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dc.contributor.authorLitvinenko, Alexanderen
dc.contributor.authorKeyes, David E.en
dc.contributor.authorKhoromskaia, Veneraen
dc.contributor.authorKhoromskij, Boris N.en
dc.contributor.authorMatthies, Hermann G.en
dc.description.abstractIn this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Mat\'ern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential $\mathcal{O}(n^d)$ to a linear scaling $\mathcal{O}(drn)$, where $d$ is the spatial dimension, $n$ is the number of mesh points in one direction, and $r$ is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, $\Vert x-y \Vert$en
dc.description.sponsorshipBayesian computational statistics & modeling group and Extreme Computing Research Center at KAUSTen
dc.subjectFourier transformen
dc.subjectMatern covarianceen
dc.subjectlow-rank tensor approximationen
dc.subjectlow-rank Tucker tensoren
dc.subjectLaplace transformen
dc.subjectLikelihood functionen
dc.subjectHilbert tensoren
dc.titleTucker Tensor analysis of Matern functions in spatial statisticsen
dc.typeTechnical Reporten
dc.contributor.departmentCEMSE Divisionen
dc.contributor.institutionMax-Planck-Institut für Mathematik in den Naturwissenschaftenen
dc.contributor.institutionMax-Planck-Institut für Mathematik in den Naturwissenschaftenen
dc.contributor.institutionTechnical University Braunschweigen

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Preprint "Tucker Tensor analysis of Matern functions in spatial statistics"

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