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First-order intrinsic autoregressions and the de Wijs process


  • Julian Besag
  • Debashis Mondal


We discuss intrinsic autoregressions for a first-order neighbourhood on a two-dimensional rectangular lattice and give an exact formula for the variogram that extends known results to the asymmetric case. We obtain a corresponding asymptotic expansion that is more accurate and more general than previous ones and use this to derive the de Wijs variogram under appropriate averaging, a result that can be interpreted as a two-dimensional spatial analogue of Brownian motion obtained as the limit of a random walk in one dimension. This provides a bridge between geostatistics, where the de Wijs process was once the most popular formulation, and Markov random fields, and also explains why statistical analysis using intrinsic autoregressions is usually robust to changes of scale. We briefly describe corresponding calculations in the frequency domain, including limiting results for higher-order autoregressions. The paper closes with some practical considerations, including applications to irregularly-spaced data. Copyright 2005, Oxford University Press.

Suggested Citation

  • Julian Besag & Debashis Mondal, 2005. "First-order intrinsic autoregressions and the de Wijs process," Biometrika, Biometrika Trust, vol. 92(4), pages 909-920, December.
  • Handle: RePEc:oup:biomet:v:92:y:2005:i:4:p:909-920

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    Cited by:

    1. Donald, Margaret & Alston, Clair L. & Young, Rick R. & Mengersen, Kerrie L., 2011. "A Bayesian analysis of an agricultural field trial with three spatial dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3320-3332, December.
    2. Peter J. Diggle & Raquel Menezes & Ting‐li Su, 2010. "Geostatistical inference under preferential sampling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 59(2), pages 191-232, March.
    3. Paulauskas, Vygantas, 2007. "On unit roots for spatial autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 209-226, January.
    4. Schmidt, Paul & Mühlau, Mark & Schmid, Volker, 2017. "Fitting large-scale structured additive regression models using Krylov subspace methods," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 59-75.
    5. Peter J. Diggle & Emanuele Giorgi, 2016. "Model-Based Geostatistics for Prevalence Mapping in Low-Resource Settings," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1096-1120, July.
    6. Sanjay Chaudhuri & Debashis Mondal & Teng Yin, 2017. "Hamiltonian Monte Carlo sampling in Bayesian empirical likelihood computation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 293-320, January.

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