Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

In 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. C...

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Vydáno v:Journal of computational methods in applied mathematics Ročník 19; číslo 1; s. 101 - 122
Hlavní autoři: Litvinenko, Alexander, Keyes, David, Khoromskaia, Venera, Khoromskij, Boris N., Matthies, Hermann G.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Minsk De Gruyter 01.01.2019
Walter de Gruyter GmbH
Témata:
60H15
60H35
65N25
Bayesian Update
Computing costs
Computing time
Convergence
Covariance matrix
Datasets
Decomposition
Economic models
Format
Fourier Transform
Geostatistical Optimal Design
Hilbert Tensor
Kalman Filter
Kriging
Loglikelihood Surrogate
Low-Rank Tensor Approximation
Matérn Covariance
Parameters
Quadratic forms
Tensor analysis
Tensors
ISSN:1609-4840, 1609-9389
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Shrnutí:In 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 three dimensions. 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 Matérn- 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 to a linear scaling , where is the spatial dimension, is the number of mesh points in one direction, and 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,
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ISSN:1609-4840
1609-9389
DOI:10.1515/cmam-2018-0022