Skip to main content

Computationally simple anisotropic lattice covariograms


When working with contemporary spatial ecological datasets, statistical modellers are often confronted by two major challenges: (I) the need for covariance models with the flexibility to accommodate directional patterns of anisotropy; and (II) the computational effort demanded by high-dimensional inverse and determinant problems involving the covariance matrix \(\vec {V}\). In the case of rectangular lattice data, the spatially separable covariogram is a longstanding but underused model that can reduce arithmetic complexity by orders of magnitude. We examine a class of covariograms for stationary data that extends the separable model through affine coordinate transformations, providing a far greater flexibility for handling anisotropy than that offered by the standard approach of using geometric anisotropy to extend an isotropic model. This motivates our development of an extremely fast estimator of the orientation of the axes of range anisotropy on spatial lattice data, and a powerful visual diagnostic for nonstationarity. In a case study, we demonstrate how these tools can be used to analyze and predict forest damage patterns caused by outbreaks of the mountain pine beetle.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


Download references


The authors thank the Lewis Research Group for providing expert advice and feedback. MAL is also grateful for support through NSERC and the Canada Research Chair Program. This research was supported by a grant to MAL from the Natural Science and Engineering Research Council of Canada (Grant No. NET GP 434810-12) to the TRIA Network, with contributions from Alberta Agriculture and Forestry, Foothills Research Institute, Manitoba Conservation and Water Stewardship, Natural Resources Canada-Canadian Forest Service, Northwest Territories Environment and Natural Resources, Ontario Ministry of Natural Resources and Forestry, Saskatchewan Ministry of Environment, West Fraser and Weyerhaeuser.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dean Koch.

Additional information

Handling Editor: Bryan F. J. Manly

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary materials 1 (PDF 132 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Koch, D., Lele, S. & Lewis, M.A. Computationally simple anisotropic lattice covariograms. Environ Ecol Stat 27, 665–688 (2020).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Bisymmetry
  • Covariogram
  • Geometric anisotropy
  • Lattice
  • Separable covariance