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On the Second‐Order Random Walk Model for Irregular Locations

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  • FINN LINDGREN
  • HÅVARD RUE

Abstract

. The second‐order random walk (RW2) model is commonly used for smoothing data and for modelling response functions. It is computationally efficient due to the Markov properties of the joint (intrinsic) Gaussian density. For evenly spaced locations the RW2 model is well established, whereas for irregularly spaced locations there is no well established construction in the literature. By considering the RW2 model as the solution of a stochastic differential equation (SDE), a discretely observed integrated Wiener process, it is possible to derive the density preserving the Markov properties by augmenting the state‐space with the velocities. Here, we derive a computationally more efficient RW2 model for irregular locations using a Galerkin approximation to the solution of the SDE without the need of augmenting the state‐space. Numerical comparison with the exact solution demonstrates that the error in the Galerkin approximation is small and negligible in applications.

Suggested Citation

  • Finn Lindgren & Håvard Rue, 2008. "On the Second‐Order Random Walk Model for Irregular Locations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 691-700, December.
    • Handle: RePEc:bla:scjsta:v:35:y:2008:i:4:p:691-700
    DOI: 10.1111/j.1467-9469.2008.00610.x
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    References listed on IDEAS

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    1. Ludwig Fahrmeir & Stefan Lang, 2001. "Bayesian inference for generalized additive mixed models based on Markov random field priors," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 50(2), pages 201-220.
    2. Chib, Siddhartha & Jeliazkov, Ivan, 2006. "Inference in Semiparametric Dynamic Models for Binary Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 685-700, June.
    3. Håvard Rue, 2001. "Fast sampling of Gaussian Markov random fields," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 325-338.
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    Cited by:

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    3. MAWANGA Freddie Festo, 2017. "Investigating A Random Walk In Air Cargo Exports Of Fresh Agricultural Products: Evidence From A Developing Country," Studies in Business and Economics, Lucian Blaga University of Sibiu, Faculty of Economic Sciences, vol. 12(1), pages 129-140, April.
    4. Van Niekerk, Janet & Krainski, Elias & Rustand, Denis & Rue, Håvard, 2023. "A new avenue for Bayesian inference with INLA," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).
    5. Yu Yue & Paul Speckman & Dongchu Sun, 2012. "Priors for Bayesian adaptive spline smoothing," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 577-613, June.
    6. Victor Medina-Olivares & Finn Lindgren & Raffaella Calabrese & Jonathan Crook, 2023. "Joint model for longitudinal and spatio-temporal survival data," Papers 2311.04008, arXiv.org.
    7. Özgür Asar & David Bolin & Peter J. Diggle & Jonas Wallin, 2020. "Linear mixed effects models for non‐Gaussian continuous repeated measurement data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 69(5), pages 1015-1065, November.
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    9. Birgit Schrödle & Leonhard Held, 2011. "A primer on disease mapping and ecological regression using $${\texttt{INLA}}$$," Computational Statistics, Springer, vol. 26(2), pages 241-258, June.
    10. Brown, Paul T. & Joshi, Chaitanya & Joe, Stephen & Rue, Håvard, 2021. "A novel method of marginalisation using low discrepancy sequences for integrated nested Laplace approximations," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).

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