Estimating a Centered Matérn (1) Process: Three Alternatives to Maximum Likelihood via Conjugate Gradient Linear Solvers

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Let denote the candidate correlation matrix of , with inverse-range , that is, the matrix whose element in the row and column is . The only difficulty regarding implementation in the case of no measurement error, of both the hybrid method and CGEM-EV, is the computation of the quadratic form , the so-called "Gibbs energy of associated with a given ".

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The computation of uses a conjugate-gradient (CG) solver preconditioned by a classical factored sparse approximate inverse (FSAI) preconditioning (see [7] for a recent survey), each product by being obtained via FFT from the standard embedding of in a circulant matrix.

It is observed here that this implementation is quite fast, even for .

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Contributed by: Didier A. Girard (June 2015)
(CNRS-LJK and Université Grenoble Alpes)
Open content licensed under CC BY-NC-SA


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References

[1] P. Whittle, "On Stationary Processes in the Plane," Biometrika, 41(3–4), 1954 pp. 434–449. doi:10.2307/2332724.

[2] H. Zhang, "Asymptotics and Computation for Spatial Statistics," in Advances and Challenges in Space-time Modelling of Natural Events (E. Porcu, J. M. Montero, and M. Schlather, eds.), New York: Springer, 2012 pp. 239–252. doi:10.1007/978-3-642-17086-7_10.

[3] C. G. Kaufman and B. A. Shaby, "The Role of the Range Parameter for Estimation and Prediction in Geostatistics," Biometrika, 100(2), 2013 pp. 473–484. doi:10.1093/biomet/ass079.

[4] N. Cressie, "Fitting Variogram Models by Weighted Least Squares," Journal of the International Association for Mathematical Geology, 17(5), 1985 pp. 563–586. doi:10.1007/BF01032109.

[5] H. Zhang and D. L. Zimmerman, "Hybrid Estimation of Semivariogram Parameters," Mathematical Geology, 39(2), 2007 pp. 247–260. doi:10.1007/s11004-006-9070-8.

[6] D. A. Girard, "Asymptotic Near-Efficiency of the ‘Gibbs-Energy and Empirical-Variance’ Estimating Functions for Fitting Matérn Models to a Dense (Noisy) Series." arxiv.org/abs/0909.1046v2.

[7] C. Janna, M. Ferronato, F. Sartoretto, and G. Gambolati, "FSAIPACK: A Software Package for High Performance FSAI Preconditioning," ACM Transactions on Mathematical Software, 41(2), forthcoming.



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