Fast and Exact Simulation of Large Gaussian Lattice Systems in ...

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α1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.951.0 1.2 1.4 1.6 1.8 2.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Figure 3: Intrinsic Embedding for the Powered Exponential Class. We consider the block circulantmatrix C(P r σ r , 2 · 2048, r/2048) where σ r is given by (12), (13) and (14), and ϕ is powered exponentialwith shape parameter α and scale parameter θ, respectively. The isolines show the smallestvalue of r that makes the matrix nonnegative definite. The white area in the (θ, α) plane marksthe parameter combinations for which no such r could be found.θν0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.251.0 1.2 1.4 1.6 1.8 2.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Figure 4: Same as Figure 3 for Matérn Covariances with Shape Parameter ν and Scale Parameter θ.θ12
4 ImplementationWe now describe strategies for choosing a simulation algorithm, and we discuss the current computationallimits of exact simulation. The cut-off embedding and intrinsic embedding techniqueshave been implemented in Version 1.2.5 of the RandomFields package (Schlather 2001) for theR language (Ihaka and Gentleman 1996), and we give an example of using this package.4.1 Choice of simulation algorithmRecall that we wish to simulate a Gaussian surface Z with cov{Z(x), Z(y)} = ϕ(|x − y|) on asquare lattice of size n × n, with spacing s/n along each coordinate. Without loss of generalitywe can assume that s = 1/ √ 2, which avoids a scaling argument in the use of the above results.Sophisticated users of the RandomFields package will specify the desired simulation techniquedirectly, and in Section 4.3 below we give an example of how this can be done. For less experiencedusers, the GaussRF function provides an automated search for a suitable simulation algorithm. Indescribing this search, we suppose that the fast Fourier transform can be applied to square matricesof size at most 2m×2m where m is a power of 2. On our 512 MB RAM machine we found m = 2 048to be a reasonable choice; m = 4 096 was still feasible but caused discontinuities in the responsetime of the system.If n is small, say n ≤ 64, the Cholesky decomposition technique applies and performs well. Forlarger systems, the standard embedding approach is the most straightforward and the most easilyapplicable among the three variants of the circulant embedding technique. Hence, the automatedsearch for an exact simulation algorithm considers the standard approach first, by checking whetherthe block circulant matrix C(P cs ϕ; 2cn, s/n) is nonnegative definite for some c ∈ [1, m/n] such thatcn is a highly composite integer. If the standard embedding approach fails, the search proceedsbased on the user’s preferences. If the GaussRF parameter stationary.only is set to FALSE,stationarity is not essential, and the intrinsic embedding technique provides a particularly efficientalternative. Let c be such that cn is the smallest power of two greater than or equal to n/s, anddefine σ cs as in (12). The GaussRF function uses the intrinsic embedding technique if cn ≤ mand the block circulant matrix C(P cs σ cs ; 2cn, s/n) is nonnegative definite, or if 2cn ≤ m andC(P 2cs σ 2cs ; 2 · 2cn, s/n) is nonnegative definite. If the intrinsic embedding approach fails, or if theparameter stationary.only is set to TRUE, the search turns to the cut-off embedding approach. If(9) or (11) yield a cut-off, r, such that r/s ≤ m/n, we define ρ by (8) or (10), respectively, find thesmallest c ≥ r/s such that cn is a highly composite integer, and check whether the block circulantmatrix C(P cs ρ; 2cn, s/n) is nonnegative definite. There is no guarantee that any of the approacheswork, and exact simulation may not be feasible.At various stages in this search, block circulant matrices need to be checked for nonnegativedefiniteness. Whenever possible, we use theoretical results or look-up tables for doing this. Aless computationally efficient alternative is to determine the eigenvalues numerically using the fastFourier transform. In this numerical test, we consider a block circulant matrix to be nonnegativedefinite if the real parts of all eigenvalues are nonnegative and the moduli of the imaginary parts aresmall (10 −8 or less, thereby allowing for numerical noise). Clearly, refinements in the automatedsearch are possible. To simulate random fields with standard covariance structures, for instance, onemight turn to the intrinsic embedding and cut-off embedding approaches directly, without attemptingto invoke the standard approach. Applications in environmental risk assessment and simulationstudies sometimes require large numbers of realizations. In these cases, parallel computing providesfurther improvements in computational feasibility and efficiency (Ševčíková 2004).13
Page 6 and 7: Table 3: Smallest Eigenvalue, λ mi
Page 8 and 9: Table 4: Criteria of the Pólya Typ
Page 14 and 15: 2048alpha = 0.52048alpha = 1.010241
Page 16 and 17: ReferencesAdler, D., Nenadić, O. a

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