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Eigenstructures of Spatial Design Matrices

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  • Gorsich, David J.
  • Genton, Marc G.
  • Strang, Gilbert
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    Abstract

    In estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 80 (2002)
    Issue (Month): 1 (January)
    Pages: 138-165

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    Handle: RePEc:eee:jmvana:v:80:y:2002:i:1:p:138-165

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    Related research

    Keywords: discrete cosine transform eigenvalue eigenvector kriging Kronecker product Matheron's estimator variogram spatial statistics;

    References

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    1. Genton, Marc G. & He, Li & Liu, Xiangwei, 2001. "Moments of skew-normal random vectors and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 319-325, February.
    2. Ali, Mukhtar M, 1987. "Durbin-Watson and Generalized Durbin-Watson Tests for Autocorrelations and Randomness," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(2), pages 195-203, April.
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    Cited by:
    1. Reinaldo Arellano-Valle & Marc Genton, 2010. "An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators," Annals of the Institute of Statistical Mathematics, Springer, vol. 62(2), pages 363-381, April.
    2. Hillier, Grant & Martellosio, Federico, 2006. "Spatial design matrices and associated quadratic forms: structure and properties," MPRA Paper 15807, University Library of Munich, Germany.
    3. Genton, Marc G. & Gorsich, David J., 2002. "Nonparametric variogram and covariogram estimation with Fourier-Bessel matrices," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 47-57, November.
    4. Genton, Mark G. & Ruiz-Gazen, Anne, 2009. "Visualizing Influential Observations in Dependent Data," TSE Working Papers 09-051, Toulouse School of Economics (TSE).

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