T. Gneiting, H. Sevcíková, D. B. Percival, M. Schlather and Y. Jiang (2006), `Summary for `Fast and Exact Simulation of Large Gaussian Lattice Systems in R^2: Exploring the Limits,' Journal of Computational and Graphical Statistics, 15, no. 3, pp. 1-19.
The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields. The method uses periodic embeddings and relies on the fast Fourier transform. However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations. This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions. Theoretical support is given to this proposal, and software for its implementation is provided. The method yields exact simulations of planar Gaussian lattice systems with up to 1,000,000 lattice points for wide classes of processes, including those with powered exponential, Matérn and Cauchy covariances.
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