelfand A (2007) Multivariate spatial process modeling using convolved covariance functions. Math Geol 39(2):225–245CrossRefGoogle Scholar
  16. Mardia K, Goodall C (1993) Spatio-temporal analyses of multivariate environmental monitoring data. In: Patil GP, Rao CR (eds) Multivariate environmental statistics. Elsevier, Amsterdam, pp 347–386Google Scholar
  17. Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5(3):437–468CrossRefGoogle Scholar
  18. Myers D (1982) Matrix formulation of co-kriging. J Int Assoc Math Geol 14(3):249–257CrossRefGoogle Scholar
  19. Myers D (1991) Pseudo-cross variograms, positive definiteness and cokriging. Math Geol 23(6):805–816CrossRefGoogle Scholar
  20. Royle J, Berliner L (1999) A hierarchical approach to multivariate spatial modeling and prediction. J Agric Biol Environ Stat 4(1):29–56CrossRefGoogle Scholar
  21. Sangalli L, Ramsay J, Ramsay T (2013) Spatial spline regression models. J R Stat Soc-Ser B 75(4):681–703CrossRefGoogle Scholar
  22. Schmidt A, Gelfand A (2003) A bayesian coregionalization approach for multivariate pollutant data. J Geophys Res - Atmos 108(24):8783Google Scholar
  23. Ver Hoef J, Barry R (1998) Constructing and fitting models for cokriging and multivariable spatial prediction. J Stat Plann Inference 69(2):275–294CrossRefGoogle Scholar
  24. Wackernagel H (2003) Multivariate Geostatistics: an introduction with applications. Springer, New York. 3 editionCrossRefGoogle Scholar
  25. Yaglom A (1987) Correlation theory of stationary and related random functions, vol 1. Springer, New YorkCrossRefGoogle Scholar
  26. Zhang H (2007) Maximum-likelihood estimation for multivariate spatial linear coregionalization models. Environmetrics 18(2):125–139CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistical ScienceDuke UniversityDurhamUSA

Personalised recommendations