An approach for valid covariance estimation via the Fourier series
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The use of kriging for construction of prediction or risk maps requires estimating the dependence structure of the random process, which can be addressed through the approximation of the covariance function. The nonparametric estimators used for the latter aim are not necessarily valid to solve the kriging system, since the positive-definiteness condition of the covariance estimator typically fails. The usage of a parametric covariance instead may be attractive at first because of its simplicity, although it may be affected by misspecification. An alternative is suggested in this paper to obtain a valid covariance from a nonparametric estimator through the Fourier series tool, which involves two issues: estimation of the Fourier coefficients and selection of the truncation point to determine the number of terms in the Fourier expansion. Numerical studies for simulated data have been conducted to illustrate the performance of this approach. In addition, an application to a real environmental data set is included, related to the presence of nitrate in groundwater in Beja District (Portugal), so that pollution maps of the region are generated by solving the kriging equations with the use of the Fourier series estimates of the covariance.
KeywordsCovariance function Fourier series Kriging Truncation point
The authors thank the helpful suggestions from the reviewers, which have been reflected in the current paper. This work has been supported in part by grant INCITE-08-PXIB-322219-PR from Consellería de Innovación e Industria (Xunta de Galicia, Spain). R. Menezes acknowledges financial support from the projects PTDC/MAT/104879/2008 and PTDC/MAT/112338/2009 (FEDER support included) of the Portuguese Ministry of Science, Technology and Higher Education. Ó. Rubiños-López’s research has also been supported by FEDER through Xunta de Galicia Researching programs (Grupos de referencia competitiva). P. García-Soidán and Ó. Rubiños-López acknowledge financial support from the project CONSOLIDER-INGENIO CSD2008-00068.
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