The problem of estimating (from n noisy observations of a single realization, at known sites) the parameters of a centered stationary Gaussian process whose autocorrelation function belongs to the Matérn class appears in many contexts (e.g. [1, 2, 3]). The recently proposed CGEM-EV method [4] only requires the computation of several conditional means, at the observation sites, corresponding to candidate values for the Matérn parameters. In dimension 1 and when the “Matérn differentiability” parameter is fixed to k + 1/2 with k integer (an often-used value is k = 0 or k = 1, see e.g. [3], [1], [6], [7]), each of these conditional means reduces to a Kalman smoothing.
An R implementation of CGEM-EV for this context is presented : it is built on the R-package dlm [5]. It proves to be quite fast, even for high-frequency sampling (e.g. n = 8196), and an empirical comparison with the classical maximum likelihood estimator confirms the near- efficiency results of [4].
References
[1] Rasmussen C.E., Williams C.K.I. (2006), Gaussian Processes for Machine Learning, Cam- bridge, MA: MIT Press. www.gaussianprocess.org/gpml/chapters.
[2] Zhang H. (2012). “Asymptotics and Computation for Spatial Statistics,” in Advances and Challenges in Space-time Modelling of Natural Events (Lecture Notes in Statistics, Vol. 207) (E. Porcu, J. M. Montero, and M. Schlather, eds.), New York: Springer, pp. 239−−252. doi:10.1007/978-3-642-17086-7_10.
[3] Kaufman C.G., Shaby, B.A. (2013). “The Role of the Range Parameter for Estimation and Prediction in Geostatistics,” Biometrika 100 (2), pp. 473−−484. doi:10.1093/biomet/ass079.
[4] Girard D.A. (2012). “Asymptotic Near-Efficiency of the 'Gibbs-Energy and Empirical- Variance' Estimating Functions for Fitting Matérn Models to a Dense (Noisy) Series,” 2012. arxiv.org/pdf/0909.1046v2.pdf.
[5] Petris Giovanni (2010). “An R Package for Dynamic Linear Models,” Journal of Statistical Software, 36(12), 1-16. URL: http://www.jstatsoft.org/v36/i12/
[6] Girard D.A. (2014) “Estimating a Centered Ornstein-Uhlenbeck Process under Measurement Errors,” Wolfram Demonstrations Project, Published: July 1, 2014. URL: demonstrations.wolfram.com/EstimatingACenteredOrnsteinUhlenbeckProcessUnderMeasurementE/
[7] Girard D.A. (2015) “Three Alternatives to the Likelihood Maximization for Estimating a Centered Matérn (3/2) Process, ” Wolfram Demonstrations Project, URL: demonstrations.wolfram.com/ThreeAlternativesToTheLikelihoodMaximizationForEstimatingACe/
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