Covariance Modeling for Multivariate Spatial Processes Based on Separable Approximations

  • Rafael S. Erbisti
  • Thais C. O. Fonseca
  • Mariane B. Alves
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 239)


The computational treatment of high dimensionality problems is a challenge. In the context of geostatistics, analyzing multivariate data requires the specification of the cross-covariance function, which defines the dependence between the components of a response vector for all locations in the spatial domain. However, the computational cost to make inference and predictions can be prohibitive. As a result, the use of complex models might be unfeasible. In this paper, we consider a flexible nonseparable covariance model for multivariate spatiotemporal data and present a way to approximate the full covariance matrix from two separable matrices of minor dimensions. The method is applied only in the likelihood computation, keeping the interpretation of the original model. We present a simulation study comparing the inferential and predictive performance of our proposal and we see that the approximation provides important gains in computational efficiency without presenting substantial losses in predictive terms.


Nonseparable covariance Likelihood approximation Kronecker product Predictive performance Multivariate spatial process 


  1. 1.
    Banerjee, S., Carlin, B.P., Gelfand, A.E.: Hierarchical Modeling and Analysis for Spatial Data. Monographs on Statistics and Applied Probability, 1st edn. Chapman & Hall/CRC, London (2004)Google Scholar
  2. 2.
    Erbisti, R., Fonseca, T., Alves, M.: Bayesian covariance modeling of multivariate spatial random fields. Version 1, 20 July (2017). arXiv:1707.06697
  3. 3.
    Apanasovich, T.V., Genton, M.G.: Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97(1), 15–30 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. Chapman & Hall/CRC, London (2000)zbMATHGoogle Scholar
  5. 5.
    Genton, M.G.: Separable Approximations Of Space-time Covariance Matrices. Environmetrics 18, 681–695 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. The Johns Hopkings University Press, Baltimore (1996)zbMATHGoogle Scholar
  7. 7.
    Plummer, M., Best, N., Cowles, K., Vines, K.: CODA: convergence diagnosis and output analysis for MCMC. R News 6, 7–11 (2006)Google Scholar
  8. 8.
    Gneiting, T., Raftery, A.E.: Strictly proper scoring rules, prediction and estimation. J. Am. Stat. Assoc. 102(477), 360–378 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ibrahim, J.G., Chen, M.-H., Sinha, D.: Bayesian Survival Analysis, 1st edn. Springer, Berlin (2001)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Rafael S. Erbisti
    • 1
  • Thais C. O. Fonseca
    • 2
  • Mariane B. Alves
    • 2
  1. 1.Department of StatisticsCentro de Tecnologia, Federal University of Rio de JaneiroRio de JaneiroBrazil
  2. 2.Department of StatisticsCentro de Tecnologia, Federal University of Rio de JaneiroRio de JaneiroBrazil

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